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27. Sobolev Inequalities 27.1 ANALYSIS TOOLS WITH APPLICATIONS 493 27. Sobolev Inequalities 27.1. Morrey’s Inequality. d 1 d Notation 27.1. Let S − be the sphere of radius one centered at zero inside R . d 1 d For a set Γ S − ,x R , and r (0, ), let ⊂ ∈ ∈ ∞ Γx,r x + sω : ω Γ such that 0 s r . ≡ { ∈ ≤ ≤ } So Γx,r = x + Γ0,r where Γ0,r is a cone based on Γ, seeFigure49below. Γ Γ Figure 49. The cone Γ0,r. d 1 Notation 27.2. If Γ S − is a measurable set let Γ = σ(Γ) be the surface “area” of Γ. ⊂ | | Notation 27.3. If Ω Rd is a measurable set and f : Rd C is a measurable function let ⊂ → 1 fΩ := f(x)dx := f(x)dx. − m(Ω) Ω ZΩ Z By Theorem 8.35, r d 1 (27.1) f(y)dy = f(x + y)dy = dt t − f(x + tω) dσ(ω) Γx,r Γ0,r 0 Z Z Z ZΓ and letting f =1in this equation implies d (27.2) m(Γx,r)= Γ r /d. | | d 1 Lemma 27.4. Let Γ S − be a measurable set such that Γ > 0. For u 1 ⊂ | | ∈ C (Γx,r), 1 u(y) (27.3) u(y) u(x) dy |∇ d | 1 dy. − | − | ≤ Γ x y − ΓZx,r | |ΓZx,r | − | 494 BRUCE K. DRIVER† d 1 Proof. Write y = x + sω with ω S − , then by the fundamental theorem of calculus, ∈ s u(x + sω) u(x)= u(x + tω) ωdt − ∇ · Z0 and therefore, s u(x + sω) u(x) dσ(ω) u(x + tω) dσ(ω)dt | − | ≤ 0 Γ |∇ | ZΓ Z Z s d 1 u(x + tω) = t − dt |∇ d | 1 dσ(ω) 0 Γ x + tω x − Z Z | − | u(y) u(y) = |∇ d | 1 dy |∇ d | 1 dy, y x − ≤ x y − ΓZx,s | − | ΓZx,r | − | wherein the second equality we have used Eq. (27.1). Multiplying this inequality d 1 by s − and integrating on s [0,r] gives ∈ d r u(y) m(Γx,r) u(y) u(y) u(x) dy |∇ d | 1 dy = |∇ d | 1 dy | − | ≤ d x y − Γ x y − ΓZx,r ΓZx,r | − | | | ΓZx,r | − | which proves Eq. (27.3). Corollary 27.5. Suppose d<p , Γ d 1 such that Γ > 0,r (0, ) and S − 1 ≤∞ ∈ B | | ∈ ∞ u C (Γx,r). Then ∈ (27.4) u(x) C( Γ ,r,d,p) u 1,p | | ≤ | | k kW (Γx,r ) where 1/p 1 1/p 1 d− p 1 − 1 d/p C( Γ ,r,d,p):= max , − r − . | | Γ 1/p r p d · | | Ã µ − ¶ ! Proof. For y Γx,r, ∈ u(x) u(y) + u(y) u(x) | | ≤ | | | − | and hence using Eq. (27.3) and Hölder’s inequality, 1 u(y) u(x) u(y) dy + |∇ d | 1 dy | | ≤ − | | Γ x y − ΓZx,r | |ΓZx,r | − | 1 1 1 (27.5) u p 1 p + u p q L (Γx,r ) L (Γx,r ) L (Γx,r ) d 1 L (Γx,r ) ≤ m(Γx,r) k k k k Γ k∇ k k x k | | | − ·| − p where q = p 1 as before. Now − r 1 q d 1 d 1 q q = dt t − t − − dσ(ω) d 1 L (Γ0,r ) k − k 0 |·| Z ZΓ ¡ ¢ r p r d 1 d 1 1 p 1 − = Γ dt t − − − = Γ dt t− p 1 | | | | − Z0 Z0 and since ¡ ¢ d 1 p d 1 − = − − p 1 p 1 − − ANALYSIS TOOLS WITH APPLICATIONS 495 we find p 1 1/q − 1 p 1 p d p 1 p d p−1 1 p (27.6) q = − Γ r − = − Γ r − . k d 1 kL (Γ0,r) p d | | p d | | |·| − µ − ¶ µ − ¶ Combining Eqs. (27.5), Eq. (27.6) along with the identity, 1 1 1/q d 1/p (27.7) 1 q = m(Γ ) = Γ r /d − , L (Γx,r ) x,r m(Γx,r) k k m(Γx,r) | | shows ¡ ¢ 1 1/p d 1/p 1 p 1 − 1 d/p u(x) u p Γ r /d − + u Lp(Γ ) − Γ r − | | ≤ k kL (Γx,r ) | | Γ k∇ k x,r p d | | | | µ − ¶ ¡ ¢ 1/p 1 1/p 1 d− p 1 − 1 d/p = u p + u Lp(Γ ) − r − . Γ 1/p k kL (Γx,r ) r k∇ k x,r p d | | " µ − ¶ # 1/p 1 1/p 1 d− p 1 − 1 d/p max , − u 1,p r − . ≤ Γ 1/p r p d k kW (Γx,r ) · | | Ã µ − ¶ ! Corollary 27.6. For d N and p (d, ] there are constants α = αd and β = βd ∈ ∈ ∞ such that if u C1(Rd) then for all x, y Rd, ∈ ∈ p 1 −p 1/p p 1 (1 d ) (27.8) u(y) u(x) 2βα − u Lp(B(x,r) B(y,r)) x y − p | − | ≤ p d k∇ k ∩ ·| − | µ − ¶ where r := x y . | − | d 1 Proof. Let r := x y ,V := Bx(r) By(r) and Γ, Λ S − be chosen so that | − | ∩ ⊂ x + rΓ = ∂Bx(r) By(r) and y + rΛ = ∂By(r) Bx(r), i.e. ∩ ∩ 1 1 Γ = (∂Bx(r) By(r) x) and Λ = (∂By(r) Bx(r) y)= Γ. r ∩ − r ∩ − − Also let W = Γx,r Λy,r, see Figure 50 below. By a scaling, ∩ Γx,r Λy,r Γx,1 Λy,1 βd := | ∩ | = | ∩ | (0, 1) Γx,r Γx,1 ∈ | | | | is a constant only depending on d, i.e. we have Γx,r = Λy,r = β W . Integrating the inequality | | | | | | u(x) u(y) u(x) u(z) + u(z) u(y) | − | ≤ | − | | − | over z W gives ∈ u(x) u(y) u(x) u(z) dz + u(z) u(y) dz | − | ≤ − | − | − | − | ZW ZW β = u(x) u(z) dz + u(z) u(y) dz Γx,r | − | | − | | | WZ WZ β u(x) u(z) dz + u(z) u(y) dz . ≤ Γx,r | − | | − | | | ΓZ ΛZ x,r y,r 496 BRUCE K. DRIVER† Λ Γ Λ Λ Γ Γ Figure 50. The geometry of two intersecting balls of radius r := x y . Here W = Γx,r Λy,r and V = B(x, r) B(y, r). | − | ∩ ∩ Hence by Lemma 27.4, Hölder’s inequality and translation and rotation invariance of Lebesgue measure, β u(z) u(z) u(x) u(y) dz + dz |∇ d | 1 |∇ d |1 | − | ≤ Γ x z − z y − | | ΓZ | − | ΛZ | − | x,r y,r β 1 1 u p q + u p q ≤ Γ k∇ kL (Γx,r )k x d 1 kL (Γx,r ) k∇ kL (Λy,r)k y d 1 kL (Λy,r) | | µ | − ·| − | − ·| − ¶ (27.9) 2β 1 u p q ≤ Γ k∇ kL (V )k d 1 kL (Γ0,r ) | | |·| − p where q = p 1 is the conjugate exponent to p. Combining Eqs. (27.9) and (27.6) − 1 gives Eq. (27.8) with α := Γ − . | | Theorem 27.7 (Morrey’s Inequality). If d<p ,u W 1,p(Rd), then there ≤∞ ∈ exists a unique version u∗ of u (i.e. u∗ = u a.e.) such that u∗ is continuous. 0,1 d d Moreover u∗ C − p (R ) and ∈ (27.10) u∗ 0,1 d C u W 1,p(Rd) k kC − p (Rd) ≤ k k where C = C(p, d) is a universal constant. Moreover, the estimates in Eqs. (27.3), (27.4) and (27.8) still hold when u is replaced by u∗. 1 d Proof. For p< and u Cc (R ), Corollaries 27.5 and 27.6 imply ∞ ∈ u(y) u(x) u d C u 1,p d and | − | C u p d BC(R ) W (R ) 1 d L (R ) k k ≤ k k x y − p ≤ k∇ k | − | ANALYSIS TOOLS WITH APPLICATIONS 497 which implies [u]1 d C u Lp(Rd) C u W 1,p(Rd)and hence − p ≤ k∇ k ≤ k k (27.11) u 0,1 d C u W 1,p(Rd). k kC − p (Rd) ≤ k k 1,p d 1 d Now suppose u W (R ), choose (using Exercise 21.2) un Cc (R ) such that 1,p d∈ ∈ un u in W (R ). Then by Eq. (27.11), un um 0,1 d 0 as m, n → k − kC − p (Rd) → →∞ 0,1 d d 0,1 d d and therefore there exists u∗ C − p (R ) such that un u∗ in C − p (R ). ∈ → Clearly u∗ = u a.e. and Eq. (27.10) holds. 1, d If p = and u W ∞ R , then by Proposition 21.29 there is a version u∗ of u which∞ is Lipschitz∈ continuous. Now in both cases, p< and p = , the ¡ ¢ d ∞ ∞ sequence um := u ηm = u∗ ηm C∞ R and um u∗ uniformly on compact ∗ ∗ ∈ → subsets of d. Using Eq. (27.3) with u replaced by u along with a (by now) R ¡ ¢ m standard limiting argument shows that Eq. (27.3) still holds with u replaced by u∗. The proofs of Eqs. (27.4) and (27.8) only relied on Eq. (27.3) and hence go through without change. Similarly the argument in the first paragraph only relied on Eqs. (27.4) and (27.8) and hence Eq. (27.10) is also valid for p = . ∞ d Corollary 27.8 (Morrey’s Inequality). Suppose Ω o R such that Ω is compact C1-manifold with boundary and d<p . Then for⊂ u W 1,p(Ω), there exists a ≤∞0,1 d d ∈ unique version u∗ of u such that u∗ C − p (R ) and we further have ∈ (27.12) u∗ 0,1 d C u W 1,p(Ω), k kC − p (Ω) ≤ k k where C = C(p, d, Ω).
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