Sobolev Spaces and the Finite Element Method

Institutionen för naturvetenskap och teknik

Sobolev Spaces and the Finite Element Method Johan Davidsson Örebro universitet Institutionen för naturvetenskap och teknik Självständigt arbete för kandidatexamen i matematik, 15 hp

Johan Davidsson

May 2018

Supervisor: Farid Bozorgnia Examiner: Marcus Sundhäll Abstract

In this essay we present the Sobolev spaces and some basic properties of them. The Sobolev spaces serve as a theoretical framework for studying solutions to partial differential equations. The finite element method is pre- sented which is a numerical method for solving partial differential equations.

Contents

1 Preliminaries 5 1.1 Notation ...... 5 1.2 Measure, Integration and Almost Everywhere ...... 5 1.3 Functional Analysis ...... 9 1.4 Distributions ...... 10

2 Sobolev Spaces 11 2.1 Hölder Spaces ...... 11 2.2 Weak Derivative ...... 14 2.3 Deﬁnitions and Properties of Sobolev Spaces ...... 16

3 Finite Element Method 23 3.1 Weak Solution ...... 23 3.2 Finite Element Method ...... 24

4 Future Work 27

Chapter 1

Preliminaries

In Section 1.1 we present some notations that will be used frequently through- out this text. Section 1.2 deals with some basic concepts in measure theory such as an extension of the length of intervals to open sets. We also present the concept that a relation can hold almost everywhere. For a full review on measure theory the reader can look up Folland [4]. With this it will make sense to consider solutions to partial diﬀerential equations satisfying a given equation almost everywhere instead of pointwise. In Section 1.3 we state some functional analysis and at the end we brieﬂy review the concept of distributions. For a review on functional analysis and distributions one can read Debnath and Mikusiński [2].

1.1 Notation

n Let Ω denote an open subset of R and ∂Ω denote the boundary of Ω. The set N will denote the natural numbers including zero. The closure of a set Ω will be denoted by Ω. The derivative will be denoted by D. A multi-index α is an n-tuple α = (α1, . . . , αn) where αi ≥ 0 for i = 1, . . . n. Let α and β be multi-indices, below we introduce some useful notations:

Sum and diﬀerence: α ± β = (α1 ± β1, . . . , αn ± βn).

Sum of components: |α| = α1 + ... + αn.

α ∂α1 ∂αn Partial derivatives: D = α1 ... αn . ∂x1 ∂xn 1.2 Measure, Integration and Almost Everywhere

Deﬁnition 1.2.1. A σ-algebra is a nonempty family M containing subsets of a set X such that ∞ S 1. If A1,A2,... ∈ M =⇒ Ai ∈ M. i=1

5 2. If A ∈ M =⇒ Ac ∈ M.

σ-algebras are used as domains for measure. A σ-algebra is intuitively a family of nice sets that we can assign a measure to. The elements of M are therefore called measurable sets.

Deﬁnition 1.2.2. Let X be a set with σ-algebra M.A measure is a function µ : M → [0, ∞] such that

1. µ(∅) = 0.

∞ 2. If {Ai}1 is a sequence of pairwise disjoint sets in M, the following must be true ∞ ∞ [ X µ( Ai) = µ(Ai). i=1 i=1 This function µ relates a set with a number that can be interpreted as the size of that set.

Deﬁnition 1.2.3. Let P(X) be the powerset of X. An outer measure is a function µ∗ : P(X) → [0, ∞] such that

1. µ∗(∅) = 0.

2. µ∗(A) ≤ µ∗(B) if A ⊂ B.

∞ ∞ ∗ S P ∗ 3. µ ( Ai) ≤ µ (Ai). i=1 i=1 Let µ be a measure deﬁned on a σ-algebra M over X. Then the pair (X, M) is called a measuable space and the triplet (X, M, µ) is called a measure space.

Deﬁnition 1.2.4. Let A ⊂ R, then we deﬁne outer Lebesgue measure of a set A as ( ∞ ∞ ) ∗ X [ m (A) = inf (bi − ai): A ⊂ [ai, bi] . i=1 i=1

Deﬁnition 1.2.5. The family of all Lebesgue measurable sets, L(R), consists of sets A such that the following holds for all E ⊂ R m∗(E) = m∗(E ∩ A) + m∗(E ∩ Ac),

∗ where m is deﬁned in Deﬁnition 1.2.4. L(R) is a σ-algebra.

Deﬁnition 1.2.6. Let A ∈ L(R) then we deﬁne the Lebesgue measure m(A) as the outer measure of A,

m(A) = m∗(A).

6 The Lebesgue measure generalizes the notion of length, area and volume n to arbitrary sets in higher dimension. When working with R one usually consider µ to be the Lebesgue measure, which is what we will do from now on. Given a measure space (Ω, M, µ), a null set is a set A ⊂ Ω such that µ(A) = 0. We state a formal deﬁnition below. Deﬁnition 1.2.7 (Null set). Let A ⊂ Ω. The set A is called a null set if and only if for every > 0 there is a countable family of n-cubes {Ai} that covers A such that ∞ X µ(Ai) = . i=1

Example 1.2.1. Let c be a point in R. Let > 0 and choose Ii = (c − 2i+1 , c + 2i+1 ). We have that ∞ [ {c} ⊂ Ii i=1 and we get the following

∞ X 1 1 2 2 µ(Ii) = + · + · ( ) + ... = = . 2 2 2 2 2 1 − 1 i=1 2 We conclude that µ(c) = 0. Deﬁnition 1.2.8 (Almost everywhere). Let (X, M, µ) be a measure space. We say that a property holds almost everywhere, when it is true everywhere except for a set A ⊂ X such that µ(A) = 0.

Example 1.2.2. Let f, g : R → R, g(x) = x and ( x if x 6= 1; f(x) = 0 if x = 1.

We can say that f = g almost everywhere since µ(1) = 0. Next we will develop the Lebesgue integral, we start off by defining a measurable function. Recall that any mapping f : X → Y induces a mapping f −1 defined by f −1(A) = {x ∈ X : f(x) ∈ A}. Definition 1.2.9. Let (X, M), (Y, N) be measure spaces. A mapping f : X → Y is called measurable if f −1(A) ∈ M for all A ∈ N.

Deﬁnition 1.2.10. Let A ⊂ X, the indicator function χA : X → {0, 1} is deﬁned as ( 1 if x ∈ A; χA(x) = 0 if x∈ / A.

7 First oﬀ we will present the simple functions, they are easy to construct an integral from. The simple functions has properties that will help us deﬁne an integral for more general functions.

Definition 1.2.11. Let A1,...,An be a sequence of disjoint measurable sets and a1, . . . , an ∈ R.A simple function f : X → R is a function of the form n X f(x) = aiχAi (x). i=1 Pn Definition 1.2.12. For a positive simple function f = i=1 aiχAi we define the integral of f with respect to the measure µ as

n Z X f dµ = aiµ(Ai). A i=1 So we have an integral for simple functions and from [4] we know that measurable functions can be approximated by simple functions.

Deﬁnition 1.2.13. Let f be a positive measurable function. We deﬁne the integral of f as Z Z f dµ = sup φ dµ, φ≤f where the supremum is taken over all simple functions φ such that φ ≤ f.

We developed the integral for positive measurable functions. For the general case you split a function f into its positive and negative parts, that is f = f + − f −. We say that a measurable function f is integrable if both R f + dµ and R f − dµ is ﬁnite. For details on this the reader can look in [4, chapter 2]. From the deﬁnition we see that if f, g are integrable and a, b ∈ R. The following properties hold Z Z Z af + bg dµ = a f dµ + b g dµ, if f ≤ g we get that Z Z f dµ ≤ g dµ.

Theorem 1.2.1. (The dominated convergence theorem) Let {fn} be a sequence of measurable functions such that fn → f pointwise almost everywhere as n → ∞. Also let |f| ≤ g where g is integrable. Then we have that f is integrable and Z Z f = lim fn. n→∞

8 1.3 Functional Analysis

Deﬁnition 1.3.1 (Normed space). A vector space Y equipped with a norm is called a normed space.

The norm generalizes the notion of length into abstract vector spaces.

Deﬁnition 1.3.2 (Bilinear functional). A functional F (x, y) on a vector space E is called a bilinear functional if the following holds:

1. F (ax1 + bx2, y) = aF (x1, y) + bF (x2, y).

2. F (x, ay1 + by2) = conj(a)F (x, y1) + conj(b)F (x, y2). for any scalars a, b and x, x1, x2, y, y1, y2 ∈ Y . Where conj denote the com- plex conjugate.

Deﬁnition 1.3.3 (Coercive functional). A bilinear functional F (x, y) on a normed space Y is called coercive if there exist a positive constant C such that F (x, x) ≥ Ckxk2 ∀x ∈ Y.

Deﬁnition 1.3.4 (Complete space). Let (Y, k·k) be a normed space. If every cauchy sequence in Y has a limit that is also in Y , then (Y, k·k) is called a complete space(or Banach space).

Deﬁnition 1.3.5 (Inner product space). A vector space equipped with an inner product h·, ·i is called an inner product space.

n The inner product space extend the inner product from R to vector spaces.

Deﬁnition 1.3.6. Let H be an inner product space. H is called a Hilbert space if it is complete with respect to the norm ||x|| = phx, xi.

Definition 1.3.7. Let f be a measurable function on Y and fix 1 ≤ p < ∞, then we define the Lp-norm of f by

Z 1/p p kfkLp(Y ) = |f| dµ . Y For p = ∞, deﬁne

kfkL∞(Y ) = inf{C > 0 : |f(x)| ≤ C for almost every x}.

For ﬁxed p the space Lp consists of all functions f that have ﬁnite Lp norm. We present two important inequalities:

9 1 1 1. Hölder’s inequality: Let 1 < p < ∞ and p + q = 1. If f, g are measurable functions on Y , then

kfgkL1(Y ) ≤ kfkLp(Y )kgkLq(Y ). For the case p = 1 we have

kfgkL1(Y ) ≤ kfkL1(Y )kgkL∞(Y ).

2. Minkowski’s inequality: Let f, g ∈ Lp and 1 ≤ p < ∞ then we have

kf + gkp ≤ kfkp + kgkp.

1.4 Distributions

Definition 1.4.1 (Support & compact support). Let Ω ⊂ R. The support of f, denoted by supp(f), is defined as supp(f) = {x ∈ Ω: f(x) 6= 0}. We say that a function has compact support in Ω if supp(f) is a compact subset of Ω. ∞ Let Cc be the set of infinitely differentiable functions that has compact support in Ω. An element of this set is called a test function and they turn out to be quite important.

Deﬁnition 1.4.2. Let φ1, φ2,... and φ be test functions. A sequence (φn) ∞ ∞ Cc is said to converge to φ ∈ Cc , denoted φn −−→ φ, if the following holds:

1. φ1, φ2 ... and φ vanish outside some bounded set S ⊂ Ω. α α 2. D φn → D φ uniformly on Ω for every multi-index α. Deﬁnition 1.4.3. A distribution T on Ω is a continuous linear functional ∞ ∞ on Cc (Ω). That is, a mapping T : Cc (Ω) → C is a distribution if the following conditions are true 1. T (aφ + bψ) = aT (φ) + bT (ψ).

∞ Cc 2. φn −−→ φ =⇒ T (φn) → T (φ). 1 A function f on Ω is said to be locally integrable on Ω if f ∈ Lloc(U) for all compact subsets U ⊂ Ω. For every locally integrable function f there is a corresponding distribution Tf deﬁned by Z ∞ Tf (φ) = fφ dx, φ ∈ Cc (Ω). (1.1) Ω If a distribution T can be expressed in the form of (1.1) we shall call it regular. Otherwise it is singular.

10 Chapter 2

Sobolev Spaces

Partial differential equations are important due to many real world applications. Sobolev spaces are vector spaces and they are interesting because solutions of partial differential equations, when they exist, belong to some Sobolev space. So understanding the Sobolev spaces will help us to under- stand partial differential equations. We start of by introducing the Hölder spaces in Section 2.1. In Section 2.2 we present the concept of a weak derivative, this is a generalization of the classical derivative. The weak derivative is important because a Sobolev space is a set of functions which has a weak derivative up to some order. In Section 2.3 we introduce the Sobolev spaces and some properties of them. This chapter is based on the material found in Adams [1], Evans [3], Shkoller [6] and Kinnunen [7].

2.1 Hölder Spaces

A function is Lipschitz continuous if there exist a real number K such that that for every pair of points from the graph of the function the absolute value of the slope connecting the line between these points is bounded by K. That is, for positive real constant K we have

|f(x) − f(y)| ≤ K|x − y|, x, y ∈ Ω.

This Lipschitz condition implies that the ﬁrst derivative exist almost everywhere. We introduce a more general condition, a function is Hölder continuous with exponent γ if there is a positive real constant C and 0 < γ ≤ 1 such that

|f(x) − f(y)| ≤ C|x − y|γ, x, y ∈ Ω.

For γ = 0 the function f need not be continuous, for γ > 0 the condition implies continuity. But for γ > 1 the condition implies that the function f is a constant, therefore we only care about the case 0 < γ ≤ 1. Let

11 m > 0 be an integer and 0 < γ ≤ 1. The Hölder space Cm,γ(Ω) consists of functions f that are m-times continuously differentiable and whose funcion f and partial derivative are bounded. Also the mth partial derivatives are Hölder continuous with exponent γ. We present the norm and state the definition below: Definition 2.1.1. For f :Ω → R bounded and continuous we define

X α X α kfkCm,γ (Ω) = kD fkC(Ω) + [D f]C0,γ (Ω) |α|≤m |α|=m is less than inﬁnity.

We show that kfkCm,γ (Ω) is a norm.

Proof. First we show that [f]C0,γ (Ω) is a seminorm: 1. Since everything inside the supremum is greater than or equal to zero

we have that [f]C0,γ (Ω) ≥ 0. 2. Let c be a constant. |cf(x) − cf(y)| [cf]C0,γ (Ω) = sup γ x,y∈Ω |x − y| x6=y |f(x) − f(y)| = |c| sup γ x,y∈Ω |x − y| x6=y

= |c|[f]C0,γ (Ω). 3. The triangle inequality: |f(x) + g(x) − f(y) − g(y)| [f + g]C0,γ (Ω) = sup γ x,y∈Ω |x − y| x6=y |f(x) − f(y)| + |g(x) − g(y)| ≤ sup γ x,y∈Ω |x − y| x6=y |f(x) − f(y)| |g(x) − g(y)| = sup γ + sup γ x,y∈Ω |x − y| x,y∈Ω |x − y| x6=y x6=y

= [f]C0,γ (Ω) + [g]C0,γ (Ω).

12 It is a seminorm because there are functions f 6= 0 such that [f]C0,γ (Ω) = 0. Consider for example a constant function f 6= 0, this will make the numerator zero and thus [f]C0,γ (Ω) = 0. Because of the above reasoning it is obvious that

kfkC(Ω) = sup |f(x)| x∈Ω is a norm, the reader is encouraged to verify it as an exercise. Now we show that kfkCm,γ (Ω) is a norm:

1. We show that kfkCm,γ (Ω) = 0 ⇐⇒ f = 0:

Assume f = 0, this implies kfkCm,γ (Ω) = 0 since k0kC(Ω) = 0 and [0]C0,γ (Ω) = 0.

Assume kfkCm,γ (Ω) = 0, this implies

X α X α kD fkC(Ω) + [D f]C0,γ (Ω) = 0. |α|≤m |α|=m

Since all terms are positive, we get that in particular

α kD fkC(Ω) = 0

should hold for all multi-indices |α| ≤ m. This certainly includes the case α = 0 and we get that f = 0.

2. Since kfkCm,γ (Ω) is a sum of norms and seminorms, it is obvious that we can bring out the constant.

3. To show the triangle inequality we use the properties we now know:

X α α X α α kf + gkCm,γ (Ω) = kD f + D gkC(Ω) + [D f + D g]C0,γ (Ω) |α|≤m |α|=m X α X α X α ≤ kD fk + [D f]C0,γ (Ω) + kD gkC(Ω) |α|≤m |α|=m |α|≤m X α + [D g]C0,γ (Ω) |α|=m

= kfkCm,γ (Ω) + kgkCm,γ (Ω).

m,γ The Hölder space C (Ω) equppied with the norm kfkCm,γ (Ω) is a Ba- nach space [3, page 241]. We will return to the Hölder continuity in the end of this chapter.

13 2.2 Weak Derivative

The classical definition of the derivative is stated pointwise since it tells us whether the derivative of a function exist in a specific point or not. In this section we state a more general definition that gives us a derivative even if the function, to some degree, is not smooth at certain points. Below follow some reasoning that motivates the new definition of the derivative. 1 ∞ Let f ∈ C (Ω) and φ ∈ Cc (Ω), since φ = 0 on the boundary of Ω integration by parts gives us the following Z ∂φ Z ∂f f dx = − φ dx. Ω ∂xi Ω ∂xi In general, for f ∈ Ck(Ω) and some multi-index α such that |α| ≤ k we have Z Z fDαφ dx = (−1)|α| Dαfφ dx (2.1) Ω Ω This is true for f ∈ Ck, but there is something interesting going on here. 1 Let us consider (2.1) for f ∈ Lloc(Ω), the left hand side still makes sense since the integral is guaranteed to exist. In the right hand side we have a problem, since f is no longer assumed to be differentiable, Dαf does not make any sense. What we will do is replace Dαf with g and ask ourselves if 1 there is a function g ∈ Lloc(Ω) that satisfies (2.1). This is our new definition of a derivative, we state it below:

1 Deﬁnition 2.2.1 (Weak derivative). Let f, g ∈ Lloc(Ω). If Z Z fDαφ dx = (−1)|α| gφ dx Ω Ω ∞ holds for all φ ∈ Cc (Ω), then we say that g is the weak derivative of order α of f.

We use the same notation for the weak and classical derivative, it will be clear from the context which interpretation is used.

Theorem 2.2.1. Let g be the weak derivative of f, then g is unique up to a set of measure zero.

1 Proof. Let f, g, h ∈ Lloc, g 6= h and assume that both g and h is the weak derivative of f. By deﬁnition we get the following, Z Z Z α |α| |α| ∞ fD φ dx = (−1) gφ dx = (−1) hφ dx, ∀φ ∈ Cc (Ω). Ω Ω Ω Which gives us Z Z gφ dx = hφ dx, Ω Ω

14 so we get that Z ∞ (g − h)φ dx = 0 =⇒ g = h a.e. ∀φ ∈ Cc (Ω). Ω

Example 2.2.1. Consider f(x) = |x| on Ω = (−1, 1). It is clear that the classical derivative does not exist at x = 0, therefore we will ﬁnd f 0 in the weak sense. Integration by parts give

Z 1 Z 0 Z 1 fφ0 dx = (−x)φ0 dx + xφ0 dx −1 −1 0 Z 0 Z 1 = − (−1)φ dx − φ dx −1 0 Z 1 = − gφ dx. −1 where ( −1 if −1 ≤ x < 0; g(x) = 1 if 0 ≤ x ≤ 1. By Deﬁnition 2.2.1 we get that f 0 = g.

It is obvious from the deﬁnition that if a function is diﬀerentiable in the k normal sense, that is f ∈ C for k ∈ N ∪ {∞}. Then the weak derivative of f will align with the classical one. Just like the classical derivative, the weak derivative do not necessarily exist which is shown in the next example.

Example 2.2.2. Consider Ω = [−1, 1] and the Heaviside step function, ( 0 if −1 ≤ x < 0; H(x) = 1 if 0 ≤ x ≤ 1.

∞ Let φ ∈ Cc (Ω), since φ = 0 on the boundary we get the following,

Z Z 1 1 0 0 Hφ dx = φ dx = φ = −φ(0). Ω 0 0 If we can ﬁnd a function g such that Z ∞ gφ dx = φ(0) ∀φ ∈ Cc , then g is the weak derivative of H. Assume that such a function g exist, we get that Z h Z lim |g| dx = lim |g|χ[−h,h](x) dx. h→0 h→0 −h R

15 The sequence of functions |g|χ[−h,h] tends to zero almost everywhere as h tends to zero. The same sequence is also dominated by |g|χ[−1,1] on the interval [−1, 1]. So by the Lebesgue’s dominated convergence theorem we have that Z h lim |g| dx = 0. h→0 −h Therefore we can choose δ > 0 such that Z δ 1 |g| dx ≤ . −δ 2

Pick an inﬁnitely diﬀerentiable φ : R → [0, 1] such that φ(0) = 1 with support contained in the interval [−δ, δ]. We get the following

Z Z δ Z δ 1 1 = φ(0) = gφ dx = gφ dx ≤ max |φ(x)| |g| dx ≤ , Ω −δ x −δ 2 a contradiction. We conclude that there is no such function g and that H has no weak derivative. The following familiar properties for the classical derivative also hold for the weak one, if we let a, b ∈ R and f, g be m times diﬀerentiable. Then 1. Dα(Dβf) = Dβ(Dαf) = Dα+βf, for all |α| + |β| ≤ m.

2. Dα(af + bg) = aDαf + bDαg, for all |α| ≤ m.

2.3 Deﬁnitions and Properties of Sobolev Spaces

Deﬁnition 2.3.1. Let m ∈ N, for multi-index α and 1 ≤ p < ∞ we deﬁne the norm !1/p X α p kfkW m,p(Ω) = kD fkLp(Ω) , |α|≤m and for p = ∞ α kfkW m,∞(Ω) = max kD fkL∞(Ω). |α|≤m Example 2.3.1. Let 1 ≤ p < ∞, for m = 1 and m = 2 we get the following norms respectively,

Z n Z p!1/p p X ∂f kfkW 1,p(Ω) = |f| + , ∂xk Ω k=1 Ω

Z n Z p n n Z p!1/p p X ∂f X X ∂f kfkW 2,p(Ω) = |f| + + . ∂xk ∂xk∂xl Ω k=1 Ω k=1 l=1 Ω

16 For p = ∞, multi-index α = (α1, . . . , αn) and m = 0, m = 1 respectively,

n (0,...,0) o kfkW 0,∞(Ω) = max ||D f||L∞ ,

n (0,...,0) (1,0,...,0) (0,...,0,1) o kfkW 1,∞(Ω) = max ||D f||L∞ , ||D f||L∞ ,..., ||D f||L∞ . Since the order m just adds more positive terms to the sum and in the case of p = ∞, m adds more alternatives to maximize, we get the following useful estimate

α kD fkLp(Ω) ≤ kfkW m,p(Ω), |α| ≤ m. (2.2)

Lemma 2.3.1. The functional k·kW m,p,(Ω) is a norm. m,p Proof. Let f, g ∈ W (Ω), m ∈ N, p ∈ [1, ∞) and λ ∈ R.

1. kfkW m,p(Ω) = 0 implies kfkLp(Ω) = 0 which implies that f = 0 almost everywhere in Ω.

2. kλfkW m,p(Ω) = |λ|kfkW m,p(Ω) is clear from the deﬁnition. 3. The triangle inequality follow from applying Minkowski’s inequality twice, !1/p X α α p kf + gkW m,p(Ω) = kD f + D gkLp(Ω) |α|≤m !1/p p X α α ≤ kD fkLp(Ω) + kD gkLp(Ω) |α|≤m !1/p !1/p X α p X α p ≤ kD fkLp(Ω) + kD gkLp(Ω) |α|≤m |α|≤m

= kfkW m,p(Ω) + kgkW m,p(Ω).

For p = ∞ the proof is similar.

Before we state the deﬁnition of the Sobolev spaces we will talk about m the space C . If we consider this space equipped with the norm k·kW m,p it will be an incomplete space. Meaning that if we have a cauchy sequence inside Cm, the limit can be found outside of Cm. We will show two examples of this to convince the reader. 2 Example 2.3.2. Let C(R) be equipped with the L -norm, consider the function  0 if x ≤ 0;  1 fm(x) = mx if 0 < x ≤ m ;  1 1 if x > m .

17 Let m > n, Z kf − f k2 = |f − f |2 dx m n L2(R) m n R Z 1/m Z 1/n 2 2 = (fm − fn) dx + (fm − fn) dx 0 1/m Z 1/m Z 1/n = (m − n)2 x2 dx + 1 − 2nx + n2x2 dx 0 1/m 1 2n 1 1 n = − + − + → 0 when m, n → ∞. 3m 3m2 3n m m2

So fm is a cauchy sequence in our space. It is clear that limm→∞ fm = H(x) and since H(x) is not continuous we have that H(x) ∈/ C(R). We can conclude that our space is not complete. f

x 1/m

Figure 2.1: Function fm 1 Example 2.3.3. Let C ([0, 1]) be equipped with the norm k·kW 1,1([0,1]) and consider the sequence r 1 f = x + . m m Let m > n Z 1 Z 1 0 0 kfn − fmkW 1,1([0,1]) = fn − fm dx + fn − fm dx 0 0 1 2 1 2/3 1 2/3 = + x − + x 3 n m 0 r r 1 1 1 + + x − + x n m 0 2 → (1 − 1) + 1 − 1 = 0 when m, n → ∞. 3 So f is a cauchy sequence in our space. But again we have that the limit m √ 0 √1 limm→∞ fm = f = x does not lie in our space since f = 2 x can not be considered continuous on [0, 1].

18 With both the weak derivative and this incompletion of Cm in our mind, we are ready to deﬁne the Sobolev spaces.

Deﬁnition 2.3.2. For 1 ≤ p ≤ ∞, m ∈ N and multi-index α we consider the following spaces equipped with the norm in deﬁnition 2.3.1 to be the Sobolev spaces. 1. W m,p(Ω) = {f ∈ Lp(Ω) : Dαf ∈ Lp(Ω) ∀|α| ≤ m}, where Dα is the weak derivative.

m 2. The completion of C (Ω) with respect to the norm k·kW m,p . So according to the first definition the elements of a Sobolev space are functions in Lp such that the weak partial derivatives up to some order are also in Lp. These spaces generalize the space Lp because W 0,p = Lp. Func- tions in Sobolev spaces are identified if they are equal almost everywhere. By definition 2 we take the completion of Cm with respect to the Sobolev norm, meaning we consider the space that has all of the elements in Cm and the limit points.

Deﬁnition 2.3.3 (Convergence). We say the the sequence {fi} converges m,p to f, written fi → f, in W (Ω) if

lim kfi − fk m,p = 0. i→∞ W (Ω) Theorem 2.3.1 (Completeness). The Sobolev space W m,p(Ω) with respect to the norm in Deﬁnition 2.3.1 is a Banach space.

m,p Proof. Let {fi} be a cauchy sequence in W (Ω), the estimate (2.2) gives us the following

α α kD fi − D fjkLp(Ω) ≤ kfi − fjkW m,p(Ω), |α| ≤ m.

α p Which implies that {D fi} is a cauchy sequence in L (Ω). Since the space p p L (Ω) is complete, this implies that there exist functions fα, f ∈ L (Ω) such that

α lim kD fi → fαk p = 0, lim kfi → fk p = 0, |α| ≤ m. i→∞ L (Ω) i→∞ L (Ω) α Now we will show that D f = fα by using the result above and the deﬁnition of the weak derivative. Z Z α α fD φ dx = lim fiD φ dx Ω i→∞ Ω Z |α| α = (−1) lim D fiφ dx i→∞ Ω Z |α| ∞ = (−1) fαφ dx ∀φ ∈ Cc (Ω). Ω

19 α α p Now we know that D f exist and that D f = fα. Since f ∈ L (Ω) and Dαf ∈ Lp(Ω) for |α| ≤ m we have that f ∈ W m,p(Ω). The only thing left to show is that f is the limit of fi, which follows by

p X α α p lim kfi − fk m,p = lim kD fi − D fk p = 0. i→∞ W (Ω) i→∞ L |α|≤m

It should be noted that W m,p(Ω) is a separable space for 1 ≤ p < ∞ [1, page 61]. The case p = 2 is of special interest because we can deﬁne an inner product, X α α hf, giW m,2(Ω) = hD f, D giL2(Ω). |α|≤m As expected, q hf, giW m,2(Ω) = kfkW m,2(Ω). Since our norm induced by the inner product yields a complete space, W m,2(Ω) is a Hilbert space and is therefore denoted by Hm(Ω).

n Example 2.3.4. Let Ω = B(0, 1), the open unit ball in R . Consider f = |x|−a for x ∈ Ω, x 6= 0.

We want to ﬁnd out for which values a > 0, n, p does the function f belong to W 1,p(Ω). First will check for which values f belong to Lp(Ω) with the help of spherical coordinates. For positive constant M1 we get Z p −ap kfkLp(Ω) = |x| dx Ω Z 2π Z π Z π Z 1 = ··· r−apdnV ξn−1=0 ξn−2=0 ξ1=0 r=0 Z 1 n−1−ap ≤ M1 r dr 0 1 0n−ap n = M − < ∞ when a < , 1 n − ap n − ap p where

n n−1 n−2 n−3 d V = r sin (ξ1) sin (ξ2) ··· sin (ξn−2) dr dξ1 dξ2 ··· dξn−1.

p n So we have that f ∈ L (Ω) for a < p . Next we must investigate the derivative. The function f is diﬀerentiable away from x = 0 with derivative

∂f(x) −axi = a+1 for x 6= 0. ∂xi |x|

20 In the same fashion we get that Z p −(a+1)p kDfkLp(Ω) = |a| |x| dx Ω Z 1 −(a+1)p+n−1 ≤ M2 r dr 0 1 0−(a+1)p+n = M − 2 −(a + 1)p + n −(a + 1)p + n n < ∞ for a < − 1. p

p n ∞ This implies that Df ∈ L (Ω) for a < p − 1. Let φ ∈ Cc (Ω) and > 0, we get that Z ∂φ Z ∂f Z f dx = − φ dx + fφvi dx, Ω−B(0,) ∂xi Ω−B(0,) ∂xi ∂B(0,) where v = (v1, . . . , vn) denote the inward pointing normal on ∂B(0, ). Let p = 1 and α < n − 1. In this case we get

Z Z −a n−1−a fφvi dS ≤ kφkL∞ dS ≤ C → 0. ∂B(0,) ∂B(0,) Thus Z Z ∂φ ∂f ∞ f dx = − φ dx ∀φ ∈ Cc (Ω). Ω ∂xi Ω ∂xi −a 1,p n We conclude that f = |x| ∈ W (Ω) if and only if a < p − 1. Whether a function belong to a certain Sobolev space or not is obviously depending on the integrability p and regularity m of the function. This example show that it can also depend on the dimension n of the domain Ω. We will show some smoothness properties that hold for certain sobolev spaces. 1,∞ Consider n = 1 and the space W (R). A function belonging to this space is Lipschitz continuous which is shown below by using Hölder’s inequality: Z x 0 |f(x) − f(y)| = f (t) dt y Z x 0 ≤ f (t) dt y 0 ≤ f (t) L∞ k1kL1 0 = f (t) L∞ |x − y|.

21 0 ∞ Since f ∈ L (R), the following holds

|f(x) − f(y)| 0 ≤ f (t) ∞ = C < ∞. |x − y| L

1 1 1,p Now consider 1 < p < ∞, p + q = 1 and f ∈ W (R). Again using Hölder’s inequality: Z x 0 |f(x) − f(y)| = f (t) dt y Z x 0 ≤ f (t) dt y 0 ≤ f (t) Lp k1kLq 1 0 q = f (t) Lp |x − y| .

0 p In the same way as before, since f ∈ L (R), the following holds

|f(x) − f(y)| 0 1 ≤ f (t) Lp = C < ∞. |x − y| q

1,p We conclude that if f ∈ W (R) then f is Hölder continuous with expo- 1 nent γ = 1 − p .

22 Chapter 3

Finite Element Method

We present the concept of a weak solution and the ﬁnite element method, Section 3.2 is based on the material in Logg and Mardal [5].

3.1 Weak Solution

A weak solution to a differential equation is a function for which the derivatives may not all exist but still satisfy the equation in some precisely defined way. Consider the partial differential operator

X α L = AαD , |α|≤m n where α is a multi-index and Aα functions in R . If a function u is sufficiently differentiable in the classical sense so that Lu is well defined pointwise, then it is called a classical solution if it satisfies the equation Lu = f, (3.1) n for some function f on R . By a weak solution of 3.1 we mean a function n u on R that does not need to be sufficiently differentiable to make Lu meaningful in a classical sense. In this case we will have some other precise way of considering u to be a solution anyway. As for example by considering the weak derivative instead of classical. n Example 3.1.1. Given a bounded open subset Ω ⊂ R , the distance function is defined as   inf kx − yk if x ∈ Ω; dist(x, ∂Ω) = y∈∂Ω 0 if x ∈ ∂Ω. Consider the problem of finding u such that ( k∇uk = 1 in Ω; (3.2) u = 0 on ∂Ω.

23 The distance function satisﬁes this problem except at the point where the slope changes from 1 to −1. To get some idea of how this looks consider the case when we have Ω = (−1, 1) and k·k = |·|. We draw the distance function in this case below:

x −1 1

Figure 3.1: The function dist(x,{-1,1}).

It is clear that dist(x,{-1,1}) does not satisfy the problem in classical sense. But if we consider using the weak derivative instead it does satisfy (3.2) in the weak sense. We show it below: Z Z 0 Z 1 dist(x, {−1, 1})φ0 dx = (x + 1)φ0 dx + (−x + 1)φ0 dx Ω −1 0 0 Z 0

= ((x + 1)φ) − 1φ dx −1 −1 1 Z 1

+ ((−x + 1)φ) − (−1)φ dx 0 0 Z = φ(0) − gφ dx − φ(0) Ω Z = − gφ dx Ω where ( 1 if −1 ≤ x < 0; g(x) = −1 if 0 ≤ x ≤ 1. We concude that |dist(x, {−1, 1})0| = 1 in weak sense on Ω.

3.2 Finite Element Method

The ﬁnite element method is a numerical method for solving partial diﬀer- ential equations. When the problem has been formulated according to the

24 method you end up with a system of algebraic equations. A partial diﬀeren- tial equation has the general form ( Lu = f in Ω; Boundary conditions on ∂Ω.

This is called the strong form of the problem, the ﬁrst thing is to turn it into the weak form. The weak form is obtained by taking the inner product of the two sides together with some test function. So let V be a Hilbert space, we have hLu, vi = hf, vi. Deﬁne a(u, v) = hLu, vi, b(v) = hf, vi.

Where a : V × Vb → R and b : Vb → R. The spaces V and Vb are some specific Sobolev spaces, the choice of which Sobolev space to use depends on the problem. In many problems these spaces will be chosen so that they are the same space, but in general they can be chosen to be different. The weak formulation of our problem is finding u ∈ V such that

a(u, v) = b(v) ∀v ∈ V.b (3.3)

We present a result regarding the existence and uniqueness of weak solutions for the case V = Vb: Theorem 3.2.1 (Lax-Milgram Theorem). Let a(x, y) be a bounded, coercive, bilinear functional on a Hilbert space H. For every bounded linear functional b(y) on H, there exist a unique xb such that

a(xb, y) = b(y) ∀y ∈ H.

So assume a(u, v) is bounded, coercive and bilinear and b(v) bounded and linear. We have that there is a unique u satisfying (3.3). There are generalizations of this theorem, for the case V 6= Vb the interested reader can look up Babuška–Lax–Milgram Theorem. The space V can be inﬁnite so when we want to solve this in a computer we need to make it ﬁnite. Pick a Vk such that

1. dim(Vk) = k.

2. Vk ⊂ Vk+1.

3. ∪Vk = V .

25 Let the vectors {φ1, . . . , φk} span Vk. We are now looking for an approximation uk ∈ Vk to our solution u ∈ V . So instead of solving (3.3) we will solve a(uk, v) = b(v) ∀v ∈ Vbk. (3.4) Since we want to put this in a computer it is nice to have it on matrix form. If we let {wk+1, . . . , wn} span V \ Vk, we can describe our solution as u = Pk Pn Pk i=1 ciφi + i=k ciwi. We will write our approximation as uk = i=1 ciφi in order to get the matrix form. Also let {ψ1, . . . , ψk} be a basis for Vk. We get the following

k X a( ciφi, ψj) = b(ψj) for j = 1, . . . , k, i=1 which can be written as

k X cia(φi, ψj) = b(ψj) for j = 1, . . . , k. i=1 Let   c1  .  c =  .  , ck and   b(ψ1)  .  b =  .  . b(ψk)

Also let A = (ai,j) where ai,j = a(φi, ψj). We now have our equation (3.4) on the form Ac = b, which is solved by a suitable method.

26 Chapter 4

Future Work

Here is an informal discussion of the possible continuations on this essay. If I had more time I would write about the relation between the two definitions of the Sobolev spaces stated in definition 2.3.2. They are in fact equivalent, which is far from trivial to show. According to [1, page 60] this fact was discovered in 1964, having in mind that people have been working on Sobolev spaces since at least late 1930’s, it is quite interesting that this elementary result was discovered so late. Another result is the Trace Theorem, which asserts that we can talk about the boundary of a function in a Sobolev space. This is not trivial since in a Sobolev space a function is defined almost everywhere and we have that the n-dimensional measure of the boundary is zero. So it is not clear how to describe values at the boundary for a function in a Sobolev space. I consider this topic relevant since when we talk about a partial differential equation we often state some criteria on the boundary values. For a more theoretical topic, there is a big result called the Sobolev Embedding Theorem stating that certain Sobolev spaces can be embedded in various spaces such as, for example, certain Hölder spaces. This could be interesting if one want to write more about the connection between Sobolev and Hölder spaces, since this relation was only mildly investigated in this essay. The last suggestion is looking into the interesting generalizations of Sobolev spaces, such as when the space W m,p is extended to include spaces where m need not be an integer. If these spaces have some relevance to partial differential equations I do not know, but that would be interesting to investigate.

27 Bibliography

[1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2 ed. Elsevier, 2005.

[2] L. Debnath and P. Mikusiński, Hilbert Spaces with Applications, 3 ed. Elsevier, 2005.

[3] L. C. Evans, Partial Diﬀerential Equations, 1 ed. Vol. 19. American Mathematical Society, 2002.

[4] G. B. Folland, Real Analysis, 2 ed. John Wiley & Sons, 1999.

[5] A. Logg and K-E. Mardal, Lectures on the Finite Element Method.

[6] S. Shkoller, Notes on Lp and Sobolev Spaces, Department of Mathematics, University of California, 2009.

[7] J. Kinnunen, Sobolev Spaces, Department of Mathematics and Systems Analysis, Aalto University, 2017.