Sobolev Spaces

Home , Capacity of a set, Sobolev inequality, Sobolev space

JUHA KINNUNEN Sobolev spaces

Department of Mathematics, Aalto University 2021 Contents

1 SOBOLEV SPACES1 1.1 Weak derivatives ...... 1 1.2 Sobolev spaces ...... 4 1.3 Properties of weak derivatives ...... 8 1.4 Completeness of Sobolev spaces ...... 9 1.5 Hilbert space structure ...... 11 1.6 Approximation by smooth functions ...... 12 1.7 Local approximation in Sobolev spaces ...... 16 1.8 Global approximation in Sobolev spaces ...... 17 1.9 Sobolev spaces with zero boundary values ...... 18 1.10 Chain rule ...... 21 1.11 Truncation ...... 23 1.12 Weak convergence methods for Sobolev spaces ...... 25 1.13 Difference quotients ...... 33 1.14 Absolute continuity on lines ...... 36

2 SOBOLEV INEQUALITIES 42 2.1 Gagliardo-Nirenberg-Sobolev inequality ...... 43 2.2 Sobolev-Poincaré inequalities ...... 49 2.3 Morrey’s inequality ...... 55 1, 2.4 Lipschitz functions and W ∞ ...... 59 2.5 Summary of the Sobolev embeddings ...... 62 2.6 Direct methods in the calculus of variations ...... 62

3 MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 70 3.1 Representation formulas and Riesz potentials ...... 71 3.2 Sobolev-Poincaré inequalities ...... 78 3.3 Sobolev inequalities on domains ...... 86 3.4 A maximal function characterization of Sobolev spaces ...... 89 3.5 Pointwise estimates ...... 92 3.6 Approximation by Lipschitz functions ...... 97 3.7 Maximal operator on Sobolev spaces ...... 102 CONTENTS ii

4 POINTWISE BEHAVIOUR OF SOBOLEV FUNCTIONS 106 4.1 Sobolev capacity ...... 106 4.2 Capacity and measure ...... 109 4.3 Quasicontinuity ...... 116 4.4 Lebesgue points of Sobolev functions ...... 119 4.5 Sobolev spaces with zero boundary values ...... 124 1 Sobolev spaces

In this chapter we begin our study of Sobolev spaces. The Sobolev space is a vector space of functions that have weak derivatives. Motivation for studying these spaces is that solutions of partial differential equations, when they exist, belong naturally to Sobolev spaces.

1.1 Weak derivatives

Notation. Let Ω Rn be open, f : Ω R and k 1,2,.... Then we use the following ⊂ → = notations:

C(Ω) {f : f continuous in Ω} = supp f {x Ω : f (x) 0} the support of f = ∈ 6= = C0(Ω) {f C(Ω) : supp f is a compact subset of Ω} = ∈ Ck(Ω) {f C(Ω) : f is k times continuously diferentiable} = ∈ k k C (Ω) C (Ω) C0(Ω) 0 = ∩ \∞ k C∞ C (Ω) smooth functions = k 1 = = C∞(Ω) C∞(Ω) C0(Ω) 0 = ∩ compactly supported smooth functions = test functions =

■

W A R N I N G : In general, supp f * Ω. Examples 1.1: (1) Let u : B(0,1) R, u(x) 1 x . Then supp u B(0,1). → = − | | =

1 CHAPTER 1. SOBOLEV SPACES 2

(2) Let f : R R be →  x2, x 0, f (x) Ê =  x2, x 0. − < Now f C1(R) \ C2(R) although the graph looks smooth. ∈ (3) Let us deﬁne ϕ : Rn R, →  1 e x 2 1 , x B(0,1), ϕ(x) | | − ∈ = 0, x Rn \ B(0,1). ∈ n Now ϕ C∞(R ) and suppϕ B(0,1) (exercise). ∈ 0 = Let us start with a motivation for deﬁnition of weak derivatives. Let Ω Rn 1 ⊂ be open, u C (Ω) and ϕ C∞(Ω). Integration by parts gives ∈ ∈ 0 Çϕ Çu u dx ϕ dx. ˆΩ Çx j = −ˆΩ Çx j There is no boundary term, since ϕ has a compact support in Ω and thus vanishes near ÇΩ. k n Let then u C (Ω), k 1,2,..., and let α (α1,α2,...,αn) N (we use the ∈ = = ∈ convention that 0 N) be a multi-index such that the order of multi-index α ∈ | | = α1 ... αn is at most k. We denote + + Ç α u Çα1 Çαn Dαu | | ... u. α1 αn α1 αn = Çx1 ...Çxn = Çx1 Çxn

T HEMORAL : A coordinate of a multi-index indicates how many times a function is differentiated with respect to the corresponding variable. The order of a multi-index tells the total number of differentiations.

Successive integration by parts gives

α α α uD ϕ dx ( 1)| | D uϕ dx. ˆΩ = − ˆΩ Notice that the left-hand side makes sense even under the assumption u L1 (Ω). ∈ loc Deﬁnition 1.2. Assume that u L1 (Ω) and let α Nn be a multi-index. Then ∈ loc ∈ v L1 (Ω) is the αth weak partial derivative of u, written Dαu v, if ∈ loc = α α u D ϕ dx ( 1)| | vϕ dx ˆΩ = − ˆΩ 0 (0,...,0) for every test function ϕ C∞(Ω). We denote D u D u. If α 1, then ∈ 0 = = | | =

Du (D1u,D2u...,Dnu) = is the weak gradient of u. Here

Çu (0,...,1,...,0) D j u D u, j 1,..., n, = Çx j = = (the jth component is 1). CHAPTER 1. SOBOLEV SPACES 3

T HEMORAL : Classical derivatives are deﬁned as pointwise limits of difference quotients, but the weak derivatives are deﬁned as a functions satisfying the integration by parts formula. Observe, that changing the function on a set of measure zero does not affect its weak derivatives.

W ARNING We use the same notation for the weak and classical derivatives. It should be clear from the context which interpretation is used.

Remarks 1.3: (1) If u Ck(Ω), then the classical partial derivatives up to order k are also ∈ the corresponding weak derivatives of u. In this sense, weak derivatives generalize classical derivatives. (2) If u 0 almost everywhere in an open set, then Dαu 0 almost everywhere = = in the same set.

Lemma 1.4. A weak αth partial derivative of u, if it exists, is uniquely deﬁned up to a set of measure zero.

Proof. Assume that v,v L1 (Ω) are both weak αth partial derivatives of u, that e ∈ loc is, α α α uD ϕ dx ( 1)| | vϕ dx ( 1)| | veϕ dx ˆΩ = − ˆΩ = − ˆΩ

for every ϕ C∞(Ω). This implies that ∈ 0

(v ve)ϕ dx 0 for every ϕ C0∞(Ω). (1.1) ˆΩ − = ∈ Claim: v v almost everywhere in Ω. = e

Reason. Let Ω0 b Ω (i.e. Ω0 is open and Ω0 is a compact subset of Ω). The p space C0∞(Ω0) is dense in L (Ω0) (we shall return to this later). There exists a sequence of functions ϕi C∞(Ω0) such that ϕi 2 in Ω0 and ϕi sgn(v v) ∈ 0 | | É → − e almost everywhere in Ω0 as i . Here sgn is the signum function. → ∞ Identity (1.1) and the dominated convergence theorem, with the majorant 1 (v v)ϕi 2( v v ) L (Ω0), give | − e | É | | + |e| ∈

0 lim (v v)ϕi dx lim (v v)ϕi dx = i ˆ − e = ˆ i − e →∞ Ω0 Ω0 →∞ (v v)sgn(v v) dx v v dx = ˆ − e − e = ˆ | − e| Ω0 Ω0

This implies that v v almost everywhere in Ω0 for every Ω0 Ω. Thus v v = e b = e almost everywhere in Ω. ■ From the proof we obtain a very useful corollary. CHAPTER 1. SOBOLEV SPACES 4

Corollary 1.5 (Fundamental lemma of the calculus of variations). If f L1 (Ω) ∈ loc satisﬁes f ϕ dx 0 ˆΩ =

for every ϕ C∞(Ω), then f 0 almost everywhere in Ω. ∈ 0 =

T HEMORAL : This is an integral way to say that a function is zero almost everywhere.

Example 1.6. Let n 1 and Ω (0,2). Consider = =  x, 0 x 1, u(x) < < = 1, 1 x 2, É < and  1, 0 x 1, v(x) < < = 0, 1 x 2. É <

We claim that u0 v in the weak sense. To see this, we show that = 2 2 uϕ0 dx vϕ dx ˆ0 = −ˆ0

for every ϕ C∞((0,2)). ∈ 0 Reason. An integration by parts and the fundamental theorem of calculus give

2 1 2 u(x)ϕ0(x) dx xϕ0(x) dx ϕ0(x) dx ˆ0 = ˆ0 + ˆ1 ¯1 1 ¯ xϕ(x)¯ ϕ(x) dx ϕ(2) ϕ(1) = ¯0 −ˆ0 + |{z}− | {z } 0 ϕ(1) = = 1 2 ϕ(x) dx vϕ(x) dx = −ˆ0 = −ˆ0

for every ϕ C0∞((0,2)). ∈ ■

1.2 Sobolev spaces

Deﬁnition 1.7. Assume that Ω is an open subset of Rn. The Sobolev space W k,p(Ω) consists of functions u Lp(Ω) such that for every multi-index α with ∈ α k, the weak derivative Dαu exists and Dαu Lp(Ω). Thus | | É ∈ W k,p(Ω) {u Lp(Ω) : Dαu Lp(Ω), α k}. = ∈ ∈ | | É CHAPTER 1. SOBOLEV SPACES 5

If u W k,p(Ω), we deﬁne its norm ∈ 1 Ã ! p X α p u W k,p(Ω) D u dx , 1 p , k k = α k ˆΩ | | É < ∞ | |É and X α u k, esssup D u . W ∞(Ω) k k = α k Ω | | | |É 0 (0,...,0) Notice that D u D u u. Assume that Ω0 is an open subset of Ω. We say = = that Ω0 is compactly contained in Ω, denoted Ω0 b Ω, if Ω0 is a compact subset of k,p k,p Ω. A function u W (Ω), if u W (Ω0) for every Ω0 Ω. ∈ loc ∈ b

T HEMORAL : Thus Sobolev space W k,p(Ω) consists of functions in Lp(Ω) that have weak partial derivatives up to order k and they belong to Lp(Ω).

Remarks 1.8: (1) As in Lp spaces we identify W k,p functions which are equal almost everywhere. k,p (2) There are several ways to deﬁne a norm on W (Ω). The norm k,p k · kW (Ω) is equivalent, for example, with the norm

X α D u Lp(Ω), 1 p . α k k k É É ∞ | |É

and k, is also equivalent with k · kW ∞(Ω) α max D u L (Ω). α k k k ∞ | |É (3) For k 1 we use the norm = 1 ³ p p ´ p u 1,p u Du k kW (Ω) = k kLp(Ω) + k kLp(Ω) 1 µ ¶ p u p dx Du p dx , 1 p , = ˆΩ | | + ˆΩ | | É < ∞ and

u 1, esssup u esssup Du . W ∞(Ω) k k = Ω | | + Ω | | We may also consider equivalent norms

1 Ã n ! p p X ° °p u 1,p u D u , W (Ω) Lp(Ω) ° j °Lp(Ω) k k = k k + j 1 = n X ° ° p u W1,p(Ω) u L (Ω) °D j u°Lp(Ω) , k k = k k + j 1 = and

u 1,p u Lp( ) Du Lp( ) k kW (Ω) = k k Ω + k k Ω when 1 p and É < ∞ © ª u 1, max u L (Ω), D1 u L (Ω) ,..., Dnu L ( ) . k kW ∞(Ω) = k k ∞ k k ∞ k k ∞ Ω CHAPTER 1. SOBOLEV SPACES 6

α Example 1.9. Let u : B(0,1) [0, ], u(x) x − , α 0. Clearly u C∞(B(0,1) \ → ∞ = | | > ∈ {0}), but u is unbounded in any neighbourhood of the origin. We start by showing that u has a weak derivative in the entire unit ball. When x 0 , we have 6= Çu x j x j (x) α x α 1 α , j 1,..., n. − − α 2 Çx j = − | | x = − x = | | | | + Thus x Du(x) α . = − x α 2 | | + Gauss’ theorem gives

D j(uϕ) dx uϕν j dS, ˆB(0,1)\B(0,ε) = ˆÇ(B(0,1)\B(0,ε)) where ν (ν1,...,νn) is the outward pointing unit ( ν 1) normal of the boundary = | | = and ϕ C∞(B(0,1)). As ϕ 0 on ÇB(0,1), this can be written as ∈ 0 =

D j uϕ dx uD jϕ dx uϕν j dS. ˆB(0,1)\B(0,ε) + ˆB(0,1)\B(0,ε) = ˆÇB(0,ε)

By rearranging terms, we obtain

uD jϕ dx D j uϕ dx uϕν j dS. (1.2) ˆB(0,1)\B(0,ε) = −ˆB(0,1)\B(0,ε) + ˆÇB(0,ε)

x Let us estimate the last term on the right-hand side. Since ν(x) x , we have x j = − | | ν j(x) x , when x ÇB(0,ε). Thus = − | | ∈ ¯ ¯ ¯ u dS¯ α dS ¯ ϕν j ¯ ϕ L∞(B(0,1)) ε− ¯ˆÇB(0,ε) ¯ É k k ˆÇB(0,ε) n 1 α ϕ L (B(0,1))ωn 1ε − − 0 as ε 0, = k k ∞ − → → n 1 if n 1 α 0. Here ωn 1 H − (ÇB(0,1)) is the (n 1)-dimensional measure of − − > − = − the sphere ÇB(0,1).

Next we study integrability of D j u. We need this information in order to be able to use the dominated convergence theorem. A straightforward computation gives

¯ ¯ α 1 ¯D j u¯ dx Du dx α x − − dx ˆB(0,1) É ˆB(0,1) | | = ˆB(0,1) | | 1 1 α 1 α 1 n 1 α x − − dS dr αωn 1 r− − + − dr = ˆ0 ˆÇB(0,r) | | = − ˆ0 1 ¯1 n α 2 αωn 1 n α 1¯ αωn 1 r − − dr − r − − ¯ , = − ˆ = n α 1 ¯ < ∞ 0 − − 0 if n 1 α 0. − − > CHAPTER 1. SOBOLEV SPACES 7

The following argument shows that D j u is a weak derivative of u also in a neighbourhood of the origin. By the dominated convergence theorem ³ ´ uD jϕ dx lim uD jϕχ dx ˆ = ˆ ε 0 B(0,1)\B(0,ε) B(0,1) B(0,1) →

lim uD jϕ dx = ε 0 ˆ → B(0,1)\B(0,ε)

lim D j uϕ dx lim uϕν j dS = − ε 0 ˆ + ε 0 ˆ → B(0,1)\B(0,ε) → ÇB(0,ε)

lim D j uϕχ dx = −ˆ ε 0 B(0,1)\B(0,ε) B(0,1) →

D j uϕ dx. = −ˆB(0,1) Here we used the dominated convergence theorem twice: First to the function

uD , jϕχB(0,1)\B(0,ε) which is dominated by u Dϕ L1(B(0,1)), and then to the function | |k k∞ ∈ D u , j ϕχB(0,1)\B(0,ε) which is dominated by Du ϕ L1(B(0,1)). We also used (1.2) and the fact that | |k k∞ ∈ the last term there converges to zero as ε 0. → Now we have proved that u has a weak derivative in the unit ball. We note p n that u L (B(0,1)) if and only if pα n 0, or equivalently, α p . On the ∈ p − + > < n p other hand, Du L (B(0,1), if p(α 1) n 0, or equivalently, α −p . Thus 1,p | | ∈ − n p + + > < u W (B(0,1)) if and only if α − . ∈ < p Let (ri) be a countable and dense subset of B(0,1) and deﬁne u : B(0,1) [0, ], → ∞ 1 u(x) X∞ x r α. i i − = i 1 2 | − | = 1,p n p Then u W (B(0,1)) if α − . ∈ < p Reason.

X∞ 1 ° α° u 1,p x r W (B(0,1)) i ° i − °W1,p(B(0,1)) k k É i 1 2 | − | = 1 X∞ ° x α° i ° − °W1,p(B(0,1)) É i 1 2 | | = ° α° ° x − °W1,p(B(0,1)) . = | | < ∞ ■ Note that if α 0, then u is unbounded in every open subset of B(0,1) and not > differentiable in the classical sense in a dense subset.

T HEMORAL : Functions in W1,p, 1 p n, n 2, may be unbounded in every É < Ê open subset. CHAPTER 1. SOBOLEV SPACES 8

α 1,n Example 1.10. Observe, that u(x) x − , α 0, does not belong to W (B(0,1). = | | > However, there are unbounded functions in W1,n, n 2. Let u : B(0,1) R, Ê →  ³ ³ 1 ´´ log log 1 x , x 0, u(x) + | | 6= = 0, x 0. = 1,n Then u W (B(0,1)) when n 2, but u L∞(B(0,1)). This can be used to ∈ Ê ∉ construct a function in W1,n(B(0,1) that is unbounded in every open subset of B(0,1) (exercise).

T HEMORAL : Functions in W1,p, 1 p n, n 2, are not continuous. Later É É Ê we shall see, that every W1,p function with p n coincides with a continuous > function almost everywhere.

Example 1.11. The function u : B(0,1) R, →  1, xn 0, u(x) u(x1,..., xn) > = = 0, xn 0, < does not belong to W1,p(B(0,1) for any 1 p (exercise). É É ∞

1.3 Properties of weak derivatives

The following general properties of weak derivatives follow rather directly from the deﬁnition.

Lemma 1.12. Assume that u,v W k,p(Ω) and α k. Then ∈ | | É α k α ,p (1) D u W −| | (Ω), ∈ (2) Dβ(Dαu) Dα(Dβu) for all multi-indices α,β with α β k, = | | + | | É (3) for every λ,µ R, λu µv W k,p(Ω) and ∈ + ∈ Dα(λu µv) λDαu µDαv, + = + k,p (4) if Ω0 Ω is open, then u W (Ω0), ⊂ ∈ k,p (5) (Leibniz’s formula) if η C∞(Ω), then ηu W (Ω) and ∈ 0 ∈ Ã ! α X α β α β D (ηu) D η D − u, = β α β É where Ã ! α α! , α! α1!...αn! β = β!(α β)! = − and β α means that β j α j for every j 1,..., n. É É = CHAPTER 1. SOBOLEV SPACES 9

T HEMORAL : Weak derivatives have the same properties as classical derivatives of smooth functions.

Proof. (1) Follows directly from the deﬁnition of weak derivatives. See also (2). β (2) Let ϕ C∞(Ω). Then D ϕ C∞(Ω). Therefore ∈ 0 ∈ 0

for all test functions ϕ C∞(Ω). Notice that ∈ 0

α α β α1 ... αn (α1 β1) ... (αn βn) | | + | + | = + + + + + + + 2(α1 ... αn) β1 ... βn = + + + + + 2 α β . = | | + | | As 2 α is an even number, the estimate above, together with the uniqueness | | β α α β results Lemma 1.4 and Corollary 1.5, implies that D (D u) D + u. = (3) and (4) Clear.

(5) First we consider the case α 1. Let ϕ C∞(Ω). By Leibniz’s rule for | | = ∈ 0 differentiable functions and the deﬁnition of weak derivative

ηuDαϕ dx (uDα(ηϕ) u(Dαη)ϕ) dx ˆΩ = ˆΩ −

(ηDαu uDαη)ϕ dx = −ˆΩ +

for all ϕ C∞(Ω). The case α 1 follows by induction (exercise). ∈ 0 | | > ä

1.4 Completeness of Sobolev spaces

One of the most useful properties of Sobolev spaces is that they are complete. Thus Sobolev spaces are closed under limits of Cauchy sequences. k,p k,p A sequence (ui) of functions ui W (Ω), i 1,2,..., converges in W (Ω) to ∈ = a function u W k,p(Ω), if for every ε 0 there exists i such that ∈ > ε

ui u k,p ε when i i . k − kW (Ω) < Ê ε Equivalently,

lim ui u W k,p( ) 0. i k − k Ω = →∞ k,p A sequence (ui) is a Cauchy sequence in W (Ω), if for every ε 0 there exists > iε such that

ui u j k,p ε when i, j i . k − kW (Ω) < Ê ε CHAPTER 1. SOBOLEV SPACES 10

W ARNING : This is not the same condition as

ui 1 ui W k,p(Ω) ε when i iε. k + − k < Ê Indeed, the Cauchy sequence condition implies this, but the converse is not true (exercise).

Theorem 1.13 (Completeness). The Sobolev space W k,p(Ω), 1 p , k É É ∞ = 1,2,..., is a Banach space.

T HEMORAL : The spaces Ck(Ω), k 1,2,..., are not complete with respect = to the Sobolev norm, but Sobolev spaces are. This is important in existence arguments for PDEs.

Proof. Step 1: k,p is a norm. k · kW (Ω)

Reason. (1) u k,p 0 u 0 almost everywhere in Ω. k kW (Ω) = ⇐⇒ = u k,p 0 implies u Lp( ) 0, which implies that u 0 almost every- =⇒ k kW (Ω) = k k Ω = = where in Ω. u 0 almost everywhere in Ω implies ⇐= =

α α α D uϕ dx ( 1)| | uD ϕ dx 0 ˆΩ = − ˆΩ = α for all ϕ C∞(Ω). This together with Corollary 1.5 implies that D u 0 almost ∈ 0 = everywhere in Ω for all α, α k. | | É (2) λu k,p λ u k,p , λ R. Clear. k kW (Ω) = | |k kW (Ω) ∈ (3) The triangle inequality for 1 p follows from the elementary inequal- É < ∞ ity (a b)α aα bα, a, b 0, 0 α 1, and Minkowski’s inequality, since + É + Ê < É 1 Ã ! p X α α p u v k,p D u D v W (Ω) Lp(Ω) k + k = α k k + k | |É 1 Ã ! p X ¡ α α ¢p D u Lp(Ω) D v Lp(Ω) É α k k k + k k | |É 1 1 Ã ! p Ã ! p X Dαu p X Dαv p Lp(Ω) Lp(Ω) É α k k k + α k k k | |É | |É u W k,p(Ω) v W k,p(Ω). = k k + k k ■ k,p Step 2: Let (ui) be a Cauchy sequence in W (Ω). As

α α D ui D u j Lp( ) ui u j k,p , α k, k − k Ω É k − kW (Ω) | | É α p it follows that (D ui) is a Cauchy sequence in L (Ω), α k. The completeness | | É p p α p of L (Ω) implies that there exists uα L (Ω) such that D ui uα in L (Ω). In p ∈ → particular, ui u(0,...,0) u in L (Ω). → = CHAPTER 1. SOBOLEV SPACES 11

Step 3: We show that Dαu u , α k. We would like to argue = α | | É

α α uD ϕ dx lim ui D ϕ dx ˆ = i ˆ Ω →∞ Ω α α lim ( 1)| | D uiϕ dx = i − ˆ →∞ Ω α ( 1)| | uαϕ dx = − ˆΩ

for every ϕ C∞(Ω). On the second line we used the deﬁnition of the weak ∈ 0 derivative. Next we show how to conclude the ﬁrst and last equalities above.

1 p Let ϕ C∞(Ω). By Hölder’s inequality we have < < ∞ ∈ 0 ¯ ¯ ¯ ¯ ¯ α α ¯ ¯ α ¯ ¯ ui D ϕ dx uD ϕ dx¯ ¯ (ui u)D ϕ dx¯ ¯ˆΩ − ˆΩ ¯ = ¯ˆΩ − ¯ α ui u Lp( ) D ϕ p 0 É k − k Ω k kL 0 (Ω) → and consequently we obtain the ﬁrst inequality above. The last inequality follows in the same way, since ¯ ¯ ¯ α ¯ α D u ϕ dx u ϕ dx D u u p ϕ 0. ¯ i α ¯ i α L (Ω) Lp0 (Ω) ¯ˆΩ − ˆΩ ¯ É k − k k k →

p 1, p A similar argument as above (exercise). = = ∞ This means that the weak derivatives Dαu exist and Dαu u , α k. As = α | | É α α we also know that D ui uα D u, α k, we conclude that ui u W k,p(Ω) 0. k,p → = | | É k − k → Thus ui u in W (Ω). → ä Remark 1.14. W k,p(Ω), 1 p is separable. In the case k 1 consider the É < ∞ = mapping u (u,Du) from W1,p(Ω) to Lp(Ω) Lp(Ω)n and recall that a subset of a 7→ 1, × separable space is separable. However, W ∞(Ω) is not separable (exercise).

1.5 Hilbert space structure

The space W k,2(Ω) is a Hilbert space with the inner product

X α α u,v W k,2(Ω) D u,D v L2(Ω), 〈 〉 = α k〈 〉 | |É where α α α α D u,D v L2(Ω) D uD v dx. 〈 〉 = ˆΩ Observe that 1 2 u k,2 u, u . k kW (Ω) = 〈 〉W k,2(Ω) CHAPTER 1. SOBOLEV SPACES 12

1.6 Approximation by smooth functions

This section deals with the question whether every function in a Sobolev space can be approximated by a smooth function. n Deﬁne φ C∞(R ) by ∈ 0  1 c e x 2 1 , x 1, φ(x) | | − | | < = 0, x 1, | | Ê where c 0 is chosen so that > φ(x) dx 1. ˆRn =

For ε 0, set > 1 ³ x ´ φε(x) φ . = εn ε The function φ is called the standard molliﬁer or Friedrich’s molliﬁer. Observe that φ 0, suppφ B(0,ε) and ε Ê ε = 1 ³ x ´ 1 x dx dx y n d y x dx φε( ) n φ n φ( )ε φ( ) 1 ˆRn = ε ˆRn ε = ε ˆRn = ˆRn =

for all ε 0. Here we used the change of variable y x , dx εn d y. > = ε = Notation. If Ω Rn is open with ÇΩ , we write ⊂ 6= ; Ω {x Ω : dist(x,ÇΩ) ε}, ε 0. ε = ∈ > > If f L1 (Ω), we obtain its standard convolution molliﬁcation f : Ω [ , ], ∈ loc ε ε → −∞ ∞

fε(x) (f φε)(x) f (y)φε(x y) d y. = ∗ = ˆΩ − ■

T HEMORAL : Since the convolution is a weighted integral average of f over the ball B(x,ε) for every x, instead of Ω it is well deﬁned only in Ω . If Ω Rn, we ε = do not have this problem.

Remarks 1.15: (1) For every x Ω , we have ∈ ε

fε(x) f (y)φε(x y) d y f (y)φε(x y) d y. = ˆΩ − = ˆB(x,ε) −

(2) By a change of variables z x y we have = −

f (y)φε(x y) d y f (x z)φε(z) dz ˆΩ − = ˆΩ − CHAPTER 1. SOBOLEV SPACES 13

(3) For every x Ω , we have ∈ ε ¯ ¯ ¯ ¯ fε(x) ¯ f (y)φε(x y) d y¯ φε f (y) d y . | | É ¯ˆB(x,ε) − ¯ É k k∞ ˆB(x,ε) | | < ∞

(4) If f C0(Ω), then f C0(Ω ), whenever ∈ ε ∈ ε 1 0 ε ε0 dist(supp f ,ÇΩ). < < = 2

Reason. If x Ω s.t. dist(x,supp f ) ε0 (in particular, for every x Ω \ ∈ ε > ∈ ε Ωε0 ) then B(x,ε) supp f , which implies that fε(x) 0. ∩ = ; = ■ Lemma 1.16 (Properties of molliﬁers).

(1) f C∞(Ω ). ε ∈ ε (2) f f almost everywhere as ε 0. ε → → (3) If f C(Ω), then f f uniformly in every Ω0 Ω. ∈ ε → b p p (4) If f L (Ω), 1 p , then f f in L (Ω0) for every Ω0 Ω. ∈ loc É < ∞ ε → b WARNING: Assertion (4) does not hold for p , since there are functions = ∞ in L∞(Ω) that are not continuous.

Proof. (1) Let x Ω , j 1,..., n, e j (0,...,1,...,0) (the jth component is 1). Let ∈ ε = = h0 0 such that B(x, h0) Ω and let h R, h h0. Then > ⊂ ε ∈ | | < · µ ¶ ¸ fε(x he j) fε(x) 1 1 x he j y ³ x y ´ + − + − f y d y n φ φ − ( ) . h = ε ˆB(x he j,ε) B(x,ε) h ε − ε + ∪

Let Ω0 B(x, h0 ε). Then Ω0 Ω and B(x he j,ε) B(x,ε) Ω0. = + b + ∪ ⊂ Claim:

1 · µ x he j y ¶ ³ x y ´¸ 1 Çφ ³ x y ´ φ + − φ − − for every y Ω0 as h 0. h ε − ε → ε Çx j ε ∈ →

¡ x y ¢ Reason. Let ψ(x) φ − . Then = ε Çψ 1 Çφ ³ x y ´ (x) − , j 1,..., n Çx j = ε Çx j ε = and h Ç h ψ(x he j) ψ(x) (ψ(x te j)) dt Dψ(x te j) e j dt. + − = ˆ0 Çt + = ˆ0 + · ■ Thus h | | ψ(x he j) ψ(x) Dψ(x te j) e j dt | + − | É ˆ0 | + · | h ¯ µ ¶¯ 1 | | ¯ x te j y ¯ ¯Dφ + − ¯ dt É ε ˆ0 ¯ ε ¯ h | | Dφ L (Rn). É ε k k ∞ CHAPTER 1. SOBOLEV SPACES 14

This estimate shows that we can use the Lebesgue dominated convergence theorem (on the third row) to obtain

Çf fε(x he j) fε(x) ε (x) lim + − Çx j = h 0 h → 1 1 · µ x he j y ¶ ³ x y ´¸ lim φ + − φ − f (y) d y = h 0 εn ˆ h ε − ε → Ω0 1 1 Çφ ³ x y ´ − f (y) d y = εn ˆ ε Çx ε Ω0 j Çφ ε (x y) f (y) d y = ˆ Çx − Ω0 j µ Çφ ¶ ε f (x). = Çx j ∗

α A similar argument shows that D fε exists and

Dα f Dαφ f in Ω ε = ε ∗ ε for every multi-index α.

(2) Recall that B(x,ε) φε(x y) d y 1. Therefore we have ´ − = ¯ ¯ ¯ ¯ fε(x) f (x) ¯ φε(x y)f (y) d y f (x) φε(x y) d y¯ | − | = ¯ˆB(x,ε) − − ˆB(x,ε) − ¯ ¯ ¯ ¯ ¯ ¯ φε(x y)(f (y) f (x)) d y¯ = ¯ˆB(x,ε) − − ¯ 1 ³ x y ´ f y f x d y n φ − ( ) ( ) É ε ˆB(x,ε) ε | − | 1 Ωn φ L (Rn) f (y) f (x) d y 0 É k k ∞ B(x,ε) ˆ | − | → | | B(x,ε) for almost every x Ω as ε 0. Here Ωn B(0,1) and the last convergence ∈ → = | | follows from the Lebesgue’s differentiation theorem.

(3) Let Ω0 Ω00 Ω, 0 ε dist(Ω0,ÇΩ00), and x Ω0. Because Ω is compact b b < < ∈ 00 and f C(Ω), f is uniformly continuous in Ω00, that is, for every ε0 0 there exists ∈ > δ 0 such that >

f (x) f (y) ε0 for every x, y Ω00 with x y δ. | − | < ∈ | − | < By combining this with an estimate from the proof of (ii), we conclude that 1 fε(x) f (x) Ωn φ L (Rn) f (y) f (x) d y Ωn φ L (Rn) ε0 | − | É k k ∞ B(x,ε) ˆ | − | < k k ∞ | | B(x,ε)

for every x Ω0 if ε δ. ∈ < (4) Let Ω0 b Ω00 b Ω. Claim: f p dx f p dx ˆ | ε| É ˆ | | Ω0 Ω00 whenever 0 ε dist(Ω0,ÇΩ00) and 0 ε dist(Ω00,ÇΩ). < < < < CHAPTER 1. SOBOLEV SPACES 15

Reason. Take x Ω0. Hölder’s inequality implies ∈ ¯ ¯ ¯ ¯ fε(x) ¯ φε(x y)f (y) d y¯ | | = ¯ˆB(x,ε) − ¯ 1 1 1 p p φε(x y) − φε(x y) f (y) d y É ˆB(x,ε) − − | | 1 1 µ ¶ µ ¶ p p0 p φε(x y) d y φε(x y) f (y) d y É ˆB(x,ε) − ˆB(x,ε) − | |

By raising the previous estimate to power p and by integrating over Ω0, we obtain

f (x) p dx φ (x y) f (y) p d y dx ˆ | ε | É ˆ ˆ ε − | | Ω0 Ω0 B(x,ε)

Here we used Fubini’s theorem and once more the fact that the integral of φε is one. ■ p Since C(Ω00) is dense in L (Ω00). Therefore for every ε0 0 there exists g C(Ω00) > ∈ such that 1 µ ¶ p ε f g p dx 0 . ˆ | − | É 3 Ω00

By (3), we have g g uniformly in Ω0 as ε 0. Thus ε → → 1 µ ¶ p 1 p ¯ ¯ ε0 g g dx sup g g ¯Ω0¯ p , ˆ | ε − | É | ε − | < 3 Ω00 Ω0 when ε 0 is small enough. Now we use Minkowski’s inequality and the previous > claim to conclude that

1 1 µ ¶ p µ ¶ p f f p dx f g p dx ˆ | ε − | É ˆ | ε − ε| Ω0 Ω0 1 1 µ ¶ p µ ¶ p g g p dx g f p dx + ˆ | ε − | + ˆ | − | Ω0 Ω0 1 1 µ ¶ p µ ¶ p 2 g f p dx g g p dx É ˆ | − | + ˆ | ε − | Ω00 Ω0 ε0 ε0 2 ε0. É 3 + 3 =

p Thus f f in L (Ω0) as ε 0. ε → → ä CHAPTER 1. SOBOLEV SPACES 16

1.7 Local approximation in Sobolev spaces

Next we show that the convolution approximation converges locally in Sobolev spaces.

Theorem 1.17. Let u W k,p(Ω), 1 p . then ∈ É < ∞ (1) Dαu Dαu φ in Ω and ε = ∗ ε ε k,p (2) u u in W (Ω0) for every Ω0 Ω. ε → b

T HEMORAL : Smooth functions are dense in local Sobolev spaces. Thus every Sobolev function can be locally approximated with a smooth function in the Sobolev norm.

Proof. (1) Fix x Ω . Then ∈ ε

Dαu (x) Dα(u φ )(x) (u Dαφ )(x) ε = ∗ ε = ∗ ε α Dx φε(x y)u(y) d y = ˆΩ − α α ( 1)| | D y (φε(x y))u(y) d y. = − ˆΩ − Here we ﬁrst used the proof of Lemma 1.16 (1) and then the fact that

Ç ³ ³ x y ´´ Ç ³ ³ y x ´´ Ç ³ ³ x y ´´ φ − φ − φ − . Çx j ε = − Çx j ε = − Çyj ε

For every x Ω , the function ϕ(y) φ (x y) belongs to C∞(Ω). Therefore ∈ ε = ε − 0

α α α D y (φε(x y))u(y) d y ( 1)| | D u(y)φε(x y) d y. ˆΩ − = − ˆΩ − By combining the above facts, we see that

α α α α α D uε(x) ( 1)| |+| | D u(y)φε(x y) d y (D u φε)(x). = − ˆΩ − = ∗

α α Notice that ( 1)| |+| | 1. − = α (2) Let Ω0 Ω, and choose ε 0 s.t. Ω0 Ω . By (i) we know that D u b > ⊂ ε ε = α α α p D u φ in Ω0, α k. By Lemma 1.16, we have D u D u in L (Ω0) as ε 0, ∗ ε | | É ε → → α k. Consequently | | É 1 Ã ! p X α α p uε u W k,p( ) D uε D u p 0. Ω0 L (Ω0) k − k = α k k − k → | |É ä CHAPTER 1. SOBOLEV SPACES 17

1.8 Global approximation in Sobolev spaces

The next result shows that the convolution approximation converges also globally in Sobolev spaces.

Theorem 1.18 (Meyers-Serrin). If u W k,p(Ω), 1 p , then there exist k,p ∈ Ék,p < ∞ functions ui C∞(Ω) W (Ω) such that ui u in W (Ω). ∈ ∩ →

T HEMORAL : Smooth functions are dense in Sobolev spaces. Thus every Sobolev function can be approximated with a smooth function in the Sobolev norm. In particular, this holds true for the function with a dense inﬁnity set in Example 1.9.

Proof. Let Ω0 and = ; ½ 1 ¾ Ωi x Ω : dist(x,ÇΩ) B(0, i), i 1,2,.... = ∈ > i ∩ =

Then [∞ Ω Ωi and Ω1 b Ω2 b ... b Ω. = i 1 =

Claim: There exist ηi C0∞(Ωi 2 \ Ωi 1), i 1,2,..., such that 0 ηi 1 and ∈ + − = É É

X∞ ηi(x) 1 for every x Ω. i 1 = ∈ =

This is a partition of unity subordinate to the covering {Ωi}.

Reason. By using the distance function and convolution approximation we can

construct ηi C0∞(Ωi 2\Ωi 1) such that 0 ηi 1 and ηi 1 in Ωi 1\Ωi (exercise). e ∈ + − É e É e = + Then we deﬁne ηei(x) ηi(x) P , i 1,2,.... = ∞j 1 ηe j(x) = = Observe that the sum is only over four indices in a neighbourhood of a given point. ■ k,p Now by Lemma 1.12 (5), ηi u W (Ω) and ∈

supp(ηi u) Ωi 2 \ Ωi 1. ⊂ + −

Let ε 0. Choose εi 0 so small that > >

supp(φεi (ηi u)) Ωi 2 \ Ωi 1 ∗ ⊂ + − (see Remark 1.15 (4)) and ε φ (ηi u) ηi u k,p , i 1,2,.... k εi ∗ − kW (Ω) < 2i = CHAPTER 1. SOBOLEV SPACES 18

By Theorem 1.17 (2), this is possible. Deﬁne

X∞ v φεi (ηi u). = i 1 ∗ =

This function belongs to C∞(Ω), since in a neighbourhood of any point x Ω, there ∈ are at most ﬁnitely many nonzero terms in the sum. Moreover, ° ° °X∞ X∞ ° v u k,p ° φ (ηi u) ηi u° k − kW (Ω) = ° εi ∗ − ° °i 1 i 1 °W k,p = = (Ω) X∞ ° ° °φεi (ηi u) ηi u°W k,p(Ω) É i 1 ∗ − = ε X∞ . i ε É i 1 2 = ä =

Remarks 1.19: (1) The Meyers-Serrin theorem 1.18 gives the following characterization for the Sobolev spaces W k,p(Ω), 1 p : u W k,p(Ω) if and only if there Ék,p < ∞ ∈ exist functions ui C∞(Ω) W (Ω), i 1,2,..., such that ui u in k,p ∈ ∩ k,p = → W (Ω) as i . In other words, W (Ω) is the completion of C∞(Ω) in → ∞ the Sobolev norm.

Reason. Theorem 1.18. =⇒ Theorem 1.13. ⇐= ■

(2) The Meyers-Serrin theorem 1.18 is false for p . Indeed, if ui C∞(Ω) 1, 1, = ∞ 1 ∈ ∩ W ∞(Ω) such that ui u in W ∞(Ω), then u C (Ω) (exercise). Thus → ∈ 1, special care is required when we consider approximations in W ∞(Ω).

(3) Let Ω0 b Ω. The proof of Theorem 1.17 and Theorem 1.18 shows that for every ε 0 there exists v C∞(Ω) such that v u 1,p ε. > ∈ 0 k − kW (Ω0) < (4) The proof of Theorem 1.18 shows that not only C∞(Ω) but also C0∞(Ω) is dense in Lp(Ω), 1 p . É < ∞

1.9 Sobolev spaces with zero boundary values

In this section we study deﬁnitions and properties of ﬁrst order Sobolev spaces with zero boundary values in an open subset of Rn. A similar theory can be developed for higher order Sobolev spaces as well. Recall that, by Theorem 1.18, 1,p the Sobolev space W (Ω) can be characterized as the completion of C∞(Ω) with respect to the Sobolev norm when 1 p . É < ∞ CHAPTER 1. SOBOLEV SPACES 19

Deﬁnition 1.20. Let 1 p . The Sobolev space with zero boundary values 1,p É < ∞ W0 (Ω) is the completion of C0∞(Ω) with respect to the Sobolev norm. Thus 1,p u W0 (Ω) if and only if there exist functions ui C0∞(Ω), i 1,2,..., such that ∈ 1,p 1,p ∈ = ui u in W (Ω) as i . The space W (Ω) is endowed with the norm of → → ∞ 0 W1,p(Ω).

T HEMORAL : The only difference compared to W1,p(Ω) is that functions in 1,p W0 (Ω) can be approximated by C0∞(Ω) functions instead of C∞(Ω) functions, that is, 1,p W1,p(Ω) C (Ω) and W (Ω) C (Ω), = ∞ 0 = 0∞ where the completions are taken with respect to the Sobolev norm. A function 1,p in W0 (Ω) has zero boundary values in Sobolev’s sense. We may say that u, v 1,p ∈ W1,p(Ω) have the same boundary values in Sobolev’s sense, if u v W (Ω). This − ∈ 0 is useful, for example, in Dirichlet problems for PDEs.

1,p 1,p W ARNING : Roughly speaking a function in W (Ω) belongs to W0 (Ω), if it vanishes on the boundary. This is a delicate issue, since the function does not have to be zero pointwise on the boundary. We shall return to this question later.

1,p 1,p Remark 1.21. W0 (Ω) is a closed subspace of W (Ω) and thus complete (exercise).

Remarks 1.22: 1,p 1,p p (1) Clearly C∞(Ω) W (Ω) W (Ω) L (Ω). 0 ⊂ 0 ⊂ ⊂ 1,p (2) If u W (Ω), then the zero extension u : Rn [ , ], ∈ 0 e → −∞ ∞  u(x), x Ω, ue(x) ∈ = 0, x Rn \ Ω, ∈ belongs to W1,p(Rn) (exercise).

Lemma 1.23. If u W1,p(Ω) and supp u is a compact subset of Ω, then u 1,p ∈ ∈ W0 (Ω).

Proof. Let η C∞(Ω) be a cutoff function such that η 1 on the support of u. ∈ 0 = 1,p Claim: If ui C∞(Ω), i 1,2,..., such that ui u in W (Ω), then ηui C∞(Ω) ∈ = → ∈ 0 converges to ηu u in W1,p(Ω). = Reason. We observe that

1 ³ p p ´ p ηui ηu 1,p ηui ηu p D(ηui ηu) p k − kW (Ω) = k − kL (Ω) + k − kL (Ω) ηui ηu Lp( ) D(ηui ηu) Lp( ), É k − k Ω + k − k Ω CHAPTER 1. SOBOLEV SPACES 20 where

1 µ ¶ p p ηui ηu Lp(Ω) ηui ηu dx k − k = ˆΩ | − | 1 µ ¶ p p p η ui u dx = ˆΩ | | | − | 1 µ ¶ p u u p dx 0 η L∞(Ω) i É k k ˆΩ | − | → and by Lemma 1.12 (5)

1 µ ¶ p p D(ηui ηu) Lp(Ω) D(ηui ηu) dx k − k = ˆΩ | − | 1 µ ¶ p p (ui u)Dη (Dui Du)η dx = ˆΩ | − + − | 1 1 µ ¶ p µ ¶ p p p (ui u)Dη dx (Dui Du)η dx É ˆΩ | − | + ˆΩ | − | 1 µ ¶ p D u u p dx η L∞(Ω) i É k k ˆΩ | − | 1 µ ¶ p Du Du p dx 0 η L∞(Ω) i + k k ˆΩ | − | → as i . → ∞ ■ 1,p Since ηui C0∞(Ω), i 1,2,..., and ηui u in W (Ω), we conclude that 1,p ∈ = → u W (Ω). ∈ 0 ä 1,p Since W (Ω) W1,p(Ω), functions in these spaces have similar general prop- 0 ⊂ erties and they will not be repeated here. Thus we shall focus on properties that are typical for Sobolev spaces with zero boundary values.

Lemma 1.24. W1,p(Rn) W1,p(Rn) with 1 p . = 0 É < ∞

T HEMORAL : The standard Sobolev space and the Sobolev space with zero boundary value coincide in the whole space.

1,p W ARNING : W1,p(B(0,1)) W (B(0,1)), 1 p . Thus the spaces are not 6= 0 É < ∞ same in general.

1,p n Proof. Assume that u W (R ). Let ηk C0∞(B(0, k 1)) such that ηk 1 on ∈ ∈ + 1,p n = B(0, k), 0 ηk 1 and Dηk c. Lemma 1.23 implies uηk W (R ). É É | | É ∈ 0 1,p n Claim: uηk u in W (R ) as k . → → ∞ CHAPTER 1. SOBOLEV SPACES 21

Reason.

u uηk 1,p n u uηk Lp(Rn) D(u uηk) Lp(Rn) k − kW (R ) É k − k + k − k 1 1 µ ¶ p µ ¶ p p p u(1 ηk) dx D(u(1 ηk)) dx = ˆRn | − | + ˆRn | − | 1 1 µ ¶ p µ ¶ p p p u(1 ηk) dx (1 ηk)Du uDηk) dx = ˆRn | − | + ˆRn | − − | 1 1 µ ¶ p µ ¶ p p p u(1 ηk) dx (1 ηk)Du dx É ˆRn | − | + ˆRn | − | 1 µ ¶ p p uDηk dx . + ˆRn | | p p We note that limk u(1 ηk) 0 almost everywhere and u(1 ηk) u →∞ − = | − | É | | ∈ L1(Rn) will do as an integrable majorant. The dominated convergence theorem gives 1 µ ¶ p p u(1 ηk) dx 0. ˆRn | − | → A similar argument shows that

1 µ ¶ p p (1 ηk)Du dx 0 ˆRn | − | → as k . Moreover, by the dominated convergence theorem → ∞ 1 1 µ ¶ p µ ¶ p p p uDηk dx c u dx ˆRn | | É ˆB(0,k 1)\B(0,k) | | + 1 µ ¶ p p c u χB(0,k 1)\B(0,k) dx 0 = ˆRn | | + → p p 1 n as k . Here u χB(0,k 1)\B(0,k) u L (R ) will do as an integrable majo- → ∞ | | + É | | ∈ rant. ■ 1,p n 1,p n 1,p Since uηk W0 (R ), i 1,2,..., uηk u in W (R ) as k and W0 (Ω) ∈ = 1,p → → ∞ is complete, we conclude that u W (Ω). ∈ 0 ä

1.10 Chain rule

We shall prove some useful results for the ﬁrst order Sobolev spaces W1,p(Ω), 1 p . É < ∞ 1,p 1 Lemma 1.25 (Chain rule). If u W (Ω) and f C (R) such that f 0 L∞(R) ∈ ∈ ∈ and f (0) 0, then f u W1,p(Ω) and = ◦ ∈

D j(f u) f 0(u)D j u, j 1,2,..., n ◦ = = almost everywhere in Ω. CHAPTER 1. SOBOLEV SPACES 22

1,p Proof. By Theorem 1.18, there exist a sequence of functions ui C∞(Ω) W (Ω), 1,p ∈ ∩ i 1,2,..., such that ui u in W (Ω) as i . Let ϕ C∞(Ω). = → → ∞ ∈ 0 Claim: (f u)D jϕ dx lim f (ui)D jϕ dx. ˆ ◦ = i ˆ Ω →∞ Ω Reason. 1 p By Hölder’s inequality < < ∞ ¯ ¯ ¯ ¯ ¯ f (u)D jϕ dx f (ui)D jϕ dx¯ f (u) f (ui) Dϕ dx ¯ˆΩ − ˆΩ ¯ É ˆΩ | − || | 1 1 µ ¶ p µ ¶ p p p0 0 f (u) f (ui) dx Dϕ dx É ˆΩ | − | ˆΩ | | 1 1 µ ¶ p µ ¶ p p p0 0 f 0 u ui dx Dϕ dx 0. É k k∞ ˆΩ | − | ˆΩ | | →

On the last row, we used the fact that ¯ u ¯ ¯ ¯ f (u) f (ui) ¯ f 0(t) dt¯ f 0 u ui . ∞ | − | = ¯ˆui ¯ É k k | − | ■ Finally, the convergence to zero follows, because the ﬁrst and the last term are p bounded and ui u in L (Ω). → p 1, p A similar argument as above (exercise). = = ∞ Next, we use the claim above, integration by parts for smooth functions and the chain rule for smooth functions to obtain

(f u)D jϕ dx lim f (ui)D j ϕ dx ˆ ◦ = i ˆ Ω →∞ Ω

lim D j(f (ui))ϕ dx = − i ˆ →∞ Ω

lim f 0(ui)D j uiϕ dx = − i ˆ →∞ Ω

f 0(u)D j uϕ dx = −ˆΩ

(f 0 u)D j uϕ dx, j 1,..., n, = −ˆΩ ◦ =

for every ϕ C∞(Ω). We leave it as an exercise to show the fourth inequality in ∈ 0 the display above. Finally, we need to show that f (u) and f (u) Çu are in Lp(Ω). Since 0 Çx j ¯ u ¯ ¯ ¯ f (u) f (u) f (0) ¯ f 0(t) dt¯ f 0 u , | | = | − | = ¯ˆ0 | ¯ É k k∞| | we have 1 µ ¶1/p µ ¶ p p p f (u) dx f 0 u dx , ˆΩ | | É k k∞ ˆΩ | | < ∞ CHAPTER 1. SOBOLEV SPACES 23

and similarly,

1 1 µ ¶ p µ ¶ p ¯ ¯p p ¯f 0(u)D j u¯ dx f 0 Du dx . ˆ É k k∞ ˆ | | < ∞ Ω Ω ä

1.11 Truncation

The truncation property is an important property of ﬁrst order Sobolev spaces, which means that we can cut the functions at certain level and the truncated function is still in the same Sobolev space. Higher order Sobolev spaces do not enjoy this property, see Example 1.6.

1,p 1,p Theorem 1.26. If u W (Ω), then u+ max{u,0} W (Ω), u− min{u,0} ∈ = ∈ = − ∈ W1,p(Ω), u W1,p(Ω) and | | ∈  Du almost everywhere in {x Ω : u(x) 0}, Du+ ∈ > = 0 almost everywhere in {x Ω : u(x) 0}, ∈ É  0 almost everywhere in {x Ω : u(x) 0}, Du− ∈ Ê =  Du almost everywhere in {x Ω : u(x) 0}, − ∈ < and  Du almost everywhere in {x Ω : u(x) 0},  ∈ > D u 0 almost everywhere in {x Ω : u(x) 0}, | | =  ∈ =  Du almost everywhere in {x Ω : u(x) 0}. − ∈ <

T HEMORAL : In contrast with C1, the Sobolev space W1,p are closed under taking absolute values.

Proof. Let ε 0 and let f : R R, f (t) pt2 ε2 ε. The function f has the > ε → ε = + − ε following properties: f C1(R), f (0) 0 ε ∈ ε =

lim fε(t) t for every t R, ε 0 = | | ∈ →

1 2 2 1/2 t (fε)0(t) (t ε )− 2t for every t R, = 2 + = pt2 ε2 ∈ + and (fε)0 1 for every ε 0. From Lemma 1.25, we conclude that fε u k k∞ É > ◦ ∈ W1,p(Ω) and

(fε u)D jϕ dx (fε)0(u)D j uϕ dx, j 1,..., n, ˆΩ ◦ = −ˆΩ = CHAPTER 1. SOBOLEV SPACES 24

for every ϕ C∞(Ω). We note that ∈ 0  1, t 0,  > lim(fε)0(t) 0, t 0, ε 0 = = →   1, t 0, − < and consequently

u D jϕ dx lim (fε u)D jϕ dx ˆ | | = ε 0 ˆ ◦ Ω → Ω

lim (fε)0(u)D j uϕ dx = − ε 0 ˆ → Ω

D j u ϕ dx, j 1,..., n, = −ˆΩ | | = for every ϕ C∞(Ω), where D j u is as in the statement of the theorem. We leave ∈ 0 | | it as an exercise to prove that the ﬁrst equality in the display above holds. The other claims follow from formulas 1 1 u+ (u u ) and u− ( u u). = 2 + | | = 2 | | − ä

Remarks 1.27: (1) If u,v W1,p(Ω), then max{u,v} W1,p(Ω) and min{u,v} W1,p(Ω). More- ∈ ∈ ∈ over,  Du almost everywhere in {x Ω : u(x) v(x)}, D max{u,v} ∈ Ê = Dv almost everywhere in {x Ω : u(x) v(x)}, ∈ É and  Du almost everywhere in {x Ω : u(x) v(x)}, D min{u,v} ∈ É = Dv almost everywhere in {x Ω : u(x) v(x)}. ∈ Ê 1,p 1,p 1,p If u, v W (Ω), then max{u,v} W (Ω) and min{u,v} W (Ω) (exer- ∈ 0 ∈ 0 ∈ 0 cise).

Reason. 1 1 max{u,v} (u v u v ) and min{u,v} (u v u v ). = 2 + + | − | = 2 + − | − | ■

(2) If u W1,p(Ω) and λ R, then Du 0 almost everywhere in {x Ω : u(x) ∈ ∈ = ∈ = λ} (exercise). 1,p (3) If u W1,p(Ω) and λ R, then min{u,λ} W (Ω) and ∈ ∈ ∈ loc  Du almost everywhere in {x Ω : u(x) λ}, D min{u,λ} ∈ < = 0 almost everywhere in {x Ω : u(x) λ}. ∈ Ê CHAPTER 1. SOBOLEV SPACES 25

A similar claim also holds for max{u,λ}. This implies that a function u W1,p(Ω) can be approximated by the truncated functions ∈ u max{ λ,min{u,λ}} λ = −  λ almost everywhere in {x Ω : u(x) λ},  ∈ Ê u almost everywhere in {x Ω : λ u(x) λ}, =  ∈ − < <  λ almost everywhere in {x Ω : u(x) λ}, − ∈ É − in W1,p(Ω). (Here λ 0.) > Reason. By applying the dominated convergence theorem to

p p p p p 1 p 1 u u 2 ( u u ) 2 + u L (Ω), | − λ| É | | + | λ| É | | ∈ we have p p lim u uλ dx lim u uλ dx 0, λ ˆ | − | = ˆ λ | − | = →∞ Ω Ω →∞ and by applying the dominated convergence theorem to

Du Du p Du p L1(Ω), | − λ| É | | ∈ we have

p p lim Du Duλ dx lim Du Duλ dx 0. λ ˆΩ | − | = ˆΩ λ | − | = →∞ →∞ ■

T HEMORAL : Bounded W1,p functions are dense in W1,p.

1.12 Weak convergence methods for So- bolev spaces

Let 1 p and let Ω Rn be an open set. Recall that Lp(Ω;Rm) is the space of É < ∞ ⊂ Rm-valued p-integrable functions f : Ω Rm with m N. This section discusses → ∈ weak convergence techniques for Lp(Ω;Rm) even though most of the results hold for more general Banach spaces as well.

Deﬁnition 1.28. Let 1 p and m N, and let Ω Rn be an open set. A < < ∞ p m∈ ⊂ p m sequence (fi)i N of functions in L (Ω;R ) converges weakly in L (Ω;R ) to a ∈ function f Lp(Ω;Rm), if ∈

lim fi g dx f g dx i ˆ · = ˆ · →∞ Ω Ω

p0 m p for every g L (Ω;R ) with p0 p 1 . ∈ = − CHAPTER 1. SOBOLEV SPACES 26

Next we show that weakly convergent sequences are bounded and that the Lp norm is lower semicontinuous with respect to the weak convergence.

Lemma 1.29. Let 1 p and m N, and let Ω Rn be an open set. If a < < ∞ ∈ p m ⊂ sequence (fi)i N converges to f weakly in L (Ω;R ), then (fi)i N is bounded in ∈ ∈ Lp(Ω;Rm). Moreover, we have

f Lp(Ω;Rm) liminf fi Lp(Ω;Rm). (1.3) k k É i k k →∞ Proof. The claim

sup fi Lp(Ω;Rm) . i k k < ∞ follows from the uniform boundedness principle or the closed graph theorem. In

p0 m order to prove (1.3), let g L (Ω;R ) with g p m 1 and ∈ k kL 0 (Ω;R ) =

f Lp(Ω;Rm) f (x) g(x) dx. k k = ˆΩ · The deﬁnition of weak convergence, Cauchy–Schwarz’s and Hölder’s inequalities imply

f Lp(Ω;Rm) f (x) g(x) dx lim fi(x) g(x) dx k k = ˆ · = i ˆ · Ω →∞ Ω

liminf fi(x) g(x) dx liminf fi Lp(Ω;Rm) g p m É i ˆ | || | É i k k k kL 0 (Ω;R ) →∞ Ω →∞ liminf fi Lp(Ω;Rm). = i k k ä →∞ T HEMORAL : The Lp-norm is lower semicontinuous with respect to the weak convergence.

A bounded sequence in Lp(Ω;Rm) need not have a convergent subsequence. However, the following result shows that it always has a weakly convergent subsequence if 1 p . This will be important in our applications of weak < < ∞ convergence. The following result holds, since Lp(Ω;Rm) is reﬂexive and separable when 1 p . Theorem 1.30 does not hold for p 1. This can be seen by < < ∞ = considering the standard molliﬁer that approximates the Dirac’s delta.

Theorem 1.30. Let 1 p and m N, and let Ω Rn be an open set. Assume < < ∞ ∈ p m ⊂ that (fi)i N is a bounded sequence in L (Ω;R ). There exists a subsequence ∈ p m p m (fik )k N and a function f L (Ω;R ) such that fik f weakly in L (Ω;R ) as ∈ ∈ → k . → ∞

T HEMORAL : This shows that Lp with 1 p is weakly sequentially < < ∞ compact, that is, every bounded sequence in Lp has a weakly converging subsequence. One of the most useful applications of weak convergence is in compactness arguments. A bounded sequence in Lp does not need to have any convergent subsequence with convergence interpreted in the standard Lp sense. However, there exists a weakly converging subsequence. CHAPTER 1. SOBOLEV SPACES 27

Remark 1.31. Theorem 1.30 is equivalent to the fact that Lp spaces are reﬂexive for 1 p . < < ∞ Weak convergence is often too weak mode of convergence and we need tools to upgrade it to stronger modes of convergence. We begin with the following result, which is related to Lemma 1.29. The next result holds, since Lp(Ω;Rm) is a uniformly convex Banach space.

Lemma 1.32. Let 1 p and m N, and let Ω Rn be an open set. Assume < < ∞ ∈ p⊂ m that a sequence (fi)i N converges to f weakly in L (Ω;R ) and ∈

limsup fi Lp(Ω;Rm) f Lp(Ω;Rm). (1.4) i k k É k k →∞ p m Then fi f in L (Ω;R ) as i . → → ∞ Observe that, under the assumptions in Lemma 1.32, by (1.3) and (1.4) we have

f Lp(Ω;Rm) liminf fi Lp(Ω;Rm) limsup fi Lp(Ω;Rm) f Lp(Ω;Rm), k k É i k k É i k k É k k →∞ →∞ which implies

lim fi Lp(Ω;Rm) f Lp(Ω;Rm). i k k = k k →∞ This means that the limit exists with an equality in (1.4). Next we discuss another method to upgrade weak convergence to strong convergence. Mazur’s lemma below asserts that a convex and closed subspace of a reﬂexive Banach space is weakly closed.

Theorem 1.33 (Mazur’s lemma). Assume that X is a normed space and that

xi x weakly in X as i . Then there exists a sequence of convex combinations → → ∞ Pmi Pmi xi ai, j x j, with ai, j 0 and ai, j 1, such that xi x in the norm of X e = j i Ê j i = e → as i = . = → ∞

T HEMORAL : For every weakly converging sequence, there is a sequence of convex combinations that converges strongly. Thus weak convergence is upgraded to strong convergence for a sequence of convex combinations. Observe that some

of the coefﬁcients ai may be zero so that the convex combination is essentially for a subsequence.

Remark 1.34. Since Lp(Ω;Rm) is a uniformly convex Banach space, the Banach– Saks theorem which asserts that a weakly convergent sequence has a subsequence whose arithmetic means converge in the norm. Let 1 p and m N, and n < < ∞ ∈ let Ω R be an open set. Assume that a sequence (fi)i N converges to f weakly ∈ p⊂ m in L (Ω;R ) as i . Then there exists a subsequence (fik )k N for which the → ∞ ∈ 1 Pk p m arithmetic mean k j 1 fi j converges to f in L (Ω;R ) as k . The advantage = → ∞ of the Banach–Saks theorem compared to Mazur’s lemma is that we can work with the arithmetic means instead of more general convex combinations CHAPTER 1. SOBOLEV SPACES 28

Remark 1.35. Mazur’s lemma can be used to give a proof for (1.3) (exercise).

Theorem 1.36. Let 1 p . Assume that (ui) is a bounded sequence in 1,p < < ∞ 1,p W (Ω). There exists a subsequence (uik ) and u W (Ω) such that uik u p p ∈ → weakly in L (Ω) and Duik Du weakly in L (Ω) as k . Moreover, if 1,p → 1,p → ∞ ui W (Ω), i 1,2..., then u W (Ω). ∈ 0 = ∈ 0 Proof. (1) Assume that u W1,p(Ω). Denote ∈ p n 1 fi (ui,Dui) L (Ω;R + ) = ∈ p n 1 for every i N. Then (fi)i N is a bounded sequence in L (Ω;R + ). By Theorem ∈ ∈ 1.30, there exists a subsequence (fik )k N that converges weakly to some f in ∈ p n 1 p L (Ω;R + ) as k . Consider f (u,v) with u L (Ω) and v (v1,...,vn) p n → ∞ 1,p = ∈ = ∈ L (Ω;R ). We show that u W (Ω) and that (uik ,Duik ) converges to (u,Du) p n 1 ∈ weakly in L (Ω;R + ) as k . It sufﬁces to prove that the weak gradient Du → ∞ exists in Ω and that v Du. = By using test functions of the form (g1,0,...,0) or (0, g2,..., gn 1) in the deﬁni- + p tion of weak convergence, we conclude that uik u weakly in L (Ω) and Duik v p n → → weakly in L (Ω;R ) as k . For ϕ C∞(Ω) and j 1,..., n, we have → ∞ ∈ 0 =

uD jϕ dx lim ui D jϕ dx ˆ = k ˆ k Ω →∞ Ω

lim D j ui ϕ dx. = − k ˆ k →∞ Ω p n On the other hand, since Dui v weakly in L (Ω;R ), by using the test function k → (0,...,ϕ,...,0) Lp0 (Ω;Rn), where ϕ is in the jth position, we have ∈

lim D j ui ϕ dx v jϕ dx. k ˆ k = ˆ →∞ Ω Ω This implies

uD jϕ dx v jϕ dx ˆΩ = −ˆΩ

for every ϕ C0∞(Ω) and thus Du v in Ω. This shows that the weak partial ∈ = p 1,p derivatives D j u, j 1,..., n, exist and belong to L (Ω). It follows that u W (Ω). = 1,p ∈ (2) For the second claim, we assume that ui W (Ω) for every i N and that ∈ 0 ∈ the sequence

(fik )k N ((uik ,Duik ))k N ∈ = ∈ p n 1 converges weakly to f (u,Du) in L (Ω;R + ). By Theorem 1.33, there exists a = sequence of convex combinations

m m Xk Xk hk ak, j fi j ak, j(ui j ,Dui j ) = j k = j k = = CHAPTER 1. SOBOLEV SPACES 29

p n 1 that converges to f (u,Du) in L (Ω;R + ) as k . This implies = → ∞ m m Xk Xk ak, j ui j u and ak, j Dui j Du j k → j k → = = in Lp(Ω) as k and thus → ∞ m Xk ak, j ui j u j k → = in W1,p(Ω) as k . Moreover, → ∞ mk X 1,p ak, j ui j W0 (Ω) j k ∈ = 1,p 1,p for every k N. Since W0 (Ω) is a closed subspace of W (Ω), it follows that 1,p ∈ u W (Ω). ∈ 0 ä Remarks 1.37: (1) Theorem 1.36 is equivalent to the fact that W1,p spaces are reﬂexive for 1 p . < < ∞ p p (2) Since ui u weakly in L (Ω) and Dui Du weakly in L (Ω) as k k → k → → ∞ in Theorem 1.36, by Lemma 1.29 we have

u W1,p(Ω) liminf uik W1,p(Ω). k k É k k k →∞ Thus the W1,p-norm is lower semicontinuous with respect to the weak convergence in W1,p. (3) Another way to see that W1,p spaces are reflexive for 1 p is to < < ∞ recall that a closed subspace of a reflexive space is reflexive. Thus it is enough to find an isomorphism between W1,p(Ω) and a closed subspace of p n 1 p n p n L (Ω,R + ) L (Ω,R ) L (Ω,R ). The mapping u (u,Du) will do = × ··· × 1,p 7→ for this purpose. This holds true for W0 (Ω) as well. This approach can be used to characterize elements in the dual space by the Riesz representation theorem, see [3, p. 62–65], [14, Section 11.4], [17, Section 4.3].

Example 1.38. The Sobolev space is not compact in the sense that every bounded 1,p 1,p sequence (ui) in W (Ω) has a converging subsequence (uik ) and u W (Ω) such 1,p ∈ that ui u in W (Ω). For i 1,2,..., consider ui : (0,2) R, k → = →  0, 0 x 1,  < É 1 ui(x) (x 1)i, 1 x 1 , (1.5) = i  − É É + 1, 1 1 x 2. + i < < 1,1 Then ui W ((0,2)) and ui 1,1 2 for every i 1,2,.... However, there ∈ k kW ((0,2)) É = does not exist a subsequence that converges in W1,1((0,2)). To conclude this, 1,1 assume that there exists a subsequence (uik ) that converges in W ((0,2)). In CHAPTER 1. SOBOLEV SPACES 30

1 particular, the subsequence (uik ) converges in L ((0,2)) and the limit function u L1((0,2)) is ∈  0, 0 x 1, u(x) < É = 1, 1 x 2. < < However, u W1,1((0,2)). This example also shows that Theorem 1.36 does not ∉ hold true in the case p 1 (exercise). =

Example 1.39. For i 1,2,..., consider ui : (0,2) R, = →  0, 0 x 1,  < É  1 ui(x) (x 1)pi, 1 x 1 = i  − É É +  1 , 1 1 x 2. pi + i < < 1,2 Then ui W ((0,2)), ∈

2 1 i 1 3i 2 2 ui − − , Dui 1, k kL2((0,2)) = 3i2 + i2 = 3i2 k kL2((0,2)) =

1,2 for every i 1,2,... and ui u weakly in W (Ω) as i , where u 0 (exercise). = → → ∞ = Clearly

0 u W1,2((0,2)) 1 liminf ui W1,2((0,2)). = k || < É i k k →∞ This shows that norm is only lower semicontinous in the weak topology but not 1,2 continuous. Observe carefully that ui 9 u in W ((0,2)), since since

2 lim Dui 2 1 0. i k kL ((0,2)) = 6= →∞ Remarks 1.40: (1) An open set Ω Rn is an extension domain for W1,p(Ω), if there exists a ⊂ bounded linear operator L : W1,p(Ω) W1,p(Ω) such that Lu u on Ω for → = every u W1,p(Ω). It can be shown that open sets with Lipschitz boundary ∈ are extension domains, see [8, Section 4.4] and [14, Section 13.1]. Observe 1,p that every open set Ω Rn is an extension domain for W (Ω), since we ⊂ 0 may take the zero extension to Rn \ Ω. (2) Extension domains have certain compactness results that are useful, for np example, in the existence theory for PDEs. Let 1 p n and p∗ n p . É < = − Assume that Ω Rn is an extension domain with ﬁnite measure. The ⊂ Kondrachov-Rellich compactness theorem asserts that the embedding W1,p(Ω) Lq(Ω) is compact. This means that for every bounded sequence →1,p 1,p (ui) in W (Ω), 1 p n, there exists a subsequence (uik ) and u W (Ω) É < q ∈ such that uik u in L (Ω) as k for every 1 q p∗. For the proof, → → ∞ É < 1,p see [8, Section 4.6] and [14, Theorem 12.18]. Moreover, if ui W0 (Ω), 1,p ∈ i 1,2..., then u W (Ω) for every open set Ω Rn. = ∈ 0 ⊂ CHAPTER 1. SOBOLEV SPACES 31

n Theorem 1.41. Let 1 p and let Ω R be an open set. Assume that (ui)i N ∈ < <1 ∞,p ⊂ p is a bounded sequence in W (Ω) such that ui u weakly in L (Ω) as i or → 1,p → ∞ that ui u almost everywhere in Ω as i . Then u W (Ω), ui u weakly p → p →n ∞ ∈ → 1,p in L (Ω), and Dui Du weakly in L (Ω;R ) as i . Moreover, if ui W0 (Ω) → 1,p → ∞ ∈ for every i N, then u W (Ω). ∈ ∈ 0

T HEMORAL : In order to show that u W1,p(Ω) it is enough to construct 1,p ∈ functions ui W (Ω), i 1,2,..., such that ui u almost everywhere in Ω as ∈ = → i and sup ui 1,p . → ∞ i k kW (Ω) < ∞

1,p Proof. (1) It sufﬁces to prove that u W (Ω) and that (ui,Dui) (u,Du) weakly p n 1 ∈ → in L (Ω;R + ) as i . We prove the latter claim by showing that each subse- → ∞ quence (uik )k N has a further subsequence, also denoted by (uik )k N, such that ∈ ∈

(ui ,Dui ) (u,Du) (1.6) k k → p n 1 p n 1 weakly in L (Ω;R + ) as k . To see this let g L 0 (Ω;R + ). Then → ∞ ∈

ai (ui,Dui) g dx (u,Du) g dx a = ˆΩ · → ˆΩ · = as i , since otherwise the deﬁnition of convergent real-valued sequences → ∞ implies that (ai)i N has a subsequence whose all subsequences fail to converge to ∈ a. This is a contradiction with respect to (1.6) when tested with g.

Let (uik )k N be a subsequence of (ui)i N. By Theorem 1.36, there exists a ∈ ∈ 1,p subsequence, also denoted by (uik )k N, and a function v W (Ω) such that ∈ p n 1 ∈ (uik ,Duik ) (v,Dv) weakly in L (Ω;R + ) as k . It sufﬁces to show that → →p ∞ u v almost everywhere. If ui u weakly in L (Ω), then ui u weakly in = → k → Lp(Ω) and u v almost everywhere by the uniqueness of weak limits. Hence we = may assume that ui u almost everywhere in Ω as i . → → ∞ Since ui u almost everywhere in Ω as i , by Theorem 1.33, there exists → → ∞ a sequence of convex combinations

m Xk hk ak, j(ui j ,Dui j ) = j k = p n 1 that converges to (v,Dv) in L (Ω;R + ) as k . In particular, → ∞ m Xk hk,1 ak, j ui j = j k = p converges to v in L (Ω) as k , and therefore some subsequence of (hk,1)k N → ∞ ∈ converges to v almost everywhere in Ω. On the other hand, by the assumptions,

m Xk lim hk,1 lim ak, j ui j u k = k j k = →∞ →∞ = CHAPTER 1. SOBOLEV SPACES 32

almost everywhere in Ω. This shows that u v almost everywhere in Ω, from = which we conclude that u W1,p(Ω) and that (1.6) holds. 1,p ∈ 1,p (2) If ui W (Ω) for every i N, then u W (Ω) by the ﬁrst part of the ∈ 0 ∈ ∈ 0 proof and Theorem 1.36. ä Remark 1.42. Theorem 1.36 and Theorem 1.41 do not hold when p 1 (exercise). = In order to demonstrate weak convergence techniques, we prove continuity of the operator u u in Sobolev spaces W1,p(Ω). This operator is easily shown to 7→ | | be bounded in Sobolev spaces, but continuity is not a consequence of boundedness, since the operator is not linear.

n Theorem 1.43. Let 1 p and let Ω R be an open set. Assume that (ui)i N ∈ 1,p < < ∞ ⊂ 1,p 1,p is a sequence in W (Ω) that converges to u in W (Ω). Then ui u in W (Ω) | | → | | as i . → ∞ p Proof. Since ui u in L (Ω) and →

ui(x) u(x) ui(x) u(x) || | − | || É | − | p for every x Ω, we obtain ui u in L (Ω) as i . ∈ | | → | | → ∞ Next we discuss convergence of the gradients. Since (ui)i N converges in ∈ 1,p 1,p W (Ω), it ss a bounded sequence in W (Ω). Theorem 1.26 gives D ui (x) | | | | = Dui(x) and D u (x) Du(x) for almost every x Ω. It follows that ( ui )i N ∈ | | | | | | = | 1,p| ∈ p | | is a bounded sequence in W (Ω). Since ui u weakly in L (Ω) as i , | | →p | | n → ∞ Theorem 1.41 implies D ui D u weakly in L (Ω;R ) as i . | | → | | → ∞ To upgrade weak convergence to strong convergence, we note that

Theorem 1.44. Let 1 p and m N, and let Ω Rn be an open set. Assume < < ∞ ∈ ⊂ that sequences (fi)i N and (gi)i N are such that fi converges to f weakly in ∈ ∈ p m p L (Ω;R ) and gi converges to g weakly in L (Ω) as i . If fi(x) gi(x) for → ∞ | | É almost every x Ω, then f (x) g(x) for almost every x Ω. ∈ | | É ∈ Proof. Let x Ω be a Lebesgue point of g and all the components of f , and ﬁx ∈ 0 r d(x,ÇΩ). Assume that B(x,r) f (y) d y 0. Denote < < ´ 6= ¯× ¯ 1 × ¯ ¯− m e ¯ f (y) d y¯ f (y) d y R = B(x,r) B(x,r) ∈ CHAPTER 1. SOBOLEV SPACES 33

and

1 p0 m ψ B(x, r) − χB(x,r) e L (Ω;R ). = | | ∈ By Cauchy–Schwarz’s inequality and the assumptions, we have

¯× ¯ × ¯ ¯ ¯ f (y) d y¯ e f (y) d y f (y) ψ(y) d y B(x,r) = · B(x,r) = ˆΩ ·

lim fi(y) ψ(y) d y liminf fi(y) ψ(y) d y = i ˆ · É i ˆ | || | →∞ Ω →∞ Ω × liminf gi(y) ψ(y) d y g(y) d y. É i ˆ | | = B(x,r) →∞ Ω This implies ¯× ¯ × ¯ ¯ ¯ f (y) d y¯ g(y) d y, B(x,r) É B(x,r) which clearly holds also if f (y) d y 0. Since almost every point x Ω is a B(x,r) = ∈ Lebesgue point of g and all´ components of f and the claim follows by taking r 0 → on both sides of the previous estimate. ä

1.13 Difference quotients

In this section we give a characterization of W1,p, 1 p , in terms of difference < < ∞ quotients. This approach is useful in regularilty theory for PDEs. Moreover, this characterization does not involve derivatives.

1 th Deﬁnition 1.45. Let u L (Ω) and Ω0 Ω. The j difference quotient is ∈ loc b

u(x he j) u(x) Dhu(x) + − , j 1,..., n, j = h =

for x Ω0 and h R such that 0 h dist(Ω0,ÇΩ). We denote ∈ ∈ < | | < Dhu (Dhu,...,Dhu). = 1 n

T HEMORAL : Note that the deﬁnition of the difference quotient makes sense at every x Ω whenever 0 h dist(x,ÇΩ). If Ω Rn, then the deﬁnition makes ∈ < | | < = sense for every h 0. 6=

Theorem 1.46. 1,p (1) Assume u W (Ω), 1 p . Then for every Ω0 Ω, we have ∈ É < ∞ b h D u Lp( ) c Du Lp( ) k k Ω0 É k k Ω

for some constant c c(n, p) and all 0 h dist(Ω0,ÇΩ). = < | | < CHAPTER 1. SOBOLEV SPACES 34

p (2) If u L (Ω0), 1 p , and there is a constant c such that ∈ < < ∞ h D u Lp( ) c k k Ω0 É 1,p whenever 0 h dist(Ω0,ÇΩ), then u W (Ω0) and Du Lp( ) c. < | | < ∈ k k Ω0 É (3) Let 1 p , and assume that u Lp(Rn) and that there exists a constant < < ∞ ∈ C such that h D u Lp(Rn;Rn) c k k É for every h 0. Then the weak derivative Du with respect to Rn exists, 1,p n6= u W (R ) and Du Lp(Rn;Rn) c. ∈ k k É

T HEMORAL : Pointwise derivatives are deﬁned as limit of difference quotients and Sobolev spaces can be characterized by integrated difference quotients.

W ARNING : Claim (2) does not hold for p 1 (exercise). =

1,p Proof. (1) First assume that u C∞(Ω) W (Ω). Then ∈ ∩ h Ç u(x he j) u(x) (u(x te j)) dt + − = ˆ0 Çt + h Du(x te j) e j dt = ˆ0 + · h Çu (x te j) dt, j 1,..., n, = ˆ0 Çx j + =

The last inequality follows from the fact that, for 0 h dist(Ω0,ÇΩ) and t h , < | | < | | É | | we have ¯ ¯p ¯ ¯p ¯ Çu ¯ ¯ Çu ¯ ¯ (x te j)¯ dx ¯ (x)¯ dx. ˆ ¯ Çx + ¯ É ˆ ¯ Çx ¯ Ω0 j Ω j α α α α Using the elementary inequality (a1 an) n (a a ), ai 0, α 0, + ··· + É 1 + ··· + n Ê > we obtain p Ã n ! 2 n h p X h 2 p X h p D u(x) dx D u(x) dx n 2 D u(x) dx ˆ | | = ˆ | j | É ˆ | j | Ω0 Ω0 j 1 Ω0 j 1 = = n n ¯ ¯p p X h p p p X ¯ Çu ¯ n 2 D u(x) dx 2 n 2 ¯ (x)¯ dx = ˆ | j | É ˆ ¯ Çx ¯ j 1 Ω0 j 1 Ω j = = p 1 p p 2 n + 2 Du(x) dx É ˆΩ | | The general case u W1,p(Ω) follows by an approximation, see Theorem 1.18. 1,p∈ 1,p Let ui C∞(Ω) W (Ω), i N, such that ui u in W (Ω) as i . By ∈ ∩ ∈ → → ∞ passing to a subsequence, if necessary, we may also assume that ui u pointwise → almost everywhere in Ω as i . Assume that 0 h dist(Ω0,ÇΩ). Then h h → ∞ < | | < D ui(x) D u(x) for almost every x Ω0 as i . By Fatou’s lemma and → ∈ → ∞ assumption we obtain

(2) Let ϕ C∞(Ω0). Then by a change of variables we see that, for 0 h ∈ 0 < | | < dist(suppϕ,ÇΩ0), we have

ϕ(x he j) ϕ(x) u(x he j) u(x) u(x) + − dx − − ϕ(x) dx, j 1,..., n. ˆ h = −ˆ h = Ω0 Ω0 − This shows that

h h uD ϕ dx (D− u)ϕ dx, j 1,..., n. ˆ j = −ˆ j = Ω0 Ω By assumption h sup D u p c . −j L (Ω0) 0 h dist(Ω ,ÇΩ) k k É < ∞ <| |< 0 p n Since 1 p , by Theorem 1.30 there exists f L (Ω0;R ) and a sequence < < ∞ ∈ hi p n (hi)i N converging to zero such that D− u f weakly in L (Ω0;R ) as i . ∈ → → ∞ This implies µ ¶ Çϕ hi hi u dx u lim D j ϕ dx lim uD j ϕ dx ˆ Çx j = ˆ hi 0 = hi 0 ˆ Ω0 Ω0 → → Ω0

hi lim (D−j u)ϕ dx f jϕ dx = − hi 0 ˆ = −ˆ → Ω0 Ω0 CHAPTER 1. SOBOLEV SPACES 36

for every ϕ C∞(Ω0). Here the second equality follows from the dominated conver- ∈ 0 gence theorem and the last equality is weak convergence the weak convergence tested with g (0,...,ϕ,...,0), where ϕ is in the jth position. It follows that = 1,p Du f in the weak sense in Ω0 and thus u W (Ω0). By (1.3), = ∈

hi Du Lp(Ω ;Rn) f Lp(Ω ;Rn) liminf D− u Lp(Ω ;Rn) c. k k 0 = k k 0 É i k k 0 É →∞

(3) Let Ωi B(0,2i) and Ω0 B(0, i) for every i N. Assertion (2) and the = i = ∈ assumption imply that ui u , i N, has a weak derivative Dui in Ω0 and = |Ωi ∈ i Dui Lp(Ω ;Rn) c. Since Dui 1 Dui almost everywhere in Ω0 , we see that the k k 0i É + = i limit

f (x) lim χΩ (x)Dui(x) = i 0i →∞ exists for almost every x Rn. The weak derivative of u with respect to Rn ∈ coincides with f L1 (Rn;Rn) and Fatou’s lemma implies ∈ loc 1 ³ p ´ p Du p n n f p n n lim Du dx L (R ;R ) L (R ;R ) χΩ0 i k k = k k = ˆ n i | i | R →∞ 1 ³ p ´ p liminf Dui dx c. É i ˆ | | É →∞ Ω0i

From this it also follows that u W1,p(Rn). ∈ ä

1.14 Absolute continuity on lines

In this section we relate weak derivatives to classical derivatives and give a characterization W1,p in terms of absolute continuity on lines. Recall that a function u : [a, b] R is absolutely continuous, if for every ε 0, → > there exists δ 0 such that if a x1 y1 x2 y2 ... xm ym b is a partition > = < É < É É < = of [a, b] with m X (yi xi) δ, i 1 − < = then m X u(yi) u(xi) ε. i 1 | − | < = Absolute continuity can be characterized in terms of the fundamental theorem of calculus.

Theorem 1.47. A function u : [a, b] R is absolutely continuous if and only if → there exists a function g L1((a, b)) such that ∈ x u(x) u(a) g(t) dt. = + ˆa

By the Lebesgue differentiation theorem g u0 almost everywhere in (a, b). = CHAPTER 1. SOBOLEV SPACES 37

T HEMORAL : Absolutely continuous functions are precisely those functions for which the fundamental theorem of calculus holds true.

Examples 1.48: (1) Every Lipchitz continuous function u : [a, b] R is absolutely continuous. → (2) The Cantor function u is continuous in [0,1] and differentiable almost everywhere in (0,1), but not absolutely continuous in [0,1]. Reason.

1 u(1) 1 0 u(0) u0(t) dt. = 6= = + ˆ0 |{z} 0 = ■

The next result relates weak partial derivatives with the classical partial derivatives.

1,p Theorem 1.49 (Nikodym, ACL characterization). Assume that u W (Ω), ∈ loc 1 p and let Ω0 Ω. Then there exists u∗ : Ω [ , ] such that u∗ u É É ∞ b → −∞ ∞ = almost everywhere in Ω and u∗ is absolutely continuous on (n 1)-dimensional − Lebesgue measure almost every line segments in Ω0 that are parallel to the

coordinate axes and the classical partial derivatives of u∗ coincide with the weak p partial derivatives of u almost everywhere in Ω. Conversely, if u Lloc(Ω) and p ∈ 1,p there exists u∗ as above such that Di u∗ L (Ω), i 1,..., n, then u W (Ω). ∈ loc = ∈ loc

T HEMORAL : This is a very useful characterization of W1,p, since many claims for weak derivatives can be reduced to the one-dimensional claims for absolute continuous functions. In addition, this gives a practical tool to show that a function belongs to a Sobolev space.

Remarks 1.50: (1) The ACL characterization can be used to give a simple proof of Example 1.9 (exercise). (2) In the one-dimensional case we obtain the following characterization: u W1,p((a, b)), 1 p , if u can be redeﬁned on a set of measure zero ∈ É É ∞ in such a way that u Lp((a, b)) and u is absolutely continuous on every ∈ compact subinterval of (a, b) and the classical derivative exists and belongs to u Lp((a, b)). Moreover, the classical derivative equals to the weak ∈ derivative almost everywhere. (3) A function u W1,p(Ω) has a representative that has classical partial ∈ derivatives almost everywhere. However, this does not give any information concerning the total differentiability of the function. See Theorem 2.21. CHAPTER 1. SOBOLEV SPACES 38

(4) The ACL characterization can be used to give a simple proof of the Leibniz 1,p 1,p 1,p rule. If u W (Ω) L∞(Ω) and v W (Ω) L∞(Ω), then uv W (Ω) ∈ ∩ ∈ ∩ ∈ and

D j(uv) vD j u uD j v, j 1,..., n, = + = almost everywhere in Ω (exercise), compare to Lemma 1.12 (5). (5) The ACL characterization can be used to give a simple proof for Lemma 1.25 and Theorem 1.26. The claim that if u,v W1,p(Ω), then max{u,v} ∈ ∈ W1,p(Ω) and min{u,v} W1,p(Ω) follows also in a similar way (exercise). ∈ (6) The ACL characterization can be used to show that if Ω is connected, 1,p u W (Ω) and Du 0 almost everywhere in Ω, then u is a constant ∈ loc = almost everywhere in Ω (exercise).

Proof. Since the claims are local, we may assume that Ω Rn and that u has a = compact support.

Let ui u , i 1,2,..., be a sequence of standard convolution approxima- =⇒ = εi = tions of u such that supp ui B(0,R) for every i 1,2,... and ⊂ = 1 ui u 1,1 n , i 1,2,... k − kW (R ) < 2i = By Lemma 1.16 (2), the sequence of convolution approximations converges point- n wise almost everywhere and thus the limit limi ui(x) exists for every x R \ E →∞ ∈ for some E Rn with E 0. We deﬁne ⊂ | | =  n  lim ui(x), x R \ E, i ∈ u∗(x) →∞ = 0, x E. ∈ We ﬁx a standard base vector in Rn and, without loss of generality, we may assume that it is (0,...,0,1). Let

Ã n ¯ ¯! X ¯ Çui 1 Çui ¯ fi(x1,..., xn 1) ui 1 ui ¯ + ¯ (x1,..., xn) dxn − = ˆR | + − | + j 1 ¯ Çx j − Çx j ¯ = and X∞ f (x1,..., xn 1) fi(x1,..., xn 1). − = i 1 − = By the monotone convergence theorem and Fubini’s theorem

X∞ f dx1 ... dxn 1 fi dx1 ... dxn 1 ˆ n 1 − = ˆ n 1 − R − R − i 1 = X∞ fi dx1 ... dxn 1 = ˆ n 1 − i 1 R − = µ ¯ ¯¶ X∞ ¯ Çui 1 Çui ¯ ui 1 ui ¯ + ¯ dx = i 1 ˆRn | + − | + ¯ Çx j − Çx j ¯ = 1 X∞ . i < i 1 2 < ∞ = CHAPTER 1. SOBOLEV SPACES 39

1 n 1 n 1 This shows that f L (R − ) and thus f (n 1)-almost everywhere in R − . ∈ n 1 < ∞ − Let x (x1,..., xn 1) R − such that f (x) . Denote b = − ∈ b < ∞

gi(t) ui(x, t) and g(t) u∗(x, t). = b = b

Claim: (gi) is a Cauchy sequence in C(R).

Reason. Note that

i 1 X− gi g1 (gk 1 gk), i 1,2,..., = + k 1 + − = = where ¯ t ¯ ¯ ¯ gk 1(t) gk(t) ¯ (g0k 1 g0k)(s) ds¯ | + − | = ¯ˆ + − ¯ −∞

g0k 1(s) g0k(s) ds É ˆR | + − | ¯ ¯ ¯ Çuk 1 Çuk ¯ ¯ + (bx, s) (bx, s)¯ ds fk(bx). É ˆR ¯ Çxn − Çxn ¯ É Thus ¯ ¯ ¯ i 1 ¯ i 1 ¯ X− ¯ X− ¯ (gk 1(t) gk(t))¯ gk 1(t) gk(t) ¯k 1 + − ¯ É k 1 | + − | = = X∞ fk(bx) f (bx) É k 1 = < ∞ = for every t R. This implies that (gi) is a Cauchy sequence in C(R). Since C(R) is ∈ complete, there exists g C(R) such that gi g uniformly in R. It follows that ∈ → {x} R Rn \ E. b × ⊂ ■ 1 Claim: (g0i) is a Cauchy sequence in L (R). Reason. ¯ ¯ ¯ i 1 ¯ i 1 ¯ X− ¯ X− ¯ (gk 1(t) gk(t))¯ dt gk 1(t) gk(t) dt ˆR ¯k 1 + − ¯ É k 1 ˆR | + − | = = i 1 X− fk(bx) f (bx) . É k 1 É < ∞ = 1 1 This implies that (gi) is a Cauchy sequence in L (R). Since L (R) is complete, 1 1 there exists g L (R) such that g0 g in L (R) as i . e ∈ i → e → ∞ ■ Claim: g is absolutely continuous in R. Reason. t t g(t) lim gi(t) lim g0 (s) ds g0(s) ds = i = i ˆ i = ˆ e →∞ →∞ −∞ −∞ This implies that g is absolutely continuous in R and g0 g almost everywhere in = e R. ■ CHAPTER 1. SOBOLEV SPACES 40

Claim: ge is the weak derivative of g.

Reason. Let ϕ C∞(R). Then ∈ 0

gϕ0 dt lim giϕ0 dt lim g0iϕ dt gϕ dt. ˆ = i ˆ = − i ˆ = −ˆ e ■ R →∞ R →∞ R R n Thus for every ϕ C∞(R ) we have ∈ 0 Çϕ Çu∗ u∗(bx, xn) (bx, xn) dxn (bx, xn)ϕ(bx, xn) dxn ˆR Çxn = −ˆR Çxn and by Fubini’s theorem

Çϕ Çu∗ u dx ϕ dx. ˆRn Çxn = −ˆRn Çxn n This shows that u∗ has the classical partial derivatives almost everywhere in R and that they coincide with the weak partial derivatives of u almost everywhere in Rn.

Assume that u has a representative u∗ as in the statement of the theo- ⇐= n rem. For every ϕ C∞(R ), the function u∗ϕ has the same absolute continuity ∈ 0 properties as u∗. By the fundamental theorem of calculus

Ç(u∗ϕ) (bx, t) dt 0 ˆR Çxn = n 1 for (n 1)-almost every x R − . Thus − b ∈ Çϕ Çu∗ u∗(bx, t) (bx, t) dt (bx, t)ϕ(bx, t) dt ˆR Çxn = −ˆR Çxn and by Fubini’s theorem

Çϕ Çu∗ u∗ dx ϕ dx. ˆRn Çxn = −ˆRn Çxn

n Çu∗ Since u∗ u almost everywhere in R , we see that is the nth weak partial = Çxn derivative of u. The same argument applies to all other partial derivatives Çu∗ , Çx j j 1,..., n as well. = ä x Example 1.51. The radial projection u : B(0,1) ÇB(0,1), u(x) x is discontinu- → x j = | | ous at the origin. However, the coordinate functions x , j 1,..., n, are absolutely | | = continuous on almost every lines. Moreover, x x µ ¶ δ x i j x j i j x p Di | | − | | L (B(0,1)) x = x 2 ∈ | | | | whenever 1 p n. Here É <  1, i j, δi j = = 0, i j, 6= is the Kronecker symbol. By the ACL characterization the coordinate functions of u belong to W1,p(B(0,1)) whenever 1 p n. É < CHAPTER 1. SOBOLEV SPACES 41

Remark 1.52. We say that a closed set E Ω to be removable for W1,p(Ω), if ⊂ E 0 and W1,p(Ω \ E) W1,p(Ω) in the sense that every function in W1,p(Ω \ E) | | = = can be approximated by the restrictions of functions in C∞(Ω). Theorem 1.49 1,p n 1 implies the following removability theorem for W Ω): if H − (E) 0, then E 1,p n 1 = is removable for W (Ω). Observe, that if H − (E) 0, then E is contained in a = measure zero set of lines in a ﬁxed direction (equivalently the projection of E onto n 1 a hyperplane also has H − -measure zero). This result is quite sharp. For example, let Ω B(0,1) and E {x B(0,1) : n 1 = = ∈ x2 0}. Then 0 H − (E) , but E is not removable since, using Theorem 1.49 = < < ∞ again, it is easy to see that the function which is 1 on the upper half-plane and 0 on the lower half-plane does not belong to W1,pΩ). With a little more work we can 1 1,p show that E0 E B(0, ) is not removable for W (B(0,1)). = ∩ 2 2 Sobolev inequalities

The term Sobolev inequalities refers to a variety of inequalities involving functions and their derivatives. As an example, we consider an inequality of the form

1 1 µ ¶ q µ ¶ p u q dx c Du p dx (2.1) ˆRn | | É ˆRn | |

n for every u C∞(R ), where constant 0 c and exponent 1 q are ∈ 0 < < ∞ É < ∞ independent of u. By density of smooth functions in Sobolev spaces, see Theorem 1.18, we may conclude that (2.1) holds for functions in W1,p(Rn) as well. Let u n n∈ C∞(R ), u 0, 1 p n and consider u (x) u(λx) with λ 0. Since u C∞(R ), 0 6≡ É < λ = > ∈ 0 it follows that (2.1) holds true for every u with λ 0 with c and q independent of λ > λ. Thus 1 1 µ ¶ q µ ¶ p q p uλ dx c Duλ dx ˆRn | | É ˆRn | | 1 for every λ 0. By a change of variables y λx, dx n d y, we see that > = = λ 1 1 u x q dx u x q dx u y q d y u x q dx λ( ) (λ ) ( ) n n ( ) ˆRn | | = ˆRn | | = ˆRn | | λ = λ ˆRn | | and

42 CHAPTER 2. SOBOLEV INEQUALITIES 43

for every λ 0, and equivalently, > n n 1 p q u Lq(Rn) cλ − + Du Lp(Rn). k k É k k Since this inequality has to be independent of λ, we have n n np 1 0 q . − p + q = ⇐⇒ = n p −

T HEMORAL : There is only one possible exponent q for which inequality (2.1) may hold true for all compactly supported smooth functions.

For 1 p n, the Sobolev conjugate exponent of p is É < np p∗ . = n p − Observe that

(1) p∗ p, > (2) If p n , then p∗ and → − → ∞ n (3) If p 1, then p∗ n 1 . = = −

2.1 Gagliardo-Nirenberg-Sobolev inequality

The following generalized Hölder’s inequality will be useful for us.

1 1 p Lemma 2.1. Let 1 p1,..., pk with 1 and assume fi L i (Ω), É É ∞ p1 + ··· + pk = ∈ i 1,..., k. Then = k Y f1 ... fk dx fi Lpi (Ω). ˆΩ | | É i 1 k k = Proof. Induction and Hölder’s inequality (exercise). ä Sobolev proved the following theorem in the case p 1 and Nirenberg and > Gagliardo in the case p 1. = Theorem 2.2 (Gagliardo-Nirenberg-Sobolev). Let 1 p n. There exists É < c c(n, p) such that = 1 1 µ ¶ p µ ¶ p u p∗ dx ∗ c Du p dx ˆRn | | É ˆRn | |

for every u W1,p(Rn). ∈ CHAPTER 2. SOBOLEV INEQUALITIES 44

T HEMORAL : The Sobolev-Gagliardo-Nirenberg inequality implies that W1,p(Rn) Lp∗ (Rn), when 1 p n. More precisely, W1,p(Rn) is continuously ⊂ É < imbedded in Lp∗ (Rn), when 1 p n. This is the Sobolev embedding theorem for É < 1 p n. É < n Proof. (1) We start by proving the estimate for u C∞(R ). By the fundamental ∈ 0 theorem of calculus x j Çu u(x1,..., x j,..., xn) (x1,..., t j,..., xn) dt j, j 1,..., n. = ˆ Çx j = −∞ This implies that

u(x) Du(x1,..., t j,..., xn) dt j, j 1,..., n. | | É ˆR | | = By taking product of the previous estimate for each j 1,..., n, we obtain = 1 n n µ ¶ n 1 n 1 Y − u(x) − Du(x1,..., t j,..., xn) dt j . | | É j 1 ˆR | | = We integrate with respect to x1 and then we use generalized Hölder’s inequality for the product of (n 1) terms to obtain − 1 1 n µ ¶ n 1 n µ ¶ n 1 n 1 − Y − u − dx1 Du dt1 Du dt j dx1 ˆR | | É ˆR | | ˆR j 2 ˆR | | = 1 1 µ ¶ n 1 n µ ¶ n 1 − Y − Du dt1 Du dx1 dt j . É ˆR | | j 2 ˆR ˆR | | = Next we integrate with respect to x2 and use again generalized Hölder’s inequality " 1 1 # n µ ¶ n 1 n µ ¶ n 1 n 1 − Y − u − dx1 dx2 Du dt1 Du dx1 dt j dx2 ˆR ˆR| | É ˆR ˆR | | j 2 ˆR ˆR | | = 1 µ ¶ n 1 − Du dx1 dt2 = ˆR ˆR | | " 1 1 # µ ¶ n 1 n µ ¶ n 1 − Y − Du dt1 Du dx1 dt j dx2 · ˆR ˆR | | j 3 ˆR ˆR | | = 1 µ ¶ n 1 − Du dx1 dt2 É ˆR ˆR | | 1 1 µ ¶ n 1 n µ ¶ n 1 − Y − Du dt1 dx2 Du dx1 dx2 dt j . · ˆR ˆR | | j 3 ˆR ˆR ˆR | | = Then we integrate with respect to x3,..., xn and obtain

This is the required inequality for p 1. = If 1 p n, we apply the estimate above to < < v u γ, = | | where γ 1 is to be chosen later. Since γ 1, we have v C1(Rn). Hölder’s > > ∈ inequality implies

Now we choose γ so that u has the same power on both sides. Thus | | γn p p(n 1) (γ 1) γ − . n 1 = − p 1 ⇐⇒ = n p − − − This gives γn p(n 1) n pn − p∗ n 1 = n p n 1 = n p = − − − − and consequently 1 1 µ ¶ µ ¶ p p∗ u p∗ dx γ Du p dx . ˆRn | | É ˆRn | | n This proves the claim for u C∞(R ). ∈ 0 (2) Assume then that u W1,p(Rn). By Lemma 1.24 we have W1,p(Rn) 1,p n ∈ n = W (R ). Thus there exist ui C∞(R ), i 1,2,..., such that ui u 1,p n 0 ∈ 0 = k − kW (R ) → 0 as i . In particular ui u Lp(Rn) 0, as i . Thus there exists a → ∞ k − k → → ∞n p n subsequence (ui) such that ui u almost everywhere in R and ui u in L (R ). → → p n Claim: (ui) is a Cauchy sequence in L ∗ (R ).

n Reason. Since ui u j C∞(R ), we use the Sobolev-Gagliardo-Nirenberg inequal- − ∈ 0 ity for compactly supported smooth functions and Minkowski’s inequality to conclude that

ui u j p n c Dui Du j Lp(Rn) k − kL ∗ (R ) É k − k ¡ ¢ c Dui Du Lp(Rn) Du Du j Lp(Rn) 0. É k − k + k − k → ■ p n p n p n Since L ∗ (R ) is complete, there exists v L ∗ (R ) such that ui v in L ∗ (R ) as ∈ → i . → ∞ n p n Since ui u almost everywhere in R and ui v in L ∗ (R ), we have u v → n → p n p =n almost everywhere in R . This implies that ui u in L ∗ (R ) and that u L ∗ (R ). → ∈ CHAPTER 2. SOBOLEV INEQUALITIES 46

Now we can apply Minkowski’s inequality and the Sobolev-Gagliardo-Nirenberg inequality for compactly supported smooth functions to conclude that

u p n u ui p n ui p n k kL ∗ (R ) É k − kL ∗ (R ) + k kL ∗ (R ) u ui p n c Dui Lp(Rn) É k − kL ∗ (R ) + k k ¡ ¢ u ui p n c Dui Du Lp(Rn) Du Lp(Rn) É k − kL ∗ (R ) + k − k + k k c Du Lp(Rn), → k k p n p n since ui u in L ∗ (R ) and Dui Du in L (R ). This completes the proof. → → ä Remarks 2.3: (1) The Gagliardo-Nirenberg-Sobolev inequality shows that if u W1,p(Rn)with p n p n ∈ 1 p n, then u L (R ) L ∗ (R ), with p∗ p. É < ∈ ∩ > (2) The Gagliardo-Nirenberg-Sobolev inequality shows that if u W1,p(Rn) ∈ with 1 p n and Du 0 almost everywhere in Rn, then u 0 almost É < = = everywhere in Rn. (3) The Sobolev-Gagliardo-Nirenberg inequality holds for Sobolev spaces with zero boundary values in open subsets of Rn by considering the zero exten- sions. There exists c c(n, p) 0 such that = > 1 1 µ ¶ p µ ¶ p u p∗ dx ∗ c Du p dx ˆΩ | | É ˆΩ | | 1,p for every u W (Ω), 1 p n. If Ω , by Hölder’s inequality ∈ 0 É < | | < ∞ 1 1 µ ¶ q µ ¶ p 1 q p∗ ∗ 1 u dx u dx Ω − p∗ ˆΩ | | É ˆΩ | | | | 1 1 µ ¶ p 1 p c Ω − p∗ Du dx É | | ˆΩ | |

whenever 1 q p∗. Thus for sets with ﬁnite measure all exponents below É É the Sobolev exponent will do.

1,p n p∗ n (4) The Sobolev-Gagliardo-Nirenberg inequality shows that Wloc (R ) Lloc(R ). n n ⊂ To see this, let Ω b R and choose a cutoff function η C0∞(R ) such that 1,p ∈ η 1 in Ω. Then ηu W (Rn) W1,p(Rn) and ηu u in Ω and = ∈ 0 = =

u p ηu p n c D(ηu) Lp(Rn) . k kL ∗ (Ω) É k kL ∗ (R ) É k k < ∞ (5) The Sobolev-Gagliardo-Nirenberg inequality holds for higher order Sobolev n np spaces as well. Let k N, 1 p k and p∗ n kp . There exists c ∈ É < = − = c(n, p, k) such that

1 1 µ ¶ p µ ¶ p u p∗ dx ∗ c Dk u p dx ˆRn | | É ˆRn | | for every u W k,p(Ω). Here Dk u 2 is the sum of squares of all kth order ∈ | | partial derivatives of u (exercise). CHAPTER 2. SOBOLEV INEQUALITIES 47

The Sobolev–Gagliardo–Nirenberg inequality has the following consequences.

Corollary 2.4. Let 1 p n and let Ω Rn be an open set. Assume that 1,p É < ⊂ u W (Ω) is such that Du 0 almost everywhere in Ω. Then u 0 almost ∈ 0 | | = = everywhere in Ω.

Proof. Extend u as zero outside Ω. Then then have Du 0 almost everywhere | | = in Rn. Theorem 2.2 implies

u p n c Du Lp(Rn) 0. k kL ∗ (R ) É k k = It follows that u 0 almost everywhere in Rn, and thus almost everywhere in Ω. = ä np 1,n Since p∗ n p as p n , one might expect that W (Ω) would be = − → ∞ → − n continuously embedded in L∞(Ω). This is false for n 1. Let Ω B(0,1) R . The > = ⊂ function µ µ 1 ¶¶ u(x) log log 1 = + x | | 1,n belongs to W (Ω) but not L∞(Ω) (exercise). The following result is a version of the Sobolev inequality for the full range 1 p . É < ∞ Corollary 2.5. Let 1 p , let Ω Rn be an open set with Ω , and assume 1,p É < ∞ np⊂ | | < ∞ that u W0 (Ω). Let 1 q p∗ n p , for 1 p n, and 1 q for n p . ∈ É É = − É < É < ∞ É < ∞ There exists a constant c c(n, p, q) such that = 1 1 ³ q ´ q 1 1 1 ³ p ´ p u dx c Ω n − p + q Du dx . ˆΩ| | É | | ˆΩ| |

1,p T HEMORAL : Let Ω Rn be an open set with Ω . If u W (Ω) with ⊂ | | < ∞ ∈ 0 p n, then u Lq(Ω) for every q with 1 q . Ê ∈ É < ∞ Proof. Extend u as zero outside Ω. Then Du(x) 0 for almost every x Ωc. = ∈ Assume ﬁrst that 1 p n. Hölder’s inequality and Theorem 2.2 imply É < 1 n p ³ ´ q 1 1 1 ³ np ´ np− q n p q n p u dx Ω − + u − dx ˆΩ| | É | | ˆΩ| | 1 1 1 1 ³ p ´ p c(n, p) Ω n − p + q Du dx . É | | ˆΩ| |

npe Assume then that n p . If q p, choose 1 pe n satisfying q n p . By É < ∞ > < < = − e the ﬁrst part of the proof and Hölder’s inequality, we obtain

1 1 1 1 1 ³ q ´ q ³ p ´ pe u dx c(n, p, q) Ω n − pe + q Du e dx ˆΩ| | É | | ˆΩ| | 1 1 1 1 ³ p ´ p c(n, p, q) Ω n − p + q Du dx . É | | ˆΩ| | Finally, if q p, the claim follows from the previous case for some q q and É e > Hölder’s inequality on the left-hand side. ä CHAPTER 2. SOBOLEV INEQUALITIES 48

Remark 2.6. Let 1 p n and let Ω Rn be an open set with Ω . The proof É < ⊂ | | < ∞ of Corollary 2.5 shows that the Sobolev inequality

n p 1 ³ np ´ np− ³ ´ p n p p u(x) − dx c(n, p) Du(x) dx ˆΩ| | É ˆΩ| |

1,p holds for every u W (Ω). ∈ 0 Remark 2.7. When p 1 the Sobolev-Gagliardo-Nirenberg inequality is related = to the isoperimetric inequality. Let Ω Rn be a bounded domain with smooth ⊂ boundary and set  1, x Ω,  ∈  dist(x,Ω) uε(x) 1 , 0 dist(x,Ω) ε, = ε  − < < 0, dist(x,Ω) ε. Ê Note that u can been considered as an approximation of the characteristic function of Ω. The Lipschitz constant of x dist(x,Ω) is one so that the Lipschitz constant 1 7→ 1,1 n of uε is ε− and thus this function belongs to W (R ), for example, by the ACL characterization, see Theorem 1.49, we have  1 x  ε , 0 dist( ,Ω) ε, Duε(x) < < | | É 0, otherwise.

The Sobolev-Gagliardo-Nirenberg inequality with p 1 gives = n 1 n 1 n 1 µ n ¶ −n µ n ¶ −n −n n 1 n 1 Ω uε − dx uε − dx | | = ˆΩ | | É ˆRn | | 1 c Duε dx c dx É ˆRn | | É ˆ{0 dist(x,Ω) ε} ε n < < {x R : 0 dist(x,Ω) ε} n 1 c | ∈ < < | cH − (ÇΩ) = ε → This implies n n 1 Ω n 1 cH − (ÇΩ), | | − É which is an isoperimetric inequality with the same constant c as in the Sobolev- Gagliardo-Nirenberg inequality. According to the classical isoperimetric inequality, if Ω Rn is a bounded domain with smooth boundary, then ⊂ n 1 1 − 1 n n 1 Ω n n− Ω− H − (ÇΩ), | | É n n 1 where H − (ÇΩ) stands for the (n 1)-dimensional Hausdorff measure of the − boundary ÇΩ. The isoperimetric inequality is equivalent with the statement that among all smooth bounded domains with ﬁxed volume, balls have the least surface area. CHAPTER 2. SOBOLEV INEQUALITIES 49

Conversely, the Sobolev-Gagliardo-Nirenberg inequality can be proved by the isoperimetric inequality, but we shall not consider this argument here. From these considerations it is relatively obvious that the best constant in the Sobolev- 1 1 n Gagliardo-Nirenberg when p 1 should be the isoperimetric constant n− Ω− . = n This also gives a geometric motivation for the Sobolev exponent in the case p 1. =

2.2 Sobolev-Poincaré inequalities

We begin with a Poincaré inequality for Sobolev functions with zero boundary values in open subsets.

Theorem 2.8 (Poincaré). Assume that Ω Rn is bounded and 1 p . Then ⊂ É < ∞ there is a constant c c(p) such that =

u p dx cdiam(Ω)p Du p dx ˆΩ | | É ˆΩ | |

1,p for every u W (Ω). ∈ 0

1,p T HEMORAL : This shows that W (Ω) Lp(Ω) when 1 p , if Ω Rn 0 ⊂ É < ∞ ⊂ is bounded. The main difference compared to the Gagliardo-Nirenberg-Sobolev inequality is that this applies for the whole range 1 p without the Sobolev É < ∞ exponent.

Remark 2.9. The Poincaré inequality above also shows that if Du 0 almost = everywhere, then u 0 almost everywhere. For this it is essential that the = function belongs to the Sobolev space with zero boundary values.

Proof. (1) First assume that u C∞(Ω). Let y (y1,..., yn) Ω. Then ∈ 0 = ∈ n n Y £ ¤ Y £ ¤ Ω yj diam(Ω), yj diam(Ω) a j, b j , ⊂ j 1 − + = j 1 = = where a j yj diam(Ω) and b j yj diam(Ω), j 1,..., n. As the proof of Theorem = − = + = 2.2, we obtain

The second inequality follows from Hölder’s inequality. Thus

b1 bn p p u(x) dx ... u(x) dx1 ... dxn ˆΩ | | = â1 ân | | b1 bn b1 p 1 p (2diam(Ω)) − ... Du(t1, x2,..., xn) dt1 dx1 ... dxn É â1 ân â1 | | b1 bn p p (2diam(Ω)) ... Du(t1, x2,..., xn) dt1 ... dxn É â1 ân | |

(2diam(Ω))p Du(x) p dx. = ˆΩ | |

1,p (2) The case u W (Ω) follows by approximation (exercise). ∈ 0 ä The Gagliardo-Nirenberg-Sobolev inequality in Theorem 2.2 and Poincaré’s inequality in Theorem 2.8 do not hold for functions u W1,p(Ω), at least when ∈ Ω Rn is an open set Ω , since nonzero constant functions give obvious ⊂ | | < ∞ counterexamples. However, there are several ways to obtain appropriate local estimates also in this case. Next we consider estimates in the case when Ω is a cube. Later we consider similar estimates for balls. The set

Q [a1, b1] ... [an, bn], b1 a1 ... bn an = × × − = = − is a cube in Rn. The side length of Q is

l(Q) b1 a1 b j a j, j 1,..., n, = − = − = and n n l o Q(x, l) y R : yj x j , j 1,..., n = ∈ | − | É 2 = is the cube with center x and sidelength l. Clearly,

Q(x, l) ln and diam(Q(x, l)) pn l | | = = The integral average of f L1 (Rn) over cube Q(x, l) is denoted by ∈ loc × 1 fQ(x,l) f d y f (y) d y. = Q(x,l) = Q(x, l) ˆ | | Q(x,l) Same notation is used for integral averages over other sets as well.

Theorem 2.10 (Poincaré inequality on cubes). Let Ω be an open subset of 1,p Rn. Assume that u W (Ω) with 1 p . Then there is c c(n, p) such that ∈ loc É < ∞ = µ× ¶ 1 µ× ¶ 1 p p p p u uQ(x,l) d y cl Du d y Q(x,l) | − | É Q(x,l) | |

for every cube Q(x, l) b Ω. CHAPTER 2. SOBOLEV INEQUALITIES 51

T HEMORAL : The Poincaré inequality shows that if the gradient is small in a cube, then the mean oscillation of the function is small in the same cube. In particular, if the gradient is zero, then the function is constant.

Proof. (1) First assume that u C∞(Ω). Let z, y Q Q(x, l) [a1, b1] ∈ ∈ = = × ··· × [an, bn]. Then

u(z) u(y) u(z) u(z1,..., zn 1, yn) ... u(z1, y2,..., yn) u(y) | − | É | − − | + + | − | n b X j Du(z1,..., z j 1, t, yj 1,..., yn) dt − + É j 1 â j | | = p p p By Hölder’s inequality and the elementary inequality (a1 an) n (a1 p + ··· + É + a ), ai 0, we obtain ··· + n Ê p Ã n b j ! p X u(z) u(y) Du(z1,..., z j 1, t, yj 1,..., yn) dt − + | − | É j 1 â j | | =  1 p n Ã b j ! p X p 1 1  Du(z1,..., z j 1, t, yj 1,..., yn) dt (b j a j) − p  − + É j 1 â j | | − = n b j p p 1 X p n l − Du(z1,..., z j 1, t, yj 1,..., yn) dt. − + É j 1 â j | | = By Hölder’s inequality and Fubini’s theorem ¯× ¯p p ¯ ¯ u(z) uQ dz ¯ (u(z) u(y)) d y¯ dz ˆQ | − | = ˆQ ¯ Q − ¯ µ× ¶p × u(z) u(y) d y dz u(z) u(y) p dz d y É ˆQ Q | − | É ˆQ Q | − |

p p 1 n b j n l − X p Du(z1,..., z j 1, t, yj 1,..., yn) dt d y dz − + É Q j 1 ˆQ ˆQ ˆa j | | | | = p p 1 n n l − X p (b j a j) Du(z) dz dw É Q j 1 − ˆQ ˆQ | | | | = p 1 p p n + l Du(z) dz. É ˆQ | |

1,p n (2) The case u Wloc (Ω) follows by approximation. There exist ui C∞(R ), ∈ 1,p ∈ i N, satisfying ui u in W (Q) as i . By passing to a subsequence, ∈ → → ∞ if necessary, we may in addition assume that ui u almost everywhere in Q. → Moreover, it follows from Hölder’s inequality and the Lp convergence that

× × 1 ³ p ´ p (ui)Q uQ ui(x) u(x) dx ui(x) u(x) dx 0 | − | É Q| − | É Q| − | →

and thus (ui)Q uQ as i . Fatou’s lemma and the ﬁrst part of the proof for → → ∞ CHAPTER 2. SOBOLEV INEQUALITIES 52

ui C∞(Ω) give ∈ × 1 × 1 ³ p ´ p ³ p ´ p u uQ dx liminf ui (ui)Q dx Q| − | É i Q| − | →∞ × 1 ³ p ´ p liminf c(n, p, q)l Dui dx É i Q| | →∞ 1 ³× ´ p c(n, p, q)l Du p dx , É Q| | and the proof is complete. ä Theorem 2.11 (Sobolev–Poincaré inequality on cubes). Let Ω be an open 1,p subset of Rn. Assume that u W (Ω) with 1 p n. Then there is c c(n, p) ∈ loc É < = such that 1 1 µ× ¶ p µ× ¶ p p∗ ∗ p u uQ(x,l) d y cl Du d y Q(x,l) | − | É Q(x,2l) | | for every cube Q(x,2l) b Ω.

1,p n p n T HEMORAL : The Sobolev-Poincaré inequality shows that W (R ) L ∗(R ), loc ⊂ loc when 1 p n. This is a stronger version of the Poincaré inequality on cubes in É < which we have the Sobolev exponent on the left-hand side.

n Proof. Let η C∞(R ) be a cutoff function such that ∈ 0 0 η 1, Dη c , suppη Q(x,2l) and η 1 in Q(x, l). É É | | É l ⊂ =

Notice that the constant c c(n) does not depend on the cube. Then (u uQ(x,l))η = − ∈ W1,p(Rn) and by the Gagliardo-Nirenberg-Sobolev inequality, see Theorem 2.2, and the Leibniz rule, see Theorem 1.12 (5), we have

1 1 µ ¶ p µ ¶ p p∗ ∗ p∗ ∗ u uQ(x,l) d y (u uQ(x,l))η d y ˆQ(x,l) | − | É ˆRn | − | µ ¶ 1 ¯ £ ¤¯p p c ¯D (u uQ(x,l))η ¯ d y É ˆRn − 1 1 µ ¶ p µ ¶ p p p p p c η Du d y c Dη u uQ(x,l) d y É ˆRn | | + ˆRn | | | − | 1 1 µ ¶ p µ ¶ p p c p c Du d y u uQ(x,l) d y . É ˆQ(x,2l) | | + l ˆQ(x,2l) | − | By the Poincaré inequality on cubes, see Theorem 2.10, we obtain

1 µ ¶ p p u uQ(x,l) d y ˆQ(x,2l) | − | 1 1 µ ¶ p µ ¶ p p p u uQ(x,2l) d y uQ(x,2l) uQ(x,l) d y É ˆQ(x,2l) | − | + ˆQ(x,2l) | − | 1 µ ¶ p p 1 cl Du d y uQ(x,2l) uQ(x,l) Q(x,2l) p . É ˆQ(x,2l) | | + | − || | CHAPTER 2. SOBOLEV INEQUALITIES 53

By Hölder’s inequality and Poincaré inequality on cubes, see Theorem 2.10, we have

1 n × uQ(x,2l) uQ(x,l) Q(x,2l) p (2l) p u uQ(x,2l) d y | − || | É Q(x,l) | − | µ× ¶ 1 n Q(x,2l) p p (2l) p | | u uQ(x,2l) d y É Q(x, l) Q(x,2l) | − | | | 1 µ ¶ p cl Du p d y . É ˆQ(x,2l) | |

By collecting the estimates above we obtain

1 1 µ ¶ p µ ¶ p p∗ ∗ p u uQ(x,l) d y c Du d y . ˆQ(x,l) | − | É ˆQ(x,2l) | | ä

Remark 2.12. The Sobolev-Poincaré inequality also holds in the form

1 1 µ× ¶ p µ× ¶ p p∗ ∗ p u uQ(x,l) d y cl Du d y . Q(x,l) | − | É Q(x,l) | |

Observe that there is the same cube on both sides. We shall return to this question later.

Remark 2.13. The Sobolev–Poincaré inequality in Theorem 2.11 holds with the same cubes on the both sides and it holds also for p 1. We shall not consider = these versions here.

Remark 2.14. In this remark we consider the case p n. = (1) As Example 1.10 shows, functions in W1,n(Rn) are not necessarily bounded. (2) Assume that u W1,n(Rn). The Poincaré inequality implies that ∈ × µ× ¶ 1 n n u(y) uQ d y u(y) uQ d y Q | − | É Q | − | 1 µ× ¶ n cl Du(y) n d y É Q | |

c Du Ln(Rn) É k k < ∞ for every cube Q where c c(n). Thus if u W1,n(Rn), then u is of bounded = ∈ mean oscillation, denoted by u BMO(Rn), and ∈ × u sup u(y) uQ d y c Du Ln(Rn), k k∗ = Q Rn Q | − | É k k ⊂ where c c(n). = CHAPTER 2. SOBOLEV INEQUALITIES 54

(3) Assume that u W1,n(Rn). The John-Nirenberg inequality for BMO func- ∈ tions gives × c γ u γ u(x) uQ 1 e | − | dx k k∗ 1 Q É c2 γ u + − k k∗ n c2 for every cube Q in R with 0 γ , where c1 c1(n) and c2 c2(n). < < u = = c2 k k∗ By choosing γ 2 u , we obtain = k k∗

× u(x) uQ × u(x) u c | − | c | − Q | Du n u e k k dx e k k∗ dx c Q É Q É

for every cube Q in Rn. In particular, this implies that u Lp (Rn) for ∈ loc every power p, with 1 p . This is the Sobolev embedding theorem in É < ∞ the borderline case when p n. = In fact, there is a stronger result called Trudinger’s inequality, which states that for small enough c 0, we have > n µ u(x) u ¶ n 1 × c | − Q | − Du n e k k dx c Q É

for every cube Q in Rn, n 2, but we shall not discuss this issue here Ê

p T HEMORAL : W1,n(Rn) L (Rn) for every p, with 1 p . This is the ⊂ loc É < ∞ Sobolev embedding theorem in the borderline case when p n. = The next theorem gives a general Sobolev–Poincaré inequality for Sobolev functions.

Theorem 2.15. Let 1 p , let Ω Rn be an open set, and assume that 1,p < < ∞np ⊂ u Wloc (Ω). Let 1 q p∗ n p , for 1 p n, and 1 q for n p . ∈ É É = − < < É < ∞ É < ∞ There exists a constant c c(n, p, q) such that = × 1 × 1 ³ q ´ q ³ p ´ p u uQ(x,l) d y cl Du d y (2.2) Q(x,l)| − | É Q(x,2l)| |

for every cube Q(x,2l) b Ω.

For 1 p n, inequality (2.2) follows from (2.3) and Hölder’s inequality on the < < left-hand side. CHAPTER 2. SOBOLEV INEQUALITIES 55

In the case p n we proceed as in the proof of Corollary 2.5. For q p n, Ê > Ê npe there exists 1 pe n such that q n p , and (2.2) follows from (2.3) with exponent < < = − e p and an application of Hölder’s inequality on the right-hand side. For q p, e É the claim follows from the previous case and Hölder’s inequality on the left-hand side. ä The next remark shows that it is possible to obtain a Poincaré inequality on cubes without the integral average also for functions that do not have zero boundary values. However, the functions have to vanish in a large subset.

Remark 2.16. Assume u W1,p(Rn) and u 0 in a set A Q(x, l) Q satisfying ∈ = ⊂ = A γ Q for some 0 γ 1. | | Ê | | < É This means that u 0 in a large portion of Q. By the Poincaré inequality there = exists c c(n.p) such that = µ× ¶ 1 µ× ¶ 1 µ× ¶ 1 p p p p p p u d y u uQ d y uQ d y Q | | É Q | − | + Q | | µ× ¶ 1 p p cl Du d y uQ , É Q | | + | | where ¯× ¯ × ¯ ¯ uQ ¯ u(y) d y¯ χQ\A(y) u(y) d y | | = ¯ Q ¯ É Q | | µ ¶1 1 µ ¶ 1 Q \ A − p × p | | u(y) p d y É Q Q | | | | µ× ¶ 1 1 1 p p (1 c) − p u(y) d y . É − Q | |

1 1 Since 0 (1 c) − p 1, we may absorb the integral average to the left hand side É − < and obtain µ× ¶ 1 µ× ¶ 1 1 1 p p p p (1 (1 c) − p ) u d y cl Du d y . − − Q | | É Q | | It follows that there exists c c(n, p,γ) such that = 1 1 µ× ¶ p µ× ¶ p u p d y cl Du p d y . Q | | É Q | | A similar argument can be done with the Sobolev-Poincaré inequality on cubes (exercise).

2.3 Morrey’s inequality

Let A Rn. A function u : A R is Hölder continuous with exponent 0 α 1, if ⊂ → < É there exists a constant c such that

u(x) u(y) c x y α | − | É | − | CHAPTER 2. SOBOLEV INEQUALITIES 56 for every x, y A. We deﬁne the space C0,α(A) to be the space of all bounded ∈ functions that are Hölder continuous with exponent α with the norm

u(x) u(y) u α sup u(x) sup | − | . (2.4) C (A) α k k = x A | | + x,y A,x y x y ∈ ∈ 6= | − | Remarks 2.17: (1) Every function that is Hölder continuous with exponent α 1 in the whole > space is constant (exercise). (2) There are Hölder continuous functions that are not differentiable at any point. Thus Hölder continuity does not imply any differentiability properties. (3) C0,α(A) is a Banach space with the norm deﬁned above (exercise). (4) Every Hölder continuous function on A Rn can be extended to a Hölder ⊂ continuous function on Rn with the same exponent and same constant. Moreover, if A is bounded, we may assume that the Hölder continuous extension to Rn is bounded (exercise).

The next result shows that every function in W1,p(Rn) with p n has a ¡1 n ¢- > − p Hölder continuous representative up to a set of measure zero.

Theorem 2.18 (Morrey). Assume that u W1,p(Rn) with p n. Then there is ∈ > c c(n, p) 0 such that = > n 1 p u(z) u(y) c z y − Du Lp(Rn) | − | É | − | k k for almost every z, y Rn. ∈ n 1,p n Proof. (1) Assume ﬁrst that u C∞(R ) W (R ). Let z, y Q(x, l). Then ∈ ∩ ∈ 1 Ç 1 u(z) u(y) ¡u(tz (1 t)y)¢ dt Du(tz (1 t)y) (z y) dt − = ˆ0 Çt + − = ˆ0 + − · − and ¯× ¯ ¯ ¯ u(y) uQ(x,l) ¯ (u(z) u(y)) dz¯ | − | = ¯ Q(x,l) − ¯ ¯ ¯ ¯× 1 ¯ ¯ ¯ ¯ Du(tz (1 t)y) (z y) dt dz¯ = ¯ Q(x,l) ˆ0 + − · − ¯ n 1 1 ¯ Çu ¯ X ¯ tz t y ¯ z y dt dz n ¯ ( (1 ) )¯ j j É j 1 l ˆQ(x,l) ˆ0 ¯ Çx j + − ¯| − | = n 1 1 ¯ Çu ¯ X ¯ (tz (1 t)y)¯ dz dt n 1 ¯ ¯ É j 1 l − ˆ0 ˆQ(x,l) ¯ Çx j + − ¯ = n 1 1 1 ¯ Çu ¯ X ¯ (w)¯ dw dt. n 1 n ¯ ¯ = j 1 l − ˆ0 t ˆQ(tx (1 t)y,tl) ¯ Çx j ¯ = + − CHAPTER 2. SOBOLEV INEQUALITIES 57

Here we used the fact that z j yj l, Fubini’s theorem and ﬁnally the change of | − 1 | É 1 variables w tz (1 t)y z (w (1 t)y), dz n dw. By Hölder’s inequality = + − ⇐⇒ = t − − = t n 1 1 1 ¯ Çu ¯ X ¯ (w)¯ dw dt n 1 n ¯ ¯ j 1 l − ˆ0 t ˆQ(tx (1 t)y,tl) ¯ Çx j ¯ = + − n 1 µ ¯ ¯p ¶ 1 X 1 1 ¯ Çu ¯ p 1 (w) dw Q(tx (1 t)y, tl) p0 dt n 1 n ¯ ¯ É j 1 l − ˆ0 t ˆQ(tx (1 t)y,tl) ¯ Çx j ¯ | + − | = + − n(1 1 ) 1 n(1 1 ) l − p t − p n Du p dt Q(tx (1 t)y, tl) Q(x, l)) L (Q(x,l)) n 1 n ( ) É k k l − ˆ0 t + − ⊂ np 1 n l − p Du Lp(Q(x,l)). = p n k k − Thus

u(z) u(y) u(z) uQ(x,l) uQ(x,l) u(y) | − | É | − | + | − | np 1 n 2 l − p Du Lp(Q(x,l)) (2.5) É p n k k − for every z, y Q(x, l). ∈ For every z, y Rn, there exists a cube Q(x, l) z, y such that l z y . For ∈ z y 3 = | − | example, we may choose x + . Thus = 2 n n 1 p 1 p u(z) u(y) c z y − Du Lp(Q(x,l)) c z y − Du Lp(Rn) | − | É | − | k k É | − | k k for every z, y Rn. ∈ (2) Assume then that u W1,p(Rn). Let u be the standard molliﬁcation of u. ∈ ε Then n 1 p u (z) u (y) c z y − Du Lp(Rn). | ε − ε | É | − | k εk Now by Lemma 1.16 (2) and by Theorem 1.17, we obtain

n 1 p u(z) u(y) c z y − Du Lp(Rn). | − | É | − | k k when z and y are Lebesgue points of u. The claim follows from the fact that almost every point of a locally integrable function is a Lebesgue point. ä Remarks 2.19: (1) Morrey’s inequality implies that u can be extended uniquely to Rn as a Hölder continuous function u such that

n 1 p n u(x) u(y) c x y − Du Lp(Rn) for all x, y R . | − | É | − | k k ∈

Reason. Let N be a set of zero measure such that Morrey’s inequality holds for all points in Rn \ N. Now for any x Rn, choose a sequence of points ∈ CHAPTER 2. SOBOLEV INEQUALITIES 58

n (xi) such that xi R \ N, i 1,2..., and xi x as i . By Morrey’s ∈ = → → ∞ inequality (u(xi)) is a Cauchy sequence in R and thus we can deﬁne

u(x) lim u(xi). = i →∞ Now it is easy to check that u satisﬁes Morrey’s inequality in every pair of points by considering sequences of points in Rn \ N converging to the pair of points. ■ (2) If u W1,p(Rn) with p n, then u is essentially bounded. ∈ > Reason. Let y Q(x,1). Then Morrey’s and Hölder’s inequality imply ∈

u(z) u(z) uQ(x,1) uQ(x,1) | | É | − | + | | × u(z) u(y) d y u(y) d y É Q(x,1) | − | + ˆQ(x,1) | | 1 µ ¶ p p c Du Lp(Rn) u(y) d y É k k + ˆQ(x,1) | |

c u 1,p n É k kW (R ) n for almost every z R . Thus u L (Rn) c u W1,p(Rn). ∈ k k ∞ É k k ■ This implies that

u 0,1 n c u W1,p(Rn), c c(n, p), k kC − p (Rn) É k k =

where u is the Hölder continuous representative of u. Hence W1,p(Rn) is n 0,1 n continuously embedded in C − p (R ), when p n. > (3) The proof of Theorem 2.18, see (2.5), shows that if Ω is an open subset of 1,p Rn and u W (Ω), p n, then there is c c(n, p) such that ∈ loc > = 1 n u(z) u(y) c z y − p Du Lp(Q(x,l)) | − | É | − | k k for every z, y Q(x, l), Q(x, l) Ω. This is a local version of Morrey’s ∈ b inequality.

n 1,p n 0,1 n 1,p n T HEMORAL : W (R ) C − p (R ), when p n. More precisely, W (R ) is n ⊂0,1 n > continuously embedded in C − p (R ), when p n. This is the Sobolev embedding > theorem for p n. >

Deﬁnition 2.20. A function u : Rn R is differentiable at x Rn if there exists a → ∈ linear mapping L : Rn R such that → u(y) u(x) L(x y) lim | − − − | 0. y x x y = → | − | CHAPTER 2. SOBOLEV INEQUALITIES 59

If such a linear mapping L exists at x, it is unique and we denote L Du(x) and = call Du(x) the derivative of u at x. If the derivative Du exists, it is unique and satisﬁes Du(y x) Du(x) (y x) − = · − for every y Rn, where ∈ µ Çu Çu ¶ Du(x) (x),..., (x) = Çx1 Çxn is the pointwise gradient of u at x.

Theorem 2.21. If u W1,p(Rn), n p , then u is differentiable almost every- ∈ loc < É ∞ where and its derivative equals its weak derivative almost everywhere.

T HEMORAL : By the ACL characterization, see Theorem 1.49, we know that every function in W1,p, 1 p has classical partial derivatives almost É É ∞ everywhere. If p n, then every function in W1,p is also differentiable almost > everywhere.

1, n 1,p n Proof. Since W ∞(R ) W (R ), we may assume n p . By the Lebesgue loc ⊂ loc < < ∞ differentiation theorem × lim Du(z) Du(x) p dz 0 l 0 Q(x,l) | − | = → for almost every x Rn. Let x be such a point and denote ∈ v(y) u(y) u(x) Du(x) (y x), = − − · − where y Q(x, l). Observe that v W1,p(Rn) with n p . By (2.5) in the proof ∈ ∈ loc < < ∞ of Morrey’s inequality, there is c c(n, p) such that = 1 µ× ¶ p v(y) v(x) cl Dv(z) p dz | − | É Q(x,l) | | for almost every y Q(x, l), where l x y . Since v(x) 0 and Dv(z) Du(z) ∈ = | − | = = − Du(x), we obtain

1 u(y) u(x) Du(x) (y x) µ× ¶ p | − − · − | c Du(z) Du(x) p dz 0 y x É Q(x,l) | − | → | − | as y x. → ä

1, 2.4 Lipschitz functions and W ∞

Let A Rn and 0 L . A function f : A R is called Lipschitz continuous with ⊂ É < ∞ → constant L, or an L-Lipschitz function, if

f (x) f (y) L x y | − | É | − | CHAPTER 2. SOBOLEV INEQUALITIES 60

for every x, y Rn. Observe that a function is Lipschitz continuous if it is Hölder ∈ continuous with exponent one. Moreover, C0,1(A) is the space of all bounded Lipschitz continuous functions with the norm (2.4). Examples 2.22: (1) For every y Rn the function x x y is Lipschitz continuous with ∈ 7→ | − | constant one. Note that this function is not smooth. (2) For every nonempty set A Rn the function x dist(x, A) is Lipschitz ⊂ 7→ continuous with constant one. Note that this function is not smooth when A Rn (exercise). 6= The next theorem describes the relation between Lipschitz functions and Sobolev functions.

1 n Theorem 2.23. A function u Lloc(R ) has a representative that is bounded and ∈ 1, n Lipschitz continuous if and only if u W ∞(R ). ∈

T HEMORAL : The Sobolev embedding theorem for p n shows that W1,p(Rn) n > ⊂ 0,1 p n 1, n 0,1 n C − (R ). In the limiting case p we have W ∞(R ) C (R ). This is the = ∞ = Sobolev embedding theorem for p . = ∞

1, n n 1,p n Proof. Assume that u W ∞(R ). Then u L∞(R ) and u W (R ) for ⇐= ∈ ∈ ∈ loc every p n and thus by Remark 2.19 we may assume that u is a bounded > continuous function. Moreover, we may assume that the support of u is compact. n By Lemma 1.16 (3) and by Theorem 1.17, the standard molliﬁcation u C∞(R ) ε ∈ 0 for every ε 0, u u uniformly in Rn as ε 0 and > ε → →

Du L (Rn) Du L (Rn) k εk ∞ É k k ∞ for every ε 0. Thus > ¯ ¯ ¯ 1 ¯ ¯ ¯ uε(x) uε(y) ¯ Duε(tx (1 t)y) (x y) dt¯ | − | = ¯ˆ0 + − · − ¯

Du L (Rn) x y É k εk ∞ | − | Du L (Rn) x y É k k ∞ | − | for every x, y Rn. By letting ε 0, we obtain ∈ →

u(x) u(y) Du L (Rn) x y | − | É k k ∞ | − | for every x, y Rn. ∈ Assume that u is Lipschitz continuous. Then there exists L such that =⇒ u(x) u(y) L x y | − | É | − | CHAPTER 2. SOBOLEV INEQUALITIES 61

for every x, y Rn. This implies that ∈ ¯ ¯ h ¯ u(x he j) u(x) ¯ D− u(x) ¯ − − ¯ L | j | = ¯ h ¯ É for every x Rn and h 0. This means that ∈ 6= h D− u L (Rn) L k j k ∞ É for every h 0 and thus 6= h h 1 1 D− u 2 D− u L (Rn) Ω 2 L Ω 2 , k j kL (Ω) É k j k ∞ | | É | | where Ω Rn is bounded and open. ⊂ 2 n As in the proof of Theorem 1.46, by Theorem 1.30, there exists g L (Ω0;R ) ∈ hi p n and a sequence (hi)i N converging to zero such that D− u g weakly in L (Ω0;R ) ∈ → as i . This implies → ∞ µ ¶ Çϕ hi hi u dx u lim D j ϕ dx lim uD j ϕ dx ˆ Çx j = ˆ hi 0 = hi 0 ˆ Ω Ω → → Ω0

hi lim (D−j u)ϕ dx g jϕ dx = − hi 0 ˆ = −ˆ → Ω Ω

for every ϕ C∞(Ω0). It follows that Du g in the weak sense in Ω and thus ∈ 0 = u W1,2(Ω). ∈

Claim: D j u L∞(Ω), j 1,..., n, ∈ =

hi 2 Reason. Let fi D− u, i 1,2.... Since fi D j u weakly in L (Ω) as i , by = j = → → ∞ Mazur’s lemma as in the proof of Theorem 1.36, there exists a sequence convex combinations such that m Xi fei ai,k fk D j u = k i → = in Lp(Ω) as i . Observe that → ∞ ° mi ° mi °X ° X ° hk ° fei L (Ω) ° ai,k fk° ai,k °D− u(x)° L. k k ∞ = ° ° É ° j °L (Ω) É °k i °L ( ) k i ∞ = ∞ Ω = Since there exists a subsequence that converges almost everywhere, we conclude that ¯ ¯ ¯D j u(x)¯ L, j 1,..., n, É = for almost every x Ω. ∈ ■ This shows that Du L ( ), with Du L. As u is bounded, this ∞ Ω L∞(Ω) 1, ∈ k k Én implies u W ∞(Ω) for all bounded subsets Ω R . Since the norm does not ∈ 1, n ⊂ depend on Ω, we conclude that u W ∞(R ). ∈ ä A direct combination of Theorem 2.23 and Theorem 2.21 gives a proof for Rademacher’s theorem. CHAPTER 2. SOBOLEV INEQUALITIES 62

Corollary 2.24 (Rademacher). Let f : Rn R be locally Lipschitz continuous. → Then f is differentiable almost everywhere.

WARNING: For an open subset Ω of Rn, Morrey’s inequality and the characterization of Lipschitz continuous functions holds only locally, that is, W1,p(Ω) 0,1 n ⊂ − p 1, 0,1 C (Ω), when p n and W ∞(Ω) C (Ω). loc > ⊂ loc Example 2.25. Let

2 2 2 Ω {x R : 1 x 2}\{(x1,0) R : x1 1} R . = ∈ < | | < ∈ Ê ⊂ 1, 0,α Then we can construct functions such that u W ∞(Ω), but u C (Ω), for ∈ 6∈ example, by deﬁning u(x) θ, where θ is the argument of x in polar coordinates = 1, with 0 θ 2π. Then u W ∞(Ω), but u is not Lipschitz continuous in Ω. < < ∈ However, it is locally Lipschitz continuous in Ω.

2.5 Summary of the Sobolev embeddings

We summarize the results related to Sobolev embeddings below. Assume that Ω is an open subset of Rn.

1,p n p∗ n 1,p p∗ np 1 p n W (R ) L (R ), Wloc (Ω) Lloc(Ω), p∗ n p (Theorem 2.2 É < ⊂ ⊂ = − and Theorem 2.11). p p n W1,n(Rn) BMO(Rn), W1,n(Ω) L (Ω) for every p, with 1 p = ⊂ loc ⊂ loc É < ∞ (Remark 2.14 (3)). n 0,1 n 1,p n 0,1 n 1,p p n p W (R ) C − p (R ), W (Ω) C − (Ω) (Theorem 2.18). < < ∞ ⊂ loc ⊂ loc 1, n 0,1 n 1, 0,1 p W ∞(R ) C (R ), W ∞(Ω) C (Ω) (Theorem 2.23). = ∞ = loc = loc

2.6 Direct methods in the calculus of variations

Sobolev space methods are important in existence results for PDEs. Assume that Ω Rn is a bounded open set. Consider the Dirichlet problem ⊂  ∆u 0 in Ω, = u g on ÇΩ. = Let u C2(Ω) be a classical solution to the Laplace equation ∈ n 2 X Ç u ∆u 0 = 2 = j 1 Çx j = CHAPTER 2. SOBOLEV INEQUALITIES 63

and let ϕ C∞(Ω). An integration by parts gives ∈ 0 n 2 X Ç u 0 ϕ∆u dx ϕdivDu dx ϕ dx = ˆ = ˆ = ˆ 2 Ω Ω Ω j 1 Çx j = n 2 n X Ç u X Çu Çϕ ϕ dx dx Du Dϕ dx = ˆ 2 = ˆ x x = ˆ · j 1 Ω Çx j j 1 Ω Ç j Ç j Ω = = 2 for every ϕ C∞(Ω). Conversely, if u C (Ω) and ∈ 0 ∈

Du Dϕ dx 0 for every ϕ C0∞(Ω), ˆΩ · = ∈ then by the computation above

ϕ∆u dx 0 for every ϕ C0∞(Ω). ˆΩ = ∈ By the fundamental lemma in the calculus of variations, see Corollary 1.5, we conclude ∆u 0 in Ω. = T HEMORAL : Assume u C2(Ω). Then ∆u 0 in Ω if and only if ∈ =

Du Dϕ dx 0 for every ϕ C0∞(Ω). ˆΩ · = ∈

This gives a motivation to the deﬁnition below.

Deﬁnition 2.26. A function u W1,2(Ω) is a weak solution to ∆u 0 in Ω, if ∈ =

Du Dϕ dx 0 ˆΩ · =

for every ϕ C∞(Ω). ∈ 0

T HEMORAL : There are second order derivatives in the definition of a classical solution to the Laplace equation, but in the definition above is enough to assume that only first order weak derivatives exist.

The next lemma shows that, in the deﬁnition of a weak solution, the class of test functions can be taken to be the Sobolev space with zero boundary values.

Lemma 2.27. If u W1,2(Ω) is a weak solution to the Laplace equation, then ∈

Du Dv dx 0 ˆΩ · =

for every v W1,2(Ω). ∈ 0 CHAPTER 2. SOBOLEV INEQUALITIES 64

1,2 Proof. Let vi C∞(Ω), i 1,2,..., be such that vi v in W (Ω). Then by the ∈ 0 = → Cauchy-Schwarz inequality and Hölder’s inequality, we have ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ Du Dv dx Du Dvi dx¯ ¯ Du (Dv Dvi) dx¯ ¯ˆΩ · − ˆΩ · ¯ = ¯ˆΩ · − ¯

Du Dv Dvi dx É ˆΩ | || − | 1 1 µ ¶ 2 µ ¶ 2 2 2 Du dx Dv Dvi dx 0 É ˆΩ | | ˆΩ | − | → as i . Thus → ∞ Du Dv dx lim Du Dvi dx 0. ˆ · = i ˆ · = Ω →∞ Ω ä

Remark 2.28. Assume that Ω Rn is bounded and g W1,2(Ω). If there exists a ⊂ ∈ weak solution u W1,2(Ω) to the Dirichlet problem ∈  ∆u 0 in Ω, = u g W1,2(Ω), − ∈ 0 then the solution is unique. Observe that the boundary values are taken in the Sobolev sense.

1,2 1,2 1,2 Reason. Let u1 W (Ω), with u1 g W0 (Ω), and u2 W (Ω), with u2 g 1,2 ∈ − ∈ ∈ − ∈ W0 (Ω), be solutions to the Dirichlet problem above. By Lemma 2.27

Du1 Dv dx 0 and Du2 Dv dx 0 ˆΩ · = ˆΩ · =

for every v W1,2(Ω) and thus ∈ 0 1,p (Du1 Du2) Dv dx 0 for every v W0 (Ω). ˆΩ − · = ∈ Since 1,2 u1 u2 (u1 g) (u2 g) W0 (Ω), − = | {z− }−| {z− } ∈ 1,2 1,2 W (Ω) W (Ω) ∈ 0 ∈ 0 we may choose v u1 u2 and conclude = − 2 Du1 Du2 dx (Du1 Du2) (Du1 Du2) dx 0. ˆΩ | − | = ˆΩ − · − =

This implies Du1 Du2 0 almost everywhere in Ω. By the Poincaré inequality, − = see Theorem 2.8, we have

2 2 2 u1 u2 dx cdiam(Ω) Du1 Du2 dx 0. ˆΩ | − | É ˆΩ | − | =

This implies u1 u2 0 u1 u2 almost everywhere in Ω. This is a PDE − = ⇐⇒ = proof of uniqueness and in the proof of Theorem 2.31 we shall see a variational argument for the same result. ■ CHAPTER 2. SOBOLEV INEQUALITIES 65

Next we consider a variational approach to the Dirichlet problem for the Laplace equation.

Deﬁnition 2.29. Assume that g W1,2(Ω). A function u W1,2(Ω) with u g 1,2 ∈ ∈ − ∈ W0 (Ω) is a minimizer of the variational integral

I(u) Du 2 dx = ˆΩ | | with boundary values g, if

Du 2 dx Dv 2 dx ˆΩ | | É ˆΩ | |

for every v W1,2(Ω) with v g W1,2(Ω). ∈ − ∈ 0

T HEMORAL : A minimizer u minimizes the variational integral I(u) in the class of functions with given boundary values, that is, ½ ¾ 2 2 1,2 1,2 Du dx inf Dv dx : v W (Ω), v g W0 (Ω) . ˆΩ | | = ˆΩ | | ∈ − ∈ If there is a minimizer, then inﬁmum can be replaced by minimum.

Theorem 2.30. Assume that g W1,2(Ω) and u W1,2(Ω) with u g W1,2(Ω). ∈ ∈ − ∈ 0 Then ½ ¾ 2 2 1,2 1,p Du dx inf Dv dx : v W (Ω), v g W0 (Ω) ˆΩ | | = ˆΩ | | ∈ − ∈ if and only if u is a weak solution to the Dirichlet problem  ∆u 0 in Ω, = u g W1,2(Ω). − ∈ 0

T HEMORAL : A function is a weak solution to the Dirichlet problem if and only if it is a minimizer of the corresponding variational integral with the given boundary values in the Sobolev sense.

Proof. Assume that u W1,2(Ω) is a minimizer with boundary values g 1,2 =⇒ ∈ ∈ W (Ω). We use the method of variations by Lagrange. Let ϕ C0∞(Ω) and ε R. 1,2 ∈ ∈ Then (u εϕ) g W (Ω) and + − ∈ 0

D(u εϕ) 2 dx (Du εDϕ) (Du εDϕ) dx ˆΩ | + | = ˆΩ + · +

Du 2 dx 2ε Du Dϕ dx ε2 Dϕ 2 dx = ˆΩ | | + ˆΩ · + ˆΩ | | i(ε). = CHAPTER 2. SOBOLEV INEQUALITIES 66

Since u is a minimizer, i(ε) has minimum at ε 0, which implies that i0(0) 0. = = Clearly 2 i0(ε) 2 Du Dϕ dx 2ε Dϕ dx = ˆΩ · + ˆΩ | | and thus

i0(0) 2 Du Dϕ dx 0. = ˆΩ · = This shows that Du Dϕ dx 0 ˆΩ · = for every ϕ C∞(Ω). ∈ 0 Assume that u W1,2(Ω) is a weak solution to ∆u 0 with u g W1,2(Ω) ⇐= ∈ = − ∈ 0 and let v W1,2(Ω) with v g W1,2(Ω). Then ∈ − ∈ 0 Dv 2 dx D(v u) Du 2 dx ˆΩ | | = ˆΩ | − + | (D(v u) Du) (D(v u) Du) dx = ˆΩ − + · − + D(v u) 2 dx 2 D(v u) Du dx Du 2 dx. = ˆΩ | − | + ˆΩ − · + ˆΩ | | Since 1,2 v u (v g) (u g) W0 (Ω), − = | {z− } − | {z− } ∈ 1,2 1,2 W (Ω) W (Ω) ∈ 0 ∈ 0 by Lemma 2.27 we have Du D(v u) dx 0 ˆΩ · − = and thus

Dv 2 dx D(v u) 2 dx Du 2 dx Du 2 dx ˆΩ | | = ˆΩ | − | + ˆΩ | | Ê ˆΩ | | for every v W1,2(Ω) with v g W1,2(Ω). Thus u is a minimizer. ∈ − ∈ 0 ä Next we give an existence proof using the direct methods in the calculus variations. This means that, instead of the PDE, the argument uses the variational integral.

Theorem 2.31. Assume that Ω is a bounded open subset of Rn. Then for every g W1,2(Ω) there exists a unique minimizer u W1,2(Ω) with u g W1,2(Ω), ∈ ∈ − ∈ 0 which satisﬁes ½ ¾ 2 2 1,2 1,2 Du dx inf Dv dx : v W (Ω), v g W0 (Ω) . ˆΩ | | = ˆΩ | | ∈ − ∈

T HEMORAL : The Dirichlet problem for the Laplace equation has a unique solution with Sobolev boundary values in any bounded open set.

W ARNING : It is not clear whether the solution to the variational problem attains the boundary values pointwise. CHAPTER 2. SOBOLEV INEQUALITIES 67

Proof. (1) Since I(u) 0, in particular, it is bounded from below in W1,2(Ω) and Ê since u is a minimizer, g W1,2(Ω) and g g 0 W1,2(Ω), we note that ∈ − = ∈ 0 ½ ¾ 2 1,2 1,2 2 0 m inf Du dx : u W (Ω), u g W0 (Ω) D g dx . É = ˆΩ | | ∈ − ∈ É ˆΩ | | < ∞ The deﬁnition of inﬁmum then implies that there exists a minimizing sequence 1,2 1,2 ui W (Ω) with ui g W (Ω), i 1,2,..., such that ∈ − ∈ 0 =

2 lim Dui dx m. i ˆ | | = →∞ Ω

The existence of the limit implies the sequence (I(ui)) is bounded. Thus there exists a constant M such that < ∞ 2 I(ui) Dui dx M for every i 1,2,..., = ˆΩ | | É =

(2) By the Poincaré inequality, see Theorem 2.8, we obtain

2 2 ui g dx D(ui g) dx ˆΩ| − | + ˆΩ | − | 2 2 2 cdiam(Ω) D(ui g) dx D(ui g) dx É ˆΩ | − | + ˆΩ | − | 2 2 (cdiam(Ω) 1) Dui D g dx É + ˆΩ | − | µ ¶ 2 2 2 (cdiam(Ω) 1) 2 Dui dx 2 D g dx É + ˆΩ | | + ˆΩ | | µ ¶ c(diam(Ω)2 1) M D g 2 dx É + + ˆΩ | | < ∞

1,2 for every i 1,2,... This shows that (ui g) is a bounded sequence in W0 (Ω). = 1,2 − (3) By reﬂexivity of W0 (Ω), see Theorem 1.36, there is a subsequence (uik g) 1,2 1,2 2− and a function u W (Ω), with u g W (Ω), such that ui u weakly in L (Ω) ∈ − ∈ 0 k → Çui and k Çu , j 1,..., n, weakly in L2(Ω) as k . By lower semicontinuity of Çx j → Çx j = → ∞ L2-norm with respect to weak convergence, see (1.3), we have

2 2 2 Du dx liminf Dui dx lim Dui dx. ˆ | | É k ˆ | k | = i ˆ | | Ω →∞ Ω →∞ Ω Since u W1,2(Ω), with u g W1,2(Ω), we have ∈ − ∈ 0

1,2 1,2 (4) To show uniqueness, let u1 W (Ω), with u1 g W0 (Ω) and u2 1,2 1,2 ∈ − ∈ ∈ W (Ω), with u2 g W0 (Ω) be minimizers of I(u) with the same boundary 1,2 − ∈ function g W (Ω). Assume that u1 u2, that is, {x Ω : u1(x) u2(x)} 0. By ∈ 6= | ∈ 6= | > the Poincaré inequality, see Theorem 2.8, we have

2 2 2 0 u1 u2 dx cdiam(Ω) Du1 Du2 dx < ˆΩ | − | É ˆΩ | − |

u1 u2 1,2 and thus {x Ω : Du1(x) Du2(x)} 0. Let v + . Then v W (Ω) and | ∈ 6= | > = 2 ∈

1 1 1,2 v g (u1 g) (u2 g) W0 (Ω). − = 2 | {z− }+ 2 | {z− } ∈ 1,2 1,2 W (Ω) W (Ω) ∈ 0 ∈ 0

By strict convexity of ξ ξ 2 we conclude that 7→ | |

2 1 2 1 2 Dv Du1 Du2 on {x Ω : Du1(x) Du2(x)}. | | < 2 | | + 2 | | ∈ 6=

Since {x Ω : Du1(x) Du2(x)} 0 and using the fact that both u1 and u2 are | ∈ 6= | > minimizers, we obtain

2 1 2 1 2 1 1 Dv dx Du1 dx Du2 dx m m m. ˆΩ | | < 2 ˆΩ | | + 2 ˆΩ | | = 2 + 2 =

Thus I(v) m. This is a contradiction with the fact that u1 and u2 are minimiz- < ers. ä Remarks 2.32: (1) This approach generalizes to other variational integrals as well. Indeed, the proof above is based on the following steps: (a) Choose a minimizing sequence. (b) Use coercivity

ui 1,2 I(ui) . k kW (Ω) → ∞ =⇒ → ∞ to show that the minimizing sequence is bounded in the Sobolev space. (c) Use reﬂexivity to show that there is a weakly converging subsequence. (d) Use lower semicontinuity of the variational integral to show that the limit is a minimizer. (e) Use strict convexity of the variational integral to show uniqueness. (2) If we consider C2(Ω) instead of W1,2(Ω) in the Dirichlet problem above, then we end up having the following problems. If we equip C2(Ω) with the supremum norm

2 u 2 u L ( ) Du L ( ) D u L ( ), k kC (Ω) = k k ∞ Ω + k k ∞ Ω + k k ∞ Ω where D2u is the Hessian matrix of second order partial derivatives, then the variational integral is not coercive nor the space is reﬂexive. Indeed, CHAPTER 2. SOBOLEV INEQUALITIES 69

when n 2 it is possible to construct a sequence of functions for which Ê the supremum tends to inﬁnity, but the L2 norm of the gradients tends to zero. The variationall integral is not coersive even when n 1. If we try to 2 = obtain coercivity and reﬂexivity in C (Ω) by changing norm to u 1,2 k kW (Ω) then we lose completeness, since the limit functions are not necessarily in C2(Ω). The Sobolev space seems to have all desirable properties for existence of solutions to PDEs. 3 Maximal function approach to Sobolev spaces

We recall the deﬁnition of the maximal function.

Deﬁnition 3.1. The centered Hardy-Littlewood maximal function M f : Rn → [0, ] of f L1 (Rn) is ∞ ∈ loc 1 M f (x) sup f (y) d y, = r 0 B(x, r) ˆB(x,r) | | > | | where B(x, r) {y Rn : y x r} is the open ball with the radius r 0 and the = ∈ | − | < > center x Rn. ∈

T HEMORAL : The maximal function gives the maximal integral average of the absolute value of the function on balls centered at a point.

Note that the Lebesgue differentiation theorem implies

n n Lemma 3.2. If f L∞(R ), then M f L∞(R ) and M f L (Rn) f L (Rn). ∈ ∈ k k ∞ É k k ∞

T HEMORAL : If the original function is essentially bounded, then the maximal function is essentially bounded and thus ﬁnite almost everywhere. Intuitively this

70 CHAPTER 3. MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 71

is clear, since the integral averages cannot be larger than the essential supremum n n of the function. Another way to state this is that M : L∞(R ) L∞(R ) is a → bounded operator.

Proof. For every x Rn and r 0 we have ∈ > 1 1 f (y) d y f L (Rn) B(x, r) f L (Rn). B(x, r) ˆ | | É B(x, r) k k ∞ | | = k k ∞ | | B(x,r) | | n By taking supremum over r 0, we have M f (x) f L (Rn) for every x R and > É k k ∞ ∈ thus M f L (Rn) f L (Rn). k k ∞ É k k ∞ ä The following maximal function theorem was ﬁrst proved by Hardy and Little- wood in the one-dimensional case and by Wiener in higher dimensions.

Theorem 3.3 (Hardy-Littlewood-Wiener). (1) If f L1(Rn), there exists c c(n) such that ∈ = n c {x R : M f (x) λ} f 1 n for every λ 0. (3.1) | ∈ > | É λ k kL (R ) > (2) If f Lp(Rn), 1 p , then M f Lp(Rn) and there exists c c(n, p) such ∈ < É ∞ ∈ = that

M f Lp(Rn) c f Lp(Rn). (3.2) k k É k k

T HEMORAL : The ﬁrst assertion states that the Hardy-Littlewood maximal operator maps L1(Rn) to weak L1(Rn) and the second claim shows that M : Lp(Rn) Lp(Rn) is a bounded operator for p 1. → > W ARNING : f L1(Rn) does not imply that M f L1(Rn) and thus the Hardy- ∈ ∈ Littlewood maximal operator is not bounded in L1(Rn). In this case we only have the weak type estimate.

3.1 Representation formulas and Riesz potentials

We begin with considering the one-dimensional case. If u C1(R), there exists an ∈ 0 interval [a, b] R such that u(x) 0 for every x R \ [a, b]. By the fundamental ⊂ = ∈ theorem of calculus,

x x u(x) u(a) u0(y) d y u0(y) d y, (3.3) = + ˆa = ˆ −∞ since u(a) 0. On the other hand, = b ∞ 0 u(b) u(x) u0(y) d y u(x) u0(y) d y, = = + ˆx = + ˆx CHAPTER 3. MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 72

so that ∞ u(x) u0(y) d y. (3.4) = −ˆx Equalities (3.3) and (3.4) imply

x ∞ 2u(x) u0(y) d y u0(y) d y = ˆ − ˆx −∞x u0(y)(x y) ∞ u0(y)(x y) − d y − d y = ˆ x y + ˆx x y −∞ | − | | − | u0(y)(x y) − d y, = ˆ x y R | − | from which it follows that

1 u0(y)(x y) u(x) − d y for every x R. = 2 ˆ x y ∈ R | − | Next we extend the fundamental theorem of calculus to Rn.

Lemma 3.4 (Representation formula). If u C1(Rn), then ∈ 0 1 Du(y) (x y) u x d y x n ( ) · n− for every R , = ωn 1 ˆRn x y ∈ − | − | where ωn 1 nΩn is the (n 1)-dimensional measure of ÇB(0,1). − = −

T HEMORAL : This is a representation formula for a compactly supported continuously differentiable function in terms of its gradient. A function can be integrated back from its derivative using this formula.

Proof. If x Rn and e ÇB(0,1), by the fundamental theorem of calculus ∈ ∈

∞ Ç ∞ u(x) (u(x te)) dt Du(x te) e dt. = −ˆ0 Çt + = −ˆ0 + · CHAPTER 3. MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 73

By the Fubini theorem

ωn 1u(x) u(x) 1 dS(e) − = ˆÇB(0,1) ∞ Du(x te) e dt dS(e) = −ˆÇB(0,1) ˆ0 + · ∞ Du(x te) e dS(e) dt (Fubini) = −ˆ0 ˆÇB(0,1) + · ∞ y 1 Du(x y) dS(y) dt n 1 = −ˆ0 ˆÇB(0,t) + · t t − 1 n (y te, dS(e) t − dS(y)) = = ∞ y Du(x y) dS(y) dt = −ˆ ˆ + · y n 0 ÇB(0,t) | | Du(x y) y d y +n · (polar coordinates) = −ˆ n y R | | Du(z) (z x) dz z x y d y dz · n− ( , ) = −ˆ n z x = + = R | − | Du(y) (x y) d y · n− . = ˆ n x y ä R | − |

Remark 3.5. By the Cauchy-Schwarz inequality and Lemma 3.4, we have

¯ 1 Du(y) (x y) ¯ u x ¯ d y¯ ( ) ¯ · n− ¯ | | = ¯ ωn 1 ˆRn x y ¯ − | − | 1 Du(y) x y d y | || n− | É ωn 1 ˆRn x y − | − | 1 Du(y) d y | n |1 = ωn 1 ˆRn x y − − | − | 1 I1( Du )(x), = ωn 1 | | − where I f , 0 α n, is the Riesz potential α < < f (y) I f x d y α ( ) n . = ˆ n x y α R | − | −

T HEMORAL : This gives a useful pointwise bound for a compactly supported smooth function in terms of the Riesz potential of the gradient.

Remark 3.6. A similar estimate holds almost everywhere if u W1,p(Rn) or u 1,p ∈ ∈ W0 (Ω) (exercise).

We begin with a technical lemma for the Riesz potential for α 1. = Lemma 3.7. If E Rn is a measurable set with E , then ⊂ | | < ∞ 1 1 d y c(n) E n . ˆ x y n 1 É | | E | − | − CHAPTER 3. MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 74

Proof. Let B B(x, r) be a ball with B E . Then E \ B B \ E and thus = | | = | | | | = | | 1 1 1 1 d y E \ B B \ E d y. ˆ x y n 1 É | | rn 1 = | | rn 1 É ˆ x y n 1 E\B | − | − − − B\E | − | − This implies 1 1 1 d y d y d y n 1 n 1 n 1 Ê x y − = Ê\B x y − + Ê B x y − | − | | − | ∩ | − | 1 1 d y d y n 1 n 1 É ˆB\E x y − + Ê B x y − | − | ∩ | − | 1 d y n 1 = ˆB x y − | − | 1 1 c(n)r c(n) B n c(n) E n . = = | | = || | ä

p p T HEMORAL : If Ω , then I1 : L (Ω) L (Ω) is a bounded operator for | | < ∞ → 1 p . É < ∞ Proof. If p 1, Hölder’s inequality and Lemma 3.7 give > f (y) f (y) 1 | | d y | | d y n 1 1 (n 1) 1 (n 1) ˆΩ x y − = ˆΩ p p | − | x y − x y 0 − | − | | − | p 1 1 µ f (y) ¶ p µ 1 ¶ p d y d y 0 | n| 1 n 1 É ˆΩ x y − ˆΩ x y − | − | | 1− | 1 µ f (y) p ¶ p c np0 d y Ω | n| 1 É | | ˆΩ x y − | − | 1 p 1 µ f (y) p ¶ p c Ω np− | | d y . = | | ˆ x y n 1 Ω | − | − For p 1, the inequality above is clear. Thus by Fubini’s theorem and Lemma 3.7, = we have

p 1 p p − f (y) I1( f χ )(x) dx c Ω n | | d y dx ˆ | | | Ω | É | | ˆ ˆ x y n 1 Ω Ω Ω | − | − p 1 1 p c Ω n− Ω n f (y) d y. É | | | | ˆΩ | | ä Next we show that the Riesz potential can be bounded by the Hardy-Littlewood maximal function. We shall do this for the general α although α 1 will be most = important for us.

Lemma 3.9. If 0 α n, there exists c c(n,α), such that < < = f (y) | | d y crα M f (x) ˆ x y n α É B(x,r) | − | − for every x Rn and r 0. ∈ > CHAPTER 3. MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 75

n i Proof. Let x R and denote Ai B(x, r 2− ), i 0,1,2,.... Then ∈ = =

f (y) X∞ f (y) | | d y | | d y n α n α ˆB(x,r) x y − = i 0 Âi\Ai 1 x y − | − | = + | − | µ r ¶α n X∞ − f (y) d y i 1 É i 0 2 + Âi | | = µ 1 ¶α n µ r ¶α 1 µ r ¶ n X∞ − − f (y) d y Ωn i i = i 0 2 2 Ωn 2 Âi | | = µ 1 ¶α n µ r ¶α 1 X∞ − f (y) d y Ωn i = i 0 2 2 Ai Âi | | = | | µ 1 ¶i cM f (x) rα X∞ α É i 0 2 = crα M f (x). = ä

Theorem 3.10 (Sobolev inequality for Riesz potentials). Assume that α > 0, p 1 and αp n. Then there exists c c(n, p,α), such that for every f Lp(Rn) > < = ∈ we have pn Iα f p n c f Lp(Rn), p∗ . k kL ∗ (R ) É k k = n αp −

T HEMORAL : This is the Sobolev inequality for the Riesz potentials. Observe

that of α 1, then p∗ is the Sobolev conjugate of p. =

Proof. If f 0 almost everywhere, the claim is clear. Thus we may assume that = f 0 on a set of positive measure and thus M f 0 everywhere. By Hölder’s 6= > inequality

1 1 f (y) µ ¶ p µ ¶ p d y f y p d y x y (α n)p0 d y 0 | n| ( ) − , ˆ n x y α É ˆ n | | ˆ n | − | R \B(x,r) | − | − R \B(x,r) R \B(x,r) where

(α n)p ∞ (α n)p x y − 0 d y x y − 0 dS(y) dρ ˆRn\B(x,r) | − | = ˆr ˆÇB(x,ρ) | − |

∞ (α n)p ρ − 0 1 dS(y) dρ = ˆr ˆÇB(x,ρ) | {z } n 1 ωn 1 ρ − = −

∞ (α n)p0 n 1 ωn 1 ρ − + − dρ = − ˆr ωn 1 n (n α)p − r − − 0 . = (n α)p n − 0 − The exponent can be written in the form p αp n n (n α)p0 n (n α) − , − − = − − p 1 = p 1 − − CHAPTER 3. MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 76

and thus f (y) α n d y cr p f p n | n| − L (R ). ˆ n x y α É k k R \B(x,r) | − | − Lemma 3.9 implies f (y) I f x d y α ( ) | n| | | É ˆ n x y α R | − | − f (y) f (y) d y d y | n| | n| = ˆ x y α + ˆ n x y α B(x,r) | − | − R \B(x,r) | − | − ³ n ´ α α p c r M f (x) r − f Lp(Rn) . É + k k By choosing p µ M f (x) ¶ n r − , = f Lp(Rn) k k we obtain αp αp 1 n I f (x) cM f (x) − n f . (3.5) | α | É k kLp(Rn) p np By raising both sides to the power ∗ n αp , we have = − αp p I f (x) p∗ cM f (x)p f n ∗ | α | É k kLp(Rn) The maximal function theorem, see (3.2), implies

αp p p∗ n ∗ p Iα f (x) d y c f Lp(Rn) (M f (x)) dx ˆRn | | É k k ˆRn αp p c f n ∗ M f p = k kLp(Rn)k kLp(Rn) αp p c f n ∗ f p É k kLp(Rn)k kLp(Rn)

and thus αp p n + p∗ I f p n c f p n c f Lp(Rn). k α kL ∗ (R ) É k kL (R ) = k k ä

Remark 3.11. From the proof of the previous theorem we also obtain a weak type estimate when p 1. Indeed, by (3.5) with p 1, there exists c c(n,α) such that = = = α α 1 n I f (x) cM f (x) − n f | α | É k kL1(Rn) and thus the maximal function theorem with p 1, see (3.1), implies = α n ¯n n n α o¯ {x R : I f (x) t} ¯ x R : cM f (x) −n f n t ¯ | ∈ | α | > | É ¯ ∈ k kL1(Rn) > ¯ ¯n n α n o¯ n n n α ¯ x R : M f (x) ct n α f − · − ¯ É ¯ ∈ > − k kL1(Rn) ¯ n α n α ct− n α f − f 1 n É − k kL1(Rn)k kL (R ) n n n α ct− n α f − = − k kL1(Rn) for every t 0. This also implies that > n n n n α {x R : I f (x) t} ct− n α f − | ∈ | α | Ê | É − k kL1(Rn) for every t 0. > CHAPTER 3. MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 77

This gives a second proof for the Sobolev-Gagliardo-Nirenberg inequality, see Theorem 2.2.

Corollary 3.12 (Sobolev-Gagliardo-Nirenberg inequality). If 1 p n, there É < exists a constant c c(n, p) such that = np u p n c Du Lp(Rn), p∗ , k kL ∗ (R ) É k k = n p − for every u C1(Rn). ∈ 0

T HEMORAL : The Sobolev-Gagliardo-Nirenberg inequality is a consequence of pointwise estimates for the function in terms of the Riesz potential of the gradient and the Sobolev inequality for the Riesz potentials.

Proof. 1 p By Remark 3.5 < < ∞ 1 n u(x) I1( Du )(x) for every x R , | | É ωn 1 | | ∈ − Thus Theorem 3.10 with α 1 gives =

u p n c I1( Du ) p n c Du Lp(Rn). k kL ∗ (R ) É k | | kL ∗ (R ) É k k p 1 Let = n j j 1 A j {x R : 2 u(x) 2 + }, j Z, = ∈ < | | É ∈ and let ϕ : R R, ϕ(t) max{0,min{t,1}}, be an auxiliary function. For j Z define n → = ∈ u j : R [0,1], →  j 1 0, u(x) 2 − ,  | | É 1 j 1 j j 1 j u j(x) ϕ(2 − u(x) 1) 2 u(x) 1, 2 u(x) 2 , = | | − = − −  | | − < | | É 1, u(x) 2 j. | | > 1,1 n Lemma 1.25 implies u j W (R ), j Z. Observe that Du j 0 almost everywhere n ∈ ∈ = in R \ A j 1, j Z. Then − ∈ n j A j {x R : u(x) 2 } | | É | ∈ | | > | n j 1 j {x R : u j(x) 1} ( u(x) 2 2 − u(x) 1 1) = | ∈ = | | | > =⇒ | | − > ¯ n ¯ ¯{x R : I1( Du j )(x) ωn 1}¯ (Remark3 .5) É ∈ | | Ê − n µ ¶ n 1 − c Du j(x) dx (Remark3 .11) É ˆRn | | n Ã ! n 1 − c Du j(x) dx = Â j 1 | | − n Ã ! n 1 1 j 1 j − c ϕ0(2 − u(x) 1)2 − Du(x) dx É Â j 1 | | − | | − n Ã ! n 1 n − j n 1 c2− − Du(x) dx . = Â j 1 | | − CHAPTER 3. MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 78

In the last equality we used the fact that the sets A j, j Z, are pairwise disjoint. ∈ ä Remark 3.13. The Sobolev-Gagliardo-Nirenberg inequality for u W1,p(Rn) fol- 1 n ∈ 1,p n lows from Corollary 3.12 by using the fact that C0(R ) is dense in W (R ), 1 p n. É <

3.2 Sobolev-Poincaré inequalities

Next we consider Sobolev-Poincaré inequalities in balls, compare with Theorem 2.10 and Theorem 2.11 for the corresponding estimates over cubes. First we study the one-dimensional case. Assume that u C1(R) and let ∈ y, z B(x, r) (x r, x r). By the fundamental theorem of calculus ∈ = − + y u(z) u(y) u0(t) dt. − = ˆz Thus y x r + u(z) u(y) u0(t) dt u0(t) dt u0(t) dt | − | É ˆz | | É ˆx r | | = ˆB(x,r) | | − and ¯ × ¯ ¯ ¯ u(z) uB(x,r) ¯u(z) u(y) d y¯ | − | = ¯ − B(x,r) ¯ ¯× × ¯ ¯ ¯ ¯ u(z) d y u(y) d y¯ = ¯ B(x,r) − B(x,r) ¯ × u(z) u(y) d y u0(y) d y. É B(x,r) | − | É ˆB(x,r) | | This is a pointwise estimate of the oscillation of the function. Next we generalize this to Rn.

Lemma 3.14. Let u C1(Rn) and B(x, r) Rn. There exists c c(n) such that ∈ ⊂ = ¯ ¯ Du(y) ¯u(z) uB(x,r)¯ c | | d y − É ˆ z y n 1 B(x,r) | − | − CHAPTER 3. MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 79

for every z B(x, r). ∈

T HEMORAL : This is a pointwise estimate for the oscillation of the function in terms of the Riesz potential of the gradient.

Proof. For any y, z B(x, r), we have ∈ 1 Ç 1 u(y) u(z) (u(ty (1 t)z)) dt Du(ty (1 t)z) (y z) dt. − = ˆ0 Çt + − = ˆ0 + − · − By the Cauchy-Schwarz inequality

1 u(y) u(z) y z Du(ty (1 t)z) dt. | − | É | − |ˆ0 | + − |

Let ρ 0. In the next display, we make a change of variables > 1 1 n w ty (1 t)z y (w (1 t)z), dS(y) t − dS(w). = + − ⇐⇒ = t − − = ³ ´n 1 n 1 z w − Then we have w z t y z and t − | − | , where ρ y z . We arrive at | − | = | − | = ρ = | − |

u(y) u(z) dS(y) ˆB(x,r) ÇB(z,ρ) | − | ∩ 1 ρ Du(ty (1 t)z) dS(y) dt É ˆ0 ˆB(x,r) ÇB(z,ρ) | + − | ∩ 1 1 Du(w) dS(w) dt ρ n 1 = ˆ0 t − ˆB(x,r) ÇB(z,tρ) | | ∩ 1 Du(w) n dS(w) dt ρ | n |1 = ˆ0 ˆB(x,r) ÇB(z,tρ) z w − ∩ | − | ρ Du(w) 1 n 1 dS(w) ds (s t , dt ds) ρ − | n |1 ρ = ˆ0 ˆB(x,r) ÇB(z,s) z w − = = ρ ∩ | − | Du(w) n 1 dw. (polar coordinates) ρ − | n |1 = ˆB(x,r) B(z,ρ) z w − ∩ | − | Since B(x, r) B(z,2r), an integration in polar coordinates gives ⊂ × ¯ ¯ ¯u(z) uB(x,r)¯ u(z) u(y) d y − É B(x,r) | − | 1 2r u(y) u(z) dS(y) dρ = B(x, r) ˆ0 ˆB(x,r) ÇB(z,ρ) | − | | | ∩ 1 2r Du(y) n 1 d y d ρ − | n |1 ρ É B(x, r) ˆ0 ˆB(x,r) B(z,ρ) z y − | | ∩ | − | 2r 1 n 1 Du(y) ρ − dρ | | d y É B(x, r) ˆ ˆ z y n 1 | | 0 B(x,r) | − | − Du(y) c(n) | | d y. = ˆ z y n 1 ä B(x,r) | − | − CHAPTER 3. MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 80

Remarks 3.15: (1) Assume that u C1(Rn). By Lemma 3.14 and Lemma 3.9, we have ∈ Du(y) u(z) uB(x,r) c | | d y | − | É ˆ z y n 1 B(x,r) | − | − cI1( Du χB(x,r))(z) = | | Du(y) χB(x,r)(y) c | | d y É ˆ z y n 1 B(z,2r) | − | − crM( Du χB(x,r))(z), É | | for every z B(x, r). ∈ Next we show that the corresponding inequalities hold true almost every- 1,p n 1,p where if u W (R ), 1 p . Since C∞(B(x, r)) is dense in W (B(x, r)), ∈ loc É < ∞ there exists a sequence ui C∞(B(x, r)), i 1,2,..., such that ui u in ∈ = → W1,p(B(x, r)) as i . By passing to a subsequence, if necessary, we → ∞ n obtain an exceptional set N1 R with N1 0 such that ⊂ | | =

lim ui(z) u(z) i = < ∞ →∞

for every z B(x, r)\ N1. By linearity of the Riesz potential and by Lemma ∈ 3.8, we have

° ° °I1( Dui χB(x,r)) I1( Du χB(x,r))° p | | − | | L (B(x,r)) ° ° °I1(( Dui Du )χB(x,r))° p = | | − | | L (B(x,r)) 1 ° ° c B(x, r) n ° Dui Du ° p , É | | | | − | | L (B(x,r)) which implies that

p I1( Dui χB(x,r)) I1( Du χB(x,r)) in L (B(x, r)) as i . | | → | | → ∞ By passing to a subsequence, if necessary, we obtain an exceptional set

N2 B(x, r) with N2 0 such that ⊂ | | =

lim I1( Dui χB(x,r))(z) I1( Du χB(x,r))(z) i | | = | | < ∞ →∞

for every z B(x, r) \ N2. Thus ∈

u(z) uB(x,r) lim ui(z) (ui)B(x,r) | − | = i | − | →∞ c lim I1( Dui χB(x,r))(z) É i | | →∞ cI1( Du χB(x,r))(z) = | | crM( Du χB(x,r))(z), É | |

for every z B(x, r) \ (N1 N2). ∈ ∪ CHAPTER 3. MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 81

(2) By Lemma 3.14 and (3.5), we have Du(y) u(z) uB(x,r) c | | d y | − | É ˆ z y n 1 B(x,r) | − | − cI1( Du χB(x,r))(z) = | | p p 1 n ° ° n cM( Du χB(x,r))(z) − ° Du χB(x,r)° p n É | | | | L (R ) for every z B(x, r). The corresponding inequalities hold true almost ∈ everywhere if u W1,p(Rn), 1 p . ∈ loc É < ∞ This gives a proof for the Sobolev-Poincaré inequality on balls, see Theorem 2.11 for the correspoding statement for cubes. Maximal function arguments can be used for cubes as well.

Theorem 3.16 (Sobolev-Poincaré inequality on balls). Assume that u W1,p(Rn) ∈ and let 1 p n. There exists c c(n, p) such that < < = 1 1 µ× ¶ p µ× ¶ p p∗ ∗ p u uB(x,r) d y cr Du d y B(x,r) | − | É B(x,r) | | for every B(x, r) Rn. ⊂

T HEMORAL : The Sobolev-Poincaré inequality is a consequence of pointwise estimates for the oscillation of the function in terms of the Riesz potential of the gradient and the Sobolev inequality for the Riesz potentials.

Proof. By Remark 3.15, we have

Theorem 3.17 (Poincaré inequality on balls). Assume that u W1,p(Rn)and ∈ let 1 p . There exists c c(n, p) such that < < ∞ = µ× ¶ 1 µ× ¶ 1 p p p p u uB(x,r) d y cr Du d y B(x,r) | − | É B(x,r) | | for every B(x, r) Rn. ⊂ CHAPTER 3. MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 82

Proof. By Remark 3.15, we have

u(y) uB(x,r) crM( Du χB(x,r))(y) | − | É | | for almost every y B(x, r). The maximal function theorem with p 1, see (3.2), ∈ > implies

p p p u uB(x,r) d y cr M( Du χB(x,r)) d y ˆB(x,r) | − | É ˆRn | |

p p cr ( Du χB(x,r)) d y É ˆRn | | crp Du p d y. = ˆB(x,r) | | ä

The maximal function approach to Sobolev-Poincaré inequalities is more in- volved in the case p 1, since then we only have a weak type estimate. However, = it is possible to consider that case as well, but this requires a different proof. We begin with two rather technical lemmas.

Lemma 3.18. Assume that E Rn is a measurable set and that f : E [0, ] is ⊂ → ∞ a measurable function for which

¯ ¯ E ¯{x E : f (x) 0}¯ | | . ∈ = Ê 2 Then for every a R and λ 0, we have ∈ > ¯½ ¾¯ ¯ ¯ ¯ λ ¯ ¯{x E : f (x) λ}¯ ¯ x E : f (x) a ¯. ∈ > É ¯ ∈ | − | > 2 ¯

Proof. Assume ﬁrst that a λ . If x E with f (x) λ, then | | É 2 ∈ > λ f (x) a f (x) a . | − | Ê − | | > 2

Thus {x E : f (x) λ} ©x E : f (x) a λ ª and ∈ > ⊂ ∈ | − | > 2 ¯½ ¾¯ ¯ ¯ ¯ λ ¯ ¯{x E : f (x) λ}¯ ¯ x E : f (x) a ¯. ∈ > É ¯ ∈ | − | > 2 ¯

Assume then that a λ . If x E with f (x) 0, then | | > 2 ∈ = λ f (x) a a . | − | = | | > 2 Thus ½ λ ¾ {x E : f (x) 0} x E : f (x) . ∈ = ⊂ ∈ > 2 If E , then by assumption | | = ∞ ¯ ¯ E ¯{x E : f (x) 0}¯ | | ∈ = Ê 2 = ∞ CHAPTER 3. MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 83

¯© λ ª¯ and thus ¯ x E : f (x) ¯ . On the other hand, if E , then ∈ Ê 2 = ∞ | | < ∞ ¯ ¯ ¯ ¯ ¯{x E : f (x) λ}¯ E ¯{x E : f (x) 0}¯ ∈ > É | | − ∈ = {x E : f (x) 0} É | ∈ = | ¯½ ¾¯ ¯ λ ¯ ¯ x E : f (x) a ¯. É ¯ ∈ | − | > 2 ¯

This completes the proof. ä Lemma 3.19. Assume that u C0,1(Rn), that is, u is a bounded Lipschitz contin- n ∈ n uous function in R , and let B(x, r) be a ball in R . Then there exists λ0 R for ∈ which

¯ ¯ B(x, r) ¯ ¯ B(x, r) ¯{y B(x, r) : u(y) λ0}¯ | | and ¯{y B(x, r) : u(y) λ0}¯ | | . ∈ Ê Ê 2 ∈ É Ê 2 Proof. Denote E {y B(x, r) : u(y) λ}, λ R, and set λ = ∈ Ê ∈ ½ B(x, r) ¾ λ0 sup λ R : E | | . = ∈ | λ| Ê 2

Note that λ0 u L (Rn) . Thus there exists an increasing sequence of real | | É k k ∞ < ∞ numbers (λi) such that λi λ0 and → B(x, r) E | | for every i 1,2,.... | λi | Ê 2 = T Since Eλ0 ∞i 1 Eλi , Eλ1 Eλ2 ... and Eλi B(x, r) , we conclude that = = ⊃ ⊃ | | É | | < ∞ B(x, r) Eλ lim Eλ | | . | 0 | = i | i | Ê 2 →∞ This shows that ¯ ¯ B(x, r) ¯{y B(x, r) : u(y) λ0}¯ | | . ∈ Ê Ê 2 A similar argument shows the other claim (exercise). ä The next result is Theorem 3.16 with p 1. = Theorem 3.20. Assume that u W1,1(Rn). There exists c c(n) such that ∈ loc = n 1 µ× n ¶ −n × n 1 u uB(x,r) − d y cr Du d y B(x,r) | − | É B(x,r) | |

for every B(x, r) Rn. ⊂

Proof. By Lemma 3.19 there is a number λ0 R for which ∈ ¯ ¯ B(x, r) ¯ ¯ B(x, r) ¯{y B(x, r) : u(y) λ0}¯ | | and ¯{y B(x, r) : u(y) λ0}¯ | | . ∈ Ê Ê 2 ∈ É Ê 2 Denote

v max{u λ0,0} and v min{u λ0,0}. + = − − = − − CHAPTER 3. MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 84

Both of these functions belong to W1,1(Rn). For the rest of the proof v 0 denotes loc Ê either v or v . All statements are valid in both cases. + − Let j j 1 A j {y B(x, r) : 2 v(y) 2 + }, j Z, = ∈ < É ∈ and let ϕ : R R, ϕ(t) max{0,min{t,1}}, be an auxiliary function. We deﬁne → = v j : B(x, r) [0,1], → 1 j v j(y) ϕ(2 − v(y) 1), j Z. = − ∈ 1,1 Lemma 1.25 implies v j W (B(x, r)), j Z. By Remark 3.15 (2) with p 1, we ∈ ∈ = have n 1 ° ° n 1 v j(y) (v j)B(x,r) n 1 cM( Dv j χB(x,r))(y)° Dv j χB(x,r)° − . | − | − É | | | | L1(Rn) 1 Lemma 3.18 with λ and a (v j)B(x,r) gives = 2 = j A j {y B(x, r) : v(y) 2 } | | É | ∈ > | ¯½ ¾¯ ¯ 1 ¯ ¯ y B(x, r) : v j(y) ¯ É ¯ ∈ > 2 ¯ ¯½ ¾¯ ¯ 1 ¯ ¯ y B(x, r) : v j(y) (v j)B(x,r) ¯ É ¯ ∈ | − | > 4 ¯ ¯½ 1 ¾¯ ¯ n ° ° 1 n ¯ ¯ y R : M( Dv j χB(x,r))(y) c° Dv j χB(x,r)° − ¯. É ¯ ∈ | | Ê | | L1(Rn) ¯ The last term can be estimated using the weak type estimate for the maximal function, see (3.1), and the fact that

1 j Dv j 2 − Dv χA j 1 | | = | | − almost everywhere in B(x, r). Thus we arrive at

Combining the above estimates for A j , we obtain | |

n n ( j 1)n n 1 X n 1 X n+ 1 v(y) − d y v(y) − d y 2 − A j ˆB(x,r) = j Z ˆA j = j Z | | ∈ ∈ ( j 1)n jn n X + ° ° n 1 c 2 n 1 2 n 1 Dv χ − − − − ° A j 1 B(x,r)°L1(Rn) É j Z | | − ∩ ∈ n ° ° n 1 ° ° − °X ° c Dv χA j 1 B(x,r) ° − ∩ ° É ° j Z | | ° 1 n ∈ L (R ) n ° ° n 1 c° Du χB(x,r)° − . É | | L1(Rn) CHAPTER 3. MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 85

Since u λ0 v v , we obtain | − | = + + − n 1 n 1 µ× n ¶ −n µ× n ¶ −n n 1 n 1 u uB(x,r) − d y 2 u λ0 − d y B(x,r) | − | É B(x,r) | − | n 1 n 1 µ× n ¶ −n µ× n ¶ −n n 1 n 1 2 v (y) − d y 2 v (y) − d y É B(x,r) + + B(x,r) − ° ° c° Du χB(x,r)° 1 n É | | L (R ) c Du(y) d y. = ˆB(x,r) | | ä

T HEMORAL : The proof shows that in this case a weak type estimate implies a strong type estimate. Observe carefully, that this does not hold in general. The reason why this works here is that we consider gradients, which have the property that they vanish on the set where the function itself is constant.

Next we give a maximal function proof for Morrey’s inequality, see Theorem 2.18 and Remark 2.19 (3).

Theorem 3.21 (Morrey’s inequality). Assume that u C1(Rn) and let n p ∈ < < . There exists c c(n, p) such that ∞ = 1 µ× ¶ p u(y) u(z) cr Du p dw | − | É B(x,r) | |

for every B(x, r) Rn and y, z B(x, r). ⊂ ∈ Proof. By Lemma 3.14

u(y) u(z) u(y) uB(x,r) uB(x,r) u(z) | − | É | − | + | − | Du(w) Du(w) c | | dw c | | dw É ˆ y w n 1 + ˆ z w n 1 B(x,r) | − | − B(x,r) | − | − for every y, z B(x, r). Hölder’s inequality gives ∈ 1 1 µ ¶ p µ ¶ Du(w) p (1 n)p p0 | | dw Du dw y w − 0 dw , ˆ y w n 1 É ˆ | | ˆ | − | B(x,r) | − | − B(x,r) B(x,r) where

(1 n)p (1 n)p y w − 0 dw y w − 0 dw ˆB(x,r) | − | É ˆB(y,2r) | − | 2r (1 n)p ρ − 0 dS(w) dρ = ˆ0 ˆÇB(y,ρ) 2r (1 n)p0 n 1 n (n 1)p0 ωn 1 ρ − + − dρ cr − − . = − ˆ0 = Since 1 n (n (n 1)p0) 1 , − − p0 = − p CHAPTER 3. MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 86 we have

1 µ ¶ p 1 n p u(y) u(z) cr − p Du dw . | − | É ˆB(x,r) | | ä

3.3 Sobolev inequalities on domains

In this section we study open sets Ω Rn for which the Sobolev-Poincaré inequality ⊂ 1 1 µ ¶ µ ¶ p p p∗ p np u u ∗ d y c(p, n,Ω) Du d y , 1 p n, p∗ , ˆ | − Ω| É ˆ | | É < = n p Ω Ω −

holds true for all u C∞(Ω). We already know that this inequality holds if Ω ∈ is a ball, but are there other sets for which it holds true as well? We begin by introducing an appropriate class of domains.

n Deﬁnition 3.22. A bounded open set Ω R is a John domain, if there is cJ 1 ⊂ Ê and a point x0 Ω so that every point x Ω can be joined to x0 by a path γ : [0,1] ∈ ∈ → Ω such that γ(0) x, γ(1) x0 and = = 1 dist(γ(t),ÇΩ) c− x γ(t) Ê J | − | for every t [0,1]. ∈

T HEMORAL : In a John domain every point can be connected to the distinguished point with a curve that is relatively far from the boundary.

Remarks 3.23: (1) A bounded and connected open set Ω Rn satisﬁes the interior cone condi- ⊂ tion, if there exists a bounded cone

n 2 2 2 C {x R : x1 xn 1 ax , 0 xn b} = ∈ + ··· + − É É É such that every point of Ω is a vertex of a cone congruent to C and entirely contained in Ω. Every domain with interior cone condition is a John domain (exercise). Roughly speaking the main difference between the interior cone condition and a John domain is that rigid cones are replaced by twisted cones. (2) The collection of John domains is relatively large. For example, a domain whose boundary is von Koch snowﬂake is a John domain. CHAPTER 3. MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 87

Theorem 3.24. If Ω Rn is a John domain and 1 p n, then ⊂ É < 1 1 µ ¶ p µ ¶ p p∗ ∗ p u uΩ dx c(p, n, cJ ) Du dx , 1 p n, ˆΩ | − | É ˆΩ | | < <

for all u C∞(Ω). ∈

T HEMORAL : The Sobolev-Poincaré inequality holds for many other sets than balls as well.

W ARNING : A rooms and passages example shows that the Sobolev-Poincaré inequality does not hold for all sets.

Proof. Let x0 Ω be the distinguished point in the John domain. Denote B0 ∈1 = B(x0, r0), r0 dist(x0,ÇΩ). We show that there is a constant M M(cJ , n) such = 4 = that for every x Ω there is a chain of balls Bi B(xi, ri) Ω, i 1,2,..., with the ∈ = ⊂ = properties

(1) Bi Bi 1 M Bi Bi 1 , i 1,2,..., | ∪ + | É | ∩ + | = (2) dist(x,Bi) Mri, ri 0, xi x as i and É → → → ∞ (3) no point of Ω belongs to more than M balls Bi.

To construct the chain, ﬁrst assume that x is far from x0, say x Ω \ B(x,2r0). Let ∈ γ be a John path that connects x to x0. All balls on the chain are centered on

γ. We construct the balls recursively. We have already deﬁned B0. Assume that

B0,...,Bi have been constructed. Starting from the center xi of Bi we move along

γ towards x until we leave Bi for the last time. Let xi 1 be the point on γ where + this happens and deﬁne

1 Bi 1 B(xi 1, ri 1), ri 1 x xi 1 . + = + + + = 4cJ | − + |

By construction Bi Ω. Property (1) and dist(x,Bi) Mri in (2) follow from ⊂ É the fact that the consecutive balls have comparable radii and that the radii are comparable to the distances of the centers of the balls to x.

To prove (3) assume that y Bi Bi . Observe that the radii of Bi , ∈ 1 ∩ ··· ∩ k j j 1,..., k, are comparable to x y . By construction, if i j im, the center of Bi = | − | < m does not belong to Bi j . This implies that the distances between the centers of Bi j are comparable to x y . The number of points in Rn with pairwise comparable | − | n distances is bounded, that is, if z1,..., zm R satisfy ∈ r dist(zi, z j) cr for i j, c < < 6= then m N N(c, n). Thus k is bounded by a constant depending only on n and É = cJ . This implies (3). Property (3) implies , ri 0, xi x as i . → → → ∞ The case x B(x,2r0) is left as an exercise. ∈ CHAPTER 3. MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 88

Since × uBi u(y) d y u(x) = Bi → for every x Ω as i , we obtain ∈ → ∞

X∞ u(x) uB0 uBi uBi 1 | − | É i 0 | − + | = X∞ ¡ ¢ uBi uBi Bi 1 uBi Bi 1 uBi 1 É i 0 | − ∩ + | + | ∩ + − + | = µ × × ¶ X∞ Bi Bi 1 | | u uBi d y | + | u uBi 1 d y É i 0 Bi Bi 1 Bi | − | + Bi Bi 1 Bi 1 | − + | | ∩ + | | ∩ + | + = × X∞ c u uBi d y (property (1)) É i 0 Bi | − | = × X∞ c ri Du d y (Poincaré inequality, see Theorem 3.17) É i 0 Bi | | = X∞ Du c | | d y. = ˆ n 1 i 0 Bi ri − =

Property (2) implies x y cri for every y Bi and | − | É ∈ 1 c for every y Bi. rn 1 É x y n 1 ∈ i − | − | − Thus Du(y) Du(y) u(x) u c X∞ d y c d y. B0 | n |1 | n |1 | − | É i 0 ˆBi x y − É ˆΩ x y − = | − | | − | The last inequality follows from (3). We observe that

u(x) u u(x) uB uB u , | − Ω| É | − 0 | + | 0 − Ω| where by Lemma 3.7 we have

1 uB u u(x) uB dx | 0 − Ω| É Ω ˆ | − 0 | | | Ω 1 Du(y) c | | dx d y É Ω ˆ ˆ x y n 1 | | Ω Ω | − | − 1 µ 1 ¶ c Du(y) dx d y = Ω ˆ | | ˆ x y n 1 | | Ω Ω | − | − 1 1 c Ω − + n Du(y) d y. É | | ˆΩ | | By the John condition we have

and thus n 1 c − Ω − n for every y Ω. | | É x y n 1 ∈ | − | − This implies Du(y) uB u c | | d y | 0 − Ω| É ˆ x y n 1 Ω | − | − and thus Du(y) u(x) u c | | d y cI1( Du χ )(x). | − Ω| É ˆ x y n 1 = | | Ω Ω | − | − Theorem 3.10 implies

3.4 A maximal function characterization of Sobolev spaces

Similar argument as in the proof of Sobolev-Poincaré inequality gives the following pointwise estimate.

Theorem 3.25. Assume that u C1(Rn). There exists a constant c c(n) 0 such ∈ = > that u(x) u(y) c x y (M Du (x) M Du (y)) | − | É | − | | | + | | for every x, y Rn. ∈ Proof. Let x, y Rn. Then x, y B(x,2 x y ) and B(x,2 x y ) B(y,4 x y ). By ∈ ∈ | − | | − | ⊂ | − | Remark 3.15 we obtain

u(x) u(y) u(x) uB(x,2 x y ) uB(x,2 x y ) u(y) | − | É | − | − | | + | | − | − | c x y (M Du (x) M Du (y)). É | − | | | + | | ä

Remarks 3.26: (1) If Du Lp(Rn), 1 p , then by (3.2) we have M Du Lp(Rn). | | ∈ < É ∞ | | ∈ (2) If Du L1(Rn), then by (3.1) we have M Du almost everywhere. | | ∈ | | < ∞ n (3) If Du L∞(R ), then M Du M Du L (Rn) Du L (Rn) everywhere. | | ∈ | | É k | |k ∞ É k k ∞ Thus

u(x) u(y) c Du L (Rn) x y | − | É k k ∞ | − | for every x, y Rn. In other words, u is Lipschitz continuous. ∈ CHAPTER 3. MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 90

Theorem 3.27. Assume that u W1,p(Rn), 1 p . There exists c c(n) and a ∈ < < ∞ = set N Rn with N 0 such that ⊂ | | = u(x) u(y) c x y (M Du (x) M Du (y)) | − | É | − | | | + | | for every x, y Rn \ N. ∈ n 1,p n Proof. C0∞(R ) is dense in W (R ) by Lemma 1.24. Thus there exists a sequence n 1,p n ui C0∞(R ), i 1,2,..., such that ui u in W (R ) as i . By passing to a ∈ = → →n ∞ subsequence, if necessary, we obtain an exceptional set N1 R with N1 0 such ⊂ | | = that

lim ui(x) u(x) i = < ∞ →∞ n for every x R \N1. By the sublinearity of the maximal operator and the maximal ∈ function theorem ° ° ° ° °M Dui M Du ° p n °M( Dui Du )° p n | | − | | L (R ) É | | − | | L (R ) ° ° c° Dui Du ° p n É | | − | || L (R ) c Dui Du Lp(Rn) É k − k p n which implies that M Dui M Du in L (R ) as i . By passing to a sub- | | → | | → ∞ n sequence, if necessary, we obtain an exceptional set N2 R with N2 0 such ⊂ | | = that

lim M Dui (x) M Du (x) i | | = | | < ∞ →∞ n for every x R \ N2. By Theorem 3.25 ∈

u(x) u(y) lim ui(x) ui(y) | − | = i | − | →∞ c x y lim (M Dui (x) M Dui (y)) É | − | i | | + | | →∞ c x y (M Du (x) M Du (y)) É | − | | | + | | n for every x R \ (N1 N2). ∈ ∪ ä Remark 3.28. Compare the proof above to Remark 3.15, which shows that the result holds for u W1,p(Rn), 1 p . ∈ É É ∞ The following deﬁnition motivated by Theorem 3.25.

Deﬁnition 3.29. Assume that 1 p and let u Lp(Rn). For a measurable < < ∞ ∈ function g : Rn [0, ] we denote g D(u) if there exists an exceptional set → ∞ ∈ N Rn such that N 0 and ⊂ | | = u(x) u(y) x y (g(x) g(y)) (3.6) | − | É | − | + for every x, y Rn \ N. We say that u Lp(Rn) belongs to the Hajłasz-Sobolev ∈ ∈ space M1,p(Rn), if there exists g Lp(Rn) with g D(u). This space is endowed ∈ ∈ with the norm

u M1,p(Rn) u Lp(Rn) inf g Lp(Rn). k k = k k + g D(u) k k ∈ CHAPTER 3. MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 91

T HEMORAL : The space M1,p(Rn) is deﬁned through the pointwise inequality (3.6).

Theorem 3.30. Assume that 1 p . Then M1,p(Rn) W1,p(Rn) and the asso- < < ∞ = ciate norms are equivalent, that is, there exists c such that 1 u 1,p n u 1,p n c u 1,p n c k kW (R ) É k kM (R ) É k kW (R ) for every measurable function u that belongs to M1,p(Rn) W1,p(Rn). =

T HEMORAL : This is a pointwise characterization of Sobolev spaces. This can be used as a deﬁnition of the ﬁrst order Sobolev spaces on metric measure spaces.

Proof. Assume that u W1,p(Rn). By Theorem 3.27 there exists c c(n) and a ⊃ ∈ = set N Rn with N 0 such that ⊂ | | = u(x) u(y) c x y (M Du (x) M Du (y)) | − | É | − | | | + | | for every x, y Rn \ N. Thus g cM Du D(u) Lp(Rn) and by the maximal ∈ = | | ∈ ∩ function theorem

u M1,p(Rn) u Lp(Rn) inf g Lp(Rn) k k = k k + g D(u) k k ∈ u Lp(Rn) cM Du Lp(Rn) É k k + k | |k u Lp(Rn) c Du Lp(Rn) É k k + k k c u 1,p n , É k kW (R ) where c c(n, p). = Assume then that u M1,p(Rn). Then u Lp(Rn) and there exists g Lp(Rn) ⊂ ∈ ∈ ∈ with g D(u). Then ∈ u(x h) u(x) h (g(x h) g(x)) | + − | É | | + + for almost every x, h Rn and thus ∈ u(x h) u(x) p dx h p (g(x h) g(x))p dx ˆRn | + − | É | | ˆRn + + 2p h p (g(x h)p g(x)p) dx É | | ˆRn + + p 1 p p 2 + g h . É k kLp(Rn)| | By the characterization of the Sobolev space with the integrated difference quotients, see Theorem 1.46, we conclude u W1,p(Rn) and ∈

u 1,p n c u Lp(Rn) c g Lp(Rn). k kW (R ) É k k + k k

The inequality u 1,p n c u 1,p n follows by taking inﬁmum over all g k kW (R ) É k kM (R ) ∈ D(u) Lp(Rn). ∩ ä CHAPTER 3. MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 92

Remark 3.31. The pointwise characterization of Sobolev spaces in Theorem 3.30 is very useful in studying properties of Sobolev spaces. For example, if u M1,p(Rn) ∈ and g D(u) Lp(Rn), then by the triangle inequality ∈ ∩ ¯ ¯ ¯ u(x) u(y) ¯ u(x) u(y) x y (g(x) g(y)) | | − | | É | − | É | − | + Thus g D( u ) Lp(Rn) and consequently u M1,p(Rn). ∈ | | ∩ | | ∈ The pointwise characterization of Sobolev spaces in Theorem 3.30 can be used to show a similar result as Theorem 1.41.

Lemma 3.32. The function u belongs to W1,p(Rn) if and only if u Lp(Rn) and p n ∈ there are functions ui L (R ), i 1,2,..., such that ui u almost everywhere p n ∈ = → p n and gi D(ui) L (R ) such that gi g almost everywhere for some g L (R ). ∈ ∩ → ∈ Proof. If u W1,p(Rn), then the claim of the lemma is clear. To see the converse, ∈ p n p n suppose that u, g L (R ), gi D(ui) L (R ) and ui u almost everywhere and ∈ ∈ ∩ → gi g almost everywhere. Then → ¡ ¢ ui(x) ui(y) x y gi(x) gi(y) (3.7) | − | É | − | + n n for all x, y R \ Fi with Fi 0, i 1,2,... Let A R be such that ui(x) u(x) ∈ | | = = ⊂ → n S and gi(x) g(x) for all x R \ A and A 0. Write F A ∞i 1 Fi. Then F 0. → ∈ | | = = ∪ = | | = Let x, y Rn \ F, x y. From (3.7) we obtain ∈ 6= u(x) u(y) x y ¡g(x) g(y)¢ | − | É | − | + and thus g D(u) Lp(Rn). This completes the proof. ∈ ∩ ä

3.5 Pointwise estimates

In this section we revisit pointwise inequalities for Sobolev functions.

Deﬁnition 3.33. Let 0 β and R 0. The fractional sharp maximal function < < ∞ > of a locally integrable function f is deﬁned by × # β fβ,R(x) sup r− f fB(x,r) d y, = 0 r R B(x,r) | − | < < If R we simply write f #(x). = ∞ β

T HEMORAL : The fractional sharp maximal function controls the mean oscillation of the function instead of the average of the function as in the Hardy- Littlewood maximal function.

Next we prove a more general pointwise inequality than in Theorem 3.27. CHAPTER 3. MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 93

Lemma 3.34. Suppose that f is locally integrable and let 0 β . Then there < < ∞ is c c(β, n) and a set E with E 0 such that = | | = β¡ # # ¢ f (x) f (y) c x y fβ,4 x y (x) fβ,4 x y (y) (3.8) | − | É | − | | − | + | − | for every x, y Rn \ E. ∈

T HEMORAL : This is a pointwise inequality for a function without the gradient.

Proof. Let E be the complement of the set of Lebesgue points of f . By Lebesgue’s n i theorem E 0. Fix x R \ E, 0 r and denote Bi B(x,2− r), i 0,1,... | | = ∈ < < ∞ = = Then

X∞ f (x) fB(x,r) fBi 1 fBi | − | É i 0 | + − | = × X∞ Bi | | f fBi d y É i 0 Bi 1 Bi | − | | + | = × X∞ i β i β c (2− r) (2− r)− f fBi d y É i 0 Bi | − | = crβ f # (x). É β,r Let y B(x, r) \ E. Then B(x, r) B(y,2r) and we obtain ∈ ⊂

f (y) fB(x,r) f (y) fB(y,2r) fB(y,2r) fB(x,r) | − | É | − | + | − | × β # cr fβ,2r(y) f fB(y,2r) dz É + B(x,r) | − | × β # cr fβ,2r(y) c f fB(y,2r) dz É + B(y,2r) | − | crβ f # (y). É β,2r Let x, y Rn \ E, x y and r 2 x y . Then x, y B(x, r) and hence ∈ 6= = | − | ∈

f (x) f (y) f (x) fB(x,r) f (y) fB(x,r) | − | É | − | + | − | β¡ # # ¢ c x y fβ,4 x y (x) fβ,4 x y (y) . É | − | | − | + | − | This completes the proof. ä Remark 3.35. Lemma 3.34 gives a Campanato type characterization for Hölder 1 n # n continuity. Let 0 β 1 and assume that f L (R ) with f L∞(R ). In other < É ∈ loc β ∈ words, there exists a constant M such that < ∞ × β r− f fB(x,r) d y M B(x,r) | − | É for every ball B(x, r) Rn. By Lemma 3.34, there exists a set E Rn with E 0 ⊂ ⊂ | | = such that f (x) f (y) c(n,β) x y β¡f #(x) f #(y)¢ | − | É | − | β + β CHAPTER 3. MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 94

for every x, y Rn \ E. This implies that ∈ # β u(x) u(y) c(n,β) f L (Rn) x y , | − | É k β k ∞ | − | n # n for every x, y R \ E with E 0. In other words, if f L∞(R ), then f can be ∈ | | = β ∈ redefined on a set of measure zero so that the function is Hölder continuous in Rn with exponent β. On the other hand, if f C0,β(Rn), then ∈ ¯ × ¯ ¯ ¯ f (y) fB(x,r) ¯f (y) f (z) dz¯ | − | = ¯ − B(x,r) ¯ × f (y) f (z) dz crβ É B(x,r) | − | É for every y B(x, r). Thus ∈ × # β fβ,R(x) sup r− f (y) fB(x,r) d y c = 0 r R B(x,r) | − | É < < n # n for every x R and this implies that f L∞(R ). Thus f can be redefined on a ∈ β ∈ set of measure zero so that the function is Hölder continuous with exponent β if and only if f # L ( n). In the limiting case 0 we obtain the space of bounded β ∞ R β ∈ n = 1 n mean oscillation BMO(R ), which consists of functions f Lloc(R ) satisfying # n ∈ f L∞(R ). 0 ∈ Definition 3.36. Let 0 α n and R 0. The fractional maximal function of É < > f L1 (Rn) is ∈ loc × α Mα,R f (x) sup r f d y, = 0 r R B(x,r) | | < < For R , we write Mα, Mα. If α 0, we obtain the Hardy–Littlewood = ∞ ∞ = = maximal function and we write M0 M. = If u W1,1(Rn), then by the Poincaré inequality with p 1, see Theorem 3.20, ∈ loc = there is c c(n) such that = × × u uB(x,r) d y cr Du d y B(x,r) | − | É B(x,r) | | for every ball B(x, r) Rn. It follows that ⊂ × × α 1 α r − u uB(x,r) d y cr Du d y B(x,r) | − | É B(x,r) | | and consequently # u1 α,R(x) cMα,R Du (x) − É | | for every x Rn and R 0. Thus we have proved the following useful inequality. ∈ > 1,1 Corollary 3.37. Let u W (Rn) and 0 α 1. Then there is c c(n,α) and a ∈ loc É < = set E Rn with E 0 such that ⊂ | | = 1 α¡ ¢ u(x) u(y) c x y − Mα,4 x y Du (x) Mα,4 x y Du (y) | − | É | − | | − || | + | − || | for every x, y Rn \ E. ∈ CHAPTER 3. MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 95

Remark 3.38. Corollary 3.37 gives a Morrey type condition for Hölder continuity. Compare to Remark 3.35, where Hölder continuity was characterized 1,1 n by a Campanato approach. Let 0 α 1 and assume that u Wloc (R ) with n É < ∈ M u L∞(R ). In other words, there exists a constant M such that α|∇ | ∈ < ∞ × rα Du d y M B(x,r) | | É

for every ball B(x, r) Rn. By Corollary 3.37, there exists E Rn with E 0 such ⊂ ⊂ | | = that 1 α¡ ¢ u(x) u(y) c(n,α) x y − M Du (x) M Du (y) (3.9) | − | É | − | α| | + α| | for every x, y Rn \ E. This implies that ∈ 1 α u(x) u(y) c(n,α) M Du L (Rn) x y − , | − | É k α| |k ∞ | − | n n for every x, y R \ E with E 0. In other words, if M u L∞(R ) then u can ∈ | | = α|∇ | ∈ be redefined on a set of measure zero so that the function is Hölder continuous in Rn with exponent 1 α. This shows that u is Hölder continuous with the exponent − 1 α, after a possible redefinition on a set of measure zero. − Remark 3.39. From (3.9) we recover Morrey’s inequality in Theorem 2.18. To see this, assume that u W1,p(Rn) with n p . By Hölder’s inequality we have ∈ < < ∞ 1 ¡ p ¢ p M n Du (x) c(n) Mn Du (x) c(n) Du Lp(Rn) , p | | É | | É k k < ∞ for every x Rn. Thus (3.9), with α n , implies ∈ = p n 1 p u(x) u(y) c(n, p) Du Lp(Rn) x y − | − | É k k | − | for every x, y Rn \ E with E 0. This shows that u is Hölder continuous with ∈ | | = the exponent 1 n after a possible redefinition on a set of measure zero. − p The next result shows that this gives a characterization of W1,p(Rn) for 1 < p . É ∞ Theorem 3.40. Let 1 p . Then the following four conditions are equivalent. < < ∞ (1) u W1,p(Rn). ∈ (2) u Lp(Rn) and there is g Lp(Rn), g 0, such that ∈ ∈ Ê u(x) u(y) x y (g(x) g(y)) | − | É | − | + for every x, y Rn \ E with E 0. ∈ | | = (3) u Lp(Rn) and there is g Lp(Rn), g 0, such that the Poincaré inequality ∈ ∈ Ê × × u uB(x,r) d y c r g d y B(x,r) | − | É B(x,r)

holds for every x Rn and r 0. ∈ > CHAPTER 3. MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 96

(4) u Lp(Rn) and u# Lp(Rn). ∈ 1 ∈ Proof. (1) We have already seen that (1) implies (2). (2) To prove that (2) implies (3), we integrate the pointwise inequality twice over the ball B(x, r). After the ﬁrst integration we obtain ¯ × ¯ ¯ ¯ u(y) uB(x,r) ¯u(y) u(z) dz¯ | − | = ¯ − B(x,r) ¯ × u(y) u(z) dz É B(x,r) | − | µ × ¶ 2r g(y) g(z) dz É + B(x,r) from which we have × µ× × ¶ u(y) uB(x,r) d y 2r g(y) d y g(z) dz B(x,r) | − | É B(x,r) + B(x,r) × 4r g(y) d y. É B(x,r)

(3) To show that (3) implies (4) we observe that × × # 1 u1(x) sup u uB(x,r) d y csup g d y cM g(x). = r 0 r B(x,r) | − | É r 0 B(x,r) = > > (4) Then we show that (4) implies (1). By Lemma 3.34

u(x) u(y) c x y (u#(x) u#(y)) | − | É | − | 1 + 1 for every x, y Rn \ E with E 0. If we denote g cu#, then g Lp(Rn) and ∈ | | = = 1 ∈ u(x) u(y) x y (g(x) g(y)) | − | É | − | + for every x, y Rn \ E with E 0. Then we use the characterization of Sobolev ∈ | | = spaces W1,p(Rn), 1 p , with integrated difference quotients, see Theorem < < ∞ 1.46. Let h Rn. Then ∈

uh(x) u(x) u(x h) u(x) h (gh(x) g(x)), | − | = | + − | É | | + from which we conclude that

uh u Lp(Rn) h ( gh Lp(Rn) g Lp(Rn)) 2 h g Lp(Rn). k − k É | | k k + k k = | |k k The claim follows from this. ä Remark 3.41. It can be shown that u W1,1(Rn) if and only if u L1(Rn) and there ∈ ∈ is a nonnegative function g L1(Rn) and σ 1 such that ∈ Ê ¡ ¢ u(x) u(y) x y Mσ x y g(x) Mσ x y g(y) | − | É | − | | − | + | − | for every x, y Rn \ E with E 0. Moreover, if this inequality holds, then Du ∈ | | = | | É c(n,σ)g almost everywhere. CHAPTER 3. MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 97

3.6 Approximation by Lipschitz functions

Smooth functions in C∞(Ω) and C0∞(Ω) are often used as canonical test functions in mathematical analysis. However, in many occasions smooth functions can be replaced by a more flexible class of Lipschitz functions. One highly useful property of Lipschitz functions, not shared by the smooth functions, is that the pointwise minimum and maximum over L-Lipschitz functions are still L-Lipschitz. The same is in fact true also for pointwise infimum and supremum of L-Lipschitz functions, if these are finite at a single point. In particular, it follows that if u : A R is an L-Lipschitz function, then the truncations max{u, c} and min{u, c} → with c R are L-Lipschitz. ∈ Theorem 3.42 (McShane). Assume that A Rn , 0 L and that f : A R ⊂ É < ∞ n → is an L-Lipschitz function. There exists an L-Lipschitz function f ∗ : R R such → that f ∗(x) f (x) for every x A. = ∈

T HEMORAL : Every Lipschitz continuous function deﬁned on a subset A of Rn can be extended as a Lipschitz continuous function to the whole Rn.

We claim that f ∗(b) f (b) for every b A. To see this we observe that = ∈ f (b) f (a) f (b) f (a) L b a , − É | − | É | − | which implies f (b) f (a) L b a for every a A. By taking inﬁmum over a A É + | − | ∈ ∈ we obtain f (b) f ∗(b). On the other hand, by the deﬁnition f ∗(b) f (b) for every É É b A. Thus f ∗(b) f (b) for every b A. ∈ = ∈ n n Then we claim that f ∗ is L-Lipschitz in R . Let x, y R . Then ∈ © ª f ∗(x) inf f (a) L x a : a A = + | − | ∈ inf©f (a) L( y a x y ) : a Aª É + | − | + | − | ∈ inf©f (a) L y a : a Aª L x y É + | − | ∈ + | − | f ∗(y) L x y . = + | − |

By switching the roles of x and y, we arrive at f ∗(y) f ∗(x) L x y . This implies É + | − | that L x y f ∗(x) f ∗(y) L x y . − | − | É − É | − | ä Remark 3.43. The function f : Rn R, ∗ → f (x) sup©f (a) L x a : a Aª. ∗ = − | − | ∈ CHAPTER 3. MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 98

is an L-Lipschitz extension of f as well. We can see, that f ∗ is the largest L- n Lipschitz extension of f in the sense that if g: R R is L-Lipschitz and g A f , → | = then g f ∗. Correspondingly, the function f is the smallest L-Lipschitz extension É ∗ of f .

n 1,p n Since C0∞(R ) is dense in W (R ), also compactly supported Lipschitz functions are dense in W1,p(Rn). By Theorem 3.27, we give a quantitative density result for Lipschitz functions in W1,p(Rn). The main difference of the following result to the standard molliﬁcation approximation u u as ε 0 is that the ε → → value of the function is not changed in a good set {x Rn : u (x) u(x)} and there ∈ ε = is an estimate for the measure of the bad set {x Rn : u (x) u(x)}. ∈ ε 6= Theorem 3.44. Assume that u W1,p(Rn), 1 p . Then for every ε 0 there ∈ < < ∞ > exists a Lipschitz continuous function u : Rn R such that ε → (1) {x Rn : u (x) u(x)} ε and | ∈ ε 6= | < (2) u u 1,p n ε. k − εkW (R ) < Proof. Let E {x Rn : M Du (x) λ}, λ 0. We show that u is cλ-Lipschitz in λ = ∈ | | É > Eλ. By Theorem 3.27

u(x) u(y) c x y (M Du (x) M Du (y)) cλ x y | − | É | − | | | + | | É | − | for almost every x, y E . The McShane extension theorem allows us to ﬁnd a ∈ λ cλ-Lipschitz extension v : Rn R. We truncate v and obtain a 2cλ-Lipschitz λ → λ function u max{ λ,min{v ,λ}}. λ = − λ Observe that u λ in Rn and u u almost everywhere in E . | λ| É λ = λ (1) We consider measure of the set

Rn \ E {x Rn : M Du (x) λ}. λ = ∈ | | > There exits c c(n, p) such that =

λp {x Rn : M Du (x) λ} c Du(x) p dx 0 | ∈ | | | > | É ˆ{x Rn: Du(x) λ } | | → ∈ | |> 2 as λ , since Du Lp(Rn). This follows by choosing f Du in the following → ∞ ∈ = | | general fact for the Hardy-Littlewood maximal function. Claim: If f Lp(Rn), then there exists c c(n, p) such that ∈ = c {x Rn : M f (x) λ} f (x) p dx, λ 0. | ∈ > | É λ ˆ{x Rn: f (x) λ } | | > ∈ | |> 2

Reason. Let f f1 f2, where f1 f χ{ f λ } and f2 f χ{ f λ }. Then = + = | |> 2 = | |É 2 µ ¶1 p p 1 p λ − p f1(x) dx f1(x) f1(x) − dx f Lp(Rn) . ˆRn | | = ˆ{x Rn: f (x) λ } | | | | É 2 k k < ∞ ∈ | |> 2 CHAPTER 3. MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 99

1 n λ n This shows that f1 L (R ). On the other hand, f2(x) 2 for every x R , ∈ λ n | | É p ∈ which implies f n and f L (R ). Thus every L function can be 2 L∞(R ) 2 2 ∞ k k É 1 ∈ represented as a sum of an L function and an L∞ function. By Lemma 3.2, we have λ M f2 L (Rn) f2 L (Rn) k k ∞ É k k ∞ É 2 From this we conclude using sublinearity of the maximal operator that

λ M f (x) M(f1 f2)(x) M f1(x) M f2(x) M f1(x) = + É + É + 2

n λ for every x R and thus M f (x) λ implies M f1(x) . It follows that ∈ > > 2 ¯½ ¾¯ n ¯ n λ ¯ {x R : M f (x) λ} ¯ x R : M f1(x) ¯ | ∈ > | É ¯ ∈ > 2 ¯

for every λ 0. > p 1 By the maximal function theorem on L1(Rn), see (3.1), we have = ¯½ ¾¯ ¯ n λ ¯ c c ¯ x R : M f1(x) ¯ f1 L1(Rn) f (x) dx ¯ ∈ > 2 ¯ É λ k k = λ ˆ{x Rn: f (x) λ } | | ∈ | |> 2 for every λ 0. > 1 p By Chebyshev’s inequality and by the maximal function theorem < < ∞ Lp(Rn), p 1, see (3.2), we have > ¯½ λ ¾¯ µ 2 ¶p c ¯ x n : M f x ¯ M f x p dx f x p dx ¯ R 1( ) ¯ ( 1( )) p 1( ) ¯ ∈ > 2 ¯ É λ ˆRn É λ ˆRn | | c f x dx p ( ) = λ ˆ{x Rn: f (x) λ } | | ∈ | |> 2 for every λ 0. > ■ Thus we conclude that

λp Rn \ E λp {x Rn : M Du (x) λ} | λ| É | ∈ | | | > | c Du(x) p dx É ˆ{x Rn: Du(x) λ } | | ∈ | |> 2 and consequently λp Rn \ E 0 and Rn \ E 0 as λ . | λ| → | λ| → → ∞ (2) Next we prove an estimate for u u 1,p n . Since u u in E and k − εkW (R ) λ = λ u λ in Rn, we have | λ| É

p p uλ u Lp(Rn) uλ u dx k − k = ˆ n | − | R \Eλ µ ¶ p p p 2 uλ dx u dx É ˆ n | | + ˆ n | | R \Eλ R \Eλ µ ¶ p p n p 2 λ R \ Eλ u dx 0 É | | + ˆ n | | → R \Eλ CHAPTER 3. MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 100

as λ . → ∞ To prove the corresponding estimate for the gradients, we note that

D(u u) χRn\E D(u u) χRn\E Du χRn\E Du λ − = λ λ − = λ λ − λ almost everywhere. Recall that u is cλ-Lipschitz and thus Du cλ almost λ | λ| É everywhere.

p p D(uλ u) Lp(Rn) D(uλ u) dx k − k = ˆ n | − | R \Eλ µ ¶ p p p 2 Duλ dx Du dx É ˆ n | | + ˆ n | | R \Eλ R \Eλ µ ¶ p p n p 2 (2cλ) R \ Eλ Du dx 0 É | | + ˆ n | | → R \Eλ

as λ . Thus u u 1,p n 0 as λ . Observe that → ∞ k − λkW (R ) → → ∞ {x Rn : u(x) u (x)} Ω \ E , ∈ 6= λ ⊂ λ with Rn \ E 0 as λ . This proves the claims. | λ| → → ∞ ä n Remark 3.45. Let Eλ {x R : M Du (x) λ}, λ 0. Let Qi, i 1,2,... be a = ∈n | | É > = Whitney decomposition of R \ Eλ with the following properties: each Qi is open, n n cubes Qi, i 1,2,..., are disjoint, R \ Eλ ∞i 1Qi, 4Qi R \ Eλ, i 1,2,..., = = ∪ = ⊂ =

X∞ χ2Qi N , i 1 É < ∞ = and

c1 dist(Qi,E ) diam(Qi) c2 dist(Qi,E ) λ É É λ

for some constants c1 and c2.

Then we construct a partition of unity associated with the covering 2Qi,

i 1,2,... This can be done in two steps. First, let ϕi C∞(2Qi) be such that = ∈ 0 0 ϕi 1, ϕi 1 in Qi and É É = c Dϕi , | | É diam(Qi) for i 1,2,... Then we deﬁne = ϕi(x) φi(x) P = ∞j 1 ϕ j(x) = for every i 1,2,.... Observe that the sum is over ﬁnitely many terms only since = ϕi C∞(2Qi) and the cubes 2Qi, i 1,2,..., are of bounded overlap. The functions ∈ 0 = φi have the property X∞ n φi(x) χR \Eλ (x) i 1 = = for every x Rn. ∈ CHAPTER 3. MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 101

Then we deﬁne the function uλ by  u(x), x Eλ, uλ(x) ∈ = P n  ∞i 1 φi(x)u2Qi , x R \ Eλ. = ∈ n The function u is a Whitney type extension of u E to the set R \ E . λ | λ λ First we claim that

u 1,p n c u 1,p n . (3.10) k λkW (R \Eλ) É k kW (R \Eλ)

Since the cubes 2Qi, i 1,2,..., are of bounded overlap, we have = ¯ ¯p p ¯ X∞ ¯ X∞ p uλ dx φi(x)u2Qi dx c u2Qi dx ˆ n | | = ˆ n ¯ ¯ É ˆ | | R \Eλ R \Eλ i 1 i 1 2Qi =× = X∞ p p c 2Qi u dx c u dx. É | | | | É ˆ n | | i 1 2Qi R \Eλ = Then we consider an estimate for the gradient. We recall that

X∞ Φ(x) φi(x) 1 = i 1 = = n for every x R \ Eλ. Since the cubes 2Qi, i 1,2,..., are of bounded overlap, we ∈ n = see that Φ C∞(R \ E ) and ∈ λ X∞ D jΦ(x) D jφi(x) 0, j 1,2,..., n, = i 1 = = = for every x Rn \ E . Hence we obtain ∈ λ ¯ ¯ ¯ ¯ D u (x) ¯ X∞ D (x)u ¯ ¯ X∞ D (x)(u(x) u )¯ j λ ¯ jφi 2Qi ¯ ¯ jφi 2Qi ¯ | | = i 1 = i 1 − = = X∞ 1 c diam(Qi)− u(x) u2Qi χ2Qi (x) É i 1 | − | = and consequently

p X∞ p p D j uλ(x) c diam(Qi)− u(x) u2Qi χ2Qi (x). | | É i 1 | − | =

Here we again used the fact that the cubes 2Qi, i 1,2,..., are of bounded overlap. = This implies that for every j 1,2,..., n, =

p ³ X∞ p p ´ D j uλ dx c diam(Qi)− u u2Qi χ2Qi dx ˆ n | | É ˆ n | − | R \Eλ R \Eλ i 1 = X∞ p p diam(Qi)− u u2Qi dx É i 1 ˆ2Qi | − | = c X∞ Du p dx c Du p dx. É ˆ | | É ˆ n | | i 1 2Qi R \Eλ = CHAPTER 3. MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 102

Then we show that u W1,p(Rn). We know that u W1,p(Rn \E ) and that it λ ∈ λ ∈ λ is Lipschitz continuous in Rn (exercise). Moreover u W1,p(Rn) and u u in E ∈ = λ λ by (i). This implies that w u u W1,p(Rn \ E ). and that w 0 in E . By the = − λ ∈ λ = λ ACL property, u is absolutely continuous on almost every line segment parallel to the coordinate axes. Take any such line. Now w is absolutely continuous on the part of the line segment which intersects Rn \ E . On the other hand w 0 in the λ = complement of Eλ. Hence the continuity of w in the line segment implies that w is absolutely continuous on the whole line segment. We have

u u 1,p n u u 1,p n k − λkW (R ) = k − λkW (R \Eλ) u 1,p n u 1,p n É k kW (R \Eλ) + k λkW (R \Eλ) c u 1,p n . É k kW (R \Eλ)

3.7 Maximal operator on Sobolev spaces

Assume that u is Lipschitz continuous with constant L, that is

uh(y) u(y) u(y h) u(y) L h | − | = | + − | É | | n for every y, h R , where we denote uh(y) u(y h). Since the maximal function ∈ = + commutes with translations and the maximal operator is sublinear, we have

(Mu)h(x) Mu(x) M(uh)(x) Mu(x) | − | = | − | M(uh u)(x) É − 1 sup uh(y) u(y) d y = r 0 B(x, r) ˆB(x,r) | − | > | | L h . É | | This means that the maximal function is Lipschitz continuous with the same constant as the original function if Mu is not identically inﬁnity. Observe, that this proof applies to Hölder continuous functions as well. Next we show that the Hardy-Littlewood maximal operator is bounded in Sobolev spaces.

Theorem 3.46. Let 1 p . If u W1,p(Rn), then Mu W1,p(Rn). Moreover, < < ∞ ∈ ∈ there exists c c(n, p) such that =

Mu 1,p n c u 1,p n . (3.11) k kW (R ) É k kW (R )

T HEMORAL : M : W1,p(Rn) W1,p(Rn), p 1, is a bounded operator. Thus → > the maximal operator is not only bounded on Lp(Rn) but also on W1,p(Rn) for p 1. > CHAPTER 3. MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 103

Proof. The proof is based on the characterization of W1,p(Rn) by integrated difference quotients, see Theorem 1.46. By the maximal function theorem with 1 p , see (3.2), we have Mu Lp(Rn) and < < ∞ ∈

(Mu)h Mu Lp(Rn) M(uh) Mu Lp(Rn) k − k = k − k M(uh u) Lp(Rn) É k − k c uh u Lp(Rn) É k − k c Du Lp(Rn) h É k k | | for every h Rn. Theorem 1.46 gives Mu W1,p(Rn) with ∈ ∈

DMu Lp(Rn) c Du Lp(Rn). k k É k k Thus by the maximal function theorem

1 ³ p p ´ p Mu 1,p n Mu DMu k kW (R ) = k kLp(Rn) + k kLp(Rn) Mu Lp(Rn) DMu Lp(Rn) É k k + k k ¡ ¢ c u Lp(Rn) Du Lp(Rn) É k k + k k c u 1,p n . É k kW (R ) ä A more careful analysis gives a pointwise estimate for the partial derivatives.

Theorem 3.47. Let 1 p . If u W1,p(Rn), then Mu W1,p(Rn) and < < ∞ ∈ ∈

D j Mu M(D j u), j 1,2,..., n, (3.12) | | É = almost everywhere in Rn.

T HEMORAL : Differentiation commutes with a linear operator. The sublinear maximal operator semicommutes with differentiation.

Proof. If χB(0,r) is the characteristic function of B(0, r) and

χB(0,r) χr , = B(0, r) | | then 1 1 u(y) d y u(x y) d y B(x, r) ˆ | | = B(0, r) ˆ | − | | | B(x,r) | | B(0,r) 1 χB(0,r) u(x y) d y = B(0, r) ˆ n | − | | | R ( u χr)(x), = | | ∗ 1,p n where denotes the convolution. Now u χr W (R ) and by Theorem 1.17 (1) ∗ | | ∗ ∈

D j( u χr) χr D j u , j 1,2,..., n, | | ∗ = ∗ | | = CHAPTER 3. MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 104

almost everywhere in Rn.

Let rm, m 1,2,..., be an enumeration of positive rationals. Since u is locally = integrable, we may restrict ourselves to the positive rational radii in the deﬁnition of the maximal function. Hence

Mu(x) sup( u χrm )(x). = m | | ∗ n We deﬁne functions vk : R R, k 1,2,..., by → =

vk(x) max ( u χrm )(x). = 1 m k | | ∗ É É 1,p n Now (vk) is an increasing sequence of functions in W (R ), which converges to Mu pointwise and ¯ ¯ D j vk max ¯D j( u χrm )¯ | | É 1 m k | | ∗ É É ¯ ¯ max ¯χrm D j u ¯ = 1 m k ∗ | | É É M(D j u ) M(D j u), j 1,2,..., n, É | | = = almost everywhere in Rn. Here we also used Remark 1.27 and the fact that by Theorem 1.26 ¯ ¯ ¯D j u ¯ D j u , j 1,2,..., n, | | = | | = almost everywhere. Thus

n n X X Dvk Lp(Rn) D j vk Lp(Rn) M(D j u) Lp(Rn) k k É j 1 k k É j 1 k k = = and the maximal function theorem implies

n X vk W1,p(Rn) Mu Lp(Rn) M(D j u) Lp(Rn) k k É k k + j 1 k k = n X c u Lp(Rn) c D j u Lp(Rn) c É k k + j 1 k k É < ∞ = 1,p n for every k 1,2,... Hence (vk) is a bounded sequence in W (R ) which converges = 1,p n p n to Mu pointwise. Theorem 1.41 implies Mu W (R ), vk Mu weakly in L (R ) p n ∈ → and D j vk D j Mu weakly in L (R ) as k . → → ∞ Next we prove the pointwise estimate for the gradient. By Mazur’s lemma, see Theorem 1.33, there is a sequence of convex combinations such that

m Xk wk ak,l D j vl D j Mu, j 1,..., n, = l k → = = p n in L (R ) as k . There is a subsequence of (wk) which converges almost → ∞ everywhere to D j Mu. Thus we have

m m Xk ¯ ¯ Xk wk ak,l ¯D j vl¯ ak,l M(D j u) M(D j u) | | É l k É l k = = = CHAPTER 3. MAXIMAL FUNCTION APPROACH TO SOBOLEV SPACES 105 for every l 1,2,... and ﬁnally =

D j Mu lim wk M(D j u), j 1,..., n, | | = k | | É = →∞ almost everywhere in Rn. This completes the proof. ä Remarks 3.48: (1) Estimate (3.11) also follows from (3.12). To see this, we may use the maximal function theorem, see (3.2), and (3.12) to obtain

Mu 1,p n Mu Lp(Rn) DMu Lp(Rn) k kW (R ) É k k + k k c u Lp(Rn) M Du Lp(Rn) É k k + k | |k c u 1,p n , É k kW (R ) where c is the constant in (3.2). 1, n (2) If u W ∞(R ), then a slight modiﬁcation of our proof shows that Mu ∈ 1, n belongs to W ∞(R ). Moreover,

Mu 1, n Mu L (Rn) DMu L (Rn) k kW ∞(R ) = k k ∞ + k k ∞ u L (Rn) M Du L (Rn) É k k ∞ + k | |k ∞ u 1, n . É k kW ∞(R ) Hence in this case the maximal operator is bounded with constant one. 1, n Recall, that after a redeﬁnition on a set of measure zero u W ∞(R ) is a ∈ bounded and Lipschitz continuous function, see Theorem 2.23. 4 Pointwise behaviour of Sobolev functions

In this chapter we study fine properties of Sobolev functions. By definition, Sobolev functions are defined only up to Lebesgue measure zero and thus it is not always clear how to use their pointwise properties to give meaning, for example, to boundary values.

4.1 Sobolev capacity

Capacities are needed to understand pointwise behavior of Sobolev functions. They also play an important role in studies of solutions of partial differential equations.

Deﬁnition 4.1. For 1 p , the Sobolev p-capacity of a set E Rn is deﬁned < < ∞ ⊂ by

p capp(E) inf u 1,p n = u A (E) k kW (R ) ∈ ³ p p ´ inf u Lp( n) Du Lp( n) = u A (E) k k R + k k R ∈ inf ¡ u p Du p¢ dx, = u A (E) ˆ n | | + | | ∈ R © 1,p n ª where A (E) u W (R ) : u 1 on a neighbourhood of E . If A (E) , we set = ∈ Ê = ; cap (E) . Functions in A (E) are called admissible functions for E. p = ∞

T HEMORAL : Capacity measures the size of exceptional sets for Sobolev functions. Lebesgue measure is the natural measure for functions in Lp(Rn) and the Sobolev p-capacity is the natural outer measure for functions in W1,p(Rn).

106 CHAPTER 4. POINTWISE BEHAVIOUR OF SOBOLEV FUNCTIONS 107

Remark 4.2. In the deﬁnition of capacity we can restrict ourselves to the admissible functions u for which 0 u 1. Thus É É p capp(E) inf u 1,p n , = u A (E) k kW (R ) ∈ 0 © 1,p n ª where A 0(E) u W (R ) : 0 u 1, u 1 on a neighbourhood of E . = ∈ É É =

Reason. (1) Since A 0(E) A (E), we have ⊂ p capp(E) inf u 1,p n . É u A (E) k kW (R ) ∈ 0

(2) For the reverse inequality, let ε 0 and let u A (E) such that > ∈ p u cap (E) ε. k kW1,p(Rn) É p +

Then v max{0,min{u,1}} A 0(E), v u and by Remark 1.27 we have Dv = ∈ | | É | | | | É Du almost everywhere. Thus | | p p p inf u 1,p n v 1,p n u 1,p n capp(E) ε u A (E) k kW (R ) É k kW (R ) É k kW (R ) É + ∈ 0 and by letting ε 0 we obtain → p inf u 1,p n capp(E). u A (E) k kW (R ) É ∈ 0

Remarks 4.3: (1) There are several alternative deﬁnitions for capacity and, in general, it does not matter which one we choose. For example, when 1 p n, we < < may consider the deﬁnition

p capp(E) inf Du dx, = ˆRn | |

p n p n where the infimum is taken over all u L ∗(R ) with Du L (R ), u 0 ∈ | | ∈ Ê and u 1 on a neighbourhood of E. Some estimates and arguments may Ê become more transparent with this definition, but we stick to our original definition. (2) The definition of Sobolev capacity applies also for p 1, but we shall not = discuss this case here.

The Sobolev p-capacity enjoys many desirable properties, one of the most important of which says that it is an outer measure.

Theorem 4.4. The Sobolev p-capacity is an outer measure, that is,

(1) cap ( ) 0, p ; = (2) if E1 E2, then cap (E1) cap (E2) and ⊂ p É p CHAPTER 4. POINTWISE BEHAVIOUR OF SOBOLEV FUNCTIONS 108

¡S ¢ P n (3) capp ∞i 1 Ei ∞i 1 capp(Ei) whenever Ei R , i 1,2,.... = É = ⊂ =

T HEMORAL : Capacity is an outer measure, but measure theory is useless since there are very few measurable sets.

Proof. (1) Clearly cap ( ) 0. p ; = (2) A (E2) A (E1) implies cap (E1) cap (E2). ⊂ p É p P (3) Let ε 0. We may assume that ∞i 1 capp(Ei) . Choose ui A (Ei) so > = < ∞ ∈ that p i ui cap (Ei) ε2− , i 1,2,.... k kW1,p(Rn) É p + = S Claim: v supi ui is admissible for ∞i 1 Ei. = = Reason. First we show that v W1,p(Rn). Let ∈

vk max ui, k 1,2,.... = 1 i k = É É

Then (vk) is an increasing sequence such that vk v pointwise as k . More- → → ∞ over

vk max ui sup ui v , k 1,2,..., | | = | 1 i k | É | i | = | | = É É and by Remark 1.27

Dvk max Dui sup Dui , k 1,2,.... | | É 1 i k | | É i | | = É É 1,p n We show that (vk) is a a bounded sequence in W (R ). To conclude this, we observe that

Since vk v almost everywhere, by weak compactness of Sobolev spaces, see → 1,p n Theorem 1.41, we conclude that v W (R ). Since ui A (Ei), there exists an ∈ ∈ open set Oi Ei such that ui 1 on Oi for every i 1,2,.... It follows that ⊃ S Ê =S v supi ui 1 on ∞i 1 Oi, which is a neighbourhood of ∞i 1 Ei. = Ê = = ■ CHAPTER 4. POINTWISE BEHAVIOUR OF SOBOLEV FUNCTIONS 109

We conclude that

[∞ p X∞ p X∞ cap ¡ E ¢ v u cap (E ) ε. p i W1,p(Rn) i W1,p(Rn) p i i 1 É k k É i 1 k k É i 1 + = = = The claim follows by letting ε 0. → ä Remark 4.5. The Sobolev p-capacity is outer regular, that is,

cap (E) inf{cap (O) : E O, O open}. p = p ⊂ Reason. (1) By monotonicity,

cap (E) inf{cap (O) : E O, O open}. p É p ⊂ (2) To see the inequality in the other direction, let ε 0 and take u A (E) > ∈ such that p u cap (E) ε. k kW1,p(Rn) É p + Since u A (E) there is an open set O containing E such that u 1 on O, which ∈ Ê implies p cap (O) u cap (E) ε. p É k kW1,p(Rn) É p + The claim follows by letting ε 0. → ■ T HEMORAL : The capacity of a set is completely determined by the capacities of open sets containing the set. The same applies to the Lebesgue outer measure.

4.2 Capacity and measure

We are mainly interested in sets of vanishing capacity, since they are in some sense exceptional sets in the theory Sobolev spaces. Our ﬁrst result is rather immediate.

Lemma 4.6. E cap (E) for every E Rn. | | É p ⊂

T HEMORAL : Sets of capacity zero are of measure zero. Thus capacity is a ﬁner measure than Lebesgue measure.

Proof. If cap (E) , there is nothing to prove. Thus we may assume that p = ∞ cap (E) . Let ε 0 and take u A (E) such that p < ∞ > ∈ p u cap (E) ε. k kW1,p(Rn) É p + There is an open O E such that u 1 in O and thus ⊃ Ê p p E O u p dx u u cap (E) ε. Lp(Rn) W1,p(Rn) p | | É | | É ˆO | | É k k É k k É + The claim follows by letting ε 0. → ä CHAPTER 4. POINTWISE BEHAVIOUR OF SOBOLEV FUNCTIONS 110

Remark 4.7. Lemma 4.6 shows that cap (B(x, r)) 0 for every x Rn, r 0. This p > ∈ > implies that capacity is nontrivial in the sense that every nonempty open set has positive capacity.

Lemma 4.8. Let x Rn and 0 r 1. Then there exists c c(n, p) such that ∈ < É = n p cap (B(x, r)) cr − p É

T HEMORAL : For the Lebesgue measure of a ball we have B(x, r) crn, but n p | | É for the Sobolev capacity of a ball we have cap (B(x, r)) cr − . Thus the natural p É scaling dimension for capacity is n p. Observe, that the dimension for capacity − is smaller than n 1. −

Proof. Define a cutoff function  1, y B(x, r),  ∈  y x u(y) 2 | − | , y B(x,2r) \ B(x, r), = r  − ∈ 0, y Rn \ B(x,2r). ∈ Observe that 0 u 1, u is 1 -Lipschitz and Du 1 almost everywhere. Thus É É r | | É r u A (B(x, r)) and ∈ p p capp(B(x, r)) u(y) d y Du(y) d y É ˆB(x,2r) | | + ˆB(x,2r) | | p p p (1 r− ) B(x,2r) (r− r− ) B(x,2r) É + | | É + | | p n p 2r− B(x,2r) cr − , = | | = with c c(n, p) = ä Remarks 4.9: (1) Lemma 4.8 shows that every bounded set has finite capacity. Thus there are plenty of sets with finite capacity.

Reason. Assume that E Rn is bounded. Then E B(0, r) for some r, ⊂ ⊂ 0 r 1, and < É n p capp(E) capp(B(0, r)) cr − . É É < ∞ ■ (2) Lemma 4.8 implies that cap ({x}) 0 for every x Rn when 1 p n. p = ∈ < < Reason.

n p cap ({x}) cap (B(x, r)) cr − , 0 r 1. p É p É < É The claim follows by letting r 0. → ■ Remark 4.10. Let x Rn and 0 r 1 . Then there exists c c(n) such that ∈ < É 2 = µ 1 ¶1 n cap (B(x, r)) c log − . n É r CHAPTER 4. POINTWISE BEHAVIOUR OF SOBOLEV FUNCTIONS 111

Reason. Use the test function

 1 ¡ 1 ¢− 1  log r log x y , y B(x,1) \ B(x, r),  | − | ∈ u(y) 1, y B(x, r), =  ∈ 0, y Rn \ B(x,1). ∈ n This implies that capp({x}) 0 for every x R when p n (exercise). = ∈ = ■ We have shown that a point has zero capacity when 1 p n. By countable < É subadditivity all countable sets have zero capacity as well. Next we show that a point has positive capacity when p n. > Lemma 4.11. If p n, then cap ({x}) 0 for every x Rn. > p > ∈

T HEMORAL : For p n every set containing at least one point has a positive > capacity. Thus there are no nontrivial sets of capacity zero. In practice this means that capacity is a useful tool only when p n. É Proof. Let x Rn and assume that u A ({x}). Then there exists 0 r 1 such that ∈ ∈ < É u(y) 1 on B(x, r). Take a cutoff function η C∞(B(x,2)) such that 0 η 1, η 1 Ê ∈ 0 É É = in B(x, r) and Dη 2. By Morrey’s inequality, see Theorem 2.18, there exists | | É c c(n, p) 0 such that = > n 1 p (ηu)(y) (ηu)(z) c y z − D(ηu) Lp(Rn) | − | É | − | k k for almost every y, z Rn. Choose y B(x, r) and z B(x,4) \ B(x,2) so that ∈ ∈ ∈ (ηu)(y) 1 and (ηu)(z) 0. Then 1 y z 5 and thus Ê = É | − | É

p p D(ηu)(y) d y D(ηu) Lp(Rn) ˆB(x,2) | | = k k n p p c y z − (ηu)(y) (ηu)(z) c 0. Ê | −{z| }| −{z |} Ê > 5n p 1 Ê − Ê On the other hand µ ¶ D(ηu)(y) p d y 2p Dη(y)u(y) p d y η(y)Du(y) p d y ˆB(x,2) | | É ˆB(x,2) | | + ˆB(x,2) | |   p p p p p 2  Dη(y) u(y) d y η(y) Du(y) d y = ˆB(x,2) | {z |}| | + ˆB(x,2) | {z |}| | 2p 1 µ É É¶ 4p u(y) p d y Du(y) p d y É ˆB(x,2) | | + ˆB(x,2) | | 4p u p . É k kW1,p(Ω) This shows that there exists c c(n, p) 0 such that u p c 0 for every = > k kW1,p(Ω) Ê > u A ({x}) and thus cap ({x}) c 0. ∈ p Ê > ä CHAPTER 4. POINTWISE BEHAVIOUR OF SOBOLEV FUNCTIONS 112

In order to study the connection between capacity and measure, we need to consider lower dimensional measures than the Lebesgue measure. We recall the deﬁnition of Hausdorff measures.

Deﬁnition 4.12. Let E Rn and s 0. For 0 δ we set ⊂ Ê < É ∞ ( ) s X∞ s : [∞ Hδ (E) inf ri E B(xi, ri), ri δ . = i 1 ⊂ i 1 É = = The (spherical) s-Hausdorff measure of E is

s s s H (E) lim Hδ (E) supHδ (E). = δ 0 = δ 0 → > The Hausdorff dimension of E is

T HEMORAL : The Hausdorff measure is the natural s-dimensional measure up to scaling and the Hausdorff dimension is the measure theoretic dimension of the set. Observe that the dimension can be any nonnegative real number less or equal than the dimension of the space.

We begin by proving a useful measure theoretic lemma. In the proof we need some tools from measure and integration theory and real analysis.

Lemma 4.13. Assume that 0 s n, f L1 (Rn) and let < < ∈ loc ½ 1 ¾ E x n : f d y R limsup s 0 . = ∈ r 0 r ˆB(x,r) | | > → Then H s(E) 0. =

T HEMORAL : Roughly speaking the lemma above says that the set where a locally integrable function blows up rapidly is of the corresponding Hausdorff measure zero.

Proof. (1) Assume ﬁrst that f L1(Rn). ∈ (2) By the Lebesgue differentiation theorem

× lim f (y) d y f (x) , r 0 B(x,r) | | = | | < ∞ → for almost every x Rn. If x is a Lebesgue point of f , then ∈ | | 1 × f y d y c rn s f y d y limsup s ( ) limsup − ( ) 0. r 0 r ˆB(x,r) | | = r 0 B(x,r) | | = → → This shows that all Lebesgue points of f belong to the complement of E and thus | | E 0. | | = CHAPTER 4. POINTWISE BEHAVIOUR OF SOBOLEV FUNCTIONS 113

Reason. Let 0 δ 1. For every x E there exists a rx with 0 rx δ such that < < ∈ ε < É 1 s f d y ε. rx ˆB(x,rx) | | >

By the Vitali covering theorem, there exists a subfamily of countably many pairwise disjoint balls B(xi, ri), i 1,2,..., such that =

[∞ Eε B(xi,5ri). ⊂ i 1 = This gives

s s s X∞ s 5 X∞ 5 H5 (Eε) (5ri) f d y f d y. δ É É ε ˆ | | = ε ˆS | | i 1 i 1 B(xi,ri) ∞i 1 B(xi,ri) = = = By disjointness of the balls ¯ ¯ ¯ ¯ ¯[∞ ¯ X∞ X∞ n ¯ B(x, ri)¯ B(xi, ri) c ri ¯i 1 ¯ = i 1 | | = i 1 = = = rn c X∞ i f d y rs ˆ É i 1 ε i B(xi,ri) | | =n s δ − c f d y 0 as δ 0. É ε ˆRn | | → → By absolute continuity of integral

f d y 0 as δ 0. ˆS | | → → ∞i 1 B(xi,ri) = Thus s s s 5 H (Eε) lim H5δ(Eε) lim f d y 0. = δ 0 É ε δ 0 ˆS B(x ,r ) | | = → → ∞i 1 i i = s This shows that H (Eε) 0 for every ε 0. = > ■ (4) By subadditivity of the Hausdorff measure

Ã ! s s [∞ X∞ s H (E) H E 1 H (E 1 ) 0. = k 1 k É k 1 k = = = This shows that H s(E) 0. = CHAPTER 4. POINTWISE BEHAVIOUR OF SOBOLEV FUNCTIONS 114

(5) Assume then that f L1 (Rn). Then ∈ loc µ½ 1 ¾¶ s E s x n : f d y H ( ) H R limsup s 0 = ∈ r 0 r ˆB(x,r) | | > Ã → ! ½ 1 ¾ s [∞ x n : f d y H R limsup s χB(0,k) 0 = k 1 ∈ r 0 r ˆB(x,r) | | > = → µ½ 1 ¾¶ X∞ s x n : f d y H R limsup s χB(0,k) 0 0. É k 1 ∈ r 0 r ˆB(x,r) | | > = ä = → Next we compare capacity to the Hausdorff measure.

Theorem 4.14. Assume that 1 p n. Then there exists c c(n, p) such that n p < n< = cap (E) cH − (E) for every E R . p É ⊂

T HEMORAL : Capacity is smaller than (n p)-dimensional Hausdorff measure. n p − In particular, H − (E) 0 implies cap (E) 0. = p =

Proof. Let B(xi, ri), i 1,2,..., be any covering of E such that the radii satisfy = ri δ. Subadditivity implies É X∞ X∞ n p capp(E) capp(B(xi, ri)) c ri − . É i 1 É i 1 = = By taking the inﬁmum over all coverings by such balls and observing that H s(E) δ É H s(E) we obtain n p n p capp(E) cH − (E) cH − (E). É δ É ä We next consider the converse of the previous theorem. We prove that sets of p-capacity zero have Hausdorff dimension at most n p. − Theorem 4.15. Assume that 1 p n. If E Rn with cap (E) 0, then H s(E) < < ⊂ p = = 0 for all s n p. > − n Proof. (1) Let E R be such that capp(E) 0. Then for every i 1,2,..., there is ⊂ p = = u A (E) such that u 2 i. Deﬁne u P u . i 0 i W1,p(Rn) − ∞i 1 i ∈ k k É = = Claim: u A (E). ∈ Pk 1,p n Reason. Let vk i 1 ui, k 1,2,.... Then vk W (R ) and = = = ∈ ° k ° k °X ° X vk 1,p n ° ui° ui 1,p n k kW (R ) = ° ° É k kW (R ) °i 1 °W1,p n i 1 = (R ) = i X∞ X∞ p ui W1,p(Rn) 2− . É i 1 k k É i 1 < ∞ = = 1,p n Thus (vk) is a bounded sequence in W (R ). Since 0 ui 1, we observe that É É (vk) is an increasing sequence and thus vk u almost everywhere. Theorem 1.41 → implies u W1,p(Rn). Moreover, u 1 almost everywhere on a neighbourhood of E ∈ Ê which shows that u A (E). ∈ ■ CHAPTER 4. POINTWISE BEHAVIOUR OF SOBOLEV FUNCTIONS 115

(2) Claim: × limsup u d y for every x E. (4.1) r 0 B(x,r) = ∞ ∈ → Reason. Let m N and x E. Then for r 0 small enough B(x, r) is contained in ∈ ∈ > an intersection of open sets Oi, i 1,..., m, with the property that ui 1 almost = P = everywhere on Oi. This implies that u ∞i 1 ui m almost everywhere in B(x, r) = = Ê and thus × u d y m. B(x,r) Ê This proves the claim. ■

T HEMORAL : This gives a method to construct a function that blows up on any set of zero capacity.

(3) Claim: If s n p, then > − 1 Du p d y x E limsup s for every . r 0 r ˆB(x,r) | | = ∞ ∈ → Reason. Let x E and, for a contradiction, assume that ∈ 1 Du p d y limsup s . r 0 r ˆB(x,r) | | < ∞ → Then there exists c such that < ∞ 1 Du p d y c limsup s . r 0 r ˆB(x,r) | | É → The we choose R 0 so small that > Du p d y crs ˆB(x,r) | | É

i for every 0 r R. Denote Bi B(x,2− R), i 1,2,.... Then by Hölder’s inequality < É = = and the Poincaré inequality, see Theorem 3.17, we have ×

uBi 1 uBi u uBi d y | + − | É Bi 1 | − | + Bi × | | u uBi d y É Bi 1 Bi | − | | + | µ× ¶ 1 p p c u uBi d y É Bi | − | µ× ¶ 1 i p p c2− R Du d y É Bi | | p n s i − + c(2− R) p . É For k j, we obtain > k 1 k 1 p n s X− X− i −p+ uBk uB j uBi 1 uBi c (2− R) | − | É i j | + − | É i j = = CHAPTER 4. POINTWISE BEHAVIOUR OF SOBOLEV FUNCTIONS 116

and thus (uB ) is a Cauchy sequence when s n p. This contradicts (4.1) and i > − thus the claim holds true. ■ (4) Thus

½ 1 ¾ E x n : Du p d y R s ⊂ ∈ r ˆB(x,r) | | = ∞ ½ 1 ¾ x n : Du p d y R s 0 . ⊂ ∈ r ˆB(x,r) | | >

Lemma 4.13 implies

µ½ 1 ¾¶ s E s x n : Du p d y H ( ) H R s 0 0. É ∈ r ˆB(x,r) | | > =

This shows that H s(E) 0 whenever n p s n. The claim follows from this, = − < < since H s(E) 0 implies H t(E) 0 for every t s. = = Ê ä n p Remark 4.16. It can be shown that even H − (E) , 1 p n, implies cap (E) < ∞ < < p = 0.

4.3 Quasicontinuity

In this section we study ﬁne properties of Sobolev functions. It turns out that Sobolev functions are deﬁned up to a set of capacity zero.

Deﬁnition 4.17. We say that a property holds p-quasieverywhere, if it holds except for a set of p-capacity zero.

T HEMORAL : Quasieverywhere is a capacitary version of almost everywhere.

Recall that by Meyers-Serrin theorem 1.18 W1,p(Rn) C(Rn) is dense in W1,p(Rn) ∩ for 1 p and, by Theorem 1.13, the Sobolev space W1,p(Rn) is complete. The É < ∞ next result gives a way to ﬁnd a quasieverywhere converging subsequence.

1,p n n Theorem 4.18. Assume that ui W (R ) C(R ), i 1,2,..., and that (ui) is a 1,p n ∈ ∩ = Cauchy sequence in W (R ). Then there is a subsequence of (ui) that converges pointwise p-quasieverywhere in Rn. Moreover, the convergence is uniform outside a set of arbitrarily small p-capacity.

T HEMORAL : This is a Sobolev space version of the result that for every Cauchy sequence in Lp(Rn), there is a subsequence that converges pointwise almost everywhere. The claim concerning uniform convergence is a Sobolev space version of Egorov’s theorem. CHAPTER 4. POINTWISE BEHAVIOUR OF SOBOLEV FUNCTIONS 117

Proof. There exists a subsequence of (ui), which we still denote by (ui), such that

X∞ 2ip u u p . i i 1 W1,p(Rn) i 1 k − + k < ∞ = For i 1,2,..., denote = © n iª [∞ Ei x R : ui(x) ui 1(x) 2− and F j Ei. = ∈ | − + | > = i j = i By continuity 2 ui ui 1 A (Ei) and thus | − + | ∈ ip p capp(Ei) 2 ui ui 1 1,p n . É k − + kW (R ) By subadditivity we obtain

cap (F ) X∞ cap (E ) X∞ 2ip u u p . p j p i i i 1 W1,p(Rn) É i j É i j k − + k = = Thus Ã ! \∞ \∞ capp F j lim capp(F j)( F j F j, F j 1 F j, j 1,2,...) j 1 É j j 1 ⊂ + ⊂ = = →∞ = lim X∞ 2ip u u p 0. i i 1 W1,p(Rn) É j i j k − + k = →∞ = Here we used the fact that the tail of a convergent series tends to zero. We observe n T that (ui) converges pointwise in R \ ∞j 1 F j. Moreover, = k 1 k 1 X− X− i 1 l ul(x) uk(x) ui(x) ui 1(x) 2− 2 − | − | É i l | − + | É i l É = = n for every x R \ F j for every k l j, which shows that the convergence is ∈n > > uniform in R \ F j. ä Deﬁnition 4.19. A function u : Rn [ , ] is p-quasicontinuous in Rn if for → −∞ ∞ every ε 0 there is a set E such that cap (E) ε and the restriction of u to Rn \ E, > p < denoted by u Rn\E, is continuous. | Remark 4.20. By outer regularity, see Remark 4.5, we may assume that E is open in the deﬁnition above.

The next result shows that a Sobolev function has a quasicontinous representative.

Corollary 4.21. For each u W1,p(Rn) there is a p-quasicontinuous function ∈ v W1,p(Rn) such that u v almost everywhere in Rn. ∈ =

T HEMORAL : Every Lp function is deﬁned almost everywhere, but every W1,p function is deﬁned quasieverywhere. CHAPTER 4. POINTWISE BEHAVIOUR OF SOBOLEV FUNCTIONS 118

Proof. By Theorem 1.18, for every function u W1,p(Rn), there are functions 1,p n n ∈ 1,p n ui W (R ) C(R ), i 1,2,..., such that ui u in W (R ) as i . By ∈ ∩ = → → ∞ Theorem 4.18 there exists a subsequence that converges uniformly outside a set of arbitrarily small capacity. Uniform convergence implies continuity of the limit function and thus the limit function is continuous outside a set of arbitrarily small p-capacity. This completes the proof. ä Next we show that the quasicontinuous representative given by Corollary 4.21 is essentially unique. We begin with a useful observation. Remarks 4.22: (1) If G Rn is open and E Rn with E 0, then cap (G) cap (G \ E). ⊂ ⊂ | | = p = p Reason. Monotonicity implies cap (G) cap (G \ E). Ê p Ê p Let ε 0 and let u A (G \ E) be such that É > ∈ p u cap (G \ E) ε. k kW1,p(Rn) É p + Then there exists an open O Rn with (G \ E) O and u 1 almost every- ⊂ ⊂ Ê where in O. Since O G is open G (O G) and u 1 almost everywhere ∪ ⊂ ∪ Ê in O (G \ E), and almost everywhere in O G since E 0, we have ∪ ∪ | | = u A (G). ∈ p cap (G) u cap (G \ E) ε. p É k kW1,p(Rn) É p +

By letting ε 0, we obtain capp(G) capp(G \ E). → É ■ (2) For any open G Rn we have G 0 cap (G) 0. ⊂ | | = ⇐⇒ p = Reason. If G 0, then (1) implies =⇒ | | = cap (G) cap (G \G) cap ( ) 0. p = p = p ; =

If capp(G) 0, then Lemma 4.6 implies G capp(G) 0. ⇐= = | | É = ■

W ARNING : It is not true in general that capacity and measure have the same zero sets.

Theorem 4.23. Assume that u and v are p quasicontinuous functions on Rn. If − u v almost everywhere in Rn, then u v p-quasieverywhere in Rn. = =

T HEMORAL : Quasicontinuous representatives of Sobolev functions are unique.

Proof. Let ε 0 and choose open G Rn such that cap (G) ε and that the > ⊂ p < restrictions of u and v to Rn \G are continuous. Thus {x Rn \G : u(x) v(x)} is ∈ 6= open in the relative topology on Rn \G, that is, there exists open U Rn with ⊂ U \G {x Rn \G : u(x) v(x)} = ∈ 6= CHAPTER 4. POINTWISE BEHAVIOUR OF SOBOLEV FUNCTIONS 119

and U \G {x Rn \G : u(x) v(x)} 0. | | = | ∈ 6= | = Moreover,

{x Rn : u(x) v(x)} G {x Rn \G : u(x) v(x)} G U. ∈ 6= ⊂ ∪ ∈ 6= = ∪ Remark 4.22 (1) with G and E replaced by U G and U \G, respectively, implies ∪ cap ({x Rn : u(x) v(x)}) cap (G U) cap (G) ε. p ∈ 6= É p ∪ = p < This completes the proof. ä Remarks 4.24: (1) The same proof gives the following local result: Assume that u and v are p quasicontinuous on an open set O Rn. If u v almost everywhere in − ⊂ = O, then u v p-quasieverywhere in O. = (2) Observe that if u and v are p quasicontinuous and u v almost ev- − É erywhere in an open set O, then max{u v,0} 0 almost everywhere − = in O and max{u v,0} is p quasicontinuous. Then Theorem 4.23 im- − − plies max{u v,0} 0 p-quasieverywhere in O and consequently u v − = É p-quasieverywhere in O. (3) The previous theorem enables us to deﬁne the trace of a Sobolev function to an arbitrary set. If u W1,p(Rn) and E Rn, then the trace of u to E is ∈ ⊂ the restriction to E of any p quasicontinuous representative of u. This − deﬁnition is useful only if cap (E) 0. p >

4.4 Lebesgue points of Sobolev functions

By the maximal function theorem with p 1, see (3.1), there exists c c(n) such = = that n c {x R : M f (x) λ} f 1 n | ∈ > | É λ k kL (R ) for every λ 0. By Chebyshev’s inequality and the maximal function theorem > with 1 p , see (3.2), there exists c c(n, p) such that < < ∞ = n 1 p c p {x R : M f (x) λ} M f p n f p n | ∈ > | É λp k kL (R ) É λp k kL (R ) for every λ 0. Thus the Hardy-Littlewood maximal function satisﬁes weak type > estimates with respect to Lebesgue measure for functions in Lp(Rn). Next we consider capacitary weak type estimates for functions in W1,p(Rn).

Theorem 4.25. Assume that u W1,p(Rn), 1 p . Then there exists c ∈ < < ∞ = c(n, p) such that

¡ n ¢ c p capp {x R : Mu(x) λ} u 1,p n . ∈ > É λp k kW (R ) for every λ 0. > CHAPTER 4. POINTWISE BEHAVIOUR OF SOBOLEV FUNCTIONS 120

T HEMORAL : This is a capacitary version of weak type estimates for the Hardy-Littlewood maximal function.

Proof. Denote E {x Rn : Mu(x) λ}. Then E is open and by Theorem 3.46 λ = ∈ > λ Mu W1,p(Rn). Thus ∈ Mu A (Eλ). λ ∈ Since the maximal operator is bounded on W1,p(Rn), see (3.11), we obtain

° Mu °p 1 c cap ¡E ¢ ° ° Mu p u p . p λ ° ° p W1,p(Rn) p W1,p(Rn) É ° λ °W1,p(Rn) = λ k k É λ k k ä This weak type inequality can be used in studying the pointwise behaviour of Sobolev functions. We recall that x Rn is a Lebesgue point for u L1 (Rn) if the ∈ ∈ loc limit × u∗(x) lim u(y) d y = r 0 B(x,r) → exists and × lim u(y) u∗(x) d y 0. r 0 B(x,r) | − | = → The Lebesgue differentiation theorem states that almost all points are Lebesgue points for a locally integrable function. If a function belongs to W1,p(Rn), then using the capacitary weak type estimate, see Theorem 4.25, we shall prove that it has Lebesgue points p-quasieverywhere. Moreover, we show that the

p-quasicontinuous representative given by Corollary 4.21 is u∗.

We begin by proving a measure theoretic result, which is analogous to Lemma 4.13.

Lemma 4.26. Let 1 p , f Lp(Rn) and < < ∞ ∈ ½ × ¾ E x Rn : limsup rp f p d y 0 . = ∈ r 0 B(x,r) | | > → Then cap (E) 0. p =

T HEMORAL : Roughly speaking the lemma above says that the set where an Lp function blows up rapidly is of capacity zero. The main difference compared to Lemma 4.13 is that the size of the set is measured by capacity instead of Hausdorff measure.

Proof. The argument is similar to the proof of Lemma 4.13, but we reproduce it here. (1) By the Lebesgue differentiation theorem

× lim f (y) p d y f (x) p , r 0 B(x,r) | | = | | < ∞ → CHAPTER 4. POINTWISE BEHAVIOUR OF SOBOLEV FUNCTIONS 121

for almost every x Rn. If x is a Lebesgue point of f p, then ∈ | | × limsup rp f (y) p d y 0. r 0 B(x,r) | | = → This shows that all Lebesgue points of f p belong to the complement of E and | | thus E 0. | | = (2) Let ε 0 and > × n n p p o Eε x R : limsup r f d y ε . = ∈ r 0 B(x,r) | | > → Since E E and E 0, we have E 0. We show that cap (E ) 0 for every ε ⊂ | | = | ε| = p ε = ε 0, then the claim follows by subadditivity. Let 0 δ 1 . For every x E there > < < 5 ∈ ε is rx with 0 rx δ such that < É × p p rx f d y ε. B(x,rx) | | >

By the Vitali covering theorem, there exists a subfamily of countably many pairwise disjoint balls B(xi, ri), i 1,2,..., such that =

[∞ Eε B(xi,5ri). ⊂ i 1 = By subadditivity of the capacity and Lemma 4.8 we have

X∞ X∞ n p capp(Eε) capp(B(xi,5ri)) c ri − É i 1 É i 1 = = c c X∞ f p d y f (y) p d y. É ε ˆ | | = ε ˆS | | i 1 B(xi,ri) ∞i 1 B(xi,ri) = = Here c c(n, p). Finally we observe that by the disjointness of the balls = ¯ ¯ rp p ¯[∞ ¯ X∞ X∞ i p δ p ¯ B(xi, ri)¯ B(xi, ri) f d y f d y 0 ¯ ¯ n ¯i 1 ¯ = i 1 | | É i 1 ε ˆB(xi,ri) | | É ε ˆR | | → = = = as δ 0. By absolute continuity of integral →

f p d y 0 ˆS | | → ∞i 1 B(xi,ri) = as δ 0. Thus → c p capp(Eε) f d y 0 É ε ˆS | | → ∞i 1 B(xi,ri) = as δ 0, which implies that cap (E ) 0 for every ε 0. → p ε = > ä Now we are ready for a version of the Lebesgue differentiation theorem for Sobolev functions. CHAPTER 4. POINTWISE BEHAVIOUR OF SOBOLEV FUNCTIONS 122

Theorem 4.27. Assume that u W1,p(Rn) with 1 p . Then there exists ∈ < < ∞ E Rn such that cap (E) 0 and ⊂ p = × lim u(y) d y u∗(x) r 0 B(x,r) = → exists for every x Rn \ E. Moreover ∈ × lim u(y) u∗(x) d y 0 r 0 B(x,r) | − | = → n for every x R \ E and the function u∗ is the p-quasicontinuous representative ∈ of u.

T HEMORAL : A function in W1,p(Rn) with 1 p has Lebesgue points < < ∞ p-quasieverywhere. Moreover, the p-quasicontinuous representative is obtained as a limit of integral averages.

n 1,p n Proof. (1) By Theorem 1.18 there exist ui C∞(R ) W (R ) such that ∈ ∩ p i(p 1) u ui 2− + , i 1,2,.... k − kW1,p(Rn) É = Denote n i Ei {x R : M(u ui)(x) 2− }, i 1,2,.... = ∈ − > = By Theorem 4.25 there exists c c(n, p) such that = ip p i cap (Ei) c2 u ui c2− , i 1,2,.... p É k − kW1,p(Rn) É = Clearly × ui(x) uB(x,r) ui(x) u(y) d y | − | É B(x,r) | − | × × ui(x) ui(y) d y ui(y) u(y) d y, É B(x,r) | − | + B(x,r) | − | which implies that

limsup ui(x) uB(x,r) r 0 | − | → × × limsup ui(x) ui(y) d y limsup ui(y) u(y) d y É r 0 B(x,r) | − | + r 0 B(x,r) | − | → → i M(ui u)(x) 2− , É − É n for every x R \ Ei. Here we used the fact that ∈ × limsup ui(x) ui(y) d y 0, i 1,2,..., r 0 B(x,r) | − | = = → since ui is continuous and × ui(y) u(y) d y M(ui u)(x) for every r 0. B(x,r) | − | É − > CHAPTER 4. POINTWISE BEHAVIOUR OF SOBOLEV FUNCTIONS 123

S Let Fk ∞i k Ei, k 1,2,.... Then by the subadditivity of capacity we have = = = X∞ X∞ i capp(Fk) capp(Ei) c 2− . É i k É i k = = n If x R \ Fk and i, j k, then ∈ Ê

ui(x) u j(x) limsup ui(x) uB(x,r) limsup uB(x,r) u j(x) | − | É r 0 | − | + r 0 | − | → → i j 2− 2− . É + n n Thus (ui) converges uniformly in R \ Fk to a continuous function v in R \ Fk. Furthermore

limsup v(x) uB(x,r) v(x) ui(x) limsup ui(x) uB(x,r) r 0 | − | É | − | + r 0 | − | → → i v(x) ui(x) 2− É | − | + n for every x R \ Fk. The right-hand side of the previous inequality tends to zero ∈ as i . Thus → ∞ limsup v(x) uB(x,r) 0 r 0 | − | = → and consequently × v(x) lim u(y) d y u∗(x) = r 0 B(x,r) = → n T for every x R \ Fk. Deﬁne F ∞k 1 Fk. Then ∈ = = X∞ i capp(F) lim capp(Fk) c lim 2− 0 É k É k i k = →∞ →∞ = and × lim u(y) d y u∗(x) r 0 B(x,r) = → exists for every x Rn \ F. This completes the proof of the ﬁrst claim. ∈ (2) To prove the second claim, consider ½ × ¾ E x Rn : limsup rp Du(y) p d y 0 . = ∈ r 0 B(x,r) | | > → Lemma 4.26 shows that cap (E) 0. By the Poincaré inequality, see Theorem p = 3.17, we have × × p p p lim u(y) uB(x,r) d y c lim r Du(y) d y 0 r 0 B(x,r) | − | É r 0 B(x,r) | | = → → for every x Rn \ E. We conclude that ∈ × lim u(y) u∗(x) d y r 0 B(x,r) | − | → µ× ¶ 1 p p lim u(y) u∗(x) d y É r 0 B(x,r) | − | → µ× ¶ 1 p p lim u(y) uB(x,r) d y lim uB(x,r) u∗(x) 0 É r 0 B(x,r) | − | + r 0 | − | = → → CHAPTER 4. POINTWISE BEHAVIOUR OF SOBOLEV FUNCTIONS 124 whenever x Rn \ (E F). Finally we observe that ∈ ∪ cap (E F) cap (E) cap (F) 0. p ∪ É p + p = ε (3) Let ε 0 and choose k large enough so that cap (Fk) . Then by the > p < 2 outer regularity of capacity, see Remark 4.5, there is an open set O containing Fk n so that cap (O) ε. Since (ui) converges uniformly to u∗ on R \ O we conclude p < that u∗ X\O is continuous. Thus u∗ is p-quasicontinuous. | ä

4.5 Sobolev spaces with zero boundary values

In this section we return to Sobolev spaces with zero boundary values started in Section 1.9. Assume that Ω is an open subset of Rn and 1 p . Recall that 1,p É < ∞ W (Ω) with 1 p is the closure of C∞(Ω) with respect to the Sobolev norm, 0 É < ∞ 0 see Definition 1.20. Using pointwise properties of Sobolev functions we discuss 1,p the definition of W0 (Ω). 1,p The first result is a W0 (Ω) version of Corollary 4.21 which states that for every u W1,p(Rn) there is a p-quasicontinuous function v W1,p(Rn) such that ∈ ∈ u v almost everywhere in Rn. = 1,p Theorem 4.28. If u W (Ω), there exists a p-quasicontinuous function v ∈ 0 ∈ W1,p(Rn) such that u v almost everywhere in Ω and v 0 p-quasieverywhere in = = Rn \ Ω.

T HEMORAL : Quasicontinuous functions in Sobolev spaces with zero boundary values are zero quasieverywhere in the complement.

1,p Proof. Since u W0 (Ω), there exist ui C0∞(Ω), i 1,2,..., such that ui u in 1,p ∈ ∈ = 1,p n → W (Ω) as i . Since (ui) is a Cauchy sequence in W (R ), by Theorem 4.18 → ∞ n it has a subsequence of (ui) that converges pointwise p-quasieverywhere in R to a function v W1,p(Rn). Moreover, the convergence is uniform outside a set ∈ of arbitrary small p-capacity and, as in Corollary 4.21, the limit function v is p-quasicontinuous. ä Theorem 4.29. If u W1,p(Rn) is p-quasicontinuous and u 0 p-quasieverywhere ∈1,p = in Rn \ Ω, then u W (Ω). ∈ 0

T HEMORAL : Quasicontinuous functions in a Sobolev space on the whole space which are zero quasieverywhere in the complement belong to the Sobolev space with zero boundary values. In particular, continuous functions in a Sobolev space on the whole space which are zero everywhere in the complement belong to the Sobolev space with zero boundary values. CHAPTER 4. POINTWISE BEHAVIOUR OF SOBOLEV FUNCTIONS 125

Proof. (1) We show that u can be approximated by W1,p(Rn) functions with compact support in Ω. If we can construct such a sequence for u max{u,0}, then we + = can do it for u min{u,0}, and we obtain the result for u u u . Thus we − = − = + − − may assume that u 0. By Theorem 1.24 we may assume that u has a compact Ê support in Rn and by considering truncations min{u,λ}, λ 0, we may assume > that u is bounded (exercise). (2) Let δ 0 and let O Rn be an open set such that cap (O) δ and the > ⊂ p < restriction of u to Rn \ O is continuous. Denote

E {x Rn \ Ω : u(x) 0}. = ∈ 6=

By assumption cap (E) 0. Let v A 0(O E) such that 0 v 1 and p = ∈ ∪ É É p v δ, k kW1,p(Rn) < see Remark 4.2. Then v 1 in an open set G containing O E. Deﬁne = ∪ u (x) max{u(x) ε,0}, 0 ε 1. ε = − < < Let x ÇΩ\G. Since u(x) 0 and the restriction of u to Rn \G is continuous, there ∈ = exists rx 0 such that u 0 in B(x, rx) \G. Thus (1 v)u 0 in B(x, rx) G for > ε = − ε = ∪ every x ÇΩ \G. This shows that (1 v)u is zero in a neighbourhood of Rn \ Ω, ∈ − ε which implies that (1 v)uε is compactly supported in Ω. Lemma 1.23 implies 1,p − (1 v)u W (Ω). We show that this kind of functions converge to u in W1,p(Rn). − ε ∈ 0 (3) Since  u ε in {x Rn : u(x) ε}, uε − ∈ Ê = 0 in {x Rn : u(x) ε}, ∈ É by Remark 1.27 we have  Du almost everywhere in {x Rn : u(x) ε}, Duε ∈ Ê = 0 almost everywhere in {x Rn : u(x) ε}. ∈ É Thus

u (1 v)u 1,p n u u 1,p n vu 1,p n . k − − εkW (R ) É k − εkW (R ) + k εkW (R ) Using the facts that u u ε and supp(u u ) supp u, we obtain − ε É − ε ⊂

u u 1,p n u u Lp(Rn) Du Du Lp(Rn) k − εkW (R ) É k − εk + k − εk ε χsupp u Lp(Rn) χ{0 u ε}Du Lp(Rn) 0 É k k + k < É k → as ε 0. Observe that, by the dominated convergence theorem, we have → 1 µ ¶ p p lim χ{0 u ε}Du Lp(Rn) lim χ{0 u ε} Du dx ε 0 k < É k = ε 0 ˆ n < É | | → → R 1 µ ¶ p p limχ{0 u ε} Du dx 0, = ˆ n ε 0 < É | | = R → CHAPTER 4. POINTWISE BEHAVIOUR OF SOBOLEV FUNCTIONS 126

p p 1 n where χ{0 u ε} Du Du L (R ) may be used as an integrable majorant. On < É | | É | | ∈ the other hand,

vu 1,p n vu Lp(Rn) D(vu ) Lp(Rn) k εkW (R ) É k εk + k ε k vu Lp(Rn) u Dv Lp(Rn) vDu Lp(Rn) É k εk + k ε k + k εk uv Lp(Rn) uDv Lp(Rn) vDu Lp(Rn) É k k + k k + k εk u L (Rn) v Lp(Rn) u L (Rn) Dv Lp(Rn) vDu Lp(Rn) É k k ∞ k k + k k ∞ k k + k k 2 u L (Rn) v 1,p n vDu Lp(Rn) É k k ∞ k kW (R ) + k k 1 p 2δ u L (Rn) vDu Lp(Rn). É k k ∞ + k k p n Since v v 0 in L (R ) as δ 0, there is a subsequence (δi) for which vi = δ → → = v 0 almost everywhere as i . By the dominated convergence theorem, we δi → → ∞ have

1 µ ¶ p p p lim vi Du Lp(Rn) lim vi Du dx i k k = i ˆ n | | | | →∞ →∞ R 1 µ ¶ p p p ( lim vi ) Du dx 0, = ˆ n i | | | | = R →∞ p p p p 1 n where vi Du Du , so that Du L (R ) may be used as an integrable | | | | É | | | | ∈ majorant. Thus we conclude that

µ 1 ¶ p lim vi uε W1,p(Rn) lim 2δ u L (Rn) vi Du Lp(Rn) 0. i k k É i i k k ∞ + k k = →∞ →∞ Thus

u (1 vi)u 1,p n 0 k − − εkW (R ) → as ε 0 and i . Since → → ∞ 1,p 1,p n (1 vi)u W (Ω) and (1 vi)u u in W (R ) − ε ∈ 0 − ε → 1,p as ε 0 and i , we conclude that u W (Ω). → → ∞ ∈ 0 ä Remark 4.30. If u W1,p(Rn) is continuous and zero everywhere in Rn \ Ω, then 1,p ∈ u W (Ω). ∈ 0 We obtain a very useful characterization of Sobolev spaces with zero boundary values on an arbitrary open set by combining the last two theorems.

1,p Corollary 4.31. u W0 (Ω) if and only if there exists a p-quasicontinuous 1,p ∈ n function u∗ W (R ) such that u∗ u almost everywhere in Ω and u 0 p- ∈ = = quasieverywhere in Rn \ Ω.

T HEMORAL : Quasicontinuous functions in Sobolev spaces with zero boundary values are precisely functions in the Sobolev space on the whole space which CHAPTER 4. POINTWISE BEHAVIOUR OF SOBOLEV FUNCTIONS 127

are zero quasieverywhere in the complement. This result can be used to show that a given function belongs to the Sobolev space with zero boundary values without constructing an approximating sequence of compactly supported smooth functions.

There is also a characterization of Sobolev spaces with zero boundary values using Lebesgue points for Sobolev functions.

Theorem 4.32. Assume that Ω Rn is an open set and u W1,p(Rn) with 1 p 1,p ⊂ ∈ < < . Then u W (Ω) if and only if ∞ ∈ 0 × lim u(y) d y 0 r 0 B(x,r) = → for p-quasievery x Rn \ Ω. ∈

T HEMORAL : A function in the Sobolev space on the whole space belongs to the Sobolev space with zero boundary values if and only if the limit of integral averages is zero quasieverywhere in the complement.

1,p Proof. If u W0 (Ω), then by Theorem 4.28 there exists a p-quasicontinuous =⇒ ∈1,p n function u∗ W (R ) such that u∗ u almost everywhere in Ω and u∗ 0 ∈ = = p-quasieverywhere in Rn \ Ω. Theorem 4.27 shows that the limit × u∗(x) lim u(y) d y = r 0 B(x,r) →

exists p-quasieverywhere and that the function u∗ is a p-quasicontinuous representative of u. This shows that × lim u(y) d y u∗(x) 0 r 0 B(x,r) = = → for p-quasievery x Rn \ Ω. ∈ Assume then that u W1,p(Rn) and ⇐= ∈ × lim u(y) d y 0 r 0 B(x,r) = → for p-quasievery x Rn \ Ω. Theorem 4.27 shows that the limit ∈ × u∗(x) lim u(y) d y = r 0 B(x,r) →

exists p-quasieverywhere and that the function u∗ is a p-quasicontinuous repre- n sentative of u. We conclude that u∗(x) 0 for p-quasievery x R \ Ω. = ∈ ä 1,p Example 4.33. Let Ω B(0,1) \{0} and u : Ω R, u(x) 1 x . Then u W0 (Ω) =1,p → = − | | ∈ for 1 p n and u W (Ω) for p n. < É ∉ 0 > CHAPTER 4. POINTWISE BEHAVIOUR OF SOBOLEV FUNCTIONS 128

T HEMORAL : A function that belongs to the Sobolev space with zero boundary values does not have to be zero at every point of the boundary.

Remark 4.34. Theorem 4.32 gives a practical tool to show that a function belongs to a Sobolev space with zero boundary values. For example, the following claims follow from Theorem 4.32 (exercise).

1,p (1) Assume that u W1,p(Ω) has a compact support, then u W (Ω). ∈ ∈ 0 1,p 1,p (2) Assume that u W (Ω). Then u W (Ω). ∈ 0 | | ∈ 0 1,p 1,p (3) Assume that u W0 (Ω). If v W (Ω) and 0 v u almost everywhere ∈1,p ∈ É É in Ω, then v W (Ω). ∈ 0 1,p 1,p (4) Assume that u W0 (Ω). If v W (Ω) and v u almost everywhere ∈ ∈ | | É | | 1,p in Ω \ K, where K is a compact subset of Ω, then v W (Ω). ∈ 0 Let E Ω be a relatively closed set, that is, there exists a closed F Rn such ⊂ 1,p 1,p ⊂ that E Ω F, with E 0. It is clear that W (Ω \ E) W (Ω). By = ∩ | | = 0 ⊂ 0 1,p 1,p W (Ω \ E) W (Ω) 0 = 0 1,p we mean that every u W0 (Ω) can be approximated by functions in C0∞(Ω \ E) 1,p ∈ or in W0 (Ω \ E). 1,p Theorem 4.35. Assume that E is a relatively closed subset of Ω. Then W0 (Ω) 1,p = W (Ω \ E) if and only if cap (E) 0. 0 p =

Proof. Assume capp(E) 0. Lemma 4.6 implies E 0 so that it is reasonable ⇐= 1,p 1=,p | | = to ask whether W (Ω) W (Ω \ E) when we consider functions deﬁned up to a 0 = 0 set of measure zero. 1,p 1,p It is clear that W0 (Ω \ E) W0 (Ω). To see reverse inclusion, let ui C0∞(Ω), ⊂ 1,p ∈ i 1,2,..., be such that ui u in W (Ω) as i . Since cap (E) 0 there are = → → ∞ p = v j A 0(E), j 1,2,..., be such that v j W1,p(Rn) 0 as j . Then (1 v j)ui 1∈,p = k k → → ∞ − ∈ W (Ω) and, since v j 1 in a neighbourhood of E, supp(1 v j)ui is a compact = − 1,p subset of Ω\E for every i, j 1,2,.... Lemma 1.23 implies (1 v j)ui W (Ω\E), = − ∈ 0 i, j 1,2,.... = Moreover, we have

u (1 v j)ui 1,p u ui 1,p v j ui 1,p , k − − kW (Ω) É k − kW (Ω) + k kW (Ω) where u ui 1,p 0 as i and k − kW (Ω) → → ∞

v j ui 1,p v j ui Lp( ) D(v j ui) Lp( ) k kW (Ω) É k k Ω + k k Ω ui L ( ) v j Lp( ) v j Dui Lp( ) ui Dv j Lp( ) É k k ∞ Ω k k Ω + k k Ω + k k Ω ui L ( ) v j Lp( ) v j Dui Lp( ) ui L ( ) Dv j Lp( ) É k k ∞ Ω k k Ω + k k Ω + k k ∞ Ω k k Ω 2 ui L ( ) v j 1,p v j Dui Lp( ). É k k ∞ Ω k kW (Ω) + k k Ω CHAPTER 4. POINTWISE BEHAVIOUR OF SOBOLEV FUNCTIONS 129

p Since v j 0 in L (Ω) as j , there is a subsequence, still denoted by (v j), → → ∞ for which v j 0 almost everywhere as j . By the dominated convergence → → ∞ theorem, we have

u (1 v j)ui 1,p 0 as i, j . k − − kW (Ω) → → ∞ Since

1,p 1,p (1 v j)ui W (Ω \ E) and (1 v j)ui u in W (Ω \ E) − ∈ 0 − → 1,p as i, j , we conclude that u W (Ω \ E). → ∞ ∈ 0 Let x0 Ω and let i0 N be large enough that =⇒ ∈ ∈

n 1 dist(x0,R \ Ω) . > i0 Deﬁne ½ ¾ n 1 Ωi x Ω : dist(x,R \ Ω) B(x0, i), i i0, i0 1,.... = ∈ > i ∩ = + S n Observe that Ωi Ωi 1 Ω and Ω Ωi. Let ui : R R, b b b ∞i i0 + ··· = = → n ui(x) dist(x,R \ Ω2i). = 1,p 1 Then ui is Lipschitz continuous, ui W0 (Ω) and ui(x) 2i for every x E Ωi, 1,p 1,p ∈ 1,Êp ∈ ∩ i 1,2,.... Since W0 (Ω) W0 (Ω \ E) we have ui W0 (Ω \ E), i 1,2,.... = = ∈ =1,p Fix i and let v j C∞(Ω \ E), j 1,2,..., such that v j ui in W (Ω \ E) as ∈ 0 = → j . Since 3i(ui v j) 1 in a neigbourhood of E Ωi, → ∞ − Ê ∩ p cap (E Ωi) 3i(ui v j) p ∩ É k − kW1,p(Ω\E) p p (3i) ui v j 0 as j . = k − kW1,p(Ω\E) → → ∞

Thus cap (E Ωi) 0, i 1,2..., and by subadditivity p ∩ = = Ã ! [∞ X∞ capp(E) capp (E Ωi) capp(E Ωi) 0. = i 1 ∩ É i 1 ∩ = = = ä CHAPTER 4. POINTWISE BEHAVIOUR OF SOBOLEV FUNCTIONS 130

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