Leray–Schauder Existence Theory for Quasilinear Elliptic

Equations

Sam Forster, Eavan Gleeson, Franca Hoffmann

March 22nd, 2014

Contents

1 Introduction 21.1 Notation and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 The Minimal Surface Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Leray–Schauder Existence Theory 42.1 Topological Fixed Point Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Leray–Schauder Fixed Point Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Leray–Schauder Existence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 A Quasilinear Maximum Principle 10

4 Gradient Estimates 12

5 Boundary Gradient Estimates 125.1 Minimal Surface Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.2 General Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

6 De Giorgi–Nash–Moser Theory 186.1 De Georgi’s Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.2 Moser’s Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.3 Global Holder Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

7 Application to the Minimal Surface Equation 34

1

1 Introduction

This project is concerned with the question of existence of classical solutions to the Dirichlet problemQu = 0 in Ωu = ϕ on ∂Ω

(1.1)

where Q is a second order quasilinear elliptic operator, ϕ is a sufficiently regular function on ∂Ω andΩ ⊂ Rn is a bounded domain.

Thus Q is an operator of the form

Qu = aij(x, u,Du)Diju+ b(x, u,Du), (1.2)

and we place certain regularity and ellipticity conditions on the coefficients aij , b. While second orderlinear elliptic equations are very well understood, their theory cannot be applied due to the nonlinearitiesin (1.2), so a new theory must be developed to cope with these nonlinearities. This new theory waspioneered by Leray and Schauder in the 1930s: at its heart is the Leray–Schauder fixed point theoremwhich allows us to establish existence of solutions to PDEs from apriori estimates. However, this is notto say that nothing can be salvaged from the linear theory. Indeed, as we shall see later, we can uselinear results such as the maximum principle to establish analogous results for the quasilinear case. Alsoapplicable is the theory of De Giorgi, Nash and Moser. These results are indispensable for establishingthe apriori bounds such as those needed to apply the Leray–Schauder theory.

The essence of the Leray–Schauder existence theorem is as follows: we embed the Dirichlet problem(1.1) into a family of related problems of the same type, depending on a parameter σ ∈ [0, 1], say

Qσu = 0 in Ωu = σϕ on ∂Ω

. (1.3)

The theorem asserts that if for some β ∈ (0, 1) there is a constant M > 0 such that, for every σ everysolution u of (1.3) satisfies the bound

‖u‖C1,β(Ω) ≤M,

then (1.1) has a solution.Thus the problem has been reduced to estimating Holder norms of solutions of second order quaslilin-

ear elliptic equations, assuming such solutions exist. So we consider the Holder norm

‖u‖C1,β(Ω) = supΩ|u|+ sup

Ωsup|γ|=1

|Dγu|+ [Du]β,Ω

≤ supΩ|u|+ sup

Ω|Du|+ [Du]β,Ω. (1.4)

Thus, in order to estimate the Holder norm, we shall estimate the three terms on the right hand side of(1.4), and we follow the following general strategy.

1. We estimate supΩ |u| in terms of the boundary data ϕ. For this, we need a generalisation of themaximum principle for quasilinear operators. This is discussed in Section 3.

2. Estimate supΩ |Du| in terms of sup∂Ω |Du|. While this can be done for general operators Q, we shallmake an additional assumption on Q when deriving this estimate, namely that Q is of divergenceform. The reasons for this additional assumption are for brevity and clarity: to gain an intuitiveunderstanding without getting lost in technical details. This step is covered in Section 4

3. From the previous step, it is apparent that we need to derive a boundary gradient estimate. Weestimate sup∂Ω |Du| in terms of supΩ |u|, and from step 1 we know that the latter is bounded.This estimate uses a barrier construction, and will rely heavily on geometric properties imposedon the boundary ∂Ω. Section 5 is devoted to a discussion of these geometric conditions and to thederivation of the estimate.

4. Finally, in Section 6, we use De Giorgi–Nash–Moser theory to bound the Holder coefficient [Du]β,Ω.We will introduce the general theory in two different ways, due to De Giorgi and Moser respectively.The De Giorgi–Nash–Moser theory can be understood as a technique rather than a collection oftheorems, and yields a toolbox of powerful results for the regularity study of elliptic PDEs such asMoser’s Harnack Inequality.

2

1.1 Notation and Preliminaries

Before delving into the Leray–Schauder theory and the associated collection of estimates, we need togive some definitions and establish our notation and conventions. Recall that we are interested in secondorder quasilinear operators, namely operators of the form

Qu = aij(x, u,Du)Diju+ b(x, u,Du),

where x = (x1, . . . , xn) ∈ Ω for some domain Ω in Rn, n ≥ 2. The function u is assumed to satisfyu ∈ C2(Ω), and consequently we shall assume for convenience that [aij ]ni,j=1 is symmetric, namely

aij = aji for all i, j ∈ 1, . . . , n. We assume that the coefficients aij(x, z, p), b(x, z, p) of Q are definedfor (x, z, p) ∈ Ω×R×Rn, and will always denote by λ(x, z, p) and Λ(x, z, p) the minimum and maximumeigenvalues of the coefficient matrix [aij(x, z, p)]i,j respectively.

Definition 1.1 (Ellipticity). Let Q be the operator defined by (1.2).

• Let U ⊆ Ω × R × Rn. We say that Q is elliptic in U if the coefficient matrix [aij(x, z, p)]i,j ispositive definite for every (x, z, p) ∈ U , namely

0 < λ(x, z, p)|ξ|2 ≤ aij(x, z, p)ξiξj ≤ Λ(x, z, p)|ξ|2

for every ξ ∈ Rn\0 and every (x, z, p) ∈ U .

• If Q is elliptic on the whole set Ω× R× Rn, we say that Q is elliptic in Ω.

• If u ∈ C1(Ω) and the matrix [aij(x, u(x), Du(x))]i,j is positive definite, we say Q is elliptic withrespect to u.

Definition 1.2 (Divergence form). We say that the operator Q is of divergence form if there is adifferentiable vector function A(x, z, p) = (A1(x, z, p), . . . , An(x, z, p)) and a scalar function B(x, z, p)such that

Qu = div A(x, u,Du) +B(x, u,Du), u ∈ C2(Ω).

In this case, using symmetry of [aij ], we find that

aij(x, z, p) =1

2

(DpiA

j(x, z, p) +DpjAi(x, z, p)

). (1.5)

Unlike the linear case, a quasililinear operator with smooth coefficients is not always expressible indivergence form.

Definition 1.3 (Variational operator). We say that the operator Q is variational if it is the Euler–Lagrange operator corresponding to the integral∫

Ω

F (x, u,Du) dx

where F is a differentiable scalar function, namely Q is of divergence form with

Ai(x, z, p) = DpiF (x, z, p), B(x, z, p) = −DxF (x, z, p).

Throughout this project, for the sake of concreteness and intuition, we will place a lot of emphasis onone particular example of a second order quasilinear elliptic equation, the minimal surface equation. Theoperator Q corresponding to this equation is in fact a variational operator. This equation is introducedin the next subsection.

1.2 The Minimal Surface Equation

The minimal surface equation arises when one tries to minimise the n-dimensional ”area“ under thegraph of twice continuously differentiable function u : Ω ⊂ Rn → R. Any such function must minimisethe so-called area functional

A(u) =

∫Ω

√1 + |Du(x)|2dx, (1.6)

3

where we suppose that u|∂Ω = ϕ is continuous. Let u be such a minimiser, then we consider variationsof the form f(t) = A(u + tv), where t is a real number and v ∈ C2(Ω) vanishes on the boundary of Ω.Then f must have a critical point at zero. Consequently

f ′(0) =

∫Ω

〈Du,Dv〉√1 + |Du|2

= −∫

Ω

n∑i=1

Di

(Diu√

1 + |Du|2

)v = 0,

where we have used integration by parts and the fact that v vanishes on ∂Ω to obtain the last equality.Since v was assumed to be continuous we may conclude that any critical point u of A must satisfy

Di

(Diu√

1 + |Du|2

)=

(1 + |Du|2)Diiu−DiuDjuDiju

(1 + |Du|2)3/2= 0,

where we have used the summation convention on repeated indices. Therefore any minimiser of the areafunctional will also solve the second-order quasilinear equation

Mu = (1 + |Du|2)∆u−DiuDjuDiju = aijDiju = 0,

where aij(x, z, p) = (1 + |p|2)δij − pipj . Let us show that this equation is elliptic. Indeed,

aij(x,Du)ξiξj = ((1 + |Du|2)δij −DiuDju)ξiξj = (1 + |Du|2)|ξ|2 − 〈Du, ξ〉2 (1.7)

Recall the Cauchy-Schwarz ineqaulity 〈v, w〉2 ≤ |v|2|w|2. Then

aij(x,Du) ≥ (1 + |Du|2)|ξ|2 − |Du|2|ξ|2 = |ξ|2.

It is also clear that

aij(x,Du) = (1 + |Du|2)|ξ|2 − 〈Du, ξ〉2 ≤ (1 + |Du|2)|ξ|2.

This proves ellipticity with λ = 1 and Λ = (1 + |Du|2) and it can be shown that λ,Λ are the smallestand largest eigenvalues of [aij(x,Du)] respectively. Note, however, that M need not be uniformly ellipticoperator in general. Indeed, Λ(x,Du)/λ(x,Du) = (1 + |Du(x)|2) may be unbounded. We shall refercome back to this example in Section 3, 5 and finally 7 to illustrate the theory we will introduce.

2 Leray–Schauder Existence Theory

2.1 Topological Fixed Point Theorems

The Brouwer fixed point theorem lies at the heart of the Leray–Schauder fixed point theorem, and hencethe Leray–Schauder existence theory. We recall the theorem below (and refer the reader to [2] for itsproof), and use it to prove a more general fixed point theorem for Banach spaces.

Theorem 2.1 (Brouwer’s fixed point theorem). Let T : B → B be a continuous map of the closed unitball B of Rn into itself. Then T has a fixed point.

Theorem 2.2. Let K be a compact convex set in a Banach space B and let T : K → K be continuous.Then T has a fixed point.

Proof. Let k ∈ N. Then, since K is compact, there exist x1, . . . , xN(k) ∈ K such that Bi = B1/k(xi) :1 ≤ i ≤ N(k) covers K, where Br(x) denotes the open ball in B of radius r around x. Define Kk to bethe convex hull of x1, . . . , xN(k). Then by convexity of K, Kk ⊆ K. We define a map Jk : K → Kk by

Jkx =

∑N(k)i=1 d(x,K\Bi)xi∑N(k)i=1 d(x,K\Bi)

for each x ∈ K. Indeed, Jk maps into Kk as

aj = aj(x) :=d(x,K\Bj)∑N(k)i=1 d(x,K\Bi)

4

satisfy 0 ≤ aj ≤ 1 for each j = 1, . . . , N(k), and∑N(k)j=1 aj = 1. Note that Jk is continuous (when k is

large enough that the denominator is always non-zero), and for any x ∈ K,

‖Jkx− x‖ =

∥∥∥∥∥∥∑N(k)i=1 d(x,K\Bi)xi∑N(k)i=1 d(x,K\Bi)

−N(k)∑i=1

aix

∥∥∥∥∥∥=

∥∥∥∥∥∑N(k)i=1 d(x,K\Bi)(xi − x)∑N(k)

i=1 d(x,K\Bi)

∥∥∥∥∥≤

N(k)∑i=1

ai(x)‖xi − x‖

<1

k, (2.1)

since ‖xi − x‖ < 1/k if ai(x) is nonzero.Now set Sk = Jk T |Kk . Then Sk : Kk → Kk is a composition of continuous maps, so is istself

continuous. Furthermore, Kk is homeomorphic to a closed ball in some Euclidean space, and hence bythe Brouwer fixed point theorem, Sk has a fixed point, xk ∈ Kk say. Now xk ∈ K for each k, so bycompactness of K, xk has a convergent subsequence (which we still denote by xk) with limit x0 ∈ K.

We claim that x0 is a fixed point of T . Indeed, applying (2.1) with x = Txk, we have

‖xk − Txk‖ = ‖Jk Txk − Txk‖ <1

k.

Sending k →∞, we deduce that Tx0 = x0 by continuity of T .

Corollary 2.3. Let K be a closed convex set in a Banach space B and T : K → K a continuous mapsuch that TK is precompact. Then T has a fixed point.

Proof. We will find a compact convex subset A ⊆ K such that T maps A into itself. Then, the previoustheorem implies that T has a fixed point in A, and hence K.

Indeed, let A be the convex hull of TK. Certainly A is convex, and since the convex hull of a compactset is itself compact, A is compact. Moreover, A ⊆ K because TK ⊆ K and K is closed, so TK ⊆ K,but K is convex by assumption so A ⊆ K. Thus

T |A : A→ TA ⊆ TK ⊆ TK ⊆ A,

so T maps A into itself and we’re done.

Before proving the Leray–Schauder fixed point theorem, we need the following lemma:

Lemma 2.4. Let B be a Banach space with open unit ball B. Suppose T : B → B is a continuous mapsuch that

1. TB is precompact, and

2. T∂B ⊆ B.

Then T has a fixed point.

Proof. Define a map T ∗ : B → B by

T ∗x =

Tx, if ‖Tx‖ ≤ 1;Tx

‖Tx‖, if ‖Tx‖ ≥ 1.

It is clear that T ∗ is continouus, and that T ∗ maps B into itself. Moreover, TB precompact =⇒ T ∗Bprecompact. Indeed,

T ∗B = Tx ∈ B : ‖Tx‖ ≤ 1 ∪ Tx

‖Tx‖: x ∈ B, ‖Tx‖ ≥ 1

= A1 ∪A2,

5

with A1 and A2 defined in the obvious way. Now T ∗B ⊆ A1 ∪ A2, and the former is closed, so to

show compactness of T ∗B, it’s enough to show that A1 ∪A2 is compact (because a closed subspace of acompact Hausdorff space is compact). As a finite union of compact sets is compact, we need only showthat A1 and A2 are compact.

• A1 is closed, and A1 ⊆ TB and the latter is compact. So A1 is compact.

• To show that A2 is compact, let yi be a sequence in A2. Two possible cases arise: either infinitelymany yi ∈ A2 or else there are only finitely many yi ∈ A2.

Should the former case arise, then we can consider a subsequence, which we still denote yi, suchthat each yi ∈ A2. Then for each i, there exists xi ∈ B such that ‖Txi‖ ≥ 1, and yi = Txi

‖Txi‖ . So

Txi ∈ TB and TB is precompact, so there is a subsequence Txik which converges to some z ∈ TB,and moreover ‖z‖ ≥ 1. So yik → z

‖z‖ , and this limit is in A2, since A2 is closed.

On the other hand, if only finitely many yi ∈ A2, then after deleting these terms, we may assumeyi ⊆ ∂A2. Now

∂A2 ⊆ Tx

‖Tx‖: x ∈ B, ‖Tx‖ = 1 ∪ Tx

‖Tx‖: x ∈ ∂B, ‖Tx‖ ≥ 1.

But by assumption T∂B ⊆ B, so that the rightmost set above is empty. So yi ⊆ Tx‖Tx‖ : x ∈

B, ‖Tx‖ = 1 ⊆ A1 ⊆ A1, which is compact by the above. So yi has a convergent subsequence,with limit y ∈ A1 say. But yi ∈ ∂A2 for each i and ∂A2 is closed, so y ∈ ∂A2 ⊆ A2.

So in either case, yi has a subsequence convergent in A2, so A2 is compact, as desired.

So we conclude that T ∗B is precompact, so by Corollary 2.3, T ∗ has a fixed point, x say. NowT∂B ⊆ B =⇒ x 6∈ ∂B =⇒ x ∈ B. Therefore, ‖T ∗x‖ = ‖x‖ < 1, so by definition of T ∗, we must have‖Tx‖ < 1, and hence Tx = T ∗x = x, so that x is a fixed point for T .

2.2 Leray–Schauder Fixed Point Theorem

In this section, we state and prove the Leray–Schauder fixed point theorem, using the fixed point theoremsintroduced in Section 2.1. But first, we require the following definition:

Definition 2.5 (Compact mapping). We say that a map between two Banach spaces is compact if it iscontinuous and it maps bounded sets to precompact sets.

Theorem 2.6 (Leray–Schauder fixed point theorem). Let B be a Banach space and T : B × [0, 1] → Ba compact map such that

• T (x, 0) = 0 for each x ∈ B:

• there exists a constant M > 0 such that for each pair (x, σ) ∈ B× [0, 1] which satisfies x = T (x, σ),we have

‖x‖ < M. (2.2)

Then x is a fixed point of the map T1 : B → B given by T1y = T (y, 1), y ∈ B.

Proof. Without loss of generality, we may assume M = 1. Otherwise just rescale the norm on B by afactor of 1/M . For 0 < ε < 1, define T ∗ε : B → B by

T ∗ε x =

T

(x

‖x‖,

1− ‖x‖ε

), if 1− ε ≤ ‖x‖ ≤ 1;

T

(x

1− ε, 1

), if ‖x‖ < 1− ε,

where B denotes the open unit ball around 0 in B as before. Certainly T ∗ε is continuous, and bycompactness of T , similarly to the proof of the previous lemma, T ∗εB is precompact. Moreover, since

6

‖x‖ = 1 for x ∈ ∂B, we have T ∗ε x = T ( x‖x‖ , 0) = 0 by hypothesis, so T ∗ε ∂B = 0 ⊂ B. So we may apply

the previous lemma to conclude that T ∗ε has a fixed point which we denote x(ε).Now take ε = 1

k for k = 2, 3, . . .. So T ∗1/k has a fixed point x( 1k ). For convenience, we write xk = x( 1

k )and also denote

σk :=

k(1− ‖xk‖), if 1− 1

k ≤ ‖xk‖ ≤ 1;1, if ‖xk‖ < 1− 1

k .

Set A = (xk, σk) : k ≥ 2. By compactness of T , we may assume there is a subsequence of A, which westill denote (xk, σk), which converges to some (x, σ) ∈ B × [0, 1].

Suppose σ < 1. Then for large enough k, σk < 1 so that ‖xk‖ ≥ 1− 1k . (In fact, the inequality must

be strict since otherwise σk = 1.) So ‖xk‖ → 1, and so ‖x‖ = 1. But ‖xk‖ ≥1

k=⇒ xk = T ∗1/kxk =

T (xk‖xk‖

, σk)→ T (x, σ) by continuity of T . So x = T (x, σ) and ‖x‖ = 1, which contradicts (2.2). Hence

σ = 1. Now, by continuity of T , we have xk = T ∗1/kxk → T (x, 1). But xk → x, so x is a fixed point ofT1, as required.

2.3 Leray–Schauder Existence Theorem

Recall that for ODEs, existence of solutions is proved by relating the ODE to a certain operator mappinga Banach space of continuous functions into itself. Solutions correspond to fixed points of this operator,which can be shown to be a contraction mapping. We then appeal to Banach’s fixed point theorem todeduce that this operator has a unique fixed point, and hence that the ODE has a unique solution. Thesame idea underpins Leray–Schauder existence theory: it relates a quasilinear elliptic PDE to an operatorwhose fixed points correspond to solutions of the PDE, and uses the fixed point theorem of the previoussection to prove existence of fixed points, and hence of solutions, of the PDE. Note, however, one keydifference to the ODE theory: the Leray–Schauder fixed point theorem does not guarantee uniquenessof the fixed point, and consequently does not imply uniqueness of solutions.

Throughout this section, Ω will denote a bounded set in Rn with boundary ∂Ω ∈ C2,α and ϕ ∈C2,α(Ω) is a given function. We define the operator Q on C2(Ω) by

Qu = aij(x, u,Du)Diju+ b(x, u,Du), (2.3)

where aij , b ∈ Cα(Ω × R × Rn) for some α ∈ (0, 1). We assume Q is elliptic in Ω, and as before denoteby λ(x, z, p), Λ(x, z, p) the minimum and maximum eigenvalues of [aij(x, z, p)]ni,j=1 respectively.

To solve the system Qu = 0 in Ωu = ϕ on ∂Ω

(2.4)

we embed it in a family of problems Qσu = 0 in Ωu = σϕ on ∂Ω

, (2.5)

where 0 ≤ σ ≤ 1 andQσu = aij(x, u,Du;σ)Diju+ b(x, u,Du;σ). (2.6)

We will impose the following assumptions:

1. Q1 = Q;

2. b(x, z, p; 0) = 0 for each (x, z, p) ∈ Ω× R× Rn;

3. Qσ is elliptic in Ω for each σ ∈ [0, 1];

4. aij(·;σ), b(·;σ) ∈ Cα(Ω× R× Rn) for each σ ∈ [0, 1], and the maps

aij(x, z, p; ·), b(x, z, p; ·) : [0, 1]→ Cα(Ω× R× Rn)

are continuous.

7

With Theorem 2.6 in mind, we want a formulation of solvability of (2.4) in terms of existence of afixed point of a compact map of B × [0, 1] into B for some Banach space B. The introduction of thefamily of problems (2.5) allows us to construct exactly this. Indeed, let β ∈ (0, 1) and choose B to bethe Banach space C1,β(Ω). Define an operator

T : C1,β(Ω)× [0, 1] → C2,αβ(Ω) ⊂ C1,β(Ω)

(v, σ) 7→ u, (2.7)

where u = T (v, σ) is the unique solution of the linear elliptic Dirichlet problemaij(x, v,Dv;σ)Diju+ b(x, v,Dv;σ) = 0 in Ω

u = σϕ on ∂Ω

. (2.8)

Note that existence of a unique C2,αβ(Ω) solution is guaranteed by the linear theory. Indeed, v ∈C1,β(Ω) =⇒ Dv ∈ Cβ(Ω) so that the coefficients

aij(x) := aij(x, v(x)Dv(x);σ),

b(x) := b(x, v(x), Dv(x);σ)

satisfy aij , b ∈ Cαβ(Ω), and since αβ < α, we have ∂Ω ∈ C2,αβ and ϕ ∈ C2,αβ(Ω). So applying thefollowing theorem, proved in [2] for instance, with γ = αβ, Aij = aij , Bi = 0, C = 0t, f = b and φ = σϕ,we see that (2.8) has a unique C2,αβ(Ω) solution.

Theorem. Let γ ∈ (0, 1) and let Ω ⊂ Rn be a bounded domain with boundary ∂Ω ∈ C2,γ . Let φ ∈ C2,γ(Ω)and define an operator L by

Lu = AijDiju+BiDiu+ Cu,

where Aij , Bi, C ∈ Cγ(Ω). Suppose f ∈ Cγ(Ω). Then the Dirichlet problemLu = f in Ωu = φ on ∂Ω

has a unique solution in C2,γ(Ω).

Remark 2.7. Notice that our operator defined in (2.8) is indeed strictly elliptic because Ω is compactand λ(x, v(x), Dv(x);σ) is continuous and positive on Ω, so attains a (positive) minimum value on Ω.

So the operator T defined in (2.7) is well-defined. From condition (1) listed above, solvability of (2.4)is equivalent to existence of a fixed point u ∈ C1,β(Ω) for T1, where T1 : C1,β(Ω)→ C1,β(Ω) is given byT1v = T (v, 1). We are now ready to prove the Leray–Schauder existence theorem.

Theorem 2.8 (Leray–Schauder Existence Theorem). Let 0 < α < 1. Suppose

• Ω ⊂ Rn is a bounded domain with ∂Ω ∈ C2,α;

• ϕ ∈ C2.α(Ω).

Let Qσ : σ ∈ [0, 1] be the family of operators defined by (2.6), satisfying conditions (1)–(4) above.Suppose for some β ∈ (0, 1), there exists a constant M > 0 such that for every σ ∈ [0, 1], every C2,α(Ω)solution u of

Qσu = 0 in Ωu = σϕ on ∂Ω

satisfies ‖u‖C1.β(Ω) < M . Then the Dirichlet problem

Qu = 0 in Ωu = ϕ on ∂Ω

has a solution in C2,α(Ω).

8

Proof. In view of the comments preceeding the theorem, it’s enough to show that the operator T definedin (2.7) satisfies the hypotheses of Theorem 2.6. It then follows that T1 has a fixed point u ∈ C1,β(Ω)and this is a C2,α(Ω) solution of the Dirichlet problem (2.4).

So we have reduced the proof to checking properties of T . Since the bound in Theorem 2.6 is assumedto hold in our hypothesis, we need only check

1. T (v, 0) = 0 for each v ∈ C1,β(Ω);

2. T is compact.

The first propery is easy to see. Indeed, let v ∈ C1,β(Ω). Condition 2 above ensures b(x, v,Dv; 0) = 0.So by definition, u = T (v, 0) is the unique solution of

aij(x, v,Dv; 0)Diju = 0 in Ωu = 0 on ∂Ω

.

But u = 0 is certainly a solution of this problem, so by uniqueness T (v, 0) = 0.To show compactness of T , we first show that T maps bounded sets in C1,β(Ω)× [0, 1] to precompact

sets in C1,β(Ω) and C2(Ω), and then use the latter to show that T is continuous. We will first useSchauder estimates to show that T maps bounded sets to bounded sets. The following theorem, whichwe state without proof, is theorem 6.6 of [2]

Theorem (Global Schauder Estimates). Let Ω ⊂ Rn be a bounded domain with ∂Ω ∈ C2,γ for someγ ∈ (0, 1). Define an operator L by

Lu = Aij(x)Diju+Bi(x)Diu+ C(x)u

. Supoose there exist λ, µ > 0 such that

1. Aij(x)ξiξj ≥ λ|ξ|2 ∀x ∈ Ω, ξ ∈ Rn;

2. |Aij |0,γ,Ω, |Bi|0,γ,Ω, |C|0,γ,Ω ≤ µ.

Let f ∈ Cγ(Ω) and φ ∈ C2,γ(Ω). Suppose u ∈ C2,γ(Ω) be a solution of the Dirichlet problemLu = f in Ωu = φ on ∂Ω

.

Then|u|2,γ,Ω ≤ C (|u|0,Ω + |φ|2,γ,Ω + |f |0,γ,Ω) ,

where C = C(n, γ, λ, µ,Ω) does not depend on u.

We will apply this theorem with γ = αβ, Aij(x) = aij(x, v(x), Dv(x);σ), Bi = C = 0, f(x) =−b(x, v(x), Dv(x);σ) and φ = σϕ. We need to check that conditions (1) and (2) hold so that we arejustified in using this theorem. Certainly (1) holds, since by ellipticity of Qσ (condition (3) above),λ(x, v(x), Dv(x);σ) is positive and continuous on the compact set Ω, so attains a positive minimumvalue there, and we may take λ = minx∈Ω λ(x, v(x), Dv(x);σ). For condition (2), since Bi = C = 0,

we only need to check that |Aij0,αβ,Ω = ‖Aij‖C(Ω) + [Aij ]αβ,Ω is finite. This is certainly the case, since

aij ∈ Cα(Ω) and v ∈ C1,β(Ω), so Aij(x) = aij(x, v(x), Dv(x);σ) ∈ Cαβ(Ω).Let v ∈ C1,β(Ω). Applying the global Schauder estimate to u = T (v, σ), we see that

|T (v, σ)|2,αβ,Ω ≤ C (|T (v, σ)|0,Ω + σ|ϕ|2,αβ,Ω + |b((·, v,Dv;σ)|0,αβ,Ω)

= C

(sup

Ω|T (v, σ)|+ σ‖ϕ‖C2,αβ(Ω) + |b((·, v,Dv;σ)|0,αβ,Ω

).

The first term on the right hand side is bounded in terms of the boundary data ϕ by the maximumprinciple (for linear elliptic operators), see Theorem 3.1. Furthermore, the second term is bounded byhypothesis since ϕ ∈ C2,α(Ω) ⊂ C2,αβ(Ω). So using condition (4) for the third term, we see that T mapsbounded sets in C1,β(Ω) × [0, 1] to bounded sets in C2,αβ(Ω). Finally, by the Arzela–Ascoli theorem,these bounded C2,αβ(Ω) sets are precompact in C1,β(Ω) and C2(Ω).

9

To prove continuity of T , we suppose (vm, σm)→ (v, σ) in C1,β(Ω), and show T (vm, σm)→ T (v, σ).Note (vm, σm)∞m=1 is convergent, hence bounded, so it follows from above that T (vm, σm)∞m=1 isprecompact in C2(Ω). Thus every subsequence of T (vm, σm)∞m=1 has a convergent subsequence. Welet T (vmk , σmk)∞k=1 denote any such convergent subsequence, and let

u := limk→∞

T (vmk , σmk).

So

aij(x, v,Dv;σ)Diju+ b(x, v,Dv;σ)

= limk→∞

aij(x, vmk , Dvmk ;σmk)DijT (vmk , σmk) + b(x, vmk , Dvmk ;σmk)

= 0,

where we have used continuity of the coefficients (condition (4) above) for the first equality. Moreover,since σmk → σ, on ∂Ω we have T (vmk , σmk) = σmkϕ→ σϕ, so that u = σϕ on ∂Ω. Hence by uniquenessof solutions to the Dirichlet problem (2.7), we have u = T (v, σ). Since this holds for every such sequencevmk , σmk), we have that T (vm, σm)∞m=1 converges to T (v, σ).

3 A Quasilinear Maximum Principle

Recall from the theory of second order linear elliptic PDEs on bounded domains that, if various signconditions are satisfied, then we can bound the solution in terms of its values on the boundary of thedomain. One such theorem we shall need, proved in [2], is stated below.

Theorem 3.1 (Linear Maximum Principle). Let Ω ⊂ Rn be a bounded domain. Suppose u ∈ C2(Ω) ∩C0(Ω) satisfies Lu ≥ f in Ω, where f ∈ C0(Ω),

Lu = Aij(x)Diju+Bi(x)Diu

and L is elliptic in Ω. Then

supΩu = sup

∂Ωu+ c sup

Ω

|f−|λ

,

where C = C(supi,Ω|Bi|λ ,diam Ω) with λ(x) the least eigenvalue of [Aij(x)]ij.

We seek a result of this nature for second order quasilinear elliptic operators Q of the form

Qu = aij(x, u,Du)Diju+ b(x, u,Du).

Recall that λ(x, z, p) denotes the smallest eigenvalue of [aij(x, z, p)]ij . We need to impose a bound onthe ratio b/λ to obtain such a result. The following theorem is proved in [2].

Theorem 3.2 (Quasilinear Maximum Principle). Let Q be elliptic in the bounded domain Ω ⊂ Rn, andsuppose there exist constants µ1, µ2 ≥ 0 such that

b(x, z, p)sign z

λ(x, z, p)≤ µ1|p|+ µ2 ∀(x, z, p) ∈ Ω× R\0 × Rn. (3.1)

Suppose u ∈ C2(Ω) ∩ C0(Ω) satisfies Qu ≥ 0 in Ω. Then

supΩu ≤ sup

∂Ωu+ + Cµ2,

where C = C(µ1,diam Ω). Furthermore, if Qu = 0 in Ω, then

supΩ|u| ≤ sup

∂Ω|u|+ Cµ2,

with C = C(µ1,diam Ω).

10

Proof. The second estimate is obtained from the proof of the first by replacing u with −u. So we onlyprove the first estimate.

If b ≡ 0, the conclusion follows directly from Theorem 3.1 with f ≡ 0, so we assume b 6≡ 0. Define

Ω+ := x ∈ Ω : u(x) > 0.

Then, in Ω+, we have

0 ≤ Qu = aijDiju+ bsignu

≤ aijDiju+ λ (µ1|Du|+ µ2) by (3.1)

≤ aijDiju+ λµ1

n∑k=1

|Dku|+ λµ2

= aijDiju+ λµ1(signDiu)Diu+ λµ2.

We now apply Theorem 3.1 with Aij(x) = aij(x, u,Du), Bi(x) = λµ1signDiu(x), C = 0 and f = −λµ2

to obtain

supΩ+

u ≤ sup∂Ω+

u+ + C supΩ+

|f−|λ

= sup∂Ω+

u+ + Cµ2, (3.2)

where C = C(supi,Ω|Bi|λ ,diam Ω). But Bi = λµ1signDiu =⇒ |Bi|

λ = µ1, so C = C(µ1,diam Ω).It remains only to extend estimate (3.2) to the whole of Ω. If Ω+ = ∅, the conclusion is trivial, so we

assume Ω+ 6= ∅. Now, note

∂Ω+ ⊆ x ∈ ∂Ω : u(x) ≥ 0 ∪ x ∈ Ω : u(x) = 0. (3.3)

Suppose that u ≡ 0 on ∂Ω+. Then, since Ω+ 6= ∅,

0 < supΩ+

u ≤ sup∂Ω+

u+ = 0,

a contradiction. Thus there exists x ∈ ∂Ω+ such that u(x) > 0, and moreover from inclusion (3.3), wesee that every x ∈ ∂Ω+ with u(x) > 0 must satisfy x ∈ ∂Ω, and hence

sup∂Ω+

u+ ≤ sup∂Ω

u+.

From this and (3.2) ,we get

supΩu = sup

Ω+

u ≤ sup∂Ω+

u+ + Cµ2

≤ sup∂Ω

u+ + Cµ2,

as required.

Example 3.3 (Minimal Surface Equation). We can apply this theorem to the minimal surface equation,

Qu = ∆u− DiuDju

1 + |Du|2Diju = 0

in a bounded domain Ω ⊂ Rn. In this case, the coefficient b ≡ 0, so triviallybsign z

λ≤ 0|p| + 0 so by

Theorem 3.2 with µ1 = µ2 = 0, we getsup

Ω|u| ≤ sup

∂Ω|u|

for every solution u ∈ C2(Ω) ∩ C0(Ω) of the minimal surface equation.

11

4 Gradient Estimates

As discussed in Section 1, we need to bound supΩ |Du| in terms of sup∂Ω |Du|. In order to obtain thisestimate, we depart from the general case of second order quasilinear operators Q and consider onlyoperators with a special divergence form (see Definition 1.2). Namely we consider operators Q of theform Qu = div A(x, u,Du) +B(x, u,Du) and we stipulate B ≡ 0 and A(x, z, p) = A(p). So Q is ellipticand

Qu = div A(Du) = 0, (4.1)

and we want to deduce an estimate for supΩ |Du|. Choose any function η ∈ C20 (Ω), multiply (4.1) by

Dkη for some k ∈ 1, . . . , n and integrate by parts to get∫Ω

A(Du) ·D(Dkη) dx = 0

=⇒∫

Ω

Ai(Du)DiDkη dx = 0

=⇒∫

Ω

Ai(Du)DkDiη dx = 0 (η ∈ C2(Ω))

=⇒∫

Ω

DkAi(Du)Diη dx = 0.

So from the chain rule, we get ∫Ω

DpjAi(Du)Dk(Dju)Diη dx = 0

=⇒∫

Ω

DpjAi(Du)DkjuDiη dx = 0.

Write w = Dku. Then ∫Ω

DpjAi(Du)DjwDiη dx = 0.

Recalling (1.5), we find ∫Ω

aij(Du)DjwDiη dx = 0 ∀η ∈ C20 (Ω).

In other words, w ∈ C1(Ω) is a weak solution of the linear elliptic equation

Di(aij(x)Djw) = 0, aij(x) = aij(Du(x)). (4.2)

So by the weak maximum principle (see, for instance, section 3.6 of [2])

supΩ|Du| = sup

∂Ω|Du|, (4.3)

as desired.

5 Boundary Gradient Estimates

In this section we introduce barriers and present an account of their use in obtaining boundary gradientestimates for the minimal surface equation before proceeding to a discussion of how these techniques canbe applied to the solutions of more general second-order quasilinear elliptic equations. This technique isused to show that in the particular case of the minimal surface equation a non-negative mean curvaturecondition on the boundary ∂Ω is sufficient for obtaining such boundary gradient estimates. In fact, itturns out that this condition is also necessary.

Let Ω ⊂ Rn and suppose that Q is an elliptic operator of the form

Qu = aij(x,Du)Diju+ b(x, u,Du),

12

where the coefficients aij , b are continuously differentiable with respect to the p variables on Ω×R×Rnand b is non-increasing in z for each (x, p) ∈ Ω × Rn. Note that these conditions are satisfied by theoperator M corresponding to the minimal surface equation. Indeed,

Mu = (1 + |Du|2)∆u−DiuDjuDiju,

thus aij(x, z, p) = (1 + |p|2)δij − pipj is independent of z and b = 0.

Definition 5.1. Consider u ∈ C2(Ω) ∩ C0(Ω) satisfying Qu = 0. Suppose that in a neighbourhoodN = Nx0

of a point x0 ∈ ∂Ω there exist functions w± ∈ C2(N ∩ Ω) ∩ C1(N ∩ Ω) such that

(i) ±Qw± < 0 in N ∩ Ω

(ii) w±(x0) = u(x0)

(iii) w−(x) ≤ u(x) ≤ w+(x) on ∂(N ∩ Ω).

Such a w+ (respectively w−) is known as an upper (lower) barrier at the point x0 for the operator Qand the function u.

Intuitively, the existence of barriers suggests that a bound on |Du(x0)| should exist as we are ”squeez-ing“ the solution u on ∂Ω by C1 functions from above and below that agree with u at the point of interest.More formally, we hav the following result:

Proposition 5.2. Suppose that u is a solution to Qu = 0 in Ω as above and that at all points x0 ∈ ∂Ωthere exists barriers w±x0

. Then there exists a constant C such that |Du|0,∂Ω = sup∂Ω |Du| ≤ C.

Proof. By the assumptions on w±, for any x ∈ Ω we have

w−(x)− w−(x0)

|x− x0|≤ u(x)− u(x0)

|x− x0|≤ w+(x)− w+(x0)

|x− x0|.

It follows that these inequalities pass to normal derivatives; let ν(x0) denote the inward pointing normalto ∂Ω at x0, then

limh→0

u(x0 + hν(x0))− u(x0)

h=∂u

∂ν(x0) = 〈Du(x0), ν(x0)〉

and the result follows.

5.1 Minimal Surface Equation

We begin by examining the special case of the minimal surface equation as this provides motivation forthe more general techniques developed later. The construction of barriers will clearly be dependent onthe regularity of ∂Ω. With this in mind we introduce the distance function related to a region.

Definition 5.3. Given a domain Ω ⊂ Rn with non-empty boundary we define the distance functiond : Rn → R by d(x) = dist (x,Ω) = infy∈Ω |x− y|.

The following properties of d will prove useful.

Proposition 5.4. d is a Lipschitz continuous function.

Proof. Let x, y ∈ Rn and pick z ∈ ∂Ω such that |y − z| = d(y). Then

d(x) ≤ |x− z| ≤ |x− y|+ |y − z|= |x− y|+ d(y),

and hence d(x)− d(y) ≤ |x− y|. Interchanging the roles of x and y gives the result.

Lemma 5.5. Suppose that ∂Ω is C2, then Ω satisfies an interior sphere condition i.e. given a pointx0 ∈ ∂Ω there exists a point x ∈ Ω and a ball BR(x) ⊂ Ω such that ∂BR(x) ∩ Ω = x0.

13

Proof. Let x0 ∈ ∂Ω. After a translation we may assume that x0 = 0. Using the notation x = (x′, xn) ∈ Rnwe may, possibly after a rotation of coordinates, write Br(0) ∩ Ω = (x′, xn) ∈ Br(0) : xn > ψ(x′),where r > 0 and ψ : Bn−1

r → R is some C2 function satisfying ψ(0) = 0, Dψ(0) = 0. Using Taylor’stheorem about 0 we see that |ψ(x′)| ≤ M |x′|2 for all x′ ∈ Br(0) and some M > 0. Letting e1, . . . , endenote the standard basis of Rn we claim that there exists an R ∈ (0, r/2) such that BR(Ren) ⊂ Ω. Ifx ∈ BR(Ren) then |x − Ren|2 = |x′|2 + (xn − R)2 < R2 and so |x′|2 + x2

n − 2xnR < 0. From above weknow that xn < ψ(x′) ≤M |x′|2 < M(2xnR− x2

n) < 2MRxn. Thus if we choose R < 1/(2M) no such xcan exist.

To discuss the geometry of ∂Ω we will need the notion of mean curvature.

Definition 5.6. Suppose that ∂Ω ∈ C2 and let ν(y), T (y) denote the inward unit normal to ∂Ω at y andthe tangent plane to ∂Ω at y respectively. By a rotation of coordinates we may assume that, for a fixedy0 ∈ ∂Ω, the xn coordinate axis is parallel to ν(y0). Thus in a neighbourhood N of y0 we may representthe boundary by a map ϕ ∈ C2(T (y0) ∩ N ). That is ∂Ω ∩ N = (y′, yn)|yn = ϕ(y′) with Dϕ(y′0) = 0.The curvature of ∂Ω at y0 is then described by the eigenvalues of the Hessian matrix [D2ϕ(y0)], which areknown as the principal curvatures κ1, . . . , κn−1 to the surface at y0 with the corresponding eigenvectorsbeing termed the principal directions to ∂Ω at y0.

Remark 5.7. The principal curvatures κ1, κ2 of a two-dimensional surface M embedded in R3 can berealised as the maximal and minimal values of the curvature of paths inM that pass through y and can beobtained by intersecting planes containing ν(y) with M. More generally, an n-dimensional hypersurfaceM → Rn+1 can be equipped with a metric g obtained by pulling back the euclidean metric from theambient space Rn+1. This allows one to consider the difference between the Riemannian connections onthe two spaces, ∇ on M and ∇ on Rn+1. In the case of hypersurfaces we have

∇XY = ∇XY + h(X,Y )ν,

where X,Y ∈ Γ(TM) are sections of the tangent bundle ofM, ν(p) denotes a unit normal to TpM1 andh is a symmetric two-tensor on TM. We can define the so-called shape operator s on M by raising anindex of h;

h(X,Y ) = g(X, sY ),

it follows that s is a self-adjoint endomorphism of TpM for any p and thus must have real eigenvaluesκ1, . . . , κn. These eigenvalues are the principal curvatures in such a context and it could be checked thattheir definition agrees with that given above.

Definition 5.8. The mean curvature of ∂Ω at y0 is given by

H(y0) =1

n− 1

n−1∑i=1

κi =1

n− 1∆ϕ(y′0),

where the last equality follows by the invariance of the trace of a matrix and y0 = (y′0, ϕ(y′0)).

To aid computations we will work in a principal coordinate system: by a further rotation of coordinateswe may assume that the x1, . . . , xn−1 axes lie along principal directions as the Hessian is a symmetricmatrix and hence a basis of its eigenvectors can be formed. In such a coordinate system we have

[D2ϕ(y0)] = diag(κ1, . . . , κn−1).

Next we wish to compute an explicit expression for the unit normal to ∂Ω at some y = (y′, ϕ(y′)) ∈ ∂Ω.Clearly the surface is realised as the zero level set of the function F (y) = yn − ϕ(y′) and thus

DF (y) = (−D1ϕ(y′), . . . ,−Dn−1ϕ(y′), 1)

is normal to the surface. Hence we may take our unit normal to have components

νi(y) =−Diϕ(y′)√

1 + |Dϕ(y′)|2, i = 1, . . . , n− 1, νn(y) =

1√1 + |Dϕ(y′)|2

.

1We have fixed an orientation on M.

14

Let us denote ν(y′) = ν(y). Then, recalling that Dϕ(y′0) = 0, for i = 1, . . . , n− 1 we can compute

Dj νi(y′0) = −Dijϕ(y′0) = −κiδij . (5.1)

With these preliminaries in hand we can now prove the following:

Proposition 5.9. Suppose that Ω is bounded and ∂Ω is C2, then there exists a µ > 0 such that d is C2

on Γµ = x ∈ Ω|d(x) < µ.

Proof. By Lemma 5.5 we know that given any x0 ∈ ∂Ω we can find a ball B depending on x0 such thatB ∩ ∂Ω = x0 with the radii of the balls being bounded from below by a positive constant, which we taketo be µ. It can be shown that 1/µ bounds the principal curvatures from above. For every x ∈ Γµ thereexists a unique y ∈ ∂Ω such that d(x) = |x− y| and clearly the points are related by

x = y + ν(y)d, (5.2)

where ν(y) is the inward unit normal at y. The idea now is to use the inverse function theorem to showthat (5.2) determines y and d as C1 functions of x. Define the function g : (T (y0) ∩N )× R→ Rn by

g(y′, d) = y + ν(y)d.

Clearly g is C1 and by (5.1) its Jacobian at the point (y′0, d(x)) takes the form

[Dg] = diag(1− κ1d, . . . , 1− κn−1d, 1). (5.3)

We conclude that det(Dg) = Πn−1i=1 (1−κid) > 0, since d(x) < µ in Γµ and therefore 1 > d(x)/µ > d(x)κi

for i = 1, . . . , n − 1. Thus the inverse function function can be applied in some neighbourhood U of x0

to conclude that y′ is locally a C1 mapping of x. Then

Dd(x) =x− y|x− y|

= ν(y′(x))

is also a C1 mapping of x and so d ∈ C2(Γµ).

As a corollary of this proposition we obtain:

Lemma 5.10. Let Ω and µ be as above and let x0 ∈ Γµ, y0 ∈ ∂Ω be such that d(x0) = |x0 − y0|. Thenin a priciple coordinate system at y0

[D2d(x0)] = diag

(−κ1

1− κ1d, . . . ,

−κn−1

1− κn−1d, 0

).

Proof. Note that Dd(x0) = ν(y0) = (0, . . . , 0, 1), which implies that Dind(x0) = 0 for i = 1, . . . , n.Moreover, by the chain rule, Dijd(x0) = Dj(νi y)(x0) = Dkνi(y0)Djyk(x0). This is zero unless i = j,in which case it takes the desired form −κi/(1− κid) by (5.3) and (5.1).

This result shows us that

∆d(x) = −n−1∑i=1

κi1− κid

≤n−1∑i=1

κi = −(n− 1)H(y). (5.4)

Let us suppose that ∂Ω is C2 so that we can find µ such that d is C2 on Γµ. We seek a barrier ofthe form v(x) = ϕ(x) + (ψ d)(x), where ϕ ∈ C2, ψ ∈ C2([0, a]) for some a < µ to be determinedlater. Taking ψ(0) = 0 implies that v(y0) = ϕ(y0). We also impose the conditions ψ′ ≥ 1, ψ′′ < 0. ThenDiv = Diϕ+ψ′(d)Did and Dijv = Dijϕ+ψ′(d)Dijd+ψ′′(d)DidDjd. After some computation we arriveat

Mv = (1 + |Dv|2)∆v −DivDjvDijv

= (1 + |Dϕ|2)∆ϕDiϕDjϕDijϕ

+ ψ′(d)[2Did∆ϕ+ (1 + |Dϕ|2)∆d−DidDjϕDijϕ−DiϕDjϕDijd]

+ ψ′(d)2[∆ϕ+ 2DiϕDid∆d−DijϕDidDjd] + ψ(d)3∆d

+ ψ′′(d)[1 + |Dϕ|2 − (DiϕDid)2],

15

having made use of the fact that |Dd| = 1 and hence DidDijd = 0. By the Cauchy-Schwarz inequalitywe find 1 + |Dϕ|2 − (ϕidi)

2) ≥ 1 + |Dϕ|2 − |Dϕ|2|Dd|2 ≥ 1 and, since ψ′′ < 0, the last term can bebounded above by ψ′′. Also ϕ and d are C2 in Γa = x ∈ Ω : d(x) < a and hence their derivatives upto and including those of second order can be bounded above by constants. Finally, we can control theψ′ term by ψ′2 as ψ′ ≥ 1. Together these remarks lead to

Qv ≤ ψ′′ + Cψ′2 + ψ′3∆d. (5.5)

Assuming our boundary surface ∂Ω to have non-negative mean curvature at every point will imply that∆d ≤ −(n − 1)H ≤ 0 by(5.4). This results in Qv ≤ ψ′′ + Cψ′2. We would like to show that Qv < 0.Solving the differential equation ψ′′(d) + Cψ′(d)2 = 0 yields

ψ(d) =1

νlog(1 + kd).

We can therefore ensure that v is an upper barrier if

ψ(a) =1

νlog(1 + ka) = sup

Ω|u| = M, or ka = eνM − 1,

ψ′(d) =k

ν(1 + kd)≥ k

ν(1 + ka)=

k

νeνM≥ 1,

which holds when k ≥ νeνM .We have thus shown the following theorem:

Theorem 5.11. Suppose that Ω is a bounded open subset of Rn such that ∂Ω ∈ C2 satisfies a non-negative mean curvature condition at each point of ∂Ω. Let u ∈ C2(Ω)∩C1(Ω) satisfy Mu = 0 in Ω andu = ϕ on ∂Ω, where ϕ ∈ C2(Ω). Then we have

|Du| ≤ C on ∂Ω

.

This result allows us to control supΩ |Du| using the gradient estimate (4.3) for the minimal surfaceequation, and will help us to satisfy the necessary conditions to apply the De Giorgi–Nash–Moser theory(see Section 6). We will now discuss how one obtains boundary gradient estimates for more generalsecond order quasilinear elliptic equations for domains that satisfy either an exterior sphere or an exteriorhyperplane condition by building on the strategies presented above.

5.2 General Domains

A domain Ω ⊂ Rn satisfies an exterior sphere condition at x0 ∈ ∂Ω if there exists a ball B = BR(y) withx0 ∈ B ∩ Ω = B ∩ ∂Ω = x0. We define the distance function in such an instance by d(x) = dist (x, ∂B).Observe that

d(x) = |x− y| −R, Did(x) =xi − yi|x− y|

, Dijd(x) = |x− y|−3(|x− y|2δij − (xi − yi)(xj − yj)). (5.6)

Similarly, a domain Ω is said to satisfy an exterior plane condition at the point x0 ∈ ∂Ω if there exists ahyperplane P with x0 ∈ P ∩ ∂Ω. Note that this will hold for convex domains in particular. We define adistance function in this case by d′(x) = dist (x,P) for x ∈ Rn, which satisfies Dijd

′ = 0. Indeed, let ussuppose that after a rotation of coordinates xn coincides with the normal direction to P. Then Did

′ = 0for i 6= n and Dnd

′ will be constant, which demonstrates that all second order derivatives must vanish.Let us focus on the homogeneous problem for a domain satisfying the exterior sphere condition for thetime being. Thus u|∂Ω = 0. As before, we propose to search for an upper barrier of the form v = ψ d, 2where ψ ∈ C2(I) for some interval I ⊂ R. We also impose the restriction ψ′ 6= 0 for reasons that willbecome apparent later. Let us pause to introduce the scalar function ε(x, z, p) = aij(x, z, p)pipj and note

2−v would automatically be a lower barrier

16

that ε > 0 by ellipticity of Q. This will prove useful in the sequel. Omitting the d’s as arguments of ψwe compute

Qv = ψ′aijDijd+ aijψ′′DidDjd+ b

= ψ′aijDijd+ψ′′

(ψ′)2ε+ b

≤ n− 1

Rψ′Λ + b+

ψ′′

(ψ′)2ε,

where the last line follows from (5.6) and ε is being evaluated at (x, ψ(d), ψ′(d)Dd). For v to be asubsolution in some neighbourhood N of the boundary point, which we will again take to be of the formΓa, we need Qv < 0. The idea is to propose a so called structure constraint on the coeffiecients of Q,that is we assume the existence of a non-decreasing function µ such that

|p|Λ + |b| ≤ µ(|z|)ε (5.7)

for all (x, z, p) ∈ Ω× R× Rn with |p| ≥ µ(|z|). Such a condition will imply that

Qv ≤(

ψ′′

(ψ′)2+ ν

)ε,

when ψ′ > µ and where ν = 1 + (n− 1)/R. Indeed, observe that |p| = |ψ′Dd| = ψ′ and

n− 1

Rψ′Λ + b ≤ max ((n− 1)/R, 1) (ψ′Λ + b) ≤ (1 + (n− 1)/R)(ψ′Λ + |b|) ≤ νε.

A judicious choice of ψ can be used to meet the barrier conditions. One necessary condition is thatψ(0) = 0 as u vanishes on the boundary and d(x0) = 0 for any boundary point. As stated above, wetake our neighbourhood to be of the form Γa and so we need ψ(a) = v|∂(N∩Ω) ≥ u. This again leads to

requiring ψ(a) = M = supΩ |u|. Such a ψ was constructed above and takes the form ψ(d) = 1ν log(1+kd)

where we suppose that k = µνeµM and ka = eµM − 1. Hence if Qu = 0 in Ω we obtain an estimate

|Du(x0| ≤ |Dv(x0)| ≤ ψ′(0) = µeνM . (5.8)

We now wish to extend these results to the non-zero boundary value case. Let ϕ ∈ C2(Ω) and supposeu|∂Ω = ϕ. Let w = v + ϕ so that

Qv = Qw = Qv + aijDijϕ.

Define the function F byF(x, z, p, q) = aij(x, z, p)(pi − qi)(pj − qj),

then for the transformed version of the operator Q we have ε(x, v,Dv) = F(x,w,Dw,Dϕ). The corre-sponding transformed version of the above structure constraint then has the form

(|p−Dϕ|+ |D2ϕ|)Λ + |b| ≤ µ(|z|)F(x, z, p,Dϕ), (5.9)

whenever |p−Dϕ| ≥ µ(|z|) for some non-increasing µ. These considerations lead to

Theorem 5.12. Let u ∈ C2(Ω)∩C1(Ω) satisfy Qu = 0 in Ω and u = ϕ on ∂Ω. Suppose that Ω satisifiesa uniform exterior sphere condition and ϕ ∈ C2(Ω). Then

|Du| ≤ C on ∂Ω,

where C = C(n,M, µ, |ϕ|2;Ω, δ), δ is the radius of the exterior spheres.

For more details we refer the reader to [GT].In the convex case mentioned above we can apply a very similar approach: Given zero boundary datawe have

Qv = ψ′aijDijd+ aijψ′′DidDjd+ b.

=ψ′′

(ψ′)2ε+ b,

17

Then we can impose a structure condition by assuming that b = O(ε) so that for some non-decreasingfunction µ we have

|b| ≤ µ(|z|)ε,

for |p| ≥ µ(|z|). To extend this to the non-zero boundary data case we require Λ|D2ϕ|, b = O(F), i.e.

Λ|D2ϕ|+ |b| ≤ µ(|z|)F, (5.10)

for |p−Dϕ| ≥ µ(|z|). We then obtain the following theorem on boundary gradient estimates

Theorem 5.13. Let u ∈ C2(Ω) ∩ C1(Ω) satisfy Qu = 0 in Ω and u = ϕ on ∂Ω and suppose that Ω isconvex. Then if the structure condition (5.10) holds, we have a uniform boundary gradient estimate

|Du| ≤ C on ∂Ω.

The results described above show that under sufficient regularity conditions on ∂Ω it is in many casespossible to bound |Du| on ∂Ω uniformly, which in turn allows one to bound |Du|0;Ω. In order to makeuse of the Leray–Schauder existence theory laid out in Section 2, it remains to show Holder continuityof Du, which we will address in the next section.

6 De Giorgi–Nash–Moser Theory

The De Giorgi–Nash–Moser theory provides Holder estimates and the Harnack inequality for uniformlyelliptic partial differential equations. The main theorem on Holder continuity was first obtained inde-pendently by Ennio De Giorgi [1] and John Nash [6] in 1957. Three years later, a different proof wasgiven by Jurgen Moser [5]. We will follow here the argument as presented in [4] with minor modificationsand simplifications to adapt to the present context. Our aim is to prove the following result:

Theorem 6.1 (Interior Holder Estimate). Suppose aij ∈ L∞(B1) is such that

λ|ξ|2 ≤ aij(x)ξiξj ≤ Λ|ξ|2, ∀x ∈ B1, ξ ∈ Rn

for some positive constants λ,Λ > 0. If u ∈ H1loc(B1) is a weak solution in B1,∫

B1

aijDiuDjϕ = 0, ∀ϕ ∈ H10 (B1),

then

supB 1

2

|u(x)|+ supx,y∈B 1

2

|u(x)− u(y)||x− y|α

≤ c(n,

Λ

λ

)‖u‖L2(B1)

with α = α(n, Λ

λ

)∈ (0, 1) and c

(n, Λ

λ

)> 0.

This theorem can be obtained in several ways. De Giorgi and Moser used two very different ap-proaches that are at the origin of a powerful theory which finds its application in the regularity study ofelliptic PDEs. Some of the intermediate results, such as Moser’s Harnack Inequality 6.10, are of greatinterest in their own right and have lead to a number of theorems on quasilinear elliptic equations as wewill see later. We will present the De Giorgi–Nash–Moser theory here as a sequence of theorems leadingto theorem 6.1, however, it should rather be understood as a technique of finding suitable test functionsand energy estimates in order to set up an iterative process which allows us to derive the desired bounds.This approach is used repeatedly in the proofs of this section.

Both De Giorgi and Moser first derived a supremum bound on the solution u ∈ H1(B1) in terms ofits Lp-norm. For completeness, we will include here both proofs as they contain conceptually differentideas that deserve to be mentioned.

18

Theorem 6.2 (Supremum bound). Suppose aij ∈ L∞(B1) such that ‖aij‖L∞(B1) ≤ Λ and

aij(x)ξiξj ≥ λ|ξ|2, ∀x ∈ B1, ξ ∈ Rn.

If u ∈ H1(B1) is a subsolution in B1 in the sense that∫B1

aijDiuDjϕ ≤ 0, ∀ϕ ∈ H10 (B1), ϕ ≥ 0, (6.1)

then u+ ∈ L∞loc(B1) and for any θ ∈ (0, 1) and any p > 0,

supBθ

u+ ≤ C

(1− θ)n/p‖u+‖Lp(B1),

with the constant C = C(n, p, λ,Λ) independant of θ.

Special Case p = 2: We will first prove the special case p = 2 in two different ways, one is due toDe Giorgi, and the other due to Moser.

Proof 1 [De Giorgi]Consider v = (u − k)+ for any k ≥ 0 and some cut-off function ζ ∈ C1

0 (B1) to be chosen later. Let ustake the test function ϕ := vζ2. Note that in u > k we have v = u − k and Dv = Du a.e., whereasv = 0, Dv = 0 a.e. in u ≤ k. In the following, all integrals are taken over u > k ∩B1. Substitutingthe test function ϕ into the equation, we get

0 ≥∫aijDiuDjϕ =

∫aijDivDjvζ

2 + 2

∫aijDivDjζvζ

≥λ∫|Dv|2ζ2 − 2Λ

∫|Dv||Dζ|vζ

≥λ2

∫|Dv|2ζ2 − 2Λ2

λ

∫|Dζ|2v2.

Hence ∫|Dv|2ζ2 ≤ 4Λ2

λ2

∫|Dζ|2v2.

and we obtain an energy estimate for v,∫|D(vζ)|2 ≤ 2

∫|Dv|2ζ2 + 2

∫|Dζ|2v2 .

∫|Dζ|2v2. (6.2)

It follows that ∫(vζ)2 ≤

(∫(vζ)2∗

) 22∗

|vζ 6= 0|1− 22∗

≤c(n)

∫|D(vζ)|2|vζ 6= 0| 2n

.∫|Dζ|2v2|vζ 6= 0| 2n ,

where we used Holder’s inequality and the Sobolev inequality(∫ψ2∗) 2

2∗

≤ c(n)

∫|Dψ|2 (6.3)

for any ψ ∈ H10 (B1), where c(n) is some positive constant and 2∗ = 2n/(n − 2) if n > 2 and 2∗ > 2

arbitrary if n = 2.

19

Next, we choose the cut-off function ζ ∈ C10 (B1) such that for 0 < r < R ≤ 1, ζ ∈ C∞0 (BR) with

ζ ≡ 1 in Br, 0 ≤ ζ ≤ 1 in BR and |Dζ| ≤ 2/(R− r) in BR. Define

A(k, r) := Br ∩ u ≥ k

From the above estimate we conclude∫A(k,r)

(u− k)2 ≤∫A(k,1)

v2ζ2

.∫A(k,1)

|Dζ|2v2 |vζ 6= 0| 2n

.1

(R− r)2

∫A(k,R)

(u− k)2|A(k,R)| 2n . (6.4)

Let k0 = ‖u+‖L2(B1). We will show that there exists a k ≥ k0 such that∫A(k,θ)

(u − k)2 = 0 for any

θ ∈ (0, 1). First, note that

|A(k,R)| ≤ 1

k

∫A(k,R)

u+ ≤ 1

k‖u+‖L2(?).

Fix k ≥ k0. For any h, r satisfying h ≥ k, 0 < r < 1, we have A(k, r) ⊃ A(h, r). It follows that∫A(h,r)

(u− h)2 ≤∫A(k,r)

(u− k)2

and

|A(h, r)| = |Br ∩ u− k > h− k| ≤ 1

(h− k)2

∫A(k,r)

(u− k)2

Let θ ∈ (0, 1). We conclude from (6.4) that for any h > k ≥ k0 and θ ≤ r < R ≤ 1,

∫A(h,r)

(u− h)2 .1

(R− r)2

∫A(h,R)

(u− h)2|A(h,R)| 2n .(

1

(R− r)(h− k)2n

)2(∫

A(k,R)

(u− k)2

)1+ 2n

.

Note that∫A(k,r)

(u− k)2 = ‖(u− k)+‖2L2(Br), and hence

‖(u− h)+‖L2(Br) .1

(R− r)(h− k)2n

‖(u− k)+‖L2(BR).

Using the notation ψ(k, r) := ‖(u− k)+‖L2(Br), we obtain

ψ(h, r) .1

(R− r)(h− k)2n

ψ(k,R). (6.5)

The idea behind De Giorgi’s proof is to take smaller and smaller radii r while taking bigger and biggerk, showing that there exist k with ψ(k, r) = 0 for a nonzero r. This is achieved by an iterative process.For k ≥ 0, θ ∈ (0, 1), define for m ∈ Z≥0

km =k0 + k

(1− 1

2m

)rm =θ +

1

2m(1− θ) ,

so km ≥ k0 for all m and km ↑ k0 + k, rm ↓ θ. Note that

km − km−1 =k

2m, rm−1 − rm =

1− θ2m

and hence by (6.5),

ψ(km, rm) .2m

1− θ

(2m

k

) 2n

ψ(km−1, rm−1)1+ 2n . (6.6)

20

We will prove inductively that for any m ∈ Z≥0,

ψ(km, rm) ≤ ψ(k0, r0)

γm(6.7)

for some γ > 1. This is clearly true for m = 0. Suppose (6.7) holds for m− 1. Then

ψ(km−1, rm−1)1+ 2n ≤ ψ(k0, r0)

2n

γ(m−1) 2n−1

ψ(k0, r0)

γm.

Using our previous estimate (6.6), we obtain that for some constant C > 0,

ψ(km, rm) ≤ C 2m( 2n+1)

(1− θ)k 2n

ψ(k0, r0)2n

γ(m−1) 2n−1

ψ(k0, r0)

γm.

First, let us choose γ2/n = 21+2/n, which indeed satisfies γ > 1. Next, we choose k = k∗(θ) ≥ 0,

(k∗(θ))2n = C

γ1+ 2n ψ(k0, r0)

2n

1− θ.

With these choices of γ and k, claim (6.7) follows by induction.

Finally, we let m→∞ in (6.7). We obtain

‖(u− k0 − k∗(θ))+‖L2(Bθ) = ψ(k0 + k∗(θ), θ) = 0,

in other words

supBθ

u+ . k0 + k∗(θ)

. k0 +1

(1− θ)n2‖(u− k0)+‖L2(B1)

.1

(1− θ)n2‖u+‖L2(B1). (6.8)

This proves the theorem for p = 2.

Proof 2 [Moser]The idea behind Moser’s approach is to make a different choice of test function, allowing us to obtaina bound on the Lp1 -norm of u inside a ball of radius r1 in terms of the Lp2-norm on a larger ball withradius r2,

‖u‖Lp1 (Br1 ) .1

r2 − r1‖u‖Lp2 (Br2 ), (6.9)

for p1 > p2 and r1 < r2. This will be the set-up for an iterative argument and with careful choices ofradii ri and exponents pi, we will arrive at the desired sup-bound for the case p = 2.

For k,m > 0, define u = u+ + k and

um =

u ifu < m,

k +m ifu ≥ m.

Then clearly k ≤ um ≤ k +m and um ≤ u. In u < 0 and u > m, um is constant and so Dum = 0.Let us choose the test function

ϕ = ζ2(uβmu− kβ+1

)∈ H1

0 (B1)

for some β ≥ 0 and some non-negative cut-off function ζ ∈ C10 (B1) to be determined later. The iterative

argument applied later to conclude the proof will be on β, starting with β = 0. Direct calculation yields

Dϕ =2ζDζ(uβmu− kβ+1

)+ ζ2βuβ−1

m Dumu+ ζ2uβmDu

=2ζDζ(uβmu− kβ+1

)+ ζ2uβm (βDum +Du)

21

since um = u on Dum 6= 0. Note further that ϕ = 0 and Dϕ = 0 in u ≤ 0. Hence all the integralswill be over B1 ∩ u > 0. Trivially u+ ≤ u and uβmu− kβ+1 ≤ uβmu. It follows that for any k > 0,∫

aijDiuDjϕ =

∫aijDiu (βDj um +Dj u) ζ2uβm + 2

∫aijDiuDjζ

(uβmu− kβ+1

)ζ

≥λβ∫ζ2uβm|Dum|2 + λ

∫ζ2uβm|Du|2 − Λ

∫|Du| |Dζ|uβmuζ

≥λβ∫ζ2uβm|Dum|2 +

λ

2

∫ζ2uβm|Du|2 −

2Λ2

λ

∫|Dζ|2uβmu2.

Since u is a subsolution to equation (6.1), we conclude

β

∫|Dum|2ζ2uβm +

∫|Du|2ζ2uβm .

∫|Dζ|2uβmu2.

Let us define w = uβ/2m u. Note that

|Dw|2 ≤ (1 + β)(βuβm|Dum|2 + uβm|Du|2

).

Hence, we obtain the following energy estimate for w:∫|Dw|2ζ2 . (1 + β)

∫|Dζ|2w2.

Let χ = n/(n− 2) > 1 if n > 2 and χ > 2 arbitrary if n = 2. Then by Sobolev inequality (6.3),(∫|ζw|2χ

) 1χ

.∫|D(wζ)|2 . (1 + β)

∫|Dζ|2w2.

Next, we choose the cut-off function ζ ∈ C10 (B1) such that for 0 < r < R ≤ 1, ζ ∈ C∞0 (BR) with ζ ≡ 1

in Br, 0 ≤ ζ ≤ 1 in BR and |Dζ| ≤ 2/(R− r) in BR. We obtain(∫Br

w2χ

) 1χ

.(1 + β)

(R− r)2

∫BR

w2.

Setting γ = β + 2 ≥ 2 and recalling that um ≤ u, we obtain(∫Br

uγχm

) 1χ

.(γ − 1)

(R− r)2

∫BR

uγ ,

provided the RHS integral is bounded. Taking the limit m→∞, we arrive at an inequality of the type(6.9) relating the Lq(Br)-norms of u for different exponents q and radii r,

‖u‖Lγχ(Br) .

((γ − 1)

(R− r)2)

) 1γ

‖u‖Lγ(BR). (6.10)

assuming ‖u‖Lγ(BR) <∞.

We conclude with an iterative argument on γ = 2, 2χ, 2χ2, ... for χ > 1. For θ ∈ (0, 1) and i ∈ Z≥0,let us define

γi = 2χi,

ri = θ +1

2i(1− θ),

so γi ↑ ∞ and ri ↓ θ as i→∞. Note that

γi = χγi−1, ri−1 − ri =1

2i(1− θ).

22

Then using (6.10), we have

‖u‖Lγi (Bri ) .(

(γi−1 − 1)4i

(1− θ)2)

) 1γi−1

‖u‖Lγi−1 (Bri−1)

. c(n, i)i∏

k=1

(1

(1− θ)2

) 1γk−1

‖u‖L2(B1),

where c(n, i) is defined as

c(n, i) =

i∏k=1

((γk−1 − 1) 4k

) 1γk−1 .

Provided c(n,∞) <∞, we can take the limit i→∞ and obtain a sup bound on u,

‖u‖L∞(Bθ) .1

(1− θ)n2‖u‖L2(B1) (6.11)

since∞∏k=0

(1− θ)−2γk =

(1

(1− θ)2

)∑∞k=0

1γk

=1

(1− θ)n2.

This completes Moser’s proof of theorem (6.2) for the case p = 2.

Generalisation p ≥ 2:

Let y ∈ B1 and R > 0. Define u(y) = u(Ry), aij(y) = aij(Ry). Assume aij satisfies the assumptionsof theorem 6.2 with subsolution u in B1. Then u is a subsolution in BR with coefficient matrix aij .Applying the result for p = 2, θ ∈ (0, 1) to u, we have

supBθR

u+ = supBθ

u+ .1

(1− θ)n2‖u+‖L2(B1).

Then for θ = 1/2 and for any p ≥ 2:

supBR/2

u+ . ‖u+‖L2(B1) .1

Rnp‖u+‖Lp(BR).

Take R = (1 − θ)r with θ ∈ (0, 1) and r > 0 and fix y ∈ Brθ. The above bound applied to B(1−θ)r(y)yields

supB(1−θ)r/2(y)

u+ .1

((1− θ)r)np‖u+‖Lp(B(1−θ)r(y)).

Covering Brθ with finitely many such balls centered at (yj)Nj=1, we take the maximum on both sides,

maxj=1,...,N

supB(1−θ)r/2(yj)

u+ . maxj=1,...,N

1

((1− θ)r)np‖u+‖Lp(B(1−θ)r(yj))

.1

((1− θ)r)np

N∑j=1

‖u+‖Lp(B(1−θ)r(yj))

.N

((1− θ)r)np‖u+‖Lp(Br).

Taking r = 1, we obtain theorem 6.2 for any p ≥ 2.

23

Generalisation p ∈ (0, 2): We will make use of the following lemmata:

Lemma 6.3. For constants a, b ≥ 0 and exponent p ∈ (0, 2), we have

a1− p2 b12 ≤

(1− p

2

)a+ b

1p (6.12)

Proof : This is an easy consequence of Young’s inequality.

Lemma 6.4. Let f ≥ 0 be a bounded function in [τ0, τ1] with τ0 ≥ 0. Let α ∈ [0, 1) and A > 0 a positiveconstant. If for all s, t such that τ0 ≤ t < s ≤ τ1,

f(t) ≤ αf(s) +A

(s− t)β,

then

f(t) ≤ c(α, β)A

(s− t)β.

for some positive constant c(α, β) > 0.

ProofFor 0 < τ < 1, consider the sequence tii≥0,

t0 = t, ti+1 = ti + (1− τ)τ i(s− t).

By iteration,

f(t) = f(t0) ≤ αkf(tk) +A

(1− τ)β(s− t)βk−1∑i=0

αiτ−iβ .

Next, choose τ < 1 such that ατ−β < 1, i.e. α < τβ < 1. Letting k →∞, we obtain

f(t) ≤ A

(1− τ)β(s− t)β

(1

1− ατ−β

),

which proves the lemma.

By rescaling inequality (6.8) we have for any θ ∈ (0, 1), 0 < R ≤ 1,

‖u+‖L∞(BθR) ≤C

((1− θ)R)n2‖u+‖L2(BR).

Note that for p ∈ (0, 2), ∫BR

(u+)2 ≤ ‖u+‖2−pL∞(BR)

∫BR

(u+)p

and so applying Lemma 6.3, we obtain

‖u+‖L∞(BθR) ≤‖u+‖1−p2

L∞(BR)

C

((1− θ)R)n2‖u+‖L2(BR)

≤(

1− p

2

)‖u+‖L∞(BR) +

C2p

((1− θ)R)np‖u+‖Lp(BR)

≤(

1− p

2

)‖u+‖L∞(BR) +

C2p

((1− θ)R)np‖u+‖Lp(B1).

Setting f(t) = ‖u+‖L∞(Bt) for any t ∈ (0, 1], this rewrites as: for any 0 < r < R ≤ 1,

f(r) ≤(

1− p

2

)f(R) +

C2p

(R− r)np‖u+‖Lp(B1).

24

By Lemma 6.4, there exists a constant c(n, p) > 0 such that

f(r) ≤ c(n, p) 1

(R− r)np‖u+‖Lp(B1).

Taking R→ 1− and renaming r = θ, we conclude

supBθ

u+ .1

(1− θ)np‖u+‖Lp(B1).

This completes the proof of theorem 6.2 for general p > 0.

Theorem 6.2 is at the heart of the De Giorgi–Nash–Moser Theory. De Giorgi used it to show theorem6.1 by bounding inf u below (see theorem 6.5) which implies control of oscillations (Oscillation Theorem6.8). Contrary to this direct method, Moser proved a weak Harnack inequlity (see theorem 6.9), whichis a much more powerful result then theorem 6.5. It not only implies the Oscillation Theorem 6.8, butalso a number of other results such as Liouville’s theorem and a weak maximum principle for quasilinearelliptic equations.

6.1 De Georgi’s Approach

We will start by presenting De Giorgi’s approach, which relies on the following theorem:

Theorem 6.5. Suppose aij ∈ L∞(B2) satisfies

λ|ξ|2 ≤ aij(x)ξiξj ≤ Λ|ξ|2, ∀x ∈ B2,∀ξ ∈ Rn,

and suppose u ∈ H1loc(B2) is a positive supersolution in B2,∫

B2

aijDiuDjϕ ≥ 0, ∀ϕ ∈ H10 (B2), ϕ ≥ 0.

If |B1 ∩ u ≥ 1| ≥ ε|B1| for some ε > 0, then there exists a positive constant C = C(ε, n, Λ

λ

)> 1 such

that

infB 1

2

u ≥ 1

C.

To prove this result, we require the following two lemmata:

Lemma 6.6. Let Φ ∈ C0,1loc (R) be convex. If u ∈ H1

loc(Ω) is a subsolution in Ω,∫Ω

aijDiuDjϕ ≤ 0, ∀ϕ ∈ H10 (Ω),

and Φ′ ≥ 0, then v = Φ(u) is also a subsolution provided v ∈ H1loc(Ω).

Proof. Assuming Φ ∈ C2loc(R) with Φ′(s) ≥ 0,Φ′′(s) ≥ 0, we consider the test function ϕ ∈ C1

0 (B1) withϕ ≥ 0. Then Φ′(u)ϕ ∈ H1

0 (B1) is non-negative and can be taken as a test function for the equation onu, ∫

B1

aijDivDjϕ =

∫B1

aijDiuDjϕΦ′(u)

=

∫B1

aijDiuDj(Φ′(u)ϕ)−

∫B1

aijDiuDjuϕΦ′′(u)

≤∫B1

aijDiuDj(Φ′(u)ϕ)− λ

∫B1

|Du|2ϕΦ′′(u) ≤ 0.

For a general convex Φ ∈ C0,1loc (R), let us instead take Φε(s) = ρε ∗ Φ(s) with ρε the standard mollifier.

Then Φε(u) is a subsolution and Φ′ε(s)→ Φ′(s) a.e. as ε→ 0. The result follows by Lebesgue’s dominantconvergence theorem.

25

Lemma 6.7. For any ε > 0, if u ∈ H1(B1) satisfies |u = 0 ∩B1| ≥ ε|B1|, then there exists a positiveconstant C = C(ε, n) such that ∫

B1

u2 ≤ C∫B1

|Du|2.

Proof. Suppose the lemma is false. Then there exists ε > 0 and a sequence (um)m≥0 ⊂ H1(B1) suchthat

|um = 0 ∩B1| ≥ ε|B1|,∫B1

u2m ≥ m

∫B1

|Dum|2.

Normalising, we get ∫B1

u2m = 1 and lim

m→∞

∫B1

|Dum|2 = 0

So the sequence (um)m≥0 converges weakly in H1(B1) for some u0 ∈ H1(B1), and since we must have‖u0‖L2(B1) = 1, it also converges strongly in L2. Clearly u0 6= 0 constant a.e. We conclude

0 = limm→

∫B1

|um − u0|2

≥ limm→

∫um=0∩B1

|um − u0|2

≥|u0|2∫m

|um = 0|

>0.

Contradiction.

Proof of Theorem 6.5: Let us assume u ≥ δ > 0. Define v = (log u)−, where the negative partof a function f is defined as f− = max0,−f. Then by Lemma 6.6, v is a subsolution, bounded bylog(1/δ). Hence, we can bound the supremum using Theorem 6.2,

supB1/2

v ≤ C‖v‖L2(B1).

Since |B1 ∩ v = 0| = |B1 ∩ u ≥ 1| ≥ ε|B1|, we can apply Lemma 6.7 to obtain

supB1/2

v ≤ C‖Dv‖L2(B1).

In order to show that the RHS is bounded, let us take the test function ϕ = ζ2/u for ζ ∈ C10 (B2),

0 ≤∫B2

aijDiuDjϕ = 2

∫B2

aijDiuDjζζ

u−∫B2

aijDiuDjuζ2

u2,

and hence ∫B2

|Du|2 ζ2

u2.∫B2

|Dζ|2,

which implies ∫B2

ζ2|D(log u)|2 .∫B2

|Dζ|2.

Next, we choose the cut-off function ζ ∈ C10 (B2) such that ζ ≡ 1 in B1, |Dζ| ≤ 2, and hence∫

B1|D(log u)|2 is bounded by a constant only depending on λ,Λ, n. It follows that

supB1/2

v . ‖Dv‖L2(B1) . ‖D(log u)‖L2(B1) ≤ C(n, λ,Λ) = C,

and so we can concludeinfB1/2

u ≥ e−C > 0.

Taking δ → 0, the result holds true for any u > 0.

26

Theorem 6.5 allows us to prove the following Oscillation Theorem:

Theorem 6.8 (Oscillation Theorem). Suppose aij ∈ L∞(B2) satisfies

λ|ξ|2 ≤ aij(x)ξiξj ≤ Λ|ξ|2, ∀x ∈ B2,∀ξ ∈ Rn.

If u ∈ H1loc(B2) is a bounded solution in B2,∫

B2

aijDiuDjϕ = 0, ∀ϕ ∈ H10 (B2),

then there exists γ = γ(n, λ,Λ) ∈ (0, 1) such that

oscB1/2

u ≤ γ oscB1

u,

where osc = sup− inf.

Proof : Let us define

α1 = supB1

u, β1 = infB1

u,

α2 = supB1/2

u, β2 = infB1/2

u,

and the solutions

v =u− β1

α1 − β1≥ 0, w =

α1 − uα1 − β1

≥ 0.

We consider the following two cases seperately:

u ≥ 1

2(α1 + β1)⇔ v ≥ 1

2,

u ≤ 1

2(α1 + β1)⇔ w ≥ 1

2.

Case 1 : If v ≥ 1/2, then

|2v ≥ 1 ∩B1| = |B1| ≥1

2|B1|.

So by Theorem 6.5 there exists C > 1 such that

β2 − β1

α1 − β1≥ 1

C.

Case 1 : If w ≥ 1/2, then

|2w ≥ 1 ∩B1| = |B1| ≥1

2|B1|.

So by theorem 6.5 there exists C > 1 such that

α1 − α2

α1 − β1≥ 1

C.

In both cases, since β2 ≥ β1 and α2 ≤ α1, we obtain

α2 − β2 ≤(

1− 1

C

)(α1 − β1).

Theorem 6.1 follows directly from Oscillation Theorem 6.8.

27

6.2 Moser’s Approach

Moser used the supremum bound derived in theorem 6.2 to show the following Harnack inequlity:

Theorem 6.9 (Harnack Inequality I). Suppose aij ∈ L∞(Ω) is uniformly elliptic,

λ|ξ|2 ≤ aij(x)ξiξj ≤ Λ|ξ|2

for some positive constants λ,Λ > 0. If u ∈ H1(Ω) is a non-negative supersolution,∫Ω

aijDiuDjϕ ≥ 0, ∀ϕ ∈ H10 (Ω), ϕ ≥ 0,

then for any BR ⊂ Ω, for any 0 < p < n/(n− 2) and any 0 < θ < τ < 1, we have

infBθR

u ≥ C

Rnp‖u‖Lp(BτR),

where C = C(n, p, λ,Λ, θ, τ) is a positive constant.

The Harnack inequality generally used in the context of quasilinear elliptic equations is a directconsequence of Harnack inequality 6.9 and theorem 6.2, known as Moser’s Harnack Inequality :

Theorem 6.10 (Harnack Inequality II). Suppose aij ∈ L∞(Ω) is uniformly elliptic,

λ|ξ|2 ≤ aij(x)ξiξj ≤ Λ|ξ|2

for some positive constants λ,Λ > 0. If u ∈ H1(Ω) is a solution,∫Ω

aijDiuDjϕ = 0, ∀ϕ ∈ H10 (Ω),

then for any BR ⊂ Ω,supBR

u ≤ C infBR

u

where C = C(n, λ,Λ) is a positive constant.

Proof (Harnack Inequality I): We will first proof that there exists a specific choice of p = p0 forwhich 6.9 holds true (step 1), and then generalise to any 0 < p < n/(n− 2) (step 2).Step 1: For k > 0, let u = u+ k and define v = 1/u. Taking the test function ϕv2 for any ϕ ∈ H1

0 (B1),we can derive an equation for v, ∫

B1

aijDivDjϕ ≤ 0, ∀ϕ ∈ H10 (B1)

So v is a subsolution to the equation, and we can thus apply theorem 6.2: for any 0 < θ < τ < 1 andany p > 0,

supBθ

≤ C‖v‖Lp(Bτ ),

in other words,

infBθu ≥ C

(∫Bτ

u−p)− 1

p

= C

(∫Bτ

u−p∫Bτ

up)− 1

p(∫

Bτ

up) 1p

,

where C = C(n, p, λ, τ, θ). The aim is to show the existence of a p0 such that∫Bτ

u−p0∫Bτ

up0 ≤ C(n, λ, τ).

This will follow from the following result:

28

Claim: For all τ ∈ (0, 1) there exist a p0 such that∫Bτ

ep0|w| ≤ C(n, λ, τ), (6.13)

where w = log u−A with A = 1|Bτ |

∫Bτ

log u.

Noting that ∫Bτ

u−p0∫Bτ

up0 ≤(∫

Bτ

epo|w|)2

,

theorem 6.10 follows after rescaling.

Proof of Claim: Writing

ep0|w| = 1 + p0|w|+1

2!(p0|w|)2

+ ...

we see that we need to estimate∫Bτ|w|β , β ∈ N. To do that, we first derive an equation for w. Taking

ϕv as a test function for any ϕ ∈ L∞(B1) ∩H10 (B1), ϕ ≥ 0, we obtain∫

B1

aijDiwDjwϕ =

∫B1

aijDiu

(Djϕ

1

u−Dj(ϕv)

)≤∫B1

aij1

uDiuDjϕ =

∫B1

aijDiwDjϕ. (6.14)

If we replace ϕ by ϕ2, then Holder’s inequality and uniform ellipticity of aij ∈ L∞(B1) yield for anyϕ ∈ L∞(B1) ∩H1

0 (B1), ∫B1

|Dw|2ϕ2 .∫B1

|Dϕ|2. (6.15)

Next, we choose the cut-off function ϕ ∈ C10 (B1) such that ϕ ≡ 1 in Bτ for τ ∈ (0, 1). Then∫

Bτ

|Dw|2 .∫B1

|Dϕ|2 = C(n, λ,Λ, τ).

Since∫Bτw = 0, we can apply Poincare’s inequality,∫

Bτ

w2 ≤ c(n, τ)

∫Bτ

|Dw|2 ≤ C(n, λ,Λ, τ).

Furthermore, we conclude from (6.15), ∫Bτ′

w2 ≤ C(n, λ,Λ, τ, τ ′) (6.16)

for any τ ′ ∈ (τ, 1). Using these inequalities, we can estimate∫Bτ|w|β for any real β ≥ 2. Let us define

the truncation of w,

wm =

−m w ≤ −mw |w| < m

m w ≥ m

Substituting the test function ϕ = ζ2|wm|2β ∈ H10 (B1) ∩ L∞(B1) into the equation on w, (6.14), we

obtain ∫B1

aijDiwDjwζ2|wm|2β ≤

∫B1

aijDiw(2ζDjζ|wm|2β + ζ22β|wm|2β−1Dj |wm|

)By Young’s inequality,

2β|wm|2β−1 ≤ 2β − 1

2β|wm|2β +

1

2β(2β)2β =

(1− 1

2β

)|wm|2β + (2β)2β−1.

29

Noting that aijDiwDj |wm| = aijDiwmDj |wm| ≤ aijDiwmDjwm a.e. in B1, it follows that∫B1

aijDiwDjwζ2|wm|2β ≤ 2

∫B1

aijDiwDjζ|wm|2βζ + 2β

∫B1

aijDiwmDjwmζ2|wm|2β−1

≤ 2

∫B1

aijDiwDjζ|wm|2βζ +

∫B1

aijDiwmDjwmζ2

((1− 1

2β)|wm|2β + (2β)2β−1

)Note that Dwm = Dw on Dwm 6= 0. Separating the integral into Dwm = 0 ∩ B1 and Dwm 6=0 ∩B1, cancelling terms and multiplying by 2β on both sides, we can write∫

Dwm 6=0aijDiwDjwζ

2|wm|2β +

∫Dwm=0

aijDiwDjwζ2|wm|2β

≤ 4β

∫B1

aijDiwDjζ|wm|2βζ + (2β)2β

∫B1

aijDiwmDjwmζ2.

Using ellipticity again, this gives∫B1

|Dw|2ζ2|wm|2β . β∫B1

|wm|2βζ|Dw||Dζ|+ (2β)2β

∫B1

ζ2|Dwm|2

The first term on the RHS can be dealt with using Cauchy’s inequality, and we obtain∫B1

|Dwm|2ζ2|wm|2β ≤∫B1

|Dw|2ζ2|wm|2β . β2

∫B1

|wm|2β |Dζ|2 + (2β)2β

∫B1

ζ2|Dwm|2. (6.17)

In what follows, we drop the m in wm and take the limit m → ∞ at the end, so the final result holdsindeed true for w. We have∫B1

|D(ζ|w|β)|2 ≤ 2

∫B1

|Dζ|2|w|2β + 2β2

∫B1

ζ2|w|2β−2|Dw|2

≤ 2

∫B1

|Dζ|2|w|2β + 2β − 1

β

∫B1

|w|2βζ2|Dw|2 + 2β2β−1

∫B1

ζ2|Dw|2

.∫B1

|Dζ|2|w|2β +β − 1

β(2β)2β

∫B1

ζ2|Dw|2 +β − 1

ββ2

∫B1

|w|2β |Dζ|2 + β2β−1

∫B1

ζ2|Dw|2

.β2

∫B1

|Dζ|2|w|2β + (2β)2β

∫B1

ζ2|Dw|2

where we used Young’s inequality in the second line and (6.17) in the third line. Then by the Sobolevinequality (6.3) applied to ζ|w|β ∈W 1,2

0 (Rn) and (6.15), we obtain(∫B1

ζ2χ|w|2βχ) 1χ

.∫B1

|D(ζ|w|β)|2

. β2

∫B1

|Dζ|2|w|2β + (2β)2β

∫B1

ζ2|Dw|2

. β2

∫B1

|Dζ|2|w|2β + (2β)2β

∫B1

|Dζ|2

where χ = n/(n − 2) > 1 if n > 2 and χ > 1 arbitrary if n = 2. Now, we can choose a suitable cut-offfunction ζ ∈ C1

0 (B1) as follows: for τ ≤ r < R ≤ 1, ζ ∈ C10 (BR) with ζ ≡ 1 in Br and |Dζ| ≤ 2/(R− r).

Then (∫Br

|w|2βχ) 1χ

.1

(R− r)2

((2β)2β + β2

∫BR

|w|2β)

(6.18)

We are now ready to derive the desired bound on∫Bτ|w|β using an iterative argument on β = 1, χ, χ2, ....

For 0 < τ < τ ′ < 1 and i ∈ Z≥0, let us define

βi = χi

ri = τ +1

2i(τ ′ − τ),

30

so βi ↑ ∞ and ri ↓ τ as i→∞. Note that

βi = χβi−1, ri−1 − ri =1

2i(τ ′ − τ).

Then using (6.18), we obtain

Ii ≤ Ci−1

χi−1

(2χi−1 + (χi−1)

1

χi−1 Ii−1

)≤ (Cχ)

i−1

χi−1(2χi−1 + Ii−1

),

where C = C(n, λ,Λ, τ, τ ′) > 0 is a positive constant and where we define

Ii = ‖w‖L2χi (Bri ).

Iterating the above inequality we obtain

Ii ≤ (Cχ)i−1

χi−1(2χi−1 + Ii−1

)≤

i∑s=1

i∏k=s

(Cχ)k−1

χ−1 2χs−1 +

i∏k=1

(Cχ)k−1

χ−1 I0

.i∑

s=1

χs−1 + I0

. χi + I0

since∑∞k=0 k/χ

k < ∞ and χ > 1. Now, for an arbitrary β ∈ N, β ≥ 2, there exists j ∈ N such that2χj−1 ≤ β ≤ 2χj . It follows, using the above estimate and boundedness of I0 given by (6.16), that(∫

Bτ

|w|β) 1β

. Ij . χj + I0 . β + I0 ≤ Cβ,

and by Stirling’s formula, ∫Bτ

|w|β ≤ Cβββ ≤ Cβeββ!,

where C = C(n, λ,Λ, τ, τ ′) is a positive constant. Choose

p0 ≤1

2Ce.

Then we arrrive at the desired bound ∫Bτ

(p0|w|)β

β!≤ 1

2β,

and hence we conclude ∫Bτ

ep0|w| ≤ 1 +1

2+

1

22+ ... = 2,

which proves the claim.

Remark 6.11. The above proof of claim (6.13) is long but very elementary in nature. There is ashorter but more indirect proof using the John-Nirenberg Lemma by showing that w has bounded meanoscillations.

31

Step 2: Let us now generalise the result from a specific p0 to any 0 < p < n/(n− 2). Our aim is toshow that for any 0 < r1 < r2 < 1 and for any 0 < p2 < p1 < n/(n− 2), we have

‖u‖Lp1 (Br1 ) ≤ C‖u‖Lp2 (Br2 ), (6.19)

where C = C(n, λ,Λ, r1, r2, p1, p2) > 0. Since Theorem 6.9 is true for exponent p0 given in step 1 with0 < p0 ≤ 1 < n/(n − 2) and for any radius τ sucht that 0 < θ < τ < 1, and since p0 can be madearbitrary small, equation (6.19) implies the more general result (6.9).

Recall that u = u+ k for some k > 0. Take ϕ = u−βζ2, β ∈ (0, 1), as a test function in theorem 6.9.Then

0 ≤ −β∫B1

aijDiuDj u ζ2u−β−1 + 2

∫B1

aijDiuDjζ u−βζ

. −β∫B1

|Du|2u−β−1ζ2 +

∫B1

|Du||Dζ|u−βζ

. −β∫B1

|Du|2u−β−1ζ2 +1

β

∫B1

|Dζ|2u1−β ,

and hence for γ = 1− β ∈ (0, 1)∫B1

|Du|2uγ−2ζ2 .1

(1− γ)2

∫B1

|Dζ|2uγ .

Setting w = uγ/2, we arrive at the energy estimate∫B1

|Dw|2ζ2 .γ2

(1− γ)2

∫B1

|Dζ|2uγ . 1

(1− γ)2

∫B1

|Dζ|2w2

and therefore ∫B1

|D(wζ)|2 . 1

(1− γ)2

∫B1

|Dζ|2w2.

Applying Sobolov inequality (6.3) with χ = n/(n − 2) if n > 2 and χ > 1 arbitrary if n = 2, andchoosing the cut-off function ζ ∈ C1

0 (B1) such that ζ ≡ 1 in Br, ζ ∈ C10 (BR) and |Dζ| ≤ 2/(R − r) for

0 < r < R < 1, we obtain(∫Br

w2χ

) 1χ

≤∫B1

|D(wζ)|2 . 1

(1− γ)2

1

(R− r)2

∫BR

w2.

So we conclude for u: (∫Br

uγχ) 1χ

.1

(1− γ)2

1

(R− r)2

∫BR

uγ . (6.20)

Using this inequality, we can do an iterative procedure similar to the approach used in step 1. Given0 < p2 < p1 < n/(n− 2), let us assume there exists i0 ∈ Q such that p1 = p2χ

i0 . If not, the result stillfollows by density of Q in R. For 0 < r1 < r2 < 1 and i ∈ Q≥0, we define

pi = p2χi

ri =2i0r1 − r2

2i0 − 1+

1

2i

(r2 − r1

1− 12i0

),

so r0 = r2, ri0 = r1 and pi ↑ ∞. From (6.20) it follows that(∫Bri

upi

) 1χ

.1

(1− pi−1)2

2i(1− 12i0

)

(r2 − r1)2

∫Bri−1

upi−1

and so after i0 iterations,

‖u‖Lp1 (Br1 ) ≤ C(n, λ,Λ, p1, p2, r1, r2)‖u‖Lp2 (Br2 )

as claimed. Taking k → ∞, the above statements hold true for any non-negative supersolution u ∈H1(Ω). This finishes the proof of Harnack inequality 6.9.

32

The Oscillation Theorem 6.8 follows as an easy consequence of Moser’s Harnack Inequality 6.10 whichin term yields Theorem 6.1. Theorem 6.10 also implies a number of other well-known results such asLiouville’s theorem and a weak maximum principle.

6.3 Global Holder Estimates

In order to apply the De Giorgi–Nash–Moser theory to the minimal surface equation and obtain abound satisfying the conditions of the Leray–Schauder Existence Theorem (2.8), we require a globalHolder estimate, extending Theorem 6.1 up to the boundary ∂Ω. We will here state, but not provethe necessary results (for more details, see [2]). Consider the following variation of the interior Holderestimate 6.1:

Theorem 6.12 (Interior Holder estimate I). Suppose aij ∈ L∞(Ω) is such that

λ|ξ|2 ≤ aij(x)ξiξj ≤ Λ|ξ|2, ∀x ∈ Ω, ξ ∈ Rn

for some positive constants λ,Λ > 0. If u ∈ H1(Ω) is a weak solution in Ω,∫Ω

aijDiuDjϕ = 0, ∀ϕ ∈ H10 (Ω),

then we have for any Ω′ ⊂⊂ Ω the estimate

‖u‖C0,α(Ω′) ≤ C‖u‖L2(Ω)

where C = C(n, λΛ , d′), d′ = dist (Ω′, ∂Ω) and α = α

(n, Λ

λ

)> 0.

The following theorem combines seperate Holder estimates in the interior (Theorem 6.12) and on theboundary into a global Holder estimate:

Theorem 6.13 (Global Holder estimate I). Suppose aij ∈ L∞(Ω) is such that

λ|ξ|2 ≤ aij(x)ξiξj ≤ Λ|ξ|2, ∀x ∈ Ω, ξ ∈ Rn

for some positive constants λ,Λ > 0. Suppose Ω satisfies a uniform exterior cone condition on theboundary ∂Ω. Then if u ∈ H1(Ω) is a weak solution in Ω,∫

Ω

aijDiuDjϕ = 0, ∀ϕ ∈ H10 (Ω),

and if there exist K,α0 > 0 such that

osc∂Ω∩BR(x0)u ≤ KRα0 , ∀x0 ∈ ∂Ω,∀R > 0,

then u ∈ Cα(Ω), and for any Ω′ ⊂⊂ Ω we have the estimate

‖u‖Cα(Ω) ≤ C supΩ|u|+K,

where C = C(n, λΛ , α0,diam Ω), α = α(n, Λ

λ , α0

)> 0.

In particular, let us suppose the quasililinear elliptic operator Q with smallest and largest eigenvaluesλ,Λ > 0 respectively is of the special divergence form (4.1),

Qu = div A(Du) = 0,

where A ∈ C1(Rn). Then we have shown in Section 4 that if u ∈ C2(Ω) satisfies Qu = 0 in Ω, then fork ∈ 1, ..., n), w = Dku ∈ C1(Ω) is a weak solution of the linear elliptic equation

Di(aij(x)Djw) = 0, aij(x) = aij(Du(x)).

33

By replacing Ω if necessary by a strictly contained subdomain, we can assume aij bounded and strictlyelliptic in Ω. The assumptions of Theorem 6.12 are therefore satisfied. Hence, choosing λK ,ΛK suchthat

0 < λk ≤ λ(Du), |aij(Du)| ≤ ΛK

for K := supΩ |Du|, we have the following interior and global Holder estimates:

Theorem 6.14 (Interior Holder estimate II). Let u ∈ C2(Ω) satisfy Qu = 0 in Ω where Q is ellipticand of the form (4.1) with A ∈ C1(Rn). Then for any Ω′ ⊂⊂ Ω we have the estimate

[Du]α,Ω′ ≤ Cd−α,

where C = C(n,K,ΛK/λK ,diam Ω), d = dist (Ω′, ∂Ω), α = α(n,ΛK/λK), K = supΩ |Du|.

Using theorems 6.8, 6.12 and 6.13, this result can be extended to the following global estimate:

Theorem 6.15 (Global Holder estimate II). Let u ∈ C2(Ω) satisfy Qu = 0 in Ω where Q is elliptic inΩ and of the form (4.1) with A ∈ C1(Rn). Then if ϕ ∈ C2(∂Ω) and u = ϕ on ∂Ω, we have the estimate

[Du]α,Ω ≤ C

where C = C(n,K,ΛK/λK ,Φ,Ω), α = α(n,ΛK/λK ,Ω), K = supΩ |Du|, Φ = |ϕ|1;Ω.

7 Application to the Minimal Surface Equation

Many of the results stated in the previous sections are more general than what is needed for the minimalsurface equation. We will now show how to apply the developed theory to this historical example. LetΩ be a sufficiently smooth bounded domain in Rn, and u ∈ C2(Ω) ∩ C0(Ω) satisfy M(u) = 0 in Ω andu = ϕ on ∂Ω, ϕ ∈ C0(∂Ω). Our goal is to show that there exists some β ∈ (0, 1) and M > 0 such that(1.4) is bounded by M ,

‖u‖C1,β(Ω) ≤ supΩ|u|+ sup

Ω|Du|+ [Du]β,Ω ≤M. (7.1)

Let us refer to the strategy outlined in Section 1. Step 1 has been completed in Example 3.3, where weobtained for the minimal surface equation that

supΩ|u| ≤ sup

∂Ω|u|,

which is bounded in terms of the boundary data ϕ. Concerning Step 2, we recall that we established theequality

supΩ|Du| = sup

∂Ω|Du|

in Section 4. If we make some further assumptions, in particular ϕ ∈ C2(Ω) and ∂Ω ∈ C2 is meanconvex, i.e. its mean curvature is non-negative at all x0 ∈ ∂Ω, then it follows from Theorem 5.11 that

sup∂Ω|Du| ≤ C.

Hence the first and second terms on the LHS of (7.1) can be controlled using the theory developped inSections 3 and 5. In order to bound the Holder Semi-norm, we apply the De Girogi–Nash–Moser theoryfrom Section 6 to |Du|. To do so, we use the gradient estimates in Section 4 to justify that we indeedsatisfy the conditions of the Holder estimate 6.1. In fact, we require here a variation of theorem 6.1 thatincludes the boundary ∂Ω. Note that the minimal surface equation is of divergence form (4.1) with

A(Du) =Du√

1 + |Du|2.

We showed in Section 1.2 that aij(x, z, p) = δij − pipj(1+|p|2) satisfies the ellipticity condition

λ(Du(x))|ξ|2 ≤ aij(Du(x))ξiξj ≤ Λ, ∀x ∈ Ω, ξ ∈ Rn

34

with λ(Du(x)) = 1/(1 + |Du|2), Λ = 1. By equation (4.3) in Section 4, we conclude

λ(Du(x)) ≥ 1

(1 + supΩ |Du|2)=

1

(1 + sup∂Ω |Du|2)≥ C > 0,

where we used again the boundary gradient estimate provided by Theorem 5.11. Hence, the coefficientmatrix aij ∈ L∞(Ω) defined in (6.3) is uniformly elliptic, and w = Dku, k ∈ 1, .., n satisfies∫

Ω

aijDiwDjψ = 0, ∀ψ ∈ H10 (Ω).

Let K = supΩ |Du|. By Theorem 6.15, there exists β = β(n,ΛK/λK ,Ω) ∈ (0, 1) such that

[Du]β,Ω ≤ C

where C = C(n,K,ΛK/λK ,Φ,Ω) and Φ = |ϕ|1;Ω. This completes Step 4. We can therefore apply theLeray-Schauder existence Theorem 2.6 to obtain existence of a solution to the minimal surface equation.It is important to note here that the assumption of mean convexity is essential for this argument to hold.

35

References

[1] De Giorgi, E., Sulla differenziabilita e l’analiticita delle estremali degli integrali multipli regolari,Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3), 1957

[2] Gilbarg, D. and Trudinger, N.S., Elliptic Partial Differential Equations of Second Order, U.S. Gov-ernment Printing Office, 2001

[3] Guisti, E., Minimal Surfaces and Functions of Bounded Variation, Birkhauser Boston, 1984

[4] Han, Q. and Lin, F., Elliptic Partial Differential Equations, Courant Institute of MathematicalSciences, New York University, 2011

[5] Moser, J., A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differ-ential equations, Comm. Pure Appl. Math., 1960

[6] Nash, J., Parabolic Equations, Proceedings of the National Academy of Sciences, 1957

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