A Nonlinear Partial Differential Equation and Its Viscosity Solutions

and Its Viscosity Solutions

Audun Reigstad

Master of Science in Physics and Mathematics Supervisor: Lars Peter Lindqvist, MATH

Department of Mathematical Sciences Submission date: June 2016

Norwegian University of Science and Technology

Preface

With this thesis, I complete my master’s degree in industrial mathematics at NTNU.

I would like to thank my supervisor Peter Lindqvist for many enlightening discussions.

His support in my work this semester has been invaluable.

Abstract

We study a nonlinear partial differential equation with Lipschitz continuous coefficient functions. Existence and uniqueness of viscosity solutions is proved by approximating with minimizers of variational integrals. The solutions are shown to satisfy a corresponding minimization property. Stability of solutions with respect to small perturbations of the coefficient functions is discussed, and proved for C²-solutions.

Sammendrag

Vi studerer en ikke-lineær partiell differensialligning med Lipschitzkontinuerlige koeff- isientfunksjoner. Eksistens og entydighet av viskositetsløsninger bevises ved ˚a approksimere med minimerere av variasjonsintegraler. Det vises at løsningene har en lignende minimer- ingsegenskap. Stabilitet av løsninger med hensyn p˚a sm˚a perturbasjoner av koeffisient- funksjonene diskuteres, og bevises for C²-løsninger.

Contents

Preface i

Abstract iii

Sammendrag v

Contents vii

1 Introduction 1

2 Preliminaries 3

2.1 Ascoli’s theorem . . . 3

2.2 Lebesgue spaces . . . 4

2.3 Sobolev spaces . . . 9

2.4 Quadratic forms and convex functions . . . 17

3 The equation 23 3.1 Variational problem . . . 23

3.2 Some properties . . . 27

4 The equation for finite p 31 4.1 Existence and uniqueness . . . 32

4.2 Comparison principle . . . 33

5 Limit of solutions as p→ ∞ 37 6 Viscosity solutions 41 6.1 Equivalent definition . . . 43

7 Auxiliary equations 47 7.1 Variational problem . . . 47

7.2 Limit procedure . . . 51

7.3 Viscosity solutions . . . 55

8 Comparison principle 59 8.1 Uniqueness . . . 64

9 Stability 67 9.1 Stability in one variable . . . 68

9.2 Stability of C²-solutions . . . 69

Bibliography 75

1 Introduction

We study the nonlinear partial differential equation

∆_∞,Au=

n X i,j,k,l=1

∂a_ij

∂x_k

∂u

∂x_i

∂u

∂x_ja_kl∂u

∂x_l+ 2a_ik ∂u

∂x_k

∂²u

∂x_i∂x_ja_jl∂u

∂x_l

= 0, (1.1)

which comes from the minimax problem minu max

x∈Ω n

X i,j=1

a_ij(x)∂u

∂x_i(x)∂u

∂x_j(x)

1/2

,

among all all admissible functionsudefined in a bounded domain Ω⊂Rⁿ, having the same boundary values. Here a_ij denotes given coefficient functions. The equation is related to the infinity-Laplace equation

∆_∞u=

n X i,j=1

∂u

∂x_i

∂u

∂x_j

∂²u

∂x_i∂x_j = 0, (1.2)

which is seen by setting aij =δij/2 in (1.1), where δij is the Kronecker delta. Thus, the equation represents a generalized infinity-Laplace equation, and our strategy for showing existence and uniqueness of viscosity solutions is based on Jensen’s work in [12] on the infinity-Laplace equation, and Juutinen’s work in [13] on more general problems. To motivate this strategy, we give a short introduction to the infinity-Laplace equation. The equation was derived by Aronsson in 1967, and provides the best Lipschitz extension of given boundary values, see [1]. The concept of solutions of (1.2) is difficult. Aronsson demonstrated that the equation does not necessarily have classical solutions. For instance, he constructed the function

u(x₁, x₂) =x^4/3₁ −x^4/3₂ ,

which satisfies the equation in the whole plane, but does not have second derivatives on the axes. Furthermore, the equation does not have a weak formulation involving only the first derivatives. The way out of these difficulties turned out to be the concept of viscosity solutions, which was developed by Crandall, Evans, Ishii, Lions and others in the 1980’s.

The breakthrough came in 1993, when Jensen established uniqueness of viscosity solutions by proving the comparison principle, where the main idea is to introduce two auxiliary equations, see [12]. In the proof, viscosity solutions are constructed as limits of weak solutions of the p-Laplace equation as p→ ∞, via some subsequence.

The layout of this thesis is as follows: In Section 2 we present some results which will be used frequently throughout the text. Lebesgue spaces, Sobolev spaces, and various compactness results are discussed, followed by fundamental properties of quadratic forms.

In Section 3 we introduce a variational integral and state the assumptions on the coef- ficient functions. We derive the Euler-Lagrange equation and establish some fundamental properties of the involved operators.

Section 4 is devoted to the Euler-Lagrange equation. We prove existence and unique- ness of weak solutions, and show that these are minimizers of the variational integral.

In Section 5 we construct the limit of weak solutions of the Euler-Lagrange equation, and show that it satisfies a minimization property.

In Section 6 we introduce viscosity solutions and show some fundamental properties of these. We show that the definition of viscosity solutions can be rephrased in terms of so-called jets. Then we state Ishii’s lemma - a deep result of the theory which will play a central part in proving uniqueness of viscosity solutions.

Modified versions of Jensen’s two auxiliary equations are presented in Section 7. We show that viscosity solutions of these can be constructed as limits of weak solutions of two Euler-Lagrange equations, and that the difference between these limits can be made arbitrarily small. Furthermore, we conclude that the limit constructed in Section 5 is a viscosity solution of (1.1).

We prove the Comparison principle in Section 8, which implies that an arbitrary viscosity solution of (1.1) lies between the two auxiliary solutions, which in turn implies uniqueness of viscosity solutions.

In Section 9 we perturb the coefficient functions by constants, and state the assump- tions these has to satisfy such that there is a unique viscosity solution of the perturbed equation. Then we show stability with respect to small perturbations for solutions in one variable, and forC²-solutions.

2 Preliminaries

In the following we assume that the domain Ω is an open and connected subset of Rⁿ with boundary ∂Ω.

2.1 Ascoli’s theorem

We start out with a version of Ascoli’s theorem.

Definition 2.1. We say that a sequence of functions u_k: Ω→Ris equibounded, if sup

x∈Ω

|u_k(x)| ≤M <∞ for all k∈N, and equicontinuous, if for x, y∈Ω,

|u_k(x)−u_k(y)| ≤C|x−y|^α for all k∈N, where 0< α≤1 and C is a constant.

Theorem 2.2(Ascoli’s theorem).Let(u_k)be an equibounded and equicontinuous sequence of functions. Then there exist a subsequence (u_kj) and a continuous function u: Ω→R such that u_kj →u locally uniformly in Ω. If Ω is bounded, the functions can be extended to be continuous in the closure Ω, where the convergence is uniform.

Proof. Let (q_k)_k∈N be an enumeration of the rational points in Ω. By assumption, (u_k(q₁))_k is bounded, and by the Bolzano–Weierstrass theorem¹ has a subsequence, de- noted by (u_1j(q₁))_j converging at q₁. Similary, the sequence (u_1j(q₂))_j is bounded, so it has a subsequence (u_2j(q₂))_j which converges at q₁ and q₂. Continuing in this fashion, extracting subsequences of subsequences, we obtain sequences (u_kj(q_k))_j for all k ∈N, converging at q₁, q₂, . . ., q_k. Then the diagonal sequence, (u_jj(q_k))_jj, which we simply denote by (u_j(q_k))_j, converges at every rational point in Ω. Thus, for everyε >0 there is anN ∈N such that

|u_j(q_k)−u_i(q_k)|< ε/2 for all i, j > N.

Now we show that the constructed diagonal sequence converges at each point in Ω, not just the rational ones. Consider an arbitrary x∈Ω. By the density of the rational points in Ω, given ε >0 there is a rational point q∈Ω such that

2C|x−q|^α< ε/2.

Then by the equicontinuity,

|u_j(x)−u_i(x)| ≤ |u_j(x)−u_j(q)|+|u_j(q)−u_i(q)|+|u_i(q)−u_i(x)|

≤2C|x−q|^α+|u_j(q)−u_i(q)|

< ε/2 +ε/2 =ε,

1The proof can be found in most analysis books, see e.g. [6].

wheneveri, j > N. Thus, (u_j) is a Cauchy sequence, so the sequence converges pointwise to a function denoted byu:

u(x) = lim

j→∞uj(x).

It remains to show that the convergence is uniform. First suppose that Ω is bounded.

Then the closure Ω is compact, and we can cover it by a finite number of open balls B(x_m, r), centered at x_m with diameter 2r=ε^1/α, that is

Ω⊂

n [ m=1

B(xm, r).

Choose a rational point r_m from each ball. Since there is only a finite number of these points, givenε >0 there is anN_ε∈Nsuch that

maxm |u_j(r_m)−u_i(r_m)|< ε, for all i, j > N_ε.

Consider an arbitrary x∈Ω, which must belong to some ball, say B(x_m, r). Then

|u_j(x)−u_i(x)| ≤2C|x−r_m|^α+|u_j(r_m)−u_i(r_m)|

≤2C(2r)^α+ max

m |uj(r_m)−ui(r_m)|

≤2Cε+ε, for all i, j > N_ε.

Notice thatN_ε is independent of how we chose the pointx, so the convergence is uniform in Ω. Thus, the limit function u is continuous.

If Ω is unbounded, the proof above holds for any fixed, bounded subdomain of Ω, so the convergence is locally uniform.

The proof is based on Theorem 1 in [18].

2.2 Lebesgue spaces

Now we derive some properties of the Lebsegue spaces.

Definition 2.3. For any Lebesgue measurable function u: Ω→R we define

||u||_p,Ω=





 Z

Ω|u(x)|^pdx

1/p

if p∈[1,∞) ess sup_x∈Ω|u(x)| if p=∞,

(2.1)

where the essential supremum is

ess sup_x∈Ω|u(x)|= inf{M:u(x)≤M for a.e. x∈Ω}.

We say that u∈L^p(Ω) if ||u||_p,Ω<∞. If u∈L^p(D) for each open set D⊂⊂Ω, we say that u∈L^p_loc(Ω).²

2The notation ”⊂⊂” is explained in Definition 2.27.

When the domain Ω is evident from the context we simply write || · ||_p,Ω =|| · ||p. Observe that by the Lebesgue integral we have ||u||_p,Ω = 0 if and only if u= 0 almost everywhere in Ω, so the axiom of normed spaces is not satisfied on sets of measure zero.

To get a proper normed space we have to consider equivalence classes of functions, that is, each equivalence class consisits of functions which coincide a.e. We say that a function u has a version u, if˜ u and ˜u belong to the same equivalence class.

A fundamental property of L^p(Ω) for 1 ≤p≤ ∞ is that it is a Banach space with respect to the norm defined in (2.1).

Our next result shows that the Lebesgue spaces are nested when Ω is of finite measure:

L¹(Ω)⊇L²(Ω)⊇ · · · ⊇L^p(Ω)⊇L^q(Ω)⊇ · · · ⊇L^∞(Ω), p≤q.

First we introduce the notation

−

Z

Ωu dx= 1 µ(Ω)

Z Ωu dx

for the average of a function uover a bounded domain Ω, whereµdenotesn-dimensional Lebesgue measure.

Proposition 2.4. If µ(Ω)<∞ and u∈L^q(Ω), then

−

Z

Ω|u|^pdx

1/p

≤

−

Z

Ω|u|^qdx

1/q

when 1≤p≤q. (2.2)

Proof. By H¨older’s inequality we have

||u||^p_p=||1· |u|^p||₁≤ ||1||_q/(q−p)|||u|^p||_q/p=µ(Ω)^(q−p)/q||u||^p_q, where p≥1. This yields inequality (2.2) if p≤q.

In many limit procedures we rely on the fact that the norm is continuous as p→ ∞:

Proposition 2.5. If µ(Ω)<∞ and u∈L^∞(Ω), then

p→∞lim ||u||_p=||u||_∞. Proof. Letε >0 and define the set

A={x∈Ω :|u(x)|>||u||_∞−ε}.

Then _Z

Ω|u|^pdx≥

Z

A|u|^pdx≥(||u||_∞−ε)^pµ(A), which implies that

lim inf

p→∞ ||u||_p≥ ||u||_∞−ε.

On the other hand, by Proposition 2.4,

||u||_p≤µ(Ω)^1/p||u||_∞,

thus

lim sup

p→∞ ||u||_p≤ ||u||_∞, and the conclusion

p→∞lim ||u||_p=||u||_∞ follows sinceε >0 was arbitrarily small.

We mention a fundamental result in the calculus of variations³. Lemma 2.6 (Variational lemma). Suppose that u∈L¹_loc(Ω). If

Z

Ωuφ dx= 0 for all φ∈C₀^∞(Ω), then u= 0 a.e. in Ω.

We now turn to the dual space of L^p(Ω). An explicit characterization of bounded linear functionals onL^p(Ω) is provided by Riesz’ representation theorem⁴.

Theorem 2.7 (Riesz’ representation theorem). Let 1 ≤p < ∞ and suppose that Λ : L^p(Ω)→Ris a bounded linear functional. Then there exists a unique functionv∈L^q(Ω), where 1/p+ 1/q= 1, such that

Λ(u) =

Z

Ωuv dx, for all functionsu∈L^p(Ω). Moreover, ||Λ||=||v||_q,Ω.

A consequence of this result is that we identify the dual space of L^p(Ω) as L^q(Ω) for 1/p+ 1/q= 1 when p∈[1,∞), and write L^p(Ω)⁰=L^q(Ω).

Working in Banach spaces requires various concepts of convergence, and one of the most frequently used in this text is the notion of weak convergence.

Definition 2.8. Let X be a Banach space. We say that a sequence (xn)⊂X converges weakly to x∈X, if for all x⁰ in the dual space X⁰ we have

n→∞lim x⁰(x_n) =x⁰(x), and we writex_n* x.

By Riesz’ representation theorem there is an explicit characterization of weak conver- gence in Lebesgue spaces. Indeed, let 1≤p <∞and suppose that (un)⊂L^p(Ω) converges weakly tou∈L^p(Ω), that is

n→∞lim Λ(u_n) = Λ(u)

for all bounded linear functionals Λ onL^p(Ω). This is equivalent to

n→∞lim

Z

Ωunv dx=

Z

Ωuv dx, (2.3)

for all v∈L^q(Ω) such that 1/p+ 1/q= 1.

Now we present som key properties of weak convergence.

3We refer to Theorem 3.40 in [5] for a proof.

4The proof can be found in [6], Theorem 13.1.

Proposition 2.9. Let X be a Banach space and suppose that the sequence (x_n)⊂X converges weakly to x∈X. Then the sequence is uniformly bounded:

sup

n ||x_n||_X ≤M <∞, and the norm is lower semicontinuous:

||x||_X ≤lim inf

n→∞ ||x_n||_X.

Proposition 2.10. Let1< p <∞and assume that the sequence(u_n)⊂L^p(Ω)is uniformly bounded:

sup

n ||u_n||_p≤M <∞.

Then there is a subsequence (u_nk) and a function u∈L^p(Ω) such that u_nk * u weakly in L^p(Ω).

Proof. Let 1/p+ 1/q= 1. Since q∈(1,∞), L^q(Ω) is separable⁵. Let (v_n) be a countable collection of simple functions, which is dense in L^q(Ω). Set

Λ_n(v_j) =

Z

Ωv_ju_ndx for each j∈N. By H¨older’s inequality and the uniform boundedness we have

|Λ_n(v_j)| ≤M||v_j||_q,

so the sequence (Λ_n(v₁))_n is bounded, and by the Bolzano–Weierstrass theorem, we can extract a subsequence, denoted by (Λ_1j(v₁))_j converging at v₁. Similary, (Λ_1j(v₂))_j is bounded, so we can extract a subsequence, denoted by (Λ_2j(v₂))_j converging at v1 and v₂. Continuing this procedure, we see that the diagonal sequence (Λ_jj(v_n))_jj converges at everyv_n. To ease the notation we denote the constructed diagonal sequence by (Λ_j(v_n))_j. Thus, for every ε >0 there is an N such that

|Λ_j(v_n)−Λ_i(v_n)|< ε, for all i, j > N.

Fixv ∈L^q(Ω). By density there is a simple function v_n such that

||v−v_n||_q< ε.

It follows that

|Λ_j(v)−Λ_i(v)| ≤ |Λ_j(v−v_n)|+|Λ_i(v_n−v)|+|Λ_j(v_n)−Λ_i(v_n)|

≤2M||v−v_n||_q+ε

≤2M ε+ε for all i, j > N,

which shows that (Λ_j(v))_j is a Cauchy sequence for allv∈L^q(Ω). We denote the limit by Λ(v) = lim

n→∞Λ_n(v) for all v∈L^q(Ω),

5Consult for instance Theorem 18.1 in [6] for a proof.

which defines a bounded linear functional on L^q(Ω). Then by Riesz’ representation theo- rem there exists a unique u∈L^p(Ω) such that

Λ(v) =

Z

Ωvu dx for all v∈L^q(Ω), thus

n→∞lim

Z

Ωvu_ndx=

Z

Ωvu dx for all v∈L^q(Ω), and we conclude that u_n* uweakly in L^p(Ω).

Another concept of convergence in Banach spaces is weak-star convergence.

Definition 2.11. Let X and Y be Banach spaces such that X =Y⁰. We say that a sequence (x_n)⊂X convergesweak-star tox∈X, if for all y∈Y we have

n→∞lim x_n(y) =x(y), and we writex_n* x.^∗

By Riesz’ representation theorem we find that the notions of weak convergence and weak-star convergence coincide inL^p(Ω) when p∈(1,∞). Furthermore,u_n* u^∗ inL^∞(Ω) if and only if

n→∞lim

Z

Ωu_nv dx=

Z

Ωuv dx for all v∈L¹(Ω). (2.4) We have the following analogous results of Proposition 2.9 and Proposition 2.10.

Proposition 2.12. Let X andY be a Banach spaces such that X=Y⁰. Suppose that the sequence(x_n)⊂X converges weak-star tox∈X. Then the sequence is uniformly bounded:

sup

n ||x_n||_X ≤M <∞, and the norm is lower semicontinuous:

||x||_X ≤lim inf

n→∞ ||xn||_X.

Theorem 2.13 (Helly’s theorem). Let X and Y be Banach spaces. Suppose that X=Y⁰ and thatY is separable. Assume that the sequence (x_n)⊂X is uniformly bounded:

sup

n ||x_n||_X ≤M <∞.

Then (x_n) has a weak-star convergent subsequence.

We refer to Theorem 2.13 in [11] for a proof of Helly’s theorem. The proofs of Propo- sition 2.9 and Proposition 2.12 can be found in most functional analysis books, see e.g.

[15].

2.3 Sobolev spaces

Now we introduce Sobolev spaces and derive some important properties of these. We begin with some definitions. Recall that Ω is a domain in Rⁿ.

Definition 2.14. Consider a functionφ: Ω→Rwhich belongs toC^∞(Ω). We define the support of φ as

supp(φ) ={x∈Ω :φ(x)6= 0}.

If supp(φ) is bounded we define

C₀^∞(Ω) ={φ∈C^∞(Ω) : supp(φ)⊂Ω}. Forφ∈C₀^∞(Ω) we define φ(x) = 0 when x∈Rⁿ\Ω.

We make the following definition motivated by the integration by parts formula for continuously differentiable functions.

Definition 2.15. Let u∈L¹_loc(Ω). If there is a functionwj∈L¹_loc(Ω) such that

Z Ωu ∂φ

∂x_jdx=−

Z

Ωwjφ dx for all φ∈C₀^∞(Ω),

then we say that w_j is theweak partial derivative of u with respect tox_j in Ω. We write w_j=_∂x^∂u

j and ∇u=_∂x^∂u

1, . . . ,_∂x^∂u

n

, provided that the weak derivatives exist.

Definition 2.16. Let 1≤p≤ ∞. We say thatu∈W^1,p(Ω) ifuand all its weak derivatives

∂u

∂x_j, j= 1, . . . , n, belong to L^p(Ω).

Then

||u||_1,p,Ω=







||u||^p_p,Ω+||∇u||^p_p,Ω^1/p if p∈[1,∞)

||u||_∞,Ω+||∇u||_∞,Ω if p=∞ (2.5) defines a norm on W^1,p(Ω), where

||∇u||_p,Ω=

Z Ω

|∇u(x)|^pdx

1/p

, ||∇u||_∞,Ω= ess sup_x∈Ω|∇u(x)|.

The space W^1,p(Ω) possesses many properties similar to the space L^p(Ω), the most fundamental being that it is a Banach space with respect to the norm defined in (2.5).

Definition 2.17. We define the following spaces for 1≤p≤ ∞:

i) Let W₀^1,p(Ω) denote the closure of C₀^∞(Ω) in the space W^1,p(Ω), i.e. the closure of C₀^∞(Ω) with respect to the norm || · ||_1,p,Ω.

ii) We say that u∈W_loc^1,p(Ω) if u∈W^1,p(D) for each open set D⊂⊂Ω.

We mention that if u∈C(Ω)∩W^1,p(Ω) and u|_∂Ω= 0, then u∈W₀^1,p(Ω). In addition, if u, v∈W₀^1,p(Ω), then max{u, v},min{u, v} ∈W₀^1,p(Ω).⁶

6Consult for instance [9].

Remark 2.18. The notation

∇u_j*∇u weakly inL^p(Ω) means that

∂u_j

∂x_k * ∂u

∂x_k weakly in L^p(Ω) for each k= 1,2, . . . , n. If

u_j* u, ∇u_j* w weakly in L^p(Ω)

for somew= (w₁, w₂, . . . , w_n)∈Rⁿ, thenw=∇u. Indeed, by (2.3) the weak convergence means

j→∞lim

Z

Ωηu_jdx=

Z

Ωηu dx,

j→∞lim

Z

Ωψ∂u_j

∂x_kdx=

Z

Ωψw_kdx

for allη, ψ∈L^q(Ω) such that 1/p+ 1/q= 1. Furthermore, since∇u_j is the weak gradient of u_j we have

Z

Ωu_j ∂φ

∂x_kdx=−

Z

Ωφ∂u_j

∂x_kdx for all φ∈C₀^∞(Ω).

Letφ∈C₀^∞(Ω). Then by the above and since φ, ∂φ

∂x_k ∈L^q(Ω) we obtain

Z Ωu ∂φ

∂x_kdx= lim

j→∞

Z

Ωu_j ∂φ

∂x_kdx=− lim

j→∞

Z

Ωφ∂u_j

∂x_k dx=−

Z

Ωφw_kdx.

Thusw=∇u.

The following variant of Morrey’s inequality is useful.

Lemma 2.19 (Morrey’s inequality). Let Ωbe a bounded domain and suppose that p > n.

If u∈W₀^1,p(Ω), then

|u(x)−u(y)| ≤C_p|x−y|^1−n/p||∇u||_p,Ω for a.e. x, y∈Ω,

where C_p depends on pand n, and is such that C_p→2ⁿ⁺¹ as p→ ∞. One can redefine u in a set of measure zero and extend it to the boundary such thatu∈C(Ω) and u|_∂Ω= 0.

Proof. We first show the inequality for functions inC₀^∞(Ω). Letu∈C₀^∞(Ω) and setr >0.

Fixq, z∈Ω such that

|q−z|=r.

Letξ∈B(z, r), where B=B(z, r) is the open ball centered at z with radius r. We have u(ξ)−u(q) =

Z 1 0

d

dtu(q+t(ξ−q))dt=

Z 1

0 h∇u(q+t(ξ−q)), ξ−qidt,

where h·,·idenotes the inner product on Rⁿ. By writing u_B=−

Z

Bu(ξ)dξ, and integrating over B with respect to ξ we find

ωnrⁿ(u_B−u(q)) =

Z B(z,r)

Z 1

0 h∇u(q+t(ξ−q)), ξ−qidt dξ,

where ω_n denotes the volume of the unit ball in Rⁿ. By the Cauchy–Schwarz inequality we obtain

ω_nrⁿ|u_B−u(q)| ≤

Z B(z,r)

Z 1

0 |∇u(q+t(ξ−q))||ξ−q|dt dξ

=

Z 1 0

Z

B(z,r)|∇u(q+t(ξ−q))||ξ−q|dξ dt,

where we used Tonelli’s theorem to change the order of integration. By changing the variables to

η=q+t(ξ−q), dη=tⁿdξ,

we find that the new domain of integration is contained in the ball B(q,2rt). Then by H¨older’s inequality,

ω_nrⁿ|u_B−u(q)|

≤

Z 1 0

Z

B(z,r)|∇u(q+t(ξ−q))||ξ−q|dξ dt

≤

Z 1 0 t⁻¹⁻ⁿ

Z

B(q,2rt)∩Ω|∇u(η)||η−q|dη dt

≤

Z 1 0 t⁻¹⁻ⁿ

Z

B(q,2rt)∩Ω|∇u(η)|^pdη

1/p Z

B(q,2rt)∩Ω|η−q|^p/(p−1)dη

(p−1)/p

dt

≤ ||∇u||_p,Ω

Z 1 0 t⁻¹⁻ⁿ

Z

B(q,2rt)(2rt)^p/(p−1) dη

(p−1)/p

dt

=||∇u||_p,Ω

Z 1

0 t⁻¹⁻ⁿ2rt(ω_n(2rt)ⁿ)^(p−1)/pdt

=ω_n^(p−1)/p(2r)^1+n(p−1)/p||∇u||_p,Ω

Z 1

0 t^−n/pdt

=ω_n^(p−1)/p(2r)^1+n(p−1)/p||∇u||_p,Ω p p−n.

We evaluated the last integral by using the Monotone convergence theorem, where it was needed that p > n. Now we have

|u_B(z,r)−u(q)| ≤2^1+n(p−1)/pω_n^−1/p p

p−nr^1−n/p||∇u||_p,Ω, for z, q∈Ω such that |q−z|=r. Fix x, y∈Ω and let

z=1

2(x+y), r=|x−z|=|y−z|= 1

2|x−y|.

Then

|u(x)−u(y)| ≤ |u(x)−u_B(z,r)|+|u_B(z,r)−u(y)|

≤2·2^1+n(p−1)/pω^−1/p_n p p−n

1 2|x−y|

1−n/p

||∇u||_p,Ω

= 2ⁿ⁺¹ω_n^−1/p p

p−n|x−y|^1−n/p||∇u||_p,Ω,

(2.6)

which concludes the proof when u∈C₀^∞(Ω).

Now let u∈W₀^1,p(Ω). Then there is a sequence u_j ∈C₀^∞(Ω) such that u_j →u in W^1,p(Ω). We claim that there is a subsequence such that u_j →u a.e. in Ω. Indeed, for ε >0 we have

||u_j−u||^p_p,Ω=ε^p

Z Ω

uj−u ε

p

dx

≥ε^p

Z

{^|uj−u|≥ε}

uj−u ε

p

dx

≥ε^pµ({x∈Ω :|u_j(x)−u(x)| ≥ε}),

which shows that the sequence converges in measure to the functionu. Then it is known that there is a subsequence such that u_j→u a.e. in Ω. The strong convergence assures that

||u_j||_p,Ω≤ ||u||_p,Ω+ 1, ||∇u_j||_p,Ω≤ ||∇u||_p,Ω+ 1 for sufficiently largej, and by (2.6) we find

|uj(x)−uj(y)| ≤2ⁿ⁺¹ω_n^−1/p p

p−n|x−y|^1−n/p(||∇u||_p,Ω+ 1).

Hence (u_j) is equibounded and equicontinuous for large j. Then by Ascoli’s theorem 2.2 there is a further subsequence and a continuous function v∈C(Ω) such that u_j →v uniformly in Ω. Thus,v is a continuous version ofu, and we redefineuto bev in Ω. Then

|u(x)−u(y)| ≤ |u(x)−u_j(x)|+|u_j(x)−u_j(y)|+|u_j(y)−u(y)|

≤ |u(x)−u_j(x)|+|u_j(y)−u(y)|

+ 2ⁿ⁺¹ω^−1/p_n p

p−n|x−y|^1−n/p(||∇uj− ∇u||_p,Ω+||∇u||_p,Ω).

Lettingj → ∞we obtain

|u(x)−u(y)| ≤2ⁿ⁺¹ω^−1/p_n p

p−n|x−y|^1−n/p||∇u||_p,Ω

=C_p|x−y|^1−n/p||∇u||_p,Ω whereC_p is such that

C_p= 2ⁿ⁺¹ω_n^−1/p p

p−n →2ⁿ⁺¹ as p→ ∞.

By redefining u in a set of measure zero we can extend it to the boundary such that u∈C(Ω) and u|_∂Ω= 0.

Remark 2.20. We shall mostly encounter the situation when the domain Ω is bounded and p > n. Then since every function in W₀^1,p(Ω) has a continuous version, we always assume when given such a function that it is its continuous version.

Morrey’s inequality suggests a connection between functions in W^1,p(Ω) and H¨older continuous functions. For more details see the Rellich–Kondrachov compactness theorem 2.28 later in this section.

We note a convenient inequality.

Lemma 2.21 (Friedrichs’ inequality). Let Ω be a bounded domain. Suppose that u∈ W₀^1,p(Ω), where 1≤p≤ ∞. Then

||u||_p≤diam(Ω)||∇u||_p.

Proof. It suffices to show the inequality for u∈C₀^∞(Ω). We begin with the case when 1≤p <∞. Let u∈C₀^∞(Ω). Since Ω is bounded there are numbers η_i< ξ_i, i= 1,2, . . . , n, such that Ω⊂⊂Q, where

Q={x= (x₁, x₂, . . . , x_n)∈Rⁿ:η_i< x_i< ξ_i for each 1≤i≤n}. Then u∈C₀^∞(Q). We have

u(x₁, x2, . . . , xn) =u(η₁, x2, . . . , xn) +

Z x₁ η1

ut(t, x₂, . . . , xn)dt

=

Z x1

η1

u_t(t, x₂, . . . , x_n)dt, thus by Proposition 2.4,

|u(x)| ≤

Z x1

η₁ |u_t(t, x₂, . . . , x_n)|dt

≤

Z ξ₁ η1

|ux₁(x1, x2, . . . , xn)|dx1

≤ −

Z ξ1

η1

|∇u(x)||ξ₁−η₁|dx₁

≤

−

Z ξ1

η1

|∇u(x)|^p|ξ₁−η₁|^pdx₁

1/p

. This implies that

|u(x)|^p≤ |ξ₁−η₁|^p−1

Z ξ1

η1

|∇u(x)|^pdx₁.

Observe that the right-hand side only depends on (x₂, x₃, . . . , x_n), while the left-hand side depends on (x₁, x₂, . . . , x_n). Integrating with respect tox₁ we find

Z ξ₁ η1

|u(x)|^pdx1≤ |ξ1−η1|^p

Z ξ₁ η1

|∇u(x)|^pdx1. Now integrate over the other variables to obtain

Z

Q|u(x)|^pdx≤ |ξ₁−η₁|^p

Z

Q|∇u(x)|^pdx,

hence _Z

Ω|u(x)|^pdx≤diam(Ω)^p

Z

Ω|∇u(x)|^pdx, where diam(Ω) := sup_x,y∈Ω|x−y|.

By the continuity of the norm in Proposition 2.5, we find that the inequality is also valid for p=∞.

Now we clarify the relationship between the weak derivatives and the derivatives from calculus.

Definition 2.22. We say that u: Ω→R is differentiable at x∈Ω if there exists η∈Rⁿ such that

u(y) =u(x) +hη, y−xi+o(|y−x|) asy→x.

Ifη exists it is unique, and we denote it by ∇u(x).

So far we have denoted the weak derivatives with the same notation as the usual derivatives from calculus. Let us verify that these actually coincide when n < p≤ ∞, so that the notation is consistent.

Theorem 2.23. Let n < p≤ ∞ and suppose that u∈W_loc^1,p(Ω). Then u is differentiable a.e. in Ω and its weak gradient equals its gradient a.e.

Proof. First we consider the case whenn < p <∞. Let ∇denote the weak gradient. We need the following version of Lebegue’s differentiation theorem, see [7] for more details.

For a.e. x∈Ω we have

−

Z

B(x,r)|∇u(z)− ∇u(x)|^pdz→0 asr→0.

Fix any such x and define

v(y) =u(y)−u(x)− h∇u(x), y−xi, y∈Ω.

By consulting the proof of Morrey’s inequality 2.19 we find that the inequality is applicable to the functionv in the ball B(x, r)⊂⊂Ω. With r=|x−y| we find

|u(y)−u(x)− h∇u(x), y−xi|

=|v(y)−v(x)|

≤C_pr^1−n/p

Z

B(x,r)|∇v(z)|^pdz

1/p

=Cpr^1−n/p

ωnrⁿ−

Z

B(x,r)|∇u(z)− ∇u(x)|^pdz

1/p

=Cpω_n^1/pr

−

Z

B(x,r)|∇u(z)− ∇u(x)|^pdz

1/p

=o(r) =o(|x−y|),

whereω_n denotes the volume of the unit ball in Rⁿ. If p=∞, we have that W_loc^1,∞(Ω)⊂ W_loc^1,p(Ω) for any n < p <∞, so we can apply the argument above.

We now turn our attention to Lipschitz continuity.

Definition 2.24. A function u: Ω→R is said to be Lipschitz continuous if

|u(x)−u(y)| ≤L|x−y| when x, y∈Ω, for some constant L.

The following result provides an interesting characterization of the space W_loc^1,∞(Ω).

Theorem 2.25. A function u: Ω→R is locally Lipschitz continuous if and only if u∈ W_loc^1,∞(Ω).

Proof. Assume that u∈W_loc^1,∞(Ω). Thenu also belongs to W_loc^1,p(Ω) for some finite p > n.

Letε >0 and define the function

u_ε=ρ_ε∗u,

where ρ_ε is Friedrichs’ mollifier. Then u_ε ∈C^∞(Ω) and u_ε →u in W_loc^1,p(Ω) as ε→0.

Actually, the convergence is locally uniform⁷. Let B be a subdomain such that B⊂⊂Ω.

Then we have

||∇u_ε||_p,B≤ ||∇u||_p,B, which implies that

sup

0<ε<δ

||∇u_ε||_∞,B ≤C <∞,

for sufficiently small δ >0, where C is a constant that is independent of ε. For x, y∈B we have

u_ε(x)−u_ε(y) =

Z ₁ 0

d

dtu_ε(y+t(x−y))dt=

Z ₁

0 h∇u_ε(y+t(x−y)), x−yidt, which leads to

|uε(x)−uε(y)| ≤C|x−y|, by the Cauchy–Schwarz inequality and the above. Observe that

|u(x)−u(y)| ≤ |u(x)−uε(x)|+C|x−y|+|uε(y)−u(y)|.

Letting ε→0 we obtain

|u(x)−u(y)| ≤C|x−y| x, y∈B,

by the uniform convergence. Thus, u is locally Lipschitz continuous.

Now suppose that u is locally Lipschitz continuous. Once again we let B⊂⊂Ω. We have that u∈L^∞(B), so we only have to show that the weak first partial derivatives are bounded. For i= 1,2, . . . , n write

D^h_iu(x) = u(x+he_i)−u(x)

h ,

D_i^−hu(x) = u(x)−u(x−he_i)

h ,

7See Theorem 4.40 and Theorem 5.3 in [11].

whereh >0 and ei is the ith unit vector. Since u is locally Lipschitz continuous, sup

0<h<ε

||D_i^−hu||_∞,B≤M <∞,

for sufficiently small ε >0. Since L¹(B) is separable, Helly’s theorem 2.13 implies that there is a subsequenceh_k→0 and a function w_i∈L^∞(B) such that

D^−h_i ^ku* w^∗ i weak-star in L^∞(B) for every i= 1,2, . . . , n. By (2.4) this is equivalent to

k→∞lim

Z

BψD_i^−hku dx=

Z

Bψwidx for all ψ∈L¹(B).

We can approximate functions in L¹(B) by functions in C₀^∞(B) since C₀(B) is dense in L¹(B).⁸ Let φ∈C₀^∞(B). By the Dominated convergence theorem and the above we find

Z

Buφx_idx= lim

k→∞

Z

BuD^h_i^kφ dx=− lim

k→∞

Z

BφD^−h_i ^ku dx=−

Z

Bφwidx,

which holds for anyφ∈C₀^∞(B). This shows thatu_xi=w_iin the sense of weak derivatives, fori= 1,2, . . . , n. Furthermore, by the weak-star lower semicontinuity in Proposition 2.12,

||ux_i||_∞,B =||wi||_∞,B ≤lim inf

k→∞ ||D_i^−hku||_∞,B <∞,

which implies that||∇u||_∞,B <∞ for any B⊂⊂Ω, and we conclude that u∈W_loc^1,∞(Ω).

We immediately obtain the following important result.

Theorem 2.26 (Rademacher’s theorem). If u: Ω→R is locally Lipschitz continuous, then u is differentiable a.e. in Ω.

Proof. Since u is locally Lipschitz continuous, u∈W_loc^1,∞(Ω) by Theorem 2.25. Then it follows from Theorem 2.23 thatu is differentiable a.e. in Ω.

We introduce some notation, and seize the opportunity to mention a Sobolev embed- ding result. We refer to [9] for more details.

Definition 2.27. LetX and Y be Banach spaces. We say thatX iscompactly embedded inY, and write

X⊂⊂Y if

i) there exists a linear, continuous, and injective map Ψ :X→Y;

ii) the map Ψ(B) is precompact in Y for any bounded set B ⊂X, that is Ψ(B) is compact in Y.

8See for instance Lemma 4.38 in [11].

Theorem 2.28 (Rellich–Kondrachov compactness theorem). Suppose that p > n and let Ω be a bounded domain in Rⁿ with a smooth boundary ∂Ω∈C¹. Then

W^1,p(Ω)⊂⊂C^γ(Ω), where γ= 1−ⁿ_p.

Here C^γ(Ω) refers to the H¨older space:

C^γ(Ω) ={u∈C(Ω) :||u||_Cγ(Ω)<∞}, consisting of H¨older continuous functions u: Ω→R:

|u(x)−u(y)| ≤C|x−y|^γ,

where C is a constant and 0< γ ≤1. The H¨older space is in fact a Banach space with respect to the norm

||u||_Cγ(Ω)=||u||_∞,Ω+ sup

x,y∈Ω x6=y

|u(x)−u(y)|

|x−y|^γ . (2.7)

Notice that in the above theorem, γ= 1 whenp=∞, so in that case the H¨older space consists of Lipschitz continuous functions, which agrees with what we found in Theorem 2.25.

The proof of Morrey’s inequality 2.19 is based on Theorem 8.1 in [6], and the proof of Theorem 2.23 follows Theorem 5 in section 5.8.3 of [7]. The proof of Theorem 2.25 is based on Theorem 5 in section 4.2.3 of [8].

2.4 Quadratic forms and convex functions

We denote the space of all n×n real-valued symmetric matrices by Sⁿ. For B, C∈Sⁿ the notation B≥C means

hBξ, ξi ≥ hCξ, ξi for all ξ∈Rⁿ.

In the following section we assume thatA∈Sⁿ is such that for constants 0< α≤β <∞, α|η|²≤ hAη, ηi ≤β|η|²,

for all η∈Rⁿ. Thus, the matrixA is positive definite:

hAη, ηi>0 for all nonzero η∈Rⁿ. We begin with an inequality.

Proposition 2.29. If 2≤p <∞, then

DhAξ, ξi^p−22 Aξ− hAψ, ψi^p−22 Aψ, ξ−ψ^E≥4

√ α 2

p

|ξ−ψ|^p (2.8) for all ξ, ψ∈Rⁿ.