Leray–Schauder Existence Theory for Quasilinear Elliptic
Equations
Sam Forster, Eavan Gleeson, Franca Hoffmann
March 22nd, 2014
Contents
1 Introduction
1.1 Notation and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 The Minimal Surface Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
3
3
2 Leray–Schauder Existence Theory
2.1 Topological Fixed Point Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Leray–Schauder Fixed Point Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Leray–Schauder Existence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
4
6
7
3 A Quasilinear Maximum Principle
10
4 Gradient Estimates
12
5 Boundary Gradient Estimates
12
5.1 Minimal Surface Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.2 General Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
6 De
6.1
6.2
6.3
Giorgi–Nash–Moser Theory
De Georgi’s Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Moser’s Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Global Hölder Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7 Application to the Minimal Surface Equation
1
18
25
28
33
34
1
Introduction
This project is concerned with the question of existence of classical solutions to the Dirichlet problem
Qu = 0 in Ω
(1.1)
u = ϕ on ∂Ω
where Q is a second order quasilinear elliptic operator, ϕ is a sufficiently regular function on ∂Ω and
Ω ⊂ Rn is a bounded domain.
Thus Q is an operator of the form
Qu = aij (x, u, Du)Dij u + b(x, u, Du),
(1.2)
ij
and we place certain regularity and ellipticity conditions on the coefficients a , b. While second order
linear elliptic equations are very well understood, their theory cannot be applied due to the nonlinearities
in (1.2), so a new theory must be developed to cope with these nonlinearities. This new theory was
pioneered by Leray and Schauder in the 1930s: at its heart is the Leray–Schauder fixed point theorem
which allows us to establish existence of solutions to PDEs from apriori estimates. However, this is not
to say that nothing can be salvaged from the linear theory. Indeed, as we shall see later, we can use
linear results such as the maximum principle to establish analogous results for the quasilinear case. Also
applicable is the theory of De Giorgi, Nash and Moser. These results are indispensable for establishing
the 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
Qσ u = 0 in Ω
.
(1.3)
u = σϕ on ∂Ω
The theorem asserts that if for some β ∈ (0, 1) there is a constant M > 0 such that, for every σ every
solution u of (1.3) satisfies the bound
kukC 1,β (Ω) ≤ M,
then (1.1) has a solution.
Thus the problem has been reduced to estimating Hölder norms of solutions of second order quaslilinear elliptic equations, assuming such solutions exist. So we consider the Hölder norm
kukC 1,β (Ω)
=
sup |u| + sup sup |Dγ u| + [Du]β,Ω
Ω
Ω |γ|=1
≤ sup |u| + sup |Du| + [Du]β,Ω .
Ω
(1.4)
Ω
Thus, in order to estimate the Hölder 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 the
maximum 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 shall
make an additional assumption on Q when deriving this estimate, namely that Q is of divergence
form. The reasons for this additional assumption are for brevity and clarity: to gain an intuitive
understanding 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. We
estimate 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 imposed
on the boundary ∂Ω. Section 5 is devoted to a discussion of these geometric conditions and to the
derivation of the estimate.
4. Finally, in Section 6, we use De Giorgi–Nash–Moser theory to bound the Hölder 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 of
theorems, and yields a toolbox of powerful results for the regularity study of elliptic PDEs such as
Moser’s Harnack Inequality.
2
1.1
Notation and Preliminaries
Before delving into the Leray–Schauder theory and the associated collection of estimates, we need to
give some definitions and establish our notation and conventions. Recall that we are interested in second
order quasilinear operators, namely operators of the form
Qu = aij (x, u, Du)Dij u + b(x, u, Du),
where x = (x1 , . . . , xn ) ∈ Ω for some domain Ω in Rn , n ≥ 2. The function u is assumed to satisfy
u ∈ C 2 (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 defined
for (x, z, p) ∈ Ω × R × Rn , and will always denote by λ(x, z, p) and Λ(x, z, p) the minimum and maximum
eigenvalues 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 is
positive definite for every (x, z, p) ∈ U, namely
0 < λ(x, z, p)|ξ|2 ≤ aij (x, z, p)ξi ξj ≤ 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 ∈ C 1 (Ω) and the matrix [aij (x, u(x), Du(x))]i,j is positive definite, we say Q is elliptic with
respect to u.
Definition 1.2 (Divergence form). We say that the operator Q is of divergence form if there is a
differentiable 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 ∈ C 2 (Ω).
In this case, using symmetry of [aij ], we find that
aij (x, z, p) =
1
Dpi Aj (x, z, p) + Dpj Ai (x, z, p) .
2
(1.5)
Unlike the linear case, a quasililinear operator with smooth coefficients is not always expressible in
divergence 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
Z
F (x, u, Du) dx
Ω
where F is a differentiable scalar function, namely Q is of divergence form with
Ai (x, z, p) = Dpi F (x, z, p),
B(x, z, p) = −Dx F (x, z, p).
Throughout this project, for the sake of concreteness and intuition, we will place a lot of emphasis on
one particular example of a second order quasilinear elliptic equation, the minimal surface equation. The
operator Q corresponding to this equation is in fact a variational operator. This equation is introduced
in the next subsection.
1.2
The Minimal Surface Equation
The minimal surface equation arises when one tries to minimise the n-dimensional ”area“ under the
graph of twice continuously differentiable function u : Ω ⊂ Rn → R. Any such function must minimise
the so-called area functional
Z p
1 + |Du(x)|2 dx,
(1.6)
A(u) =
Ω
3
where we suppose that u|∂Ω = ϕ is continuous. Let u be such a minimiser, then we consider variations
of the form f (t) = A(u + tv), where t is a real number and v ∈ C 2 (Ω) vanishes on the boundary of Ω.
Then f must have a critical point at zero. Consequently
!
Z X
Z
n
Di u
hDu, Dvi
0
p
=−
v = 0,
f (0) =
Di p
1 + |Du|2
1 + |Du|2
Ω i=1
Ω
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 u
(1 + |Du|2 )Dii u − Di uDj uDij u
= 0,
Di p
=
(1 + |Du|2 )3/2
1 + |Du|2
where we have used the summation convention on repeated indices. Therefore any minimiser of the area
functional will also solve the second-order quasilinear equation
Mu = (1 + |Du|2 )∆u − Di uDj uDij u = aij Dij u = 0,
where aij (x, z, p) = (1 + |p|2 )δij − pi pj . Let us show that this equation is elliptic. Indeed,
2
aij (x, Du)ξi ξj = ((1 + |Du|2 )δ ij − Di uDj u)ξi ξj = (1 + |Du|2 )|ξ|2 − hDu, ξi
(1.7)
2
Recall the Cauchy-Schwarz ineqaulity hv, wi ≤ |v|2 |w|2 . Then
aij (x, Du) ≥ (1 + |Du|2 )|ξ|2 − |Du|2 |ξ|2 = |ξ|2 .
It is also clear that
2
aij (x, Du) = (1 + |Du|2 )|ξ|2 − hDu, ξi ≤ (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 smallest
and largest eigenvalues of [aij (x, Du)] respectively. Note, however, that M need not be uniformly elliptic
operator in general. Indeed, Λ(x, Du)/λ(x, Du) = (1 + |Du(x)|2 ) may be unbounded. We shall refer
come back to this example in Section 3, 5 and finally 7 to illustrate the theory we will introduce.
2
2.1
Leray–Schauder Existence Theory
Topological Fixed Point Theorems
The Brouwer fixed point theorem lies at the heart of the Leray–Schauder fixed point theorem, and hence
the Leray–Schauder existence theory. We recall the theorem below (and refer the reader to [2] for its
proof), 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 unit
ball 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 {B i = 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 be
the convex hull of {x1 , . . . , xN (k) }. Then by convexity of K, Kk ⊆ K. We define a map Jk : K → Kk by
PN (k)
d(x, K\B i )xi
Jk x = Pi=1
N (k)
i
i=1 d(x, K\B )
for each x ∈ K. Indeed, Jk maps into Kk as
d(x, K\B j )
aj = aj (x) := PN (k)
i
i=1 d(x, K\B )
4
PN (k)
satisfy 0 ≤ aj ≤ 1 for each j = 1, . . . , N (k), and 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,
PN (k)
(k)
i=1 d(x, K\B i )xi NX
−
ai x
kJk x − xk = PN (k)
i
i=1
i=1 d(x, K\B )
P
N (k) d(x, K\B i )(x − x) i
= i=1
PN (k)
i
i=1 d(x, K\B )
N (k)
≤
X
ai (x)kxi − xk
i=1
<
1
,
k
(2.1)
since kxi − xk < 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 by
the Brouwer fixed point theorem, Sk has a fixed point, xk ∈ Kk say. Now xk ∈ K for each k, so by
compactness 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 = T xk , we have
kxk − T xk k = kJk ◦ T xk − T xk k <
1
.
k
Sending k → ∞, we deduce that T x0 = 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 map
such that T K 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 previous
theorem implies that T has a fixed point in A, and hence K.
Indeed, let A be the convex hull of T K. Certainly A is convex, and since the convex hull of a compact
set is itself compact, A is compact. Moreover, A ⊆ K because T K ⊆ K and K is closed, so T K ⊆ K,
but K is convex by assumption so A ⊆ K. Thus
T |A : A → T A ⊆ T K ⊆ T K ⊆ 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 map
such that
1. T B is precompact, and
2. T ∂B ⊆ B.
Then T has a fixed point.
Proof. Define a map T ∗ : B → B by
T ∗x =


T x,
Tx
,

kT xk
if kT xk ≤ 1;
if kT xk ≥ 1.
It is clear that T ∗ is continouus, and that T ∗ maps B into itself. Moreover, T B precompact =⇒ T ∗ B
precompact. Indeed,
T ∗B
= T {x ∈ B : kT xk ≤ 1} ∪ {
= A1 ∪ A2 ,
5
Tx
: x ∈ B, kT xk ≥ 1}
kT xk
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 a
compact Hausdorff space is compact). As a finite union of compact sets is compact, we need only show
that A1 and A2 are compact.
• A1 is closed, and A1 ⊆ T B 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 infinitely
many 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 }, such
that each yi ∈ A2 . Then for each i, there exists xi ∈ B such that kT xi k ≥ 1, and yi = kTT xxii k . So
T xi ∈ T B and T B is precompact, so there is a subsequence T xik which converges to some z ∈ T B,
z
and moreover kzk ≥ 1. So yik → kzk
, 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 assume
{yi } ⊆ ∂A2 . Now
∂A2 ⊆ {
Tx
Tx
: x ∈ B, kT xk = 1} ∪ {
: x ∈ ∂B, kT xk ≥ 1}.
kT xk
kT xk
x
But by assumption T ∂B ⊆ B, so that the rightmost set above is empty. So {yi } ⊆ { kTT xk
:x∈
B, kT xk = 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. Now
T ∂B ⊆ B =⇒ x 6∈ ∂B =⇒ x ∈ B. Therefore, kT ∗ xk = kxk < 1, so by definition of T ∗ , we must have
kT xk < 1, and hence T x = 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 theorems
introduced 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 is
continuous 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] → B
a 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
kxk < M.
(2.2)
Then x is a fixed point of the map T1 : B → B given by T1 y = T (y, 1), y ∈ B.
Proof. Without loss of generality, we may assume M = 1. Otherwise just rescale the norm on B by a
factor of 1/M . For 0 < ε < 1, define Tε∗ : B → B by
 x 1 − kxk


,
, if 1 − ε ≤ kxk ≤ 1;
 T
kxk
ε
Tε∗ x =
x


T
,1 ,
if kxk < 1 − ε,

1−ε
where B denotes the open unit ball around 0 in B as before. Certainly Tε∗ is continuous, and by
compactness of T , similarly to the proof of the previous lemma, Tε∗ B is precompact. Moreover, since
6
x
kxk = 1 for x ∈ ∂B, we have Tε∗ x = T ( kxk
, 0) = 0 by hypothesis, so Tε∗ ∂B = {0} ⊂ B. So we may apply
∗
the previous lemma to conclude that Tε has a fixed point which we denote x(ε).
∗
has a fixed point x( k1 ). For convenience, we write xk = x( k1 )
Now take ε = k1 for k = 2, 3, . . .. So T1/k
and also denote
k(1 − kxk k), if 1 − k1 ≤ kxk k ≤ 1;
σk :=
1,
if kxk k < 1 − k1 .
Set A = {(xk , σk ) : k ≥ 2}. By compactness of T , we may assume there is a subsequence of A, which we
still denote {(xk , σk )}, which converges to some (x, σ) ∈ B × [0, 1].
Suppose σ < 1. Then for large enough k, σk < 1 so that kxk k ≥ 1 − k1 . (In fact, the inequality must
1
∗
be strict since otherwise σk = 1.) So kxk k → 1, and so kxk = 1. But kxk k ≥
=⇒ xk = T1/k
xk =
k
xk
T(
, σk ) → T (x, σ) by continuity of T . So x = T (x, σ) and kxk = 1, which contradicts (2.2). Hence
kxk k
∗
σ = 1. Now, by continuity of T , we have xk = T1/k
xk → T (x, 1). But xk → x, so x is a fixed point of
T1 , 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 mapping
a 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 to
deduce that this operator has a unique fixed point, and hence that the ODE has a unique solution. The
same idea underpins Leray–Schauder existence theory: it relates a quasilinear elliptic PDE to an operator
whose fixed points correspond to solutions of the PDE, and uses the fixed point theorem of the previous
section to prove existence of fixed points, and hence of solutions, of the PDE. Note, however, one key
difference to the ODE theory: the Leray–Schauder fixed point theorem does not guarantee uniqueness
of 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 ∈ C 2,α and ϕ ∈
2,α
C (Ω) is a given function. We define the operator Q on C 2 (Ω) by
Qu = aij (x, u, Du)Dij u + b(x, u, Du),
where aij , b ∈ C α (Ω × R × Rn ) for some α ∈ (0, 1).
by λ(x, z, p), Λ(x, z, p) the minimum and maximum
To solve the system
Qu = 0
u=ϕ
(2.3)
We assume Q is elliptic in Ω, and as before denote
eigenvalues of [aij (x, z, p)]ni,j=1 respectively.
in Ω
on ∂Ω
in Ω
on ∂Ω
(2.4)
we embed it in a family of problems
Qσ u = 0
u = σϕ
,
(2.5)
where 0 ≤ σ ≤ 1 and
Qσ u = aij (x, u, Du; σ)Dij u + b(x, u, Du; σ).
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) ∈ C α (Ω × R × Rn ) for each σ ∈ [0, 1], and the maps
aij (x, z, p; ·), b(x, z, p; ·) : [0, 1] → C α (Ω × R × Rn )
are continuous.
7
(2.6)
With Theorem 2.6 in mind, we want a formulation of solvability of (2.4) in terms of existence of a
fixed point of a compact map of B × [0, 1] into B for some Banach space B. The introduction of the
family of problems (2.5) allows us to construct exactly this. Indeed, let β ∈ (0, 1) and choose B to be
the Banach space C 1,β (Ω). Define an operator
T : C 1,β (Ω) × [0, 1]
→ C 2,αβ (Ω) ⊂ C 1,β (Ω)
(v, σ) 7→ u,
(2.7)
where u = T (v, σ) is the unique solution of the linear elliptic Dirichlet problem
ij
a (x, v, Dv; σ)Dij u + b(x, v, Dv; σ) = 0 in Ω
.
u = σϕ on ∂Ω
(2.8)
Note that existence of a unique C 2,αβ (Ω) solution is guaranteed by the linear theory. Indeed, v ∈
C 1,β (Ω) =⇒ Dv ∈ C β (Ω) so that the coefficients
ãij (x)
b̃(x)
:= aij (x, v(x)Dv(x); σ),
:= b(x, v(x), Dv(x); σ)
satisfy ãij , b̃ ∈ C αβ (Ω), and since αβ < α, we have ∂Ω ∈ C 2,αβ and ϕ ∈ C 2,αβ (Ω). So applying the
following theorem, proved in [2] for instance, with γ = αβ, Aij = ãij , B i = 0, C = 0t, f = b̃ and φ = σϕ,
we see that (2.8) has a unique C 2,αβ (Ω) solution.
Theorem. Let γ ∈ (0, 1) and let Ω ⊂ Rn be a bounded domain with boundary ∂Ω ∈ C 2,γ . Let φ ∈ C 2,γ (Ω)
and define an operator L by
Lu = Aij Dij u + B i Di u + Cu,
where Aij , B i , C ∈ C γ (Ω). Suppose f ∈ C γ (Ω). Then the Dirichlet problem
Lu = f in Ω
u=φ
on ∂Ω
has a unique solution in C 2,γ (Ω).
Remark 2.7. Notice that our operator defined in (2.8) is indeed strictly elliptic because Ω is compact
and 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 ∈ C 1,β (Ω) for T1 , where T1 : C 1,β (Ω) → C 1,β (Ω) is given by
T1 v = 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 ∈ C 2,α ;
• ϕ ∈ C 2.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 C 2,α (Ω)
solution u of
Qσ u = 0 in Ω
u = σϕ on ∂Ω
satisfies kukC 1.β (Ω) < M . Then the Dirichlet problem
Qu = 0
u=ϕ
in Ω
on ∂Ω
has a solution in C 2,α (Ω).
8
Proof. In view of the comments preceeding the theorem, it’s enough to show that the operator T defined
in (2.7) satisfies the hypotheses of Theorem 2.6. It then follows that T1 has a fixed point u ∈ C 1,β (Ω)
and this is a C 2,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 assumed
to hold in our hypothesis, we need only check
1. T (v, 0) = 0 for each v ∈ C 1,β (Ω);
2. T is compact.
The first propery is easy to see. Indeed, let v ∈ C 1,β (Ω). Condition 2 above ensures b(x, v, Dv; 0) = 0.
So by definition, u = T (v, 0) is the unique solution of
ij
a (x, v, Dv; 0)Dij u = 0 in Ω
.
u = 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 C 1,β (Ω) × [0, 1] to precompact
sets in C 1,β (Ω) and C 2 (Ω), and then use the latter to show that T is continuous. We will first use
Schauder estimates to show that T maps bounded sets to bounded sets. The following theorem, which
we state without proof, is theorem 6.6 of [2]
Theorem (Global Schauder Estimates). Let Ω ⊂ Rn be a bounded domain with ∂Ω ∈ C 2,γ for some
γ ∈ (0, 1). Define an operator L by
Lu = Aij (x)Dij u + B i (x)Di u + C(x)u
. Supoose there exist λ, µ > 0 such that
1. Aij (x)ξi ξj ≥ λ|ξ|2 ∀x ∈ Ω, ξ ∈ Rn ;
2. |Aij |0,γ,Ω , |B i |0,γ,Ω , |C|0,γ,Ω ≤ µ.
Let f ∈ C γ (Ω) and φ ∈ C 2,γ (Ω). Suppose u ∈ C 2,γ (Ω) be a solution of the Dirichlet problem
Lu = f in Ω
.
u = 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), B i = 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 are
justified 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 minimum
value there, and we may take λ = minx∈Ω λ(x, v(x), Dv(x); σ). For condition (2), since B i = C = 0,
ij
ij
we only need to check that |Aij
0,αβ,Ω = kA kC(Ω) + [A ]αβ,Ω is finite. This is certainly the case, since
ij
α
1,β
ij
ij
a ∈ C (Ω) and v ∈ C (Ω), so A (x) = a (x, v(x), Dv(x); σ) ∈ C αβ (Ω).
Let v ∈ C 1,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 |T (v, σ)| + σkϕkC 2,αβ (Ω) + |b((·, v, Dv; σ)|0,αβ,Ω .
Ω
The first term on the right hand side is bounded in terms of the boundary data ϕ by the maximum
principle (for linear elliptic operators), see Theorem 3.1. Furthermore, the second term is bounded by
hypothesis since ϕ ∈ C 2,α (Ω) ⊂ C 2,αβ (Ω). So using condition (4) for the third term, we see that T maps
bounded sets in C 1,β (Ω) × [0, 1] to bounded sets in C 2,αβ (Ω). Finally, by the Arzela–Ascoli theorem,
these bounded C 2,αβ (Ω) sets are precompact in C 1,β (Ω) and C 2 (Ω).
9
To prove continuity of T , we suppose (vm , σm ) → (v, σ) in C 1,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 is
∞
2
precompact in C (Ω). Thus every subsequence of {T (vm , σm )}m=1 has a convergent subsequence. We
let {T (vmk , σmk )}∞
k=1 denote any such convergent subsequence, and let
u := lim T (vmk , σmk ).
k→∞
So
aij (x, v, Dv; σ)Dij u + b(x, v, Dv; σ)
=
=
lim aij (x, vmk , Dvmk ; σmk )Dij T (vmk , σmk ) + b(x, vmk , Dvmk ; σmk )
k→∞
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 ) = σmk ϕ → σϕ, so that u = σϕ on ∂Ω. Hence by uniqueness
of solutions to the Dirichlet problem (2.7), we have u = T (v, σ). Since this holds for every such sequence
{vmk , 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 sign
conditions are satisfied, then we can bound the solution in terms of its values on the boundary of the
domain. 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 ∈ C 2 (Ω) ∩
C 0 (Ω) satisfies Lu ≥ f in Ω, where f ∈ C 0 (Ω),
Lu = Aij (x)Dij u + B i (x)Di u
and L is elliptic in Ω. Then
sup u = sup u + c sup
Ω
where C =
i
C(supi,Ω |Bλ | , diam Ω)
Ω
∂Ω
|f − |
,
λ
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)Dij u + 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 on
the 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 , and
suppose there exist constants µ1 , µ2 ≥ 0 such that
b(x, z, p)sign z
≤ µ1 |p| + µ2 ∀(x, z, p) ∈ Ω × R\{0} × Rn .
λ(x, z, p)
Suppose u ∈ C 2 (Ω) ∩ C 0 (Ω) satisfies Qu ≥ 0 in Ω. Then
sup u ≤ sup u+ + Cµ2 ,
Ω
∂Ω
where C = C(µ1 , diam Ω). Furthermore, if Qu = 0 in Ω, then
sup |u| ≤ sup |u| + Cµ2 ,
Ω
∂Ω
with C = C(µ1 , diam Ω).
10
(3.1)
Proof. The second estimate is obtained from the proof of the first by replacing u with −u. So we only
prove 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 = aij Dij u + bsign u
≤ aij Dij u + λ (µ1 |Du| + µ2 )
by (3.1)
n
X
≤ aij Dij u + λµ1
|Dk u| + λµ2
k=1
= aij Dij u + λµ1 (sign Di u)Di u + λµ2 .
We now apply Theorem 3.1 with Aij (x) = aij (x, u, Du), B i (x) = λµ1 sign Di u(x), C = 0 and f = −λµ2
to obtain
sup u ≤
sup u+ + C sup
Ω+
∂Ω+
=
Ω+
|f − |
λ
+
sup u + Cµ2 ,
(3.2)
∂Ω+
i
i
where C = C(supi,Ω |Bλ | , diam Ω). But B i = λµ1 sign Di u =⇒ |Bλ | = 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 < sup u ≤ sup u+ = 0,
Ω+
∂Ω+
a contradiction. Thus there exists x ∈ ∂Ω+ such that u(x) > 0, and moreover from inclusion (3.3), we
see that every x ∈ ∂Ω+ with u(x) > 0 must satisfy x ∈ ∂Ω, and hence
sup u+ ≤ sup u+ .
∂Ω+
∂Ω
From this and (3.2) ,we get
sup u = sup u ≤
Ω
Ω+
sup u+ + Cµ2
∂Ω+
≤ sup u+ + Cµ2 ,
∂Ω
as required.
Example 3.3 (Minimal Surface Equation). We can apply this theorem to the minimal surface equation,
Qu = ∆u −
Di uDj u
Dij u = 0
1 + |Du|2
in a bounded domain Ω ⊂ Rn . In this case, the coefficient b ≡ 0, so trivially
Theorem 3.2 with µ1 = µ2 = 0, we get
sup |u| ≤ sup |u|
Ω
2
∂Ω
0
for every solution u ∈ C (Ω) ∩ C (Ω) of the minimal surface equation.
11
bsign z
≤ 0|p| + 0 so by
λ
4
Gradient Estimates
As discussed in Section 1, we need to bound supΩ |Du| in terms of sup∂Ω |Du|. In order to obtain this
estimate, we depart from the general case of second order quasilinear operators Q and consider only
operators with a special divergence form (see Definition 1.2). Namely we consider operators Q of the
form 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 elliptic
and
Qu = div A(Du) = 0,
(4.1)
and we want to deduce an estimate for supΩ |Du|. Choose any
Dk η for some k ∈ {1, . . . , n} and integrate by parts to get
Z
A(Du) · D(Dk η) dx
Ω
Z
=⇒
Ai (Du)Di Dk η dx
Ω
Z
=⇒
Ai (Du)Dk Di η dx
Ω
Z
=⇒
Dk Ai (Du)Di η dx
function η ∈ C02 (Ω), multiply (4.1) by
=
0
=
0
=
0
=
0.
(η ∈ C 2 (Ω))
Ω
So from the chain rule, we get
Z
Dpj Ai (Du)Dk (Dj u)Di η dx = 0
Z
=⇒
Dpj Ai (Du)Dkj uDi η dx = 0.
Ω
Ω
Write w = Dk u. Then
Z
Dpj Ai (Du)Dj wDi η dx = 0.
Ω
Recalling (1.5), we find
Z
aij (Du)Dj wDi η dx = 0
∀η ∈ C02 (Ω).
Ω
In other words, w ∈ C 1 (Ω) is a weak solution of the linear elliptic equation
Di (ãij (x)Dj w) = 0,
ãij (x) = aij (Du(x)).
(4.2)
So by the weak maximum principle (see, for instance, section 3.6 of [2])
sup |Du| = sup |Du|,
Ω
(4.3)
∂Ω
as desired.
5
Boundary Gradient Estimates
In this section we introduce barriers and present an account of their use in obtaining boundary gradient
estimates for the minimal surface equation before proceeding to a discussion of how these techniques can
be applied to the solutions of more general second-order quasilinear elliptic equations. This technique is
used to show that in the particular case of the minimal surface equation a non-negative mean curvature
condition on the boundary ∂Ω is sufficient for obtaining such boundary gradient estimates. In fact, it
turns 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)Dij u + b(x, u, Du),
12
where the coefficients aij , b are continuously differentiable with respect to the p variables on Ω × R × Rn
and b is non-increasing in z for each (x, p) ∈ Ω × Rn . Note that these conditions are satisfied by the
operator M corresponding to the minimal surface equation. Indeed,
Mu = (1 + |Du|2 )∆u − Di uDj uDij u,
thus aij (x, z, p) = (1 + |p|2 )δ ij − pi pj is independent of z and b = 0.
Definition 5.1. Consider u ∈ C 2 (Ω) ∩ C 0 (Ω̄) satisfying Qu = 0. Suppose that in a neighbourhood
N = Nx0 of a point x0 ∈ ∂Ω there exist functions w± ∈ C 2 (N ∩ Ω) ∩ C 1 (N ∩ Ω̄) 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 Q
and the function u.
Intuitively, the existence of barriers suggests that a bound on |Du(x0 )| should exist as we are ”squeezing“ the solution u on ∂Ω by C 1 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 ∈ ∂Ω
there exists barriers wx±0 . 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
u(x) − u(x0 )
w+ (x) − w+ (x0 )
w− (x) − w− (x0 )
≤
≤
.
|x − x0 |
|x − x0 |
|x − x0 |
It follows that these inequalities pass to normal derivatives; let ν(x0 ) denote the inward pointing normal
to ∂Ω at x0 , then
u(x0 + hν(x0 )) − u(x0 )
∂u
lim
=
(x0 ) = hDu(x0 ), ν(x0 )i
h→0
h
∂ν
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 for
the more general techniques developed later. The construction of barriers will clearly be dependent on
the 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 function
d : Rn → R by d(x) = dist (x, Ω) = inf y∈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 C 2 , then Ω satisfies an interior sphere condition i.e. given a point
x0 ∈ ∂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 = (x0 , xn ) ∈ Rn
we may, possibly after a rotation of coordinates, write Br (0) ∩ Ω = {(x0 , xn ) ∈ Br (0) : xn > ψ(x0 )},
where r > 0 and ψ : Brn−1 → R is some C 2 function satisfying ψ(0) = 0, Dψ(0) = 0. Using Taylor’s
theorem about 0 we see that |ψ(x0 )| ≤ M |x0 |2 for all x0 ∈ Br (0) and some M > 0. Letting e1 , . . . , en
denote the standard basis of Rn we claim that there exists an R ∈ (0, r/2) such that BR (Ren ) ⊂ Ω. If
x ∈ BR (Ren ) then |x − Ren |2 = |x0 |2 + (xn − R)2 < R2 and so |x0 |2 + x2n − 2xn R < 0. From above we
know that xn < ψ(x0 ) ≤ M |x0 |2 < M (2xn R − x2n ) < 2M Rxn . Thus if we choose R < 1/(2M ) no such x
can exist.
To discuss the geometry of ∂Ω we will need the notion of mean curvature.
Definition 5.6. Suppose that ∂Ω ∈ C 2 and let ν(y), T (y) denote the inward unit normal to ∂Ω at y and
the tangent plane to ∂Ω at y respectively. By a rotation of coordinates we may assume that, for a fixed
y0 ∈ ∂Ω, the xn coordinate axis is parallel to ν(y0 ). Thus in a neighbourhood N of y0 we may represent
the boundary by a map ϕ ∈ C 2 (T (y0 ) ∩ N ). That is ∂Ω ∩ N = {(y 0 , yn )|yn = ϕ(y 0 )} with Dϕ(y00 ) = 0.
The curvature of ∂Ω at y0 is then described by the eigenvalues of the Hessian matrix [D2 ϕ(y0 )], which are
known as the principal curvatures κ1 , . . . , κn−1 to the surface at y0 with the corresponding eigenvectors
being 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 be
realised as the maximal and minimal values of the curvature of paths in M that pass through y and can be
obtained by intersecting planes containing ν(y) with M. More generally, an n-dimensional hypersurface
M ,→ Rn+1 can be equipped with a metric g obtained by pulling back the euclidean metric from the
ambient space Rn+1 . This allows one to consider the difference between the Riemannian connections on
˜ on Rn+1 . In the case of hypersurfaces we have
the two spaces, ∇ on M and ∇
˜ X Y = ∇X Y + h(X, Y )ν,
∇
where X, Y ∈ Γ(T M) are sections of the tangent bundle of M, ν(p) denotes a unit normal to Tp M1 and
h is a symmetric two-tensor on T M. We can define the so-called shape operator s on M by raising an
index of h;
h(X, Y ) = g(X, sY ),
it follows that s is a self-adjoint endomorphism of Tp M for any p and thus must have real eigenvalues
κ1 , . . . , κn . These eigenvalues are the principal curvatures in such a context and it could be checked that
their definition agrees with that given above.
Definition 5.8. The mean curvature of ∂Ω at y0 is given by
H(y0 ) =
n−1
1
1 X
κi =
∆ϕ(y00 ),
n − 1 i=1
n−1
where the last equality follows by the invariance of the trace of a matrix and y0 = (y00 , ϕ(y00 )).
To aid computations we will work in a principal coordinate system: by a further rotation of coordinates
we may assume that the x1 , . . . , xn−1 axes lie along principal directions as the Hessian is a symmetric
matrix and hence a basis of its eigenvectors can be formed. In such a coordinate system we have
[D2 ϕ(y0 )] = diag(κ1 , . . . , κn−1 ).
Next we wish to compute an explicit expression for the unit normal to ∂Ω at some y = (y 0 , ϕ(y 0 )) ∈ ∂Ω.
Clearly the surface is realised as the zero level set of the function F (y) = yn − ϕ(y 0 ) and thus
DF (y) = (−D1 ϕ(y 0 ), . . . , −Dn−1 ϕ(y 0 ), 1)
is normal to the surface. Hence we may take our unit normal to have components
−Di ϕ(y 0 )
νi (y) = p
, i = 1, . . . , n − 1,
1 + |Dϕ(y 0 )|2
1 We
have fixed an orientation on M.
14
νn (y) = p
1
1 + |Dϕ(y 0 )|2
.
Let us denote ν̄(y 0 ) = ν(y). Then, recalling that Dϕ(y00 ) = 0, for i = 1, . . . , n − 1 we can compute
Dj ν¯i (y00 ) = −Dij ϕ(y00 ) = −κi δ ij .
(5.1)
With these preliminaries in hand we can now prove the following:
Proposition 5.9. Suppose that Ω is bounded and ∂Ω is C 2 , then there exists a µ > 0 such that d is C 2
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 that
B̄ ∩ ∂Ω = x0 with the radii of the balls being bounded from below by a positive constant, which we take
to be µ. It can be shown that 1/µ bounds the principal curvatures from above. For every x ∈ Γµ there
exists 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 show
that (5.2) determines y and d as C 1 functions of x. Define the function g : (T (y0 ) ∩ N ) × R → Rn by
g(y 0 , d) = y + ν(y)d.
Clearly g is C 1 and by (5.1) its Jacobian at the point (y00 , d(x)) takes the form
[Dg] = diag(1 − κ1 d, . . . , 1 − κn−1 d, 1).
(5.3)
We conclude that det(Dg) = Πn−1
i=1 (1 − κi d) > 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 0 is locally a C 1 mapping of x. Then
Dd(x) =
x−y
= ν(y 0 (x))
|x − y|
is also a C 1 mapping of x and so d ∈ C 2 (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 |. Then
in a priciple coordinate system at y0
−κ1
−κn−1
2
[D d(x0 )] = diag
,...,
,0 .
1 − κ1 d
1 − κn−1 d
Proof. Note that Dd(x0 ) = ν(y0 ) = (0, . . . , 0, 1), which implies that Din d(x0 ) = 0 for i = 1, . . . , n.
Moreover, by the chain rule, Dij d(x0 ) = Dj (νi ◦ y)(x0 ) = Dk ν¯i (y0 )Dj yk (x0 ). This is zero unless i = j,
in which case it takes the desired form −κi /(1 − κi d) by (5.3) and (5.1).
This result shows us that
∆d(x) = −
n−1
X
i=1
n−1
X
κi
κi = −(n − 1)H(y).
≤
1 − κi d
i=1
(5.4)
Let us suppose that ∂Ω is C 2 so that we can find µ such that d is C 2 on Γµ . We seek a barrier of
the form v(x) = ϕ(x) + (ψ ◦ d)(x), where ϕ ∈ C 2 , ψ ∈ C 2 ([0, a]) for some a < µ to be determined
later. Taking ψ(0) = 0 implies that v(y0 ) = ϕ(y0 ). We also impose the conditions ψ 0 ≥ 1, ψ 00 < 0. Then
Di v = Di ϕ + ψ 0 (d)Di d and Dij v = Dij ϕ + ψ 0 (d)Dij d + ψ 00 (d)Di dDj d. After some computation we arrive
at
Mv = (1 + |Dv|2 )∆v − Di vDj vDij v
= (1 + |Dϕ|2 )∆ϕDi ϕDj ϕDij ϕ
+ ψ 0 (d)[2Di d∆ϕ + (1 + |Dϕ|2 )∆d − Di dDj ϕDij ϕ − Di ϕDj ϕDij d]
+ ψ 0 (d)2 [∆ϕ + 2Di ϕDi d∆d − Dij ϕDi dDj d] + ψ(d)3 ∆d
+ ψ 00 (d)[1 + |Dϕ|2 − (Di ϕDi d)2 ],
15
having made use of the fact that |Dd| = 1 and hence Di dDij d = 0. By the Cauchy-Schwarz inequality
we find 1 + |Dϕ|2 − (ϕi di )2 ) ≥ 1 + |Dϕ|2 − |Dϕ|2 |Dd|2 ≥ 1 and, since ψ 00 < 0, the last term can be
bounded above by ψ 00 . Also ϕ and d are C 2 in Γa = {x ∈ Ω : d(x) < a} and hence their derivatives up
to and including those of second order can be bounded above by constants. Finally, we can control the
ψ 0 term by ψ 02 as ψ 0 ≥ 1. Together these remarks lead to
Qv ≤ ψ 00 + Cψ 02 + ψ 03 ∆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 00 + Cψ 02 . We would like to show that Qv < 0.
Solving the differential equation ψ 00 (d) + Cψ 0 (d)2 = 0 yields
ψ(d) =
1
log(1 + kd).
ν
We can therefore ensure that v is an upper barrier if
ψ(a) =
1
log(1 + ka) = sup |u| = M, or ka = eνM − 1,
ν
Ω
ψ 0 (d) =
k
k
k
≥
= νM ≥ 1,
ν(1 + kd)
ν(1 + ka)
νe
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 ∈ C 2 satisfies a nonnegative mean curvature condition at each point of ∂Ω. Let u ∈ C 2 (Ω) ∩ C 1 (Ω̄) satisfy Mu = 0 in Ω and
u = ϕ on ∂Ω, where ϕ ∈ C 2 (Ω̄). Then we have
|Du| ≤ C on ∂Ω
.
This result allows us to control supΩ |Du| using the gradient estimate (4.3) for the minimal surface
equation, 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 general
second order quasilinear elliptic equations for domains that satisfy either an exterior sphere or an exterior
hyperplane 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) with
x0 ∈ B̄ ∩ Ω̄ = B̄ ∩ ∂Ω = x0 . We define the distance function in such an instance by d(x) = dist (x, ∂B).
Observe that
d(x) = |x − y| − R, Di d(x) =
xi − yi
, Dij d(x) = |x − y|−3 (|x − y|2 δ ij − (xi − yi )(xj − yj )).
|x − y|
(5.6)
Similarly, a domain Ω is said to satisfy an exterior plane condition at the point x0 ∈ ∂Ω if there exists a
hyperplane P with x0 ∈ P ∩ ∂Ω. Note that this will hold for convex domains in particular. We define a
distance function in this case by d0 (x) = dist (x, P) for x ∈ Rn , which satisfies Dij d0 = 0. Indeed, let us
suppose that after a rotation of coordinates xn coincides with the normal direction to P. Then Di d0 = 0
for i 6= n and Dn d0 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 the
time being. Thus u|∂Ω = 0. As before, we propose to search for an upper barrier of the form v = ψ ◦ d, 2
where ψ ∈ C 2 (I) for some interval I ⊂ R. We also impose the restriction ψ 0 6= 0 for reasons that will
become apparent later. Let us pause to introduce the scalar function ε(x, z, p) = aij (x, z, p)pi pj 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 ψ
we compute
Qv = ψ 0 aij Dij d + aij ψ 00 Di dDj d + b
ψ 00
ε+b
(ψ 0 )2
n−1 0
ψ 00
≤
ψ Λ + b + 0 2 ε,
R
(ψ )
= ψ 0 aij Dij d +
where the last line follows from (5.6) and ε is being evaluated at (x, ψ(d), ψ 0 (d)Dd). For v to be a
subsolution in some neighbourhood N of the boundary point, which we will again take to be of the form
Γa , 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
00
ψ
+
ν
ε,
Qv ≤
(ψ 0 )2
when ψ 0 > µ and where ν = 1 + (n − 1)/R. Indeed, observe that |p| = |ψ 0 Dd| = ψ 0 and
n−1 0
ψ Λ + b ≤ max ((n − 1)/R, 1) (ψ 0 Λ + b) ≤ (1 + (n − 1)/R)(ψ 0 Λ + |b|) ≤ νε.
R
A judicious choice of ψ can be used to meet the barrier conditions. One necessary condition is that
ψ(0) = 0 as u vanishes on the boundary and d(x0 ) = 0 for any boundary point. As stated above, we
take 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) = ν1 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 (0) = µeνM .
(5.8)
We now wish to extend these results to the non-zero boundary value case. Let ϕ ∈ C 2 (Ω̄) and suppose
u|∂Ω = ϕ. Let w = v + ϕ so that
Q̃v = Qw = Qv + aij Dij ϕ.
Define the function F by
F(x, z, p, q) = aij (x, z, p)(pi − qi )(pj − qj ),
then for the transformed version of the operator Q̃ we have ε̃(x, v, Dv) = F(x, w, Dw, Dϕ). The corresponding transformed version of the above structure constraint then has the form
(|p − Dϕ| + |D2 ϕ|)Λ + |b| ≤ µ̄(|z|)F(x, z, p, Dϕ),
(5.9)
whenever |p − Dϕ| ≥ µ̄(|z|) for some non-increasing µ̄. These considerations lead to
Theorem 5.12. Let u ∈ C 2 (Ω) ∩ C 1 (Ω̄) satisfy Qu = 0 in Ω and u = ϕ on ∂Ω. Suppose that Ω satisifies
a uniform exterior sphere condition and ϕ ∈ C 2 (Ω̄). 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 data
we have
Qv = ψ 0 aij Dij d + aij ψ 00 Di dDj d + b.
=
ψ 00
ε + b,
(ψ 0 )2
17
Then we can impose a structure condition by assuming that b = O(ε) so that for some non-decreasing
function µ we have
|b| ≤ µ(|z|)ε,
for |p| ≥ µ(|z|). To extend this to the non-zero boundary data case we require Λ|D2 ϕ|, b = O(F), i.e.
Λ|D2 ϕ| + |b| ≤ µ̄(|z|)F,
(5.10)
for |p − Dϕ| ≥ µ̄(|z|). We then obtain the following theorem on boundary gradient estimates
Theorem 5.13. Let u ∈ C 2 (Ω) ∩ C 1 (Ω̄) satisfy Qu = 0 in Ω and u = ϕ on ∂Ω and suppose that Ω is
convex. 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 cases
possible to bound |Du| on ∂Ω uniformly, which in turn allows one to bound |Du|0;Ω . In order to make
use of the Leray–Schauder existence theory laid out in Section 2, it remains to show Hölder continuity
of Du, which we will address in the next section.
6
De Giorgi–Nash–Moser Theory
The De Giorgi–Nash–Moser theory provides Hölder estimates and the Harnack inequality for uniformly
elliptic partial differential equations. The main theorem on Hölder continuity was first obtained independently by Ennio De Giorgi [1] and John Nash [6] in 1957. Three years later, a different proof was
given by Jürgen Moser [5]. We will follow here the argument as presented in [4] with minor modifications
and simplifications to adapt to the present context. Our aim is to prove the following result:
Theorem 6.1 (Interior Hölder Estimate). Suppose aij ∈ L∞ (B1 ) is such that
λ|ξ|2 ≤ aij (x)ξi ξj ≤ Λ|ξ|2 ,
∀x ∈ B1 , ξ ∈ Rn
1
for some positive constants λ, Λ > 0. If u ∈ Hloc
(B1 ) is a weak solution in B1 ,
Z
aij Di uDj ϕ = 0, ∀ϕ ∈ H01 (B1 ),
B1
then
Λ
|u(x) − u(y)|
≤ c n,
kukL2 (B1 )
sup |u(x)| + sup
|x − y|α
λ
x,y∈B 1
B1
2
with α = α n,
Λ
λ
∈ (0, 1) and c n,
2
Λ
λ
> 0.
This theorem can be obtained in several ways. De Giorgi and Moser used two very different approaches that are at the origin of a powerful theory which finds its application in the regularity study of
elliptic PDEs. Some of the intermediate results, such as Moser’s Harnack Inequality 6.10, are of great
interest in their own right and have lead to a number of theorems on quasilinear elliptic equations as we
will see later. We will present the De Giorgi–Nash–Moser theory here as a sequence of theorems leading
to theorem 6.1, however, it should rather be understood as a technique of finding suitable test functions
and 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 ∈ H 1 (B1 ) in terms of
its Lp -norm. For completeness, we will include here both proofs as they contain conceptually different
ideas that deserve to be mentioned.
18
Theorem 6.2 (Supremum bound). Suppose aij ∈ L∞ (B1 ) such that kaij kL∞ (B1 ) ≤ Λ and
aij (x)ξi ξj ≥ λ|ξ|2 ,
∀x ∈ B1 , ξ ∈ Rn .
If u ∈ H 1 (B1 ) is a subsolution in B1 in the sense that
Z
aij Di uDj ϕ ≤ 0, ∀ϕ ∈ H01 (B1 ), ϕ ≥ 0,
(6.1)
B1
then u+ ∈ L∞
loc (B1 ) and for any θ ∈ (0, 1) and any p > 0,
sup u+ ≤
Bθ
C
ku+ kLp (B1 ) ,
(1 − θ)n/p
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 to
De 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 ∈ C01 (B1 ) to be chosen later. Let us
take the test function ϕ := vζ 2 . Note that in {u > k} we have v = u − k and Dv = Du a.e., whereas
v = 0, Dv = 0 a.e. in {u ≤ k}. In the following, all integrals are taken over {u > k} ∩ B1 . Substituting
the test function ϕ into the equation, we get
Z
Z
Z
ij
ij
2
0 ≥ a Di uDj ϕ = a Di vDj vζ + 2 aij Di vDj ζvζ
Z
Z
≥λ |Dv|2 ζ 2 − 2Λ |Dv||Dζ|vζ
Z
Z
2Λ2
λ
|Dv|2 ζ 2 −
|Dζ|2 v 2 .
≥
2
λ
Hence
Z
4Λ2
|Dv| ζ ≤ 2
λ
2 2
Z
|Dζ|2 v 2 .
and we obtain an energy estimate for v,
Z
Z
Z
Z
|D(vζ)|2 ≤ 2 |Dv|2 ζ 2 + 2 |Dζ|2 v 2 . |Dζ|2 v 2 .
(6.2)
It follows that
Z
2
Z
2∗
22∗
2
(vζ)
|{vζ 6= 0}|1− 2∗
Z
2
≤c(n) |D(vζ)|2 |{vζ 6= 0}| n
Z
2
. |Dζ|2 v 2 |{vζ 6= 0}| n ,
(vζ) ≤
where we used Hölder’s inequality and the Sobolev inequality
Z
∗
ψ2
22∗
Z
≤ c(n)
|Dψ|2
(6.3)
for any ψ ∈ H01 (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 ∈ C01 (B1 ) such that for 0 < r < R ≤ 1, ζ ∈ C0∞ (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
Z
Z
(u − k)2 ≤
v2 ζ 2
A(k,r)
A(k,1)
Z
2
.
|Dζ|2 v 2 |{vζ 6= 0}| n
A(k,1)
Z
2
1
.
(u − k)2 |A(k, R)| n .
(6.4)
(R − r)2 A(k,R)
R
Let k0 = ku+ kL2 (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
Z
1
1
|A(k, R)| ≤
u+ ≤ ku+ kL2 (?) .
k A(k,R)
k
Fix k ≥ k0 . For any h, r satisfying h ≥ k, 0 < r < 1, we have A(k, r) ⊃ A(h, r). It follows that
Z
Z
2
(u − k)2
(u − h) ≤
A(k,r)
A(h,r)
and
1
(h − k)2
|A(h, r)| = |Br ∩ {u − k > h − k}| ≤
Z
(u − k)2
A(k,r)
Let θ ∈ (0, 1). We conclude from (6.4) that for any h > k ≥ k0 and θ ≤ r < R ≤ 1,
Z
1
(u − h) .
(R
−
r)2
A(h,r)
Note that
2
R
A(k,r)
Z
2
(u − h) |A(h, R)|
2
n
.
2
1
Z
(u − k)
2
(R − r)(h − k) n
A(h,R)
!1+ n2
2
.
A(k,R)
(u − k)2 = k(u − k)+ k2L2 (Br ) , and hence
k(u − h)+ kL2 (Br ) .
1
2
k(u − k)+ kL2 (BR ) .
2
ψ(k, R).
(R − r)(h − k) n
Using the notation ψ(k, r) := k(u − k)+ kL2 (Br ) , we obtain
ψ(h, r) .
1
(R − r)(h − k) n
(6.5)
The idea behind De Giorgi’s proof is to take smaller and smaller radii r while taking bigger and bigger
k, 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
1
km =k0 + k 1 − m
2
1
rm =θ + m (1 − θ) ,
2
so km ≥ k0 for all m and km ↑ k0 + k, rm ↓ θ. Note that
km − km−1 =
k
,
2m
rm−1 − rm =
1−θ
2m
and hence by (6.5),
ψ(km , rm ) .
2m
1−θ
2m
k
n2
20
2
ψ(km−1 , rm−1 )1+ n .
(6.6)
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
2
2
1+ n
ψ(km−1 , rm−1 )
ψ(k0 , r0 ) n ψ(k0 , r0 )
≤ (m−1) 2 −1
.
γm
n
γ
Using our previous estimate (6.6), we obtain that for some constant C > 0,
2
ψ(km , rm ) ≤ C
2
2m( n +1) ψ(k0 , r0 ) n ψ(k0 , r0 )
.
2
2
γm
(1 − θ)k n γ (m−1) n −1
First, let us choose γ 2/n = 21+2/n , which indeed satisfies γ > 1. Next, we choose k = k ∗ (θ) ≥ 0,
2
2
2
(k ∗ (θ)) n = C
γ 1+ n ψ(k0 , r0 ) n
.
1−θ
With these choices of γ and k, claim (6.7) follows by induction.
Finally, we let m → ∞ in (6.7). We obtain
k(u − k0 − k ∗ (θ))+ kL2 (Bθ ) = ψ(k0 + k ∗ (θ), θ) = 0,
in other words
sup u+ . k0 + k ∗ (θ)
Bθ
. k0 +
.
1
+
n k(u − k0 ) kL2 (B1 )
(1 − θ) 2
1
+
n ku kL2 (B1 ) .
(1 − θ) 2
(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 obtain
a bound on the Lp1 -norm of u inside a ball of radius r1 in terms of the Lp2 -norm on a larger ball with
radius r2 ,
1
kukLp2 (Br2 ) ,
(6.9)
kukLp1 (Br1 ) .
r2 − r1
for p1 > p2 and r1 < r2 . This will be the set-up for an iterative argument and with careful choices of
radii {ri } and exponents {pi }, we will arrive at the desired sup-bound for the case p = 2.
For k, m > 0, define ū = u+ + k and
(
ūm =
ū
k+m
if u < m,
if u ≥ m.
Then clearly k ≤ ūm ≤ k + m and ūm ≤ ū. In {u < 0} and {u > m}, ūm is constant and so Dūm = 0.
Let us choose the test function
ϕ = ζ 2 ūβm ū − k β+1 ∈ H01 (B1 )
for some β ≥ 0 and some non-negative cut-off function ζ ∈ C01 (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
β−1
Dϕ =2ζDζ ūβm ū − k β+1 + ζ 2 β ūm
Dūm ū + ζ 2 ūβm Dū
=2ζDζ ūβm ū − k β+1 + ζ 2 ūβm (βDūm + Dū)
21
since ūm = ū on {Dūm 6= 0}. Note further that ϕ = 0 and Dϕ = 0 in {u ≤ 0}. Hence all the integrals
will be over B1 ∩ {u > 0}. Trivially u+ ≤ ū and ūβm ū − k β+1 ≤ ūβm ū. It follows that for any k > 0,
Z
Z
Z
ij
ij
2 β
a Di uDj ϕ = a Di ū (βDj ūm + Dj ū) ζ ūm + 2 aij Di ūDj ζ ūβm ū − k β+1 ζ
Z
Z
Z
≥λβ ζ 2 ūβm |Dūm |2 + λ ζ 2 ūβm |Dū|2 − Λ |Dū| |Dζ|ūβm ūζ
Z
Z
Z
2Λ2
λ
ζ 2 ūβm |Dū|2 −
|Dζ|2 ūβm ū2 .
≥λβ ζ 2 ūβm |Dūm |2 +
2
λ
Since u is a subsolution to equation (6.1), we conclude
Z
Z
Z
β |Dūm |2 ζ 2 ūβm + |Dū|2 ζ 2 ūβm . |Dζ|2 ūβm ū2 .
β/2
Let us define w = ūm ū. Note that
|Dw|2 ≤ (1 + β) β ūβm |Dūm |2 + ūβm |Dū|2 .
Hence, we obtain the following energy estimate for w:
Z
Z
|Dw|2 ζ 2 . (1 + β) |Dζ|2 w2 .
Let χ = n/(n − 2) > 1 if n > 2 and χ > 2 arbitrary if n = 2. Then by Sobolev inequality (6.3),
Z
|ζw|2χ
χ1
Z
|D(wζ)|2 . (1 + β)
.
Z
|Dζ|2 w2 .
Next, we choose the cut-off function ζ ∈ C01 (B1 ) such that for 0 < r < R ≤ 1, ζ ∈ C0∞ (BR ) with ζ ≡ 1
in Br , 0 ≤ ζ ≤ 1 in BR and |Dζ| ≤ 2/(R − r) in BR . We obtain
Z
w
2χ
χ1
Z
(1 + β)
(R − r)2
.
Br
w2 .
BR
Setting γ = β + 2 ≥ 2 and recalling that ūm ≤ ū, we obtain
Z
ūγχ
m
χ1
Br
(γ − 1)
.
(R − r)2
Z
ūγ ,
BR
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 ū for different exponents q and radii r,
kūkLγχ (Br ) .
(γ − 1)
)
(R − r)2
γ1
kūkLγ (BR ) .
(6.10)
assuming kūkLγ (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
(1 − θ),
2i
so γi ↑ ∞ and ri ↓ θ as i → ∞. Note that
γi = χγi−1 ,
ri−1 − ri =
22
1
(1 − θ).
2i
Then using (6.10), we have
1
(γi−1 − 1)4i γi−1
)
kūkLγi−1 (Bri−1 )
.
(1 − θ)2
γ 1
i Y
k−1
1
. c(n, i)
kūkL2 (B1 ) ,
2
(1 − θ)
kūkLγi (Bri )
k=1
where c(n, i) is defined as
i
Y
c(n, i) =
(γk−1 − 1) 4k
γ
1
k−1
.
k=1
Provided c(n, ∞) < ∞, we can take the limit i → ∞ and obtain a sup bound on ū,
kūkL∞ (Bθ ) .
since
∞
Y
(1 − θ)
− γ2
k
=
k=0
1
n kūkL2 (B1 )
(1 − θ) 2
1
(1 − θ)2
P∞
k=0
1
γk
=
(6.11)
1
n .
(1 − θ) 2
This completes Moser’s proof of theorem (6.2) for the case p = 2.
Generalisation p ≥ 2:
Let y ∈ B1 and R > 0. Define ũ(y) = u(Ry), ãij (y) = aij (Ry). Assume ãij satisfies the assumptions
of theorem 6.2 with subsolution ũ in B1 . Then u is a subsolution in BR with coefficient matrix aij .
Applying the result for p = 2, θ ∈ (0, 1) to ũ, we have
1
+
n kũ kL2 (B1 ) .
(1 − θ) 2
sup u+ = sup ũ+ .
BθR
Bθ
Then for θ = 1/2 and for any p ≥ 2:
sup u+ . kũ+ kL2 (B1 ) .
BR/2
1
+
n ku kLp (BR ) .
Rp
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
1
+
sup
u+ .
n ku kLp (B(1−θ)r (y)) .
p
((1
−
θ)r)
B(1−θ)r/2 (y)
Covering Brθ with finitely many such balls centered at (yj )N
j=1 , we take the maximum on both sides,
max
sup
j=1,...,N B
(1−θ)r/2 (yj )
u+ . max
j=1,...,N
1
+
n ku kLp (B(1−θ)r (yj ))
((1 − θ)r) p
.
N
X
1
ku+ kLp (B(1−θ)r (yj ))
n
((1 − θ)r) p j=1
.
N
+
n ku kLp (Br ) .
((1 − θ)r) p
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
p 1
1
p
a + bp
a1− 2 b 2 ≤ 1 −
2
(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 positive
constant. If for all s, t such that τ0 ≤ t < s ≤ τ1 ,
f (t) ≤ αf (s) +
then
f (t) ≤ c(α, β)
A
,
(s − t)β
A
.
(s − t)β
for some positive constant c(α, β) > 0.
Proof
For 0 < τ < 1, consider the sequence {ti }i≥0 ,
t0 = t,
ti+1 = ti + (1 − τ )τ i (s − t).
By iteration,
f (t) = f (t0 ) ≤ αk f (tk ) +
k−1
X
A
αi τ −iβ .
β
β
(1 − τ ) (s − t) i=0
Next, choose τ < 1 such that ατ −β < 1, i.e. α < τ β < 1. Letting k → ∞, we obtain
A
1
f (t) ≤
,
(1 − τ )β (s − t)β 1 − ατ −β
which proves the lemma.
By rescaling inequality (6.8) we have for any θ ∈ (0, 1), 0 < R ≤ 1,
C
+
n ku kL2 (BR ) .
((1 − θ)R) 2
ku+ kL∞ (BθR ) ≤
Note that for p ∈ (0, 2),
Z
+ 2
(u ) ≤
BR
ku+ k2−p
L∞ (BR )
Z
(u+ )p
BR
and so applying Lemma 6.3, we obtain
1− p
ku+ kL∞ (BθR ) ≤ku+ kL∞2(BR )
C
+
n ku kL2 (BR )
((1 − θ)R) 2
2
p +
Cp
+
≤ 1−
ku kL∞ (BR ) +
n ku kLp (BR )
2
((1 − θ)R) p
2
p +
Cp
+
ku kL∞ (BR ) +
≤ 1−
n ku kLp (B1 ) .
p
2
((1 − θ)R)
Setting f (t) = ku+ kL∞ (Bt ) for any t ∈ (0, 1], this rewrites as: for any 0 < r < R ≤ 1,
2
p
Cp
+
f (r) ≤ 1 −
f (R) +
n ku kLp (B1 ) .
2
(R − r) p
24
By Lemma 6.4, there exists a constant c(n, p) > 0 such that
f (r) ≤ c(n, p)
1
+
n ku kLp (B1 ) .
(R − r) p
Taking R → 1− and renaming r = θ, we conclude
sup u+ .
Bθ
1
+
n ku kLp (B1 ) .
(1 − θ) p
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 theorem
6.1 by bounding inf u below (see theorem 6.5) which implies control of oscillations (Oscillation Theorem
6.8). Contrary to this direct method, Moser proved a weak Harnack inequlity (see theorem 6.9), which
is a much more powerful result then theorem 6.5. It not only implies the Oscillation Theorem 6.8, but
also a number of other results such as Liouville’s theorem and a weak maximum principle for quasilinear
elliptic 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)ξi ξj ≤ Λ|ξ|2 ,
∀x ∈ B2 , ∀ξ ∈ Rn ,
1
and suppose u ∈ Hloc
(B2 ) is a positive supersolution in B2 ,
Z
aij Di uDj ϕ ≥ 0, ∀ϕ ∈ H01 (B2 ), ϕ ≥ 0.
B2
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
1
inf u ≥ .
B1
C
2
To prove this result, we require the following two lemmata:
0,1
1
Lemma 6.6. Let Φ ∈ Cloc
(R) be convex. If u ∈ Hloc
(Ω) is a subsolution in Ω,
Z
aij Di uDj ϕ ≤ 0, ∀ϕ ∈ H01 (Ω),
Ω
0
1
and Φ ≥ 0, then v = Φ(u) is also a subsolution provided v ∈ Hloc
(Ω).
2
Proof. Assuming Φ ∈ Cloc
(R) with Φ0 (s) ≥ 0, Φ00 (s) ≥ 0, we consider the test function ϕ ∈ C01 (B1 ) with
0
1
ϕ ≥ 0. Then Φ (u)ϕ ∈ H0 (B1 ) is non-negative and can be taken as a test function for the equation on
u,
Z
Z
ij
a Di vDj ϕ =
aij Di uDj ϕΦ0 (u)
B1
B1
Z
Z
ij
0
=
a Di uDj (Φ (u)ϕ) −
aij Di uDj uϕΦ00 (u)
B1
B1
Z
Z
ij
0
≤
a Di uDj (Φ (u)ϕ) − λ
|Du|2 ϕΦ00 (u) ≤ 0.
B1
0,1
general convex Φ ∈ Cloc
(R), let
Φε (u) is a subsolution and Φ0ε (s)
For a
Then
convergence theorem.
B1
us instead take Φε (s) = ρε ∗ Φ(s) with ρε the standard mollifier.
→ Φ0 (s) a.e. as ε → 0. The result follows by Lebesgue’s dominant
25
Lemma 6.7. For any ε > 0, if u ∈ H 1 (B1 ) satisfies |{u = 0} ∩ B1 | ≥ ε|B1 |, then there exists a positive
constant C = C(ε, n) such that
Z
Z
u2 ≤ C
|Du|2 .
B1
B1
Proof. Suppose the lemma is false. Then there exists ε > 0 and a sequence (um )m≥0 ⊂ H 1 (B1 ) such
that
Z
Z
|{um = 0} ∩ B1 | ≥ ε|B1 |,
u2m ≥ m
|Dum |2 .
B1
B1
Normalising, we get
Z
Z
u2m = 1
and
|Dum |2 = 0
lim
m→∞
B1
B1
So the sequence (um )m≥0 converges weakly in H 1 (B1 ) for some u0 ∈ H 1 (B1 ), and since we must have
ku0 kL2 (B1 ) = 1, it also converges strongly in L2 . Clearly u0 6= 0 constant a.e. We conclude
Z
0 = lim
|um − u0 |2
m→ B
Z 1
≥ lim
|um − u0 |2
m→ {u =0}∩B
1
Z m
2
|{um = 0}|
≥|u0 |
m
>0.
Contradiction.
Proof of Theorem 6.5: Let us assume u ≥ δ > 0. Define v = (log u)− , where the negative part
of a function f is defined as f − = max{0, −f }. Then by Lemma 6.6, v is a subsolution, bounded by
log(1/δ). Hence, we can bound the supremum using Theorem 6.2,
sup v ≤ CkvkL2 (B1 ) .
B1/2
Since |B1 ∩ {v = 0}| = |B1 ∩ {u ≥ 1}| ≥ ε|B1 |, we can apply Lemma 6.7 to obtain
sup v ≤ CkDvkL2 (B1 ) .
B1/2
In order to show that the RHS is bounded, let us take the test function ϕ = ζ 2 /u for ζ ∈ C01 (B2 ),
Z
Z
Z
ζ
ζ2
0≤
aij Di uDj ϕ = 2
aij Di uDj ζ −
aij Di uDj u 2 ,
u
u
B2
B2
B2
and hence
Z
|Du|2
B2
ζ2
.
u2
Z
|Dζ|2 ,
B2
which implies
Z
ζ 2 |D(log u)|2 .
B2
Z
|Dζ|2 .
B2
Next,
we choose the cut-off function ζ ∈ C01 (B2 ) such that ζ ≡ 1 in B1 , |Dζ| ≤ 2, and hence
R
|D(log u)|2 is bounded by a constant only depending on λ, Λ, n. It follows that
B1
sup v . kDvkL2 (B1 ) . kD(log u)kL2 (B1 ) ≤ C(n, λ, Λ) = C,
B1/2
and so we can conclude
inf u ≥ e−C > 0.
B1/2
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)ξi ξj ≤ Λ|ξ|2 ,
1
If u ∈ Hloc
(B2 ) is a bounded solution in B2 ,
Z
aij Di uDj ϕ = 0,
∀x ∈ B2 , ∀ξ ∈ Rn .
∀ϕ ∈ H01 (B2 ),
B2
then there exists γ = γ(n, λ, Λ) ∈ (0, 1) such that
osc u ≤ γ oscu,
B1/2
B1
where osc = sup − inf.
Proof : Let us define
α1 = sup u,
β1 = inf u,
B1
B1
α2 = sup u,
β2 = inf u,
B1/2
B1/2
and the solutions
v=
u − β1
≥ 0,
α1 − β1
w=
α1 − u
≥ 0.
α1 − β1
We consider the following two cases seperately:
1
1
(α1 + β1 ) ⇔ v ≥ ,
2
2
1
1
u ≤ (α1 + β1 ) ⇔ w ≥ .
2
2
u≥
Case 1 : If v ≥ 1/2, then
|{2v ≥ 1} ∩ B1 | = |B1 | ≥
1
|B1 |.
2
So by Theorem 6.5 there exists C > 1 such that
β2 − β1
1
≥ .
α1 − β1
C
Case 1 : If w ≥ 1/2, then
|{2w ≥ 1} ∩ B1 | = |B1 | ≥
1
|B1 |.
2
So by theorem 6.5 there exists C > 1 such that
α1 − α2
1
≥ .
α1 − β1
C
In both cases, since β2 ≥ β1 and α2 ≤ α1 , we obtain
1
α2 − β2 ≤ 1 −
(α1 − β1 ).
C
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)ξi ξj ≤ Λ|ξ|2
for some positive constants λ, Λ > 0. If u ∈ H 1 (Ω) is a non-negative supersolution,
Z
aij Di uDj ϕ ≥ 0, ∀ϕ ∈ H01 (Ω), ϕ ≥ 0,
Ω
then for any BR ⊂ Ω, for any 0 < p < n/(n − 2) and any 0 < θ < τ < 1, we have
inf u ≥
BθR
C
n kukLp (Bτ R ) ,
Rp
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 direct
consequence 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)ξi ξj ≤ Λ|ξ|2
for some positive constants λ, Λ > 0. If u ∈ H 1 (Ω) is a solution,
Z
aij Di uDj ϕ = 0, ∀ϕ ∈ H01 (Ω),
Ω
then for any BR ⊂ Ω,
sup u ≤ C inf u
BR
BR
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 for
which 6.9 holds true (step 1), and then generalise to any 0 < p < n/(n − 2) (step 2).
Step 1: For k > 0, let ū = u + k and define v = 1/ū. Taking the test function ϕv 2 for any ϕ ∈ H01 (B1 ),
we can derive an equation for v,
Z
aij Di vDj ϕ ≤ 0, ∀ϕ ∈ H01 (B1 )
B1
So v is a subsolution to the equation, and we can thus apply theorem 6.2: for any 0 < θ < τ < 1 and
any p > 0,
sup ≤ CkvkLp (Bτ ) ,
Bθ
in other words,
Z
inf ū ≥ C
Bθ
−p
− p1
ū
Z
=C
Bτ
−p
Z
ū
Bτ
p
− p1 Z
ū
Bτ
ū
Bτ
where C = C(n, p, λ, τ, θ). The aim is to show the existence of a p0 such that
Z
Z
−p0
ū
ūp0 ≤ C(n, λ, τ ).
Bτ
Bτ
This will follow from the following result:
28
p
p1
,
Claim: For all τ ∈ (0, 1) there exist a p0 such that
Z
ep0 |w| ≤ C(n, λ, τ ),
(6.13)
Bτ
where w = log ū − A with A =
1
|Bτ |
R
Bτ
log ū.
Noting that
Z
−p0
Z
p0
ū
Bτ
ū
Z
≤
Bτ
e
po |w|
2
,
Bτ
theorem 6.10 follows after rescaling.
Proof of Claim: Writing
ep0 |w| = 1 + p0 |w| +
1
2
(p0 |w|) + ...
2!
R
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 ) ∩ H01 (B1 ), ϕ ≥ 0, we obtain
Z
Z
Z
Z
1
ij
ij
ij 1
a Di wDj wϕ =
a
Di ūDj ϕ =
aij Di wDj ϕ.
a Di ū Dj ϕ − Dj (ϕv) ≤
(6.14)
ū
ū
B1
B1
B1
B1
If we replace ϕ by ϕ2 , then Hölder’s inequality and uniform ellipticity of aij ∈ L∞ (B1 ) yield for any
ϕ ∈ L∞ (B1 ) ∩ H01 (B1 ),
Z
Z
|Dw|2 ϕ2 .
|Dϕ|2 .
(6.15)
B1
B1
Next, we choose the cut-off function ϕ ∈ C01 (B1 ) such that ϕ ≡ 1 in Bτ for τ ∈ (0, 1). Then
Z
Z
|Dw|2 .
|Dϕ|2 = C(n, λ, Λ, τ ).
Bτ
Since
R
Bτ
B1
w = 0, we can apply Poincaré’s inequality,
Z
w2 ≤ c(n, τ )
Bτ
Z
|Dw|2 ≤ C(n, λ, Λ, τ ).
Bτ
Furthermore, we conclude from (6.15),
Z
w2 ≤ C(n, λ, Λ, τ, τ 0 )
(6.16)
Bτ 0
R
for any τ 0 ∈ (τ, 1). Using these inequalities, we can estimate Bτ |w|β for any real β ≥ 2. Let us define
the truncation of w,

−m w ≤ −m

wm = w
|w| < m


m
w≥m
Substituting the test function ϕ = ζ 2 |wm |2β ∈ H01 (B1 ) ∩ L∞ (B1 ) into the equation on w, (6.14), we
obtain
Z
Z
ij
2
2β
a Di wDj wζ |wm | ≤
aij Di w 2ζDj ζ|wm |2β + ζ 2 2β|wm |2β−1 Dj |wm |
B1
B1
By Young’s inequality,
2β|wm |2β−1 ≤
1
2β − 1
|wm |2β +
(2β)2β =
2β
2β
29
1
1−
|wm |2β + (2β)2β−1 .
2β
Noting that aij Di wDj |wm | = aij Di wm Dj |wm | ≤ aij Di wm Dj wm a.e. in B1 , it follows that
Z
Z
Z
aij Di wDj wζ 2 |wm |2β ≤ 2
aij Di wDj ζ|wm |2β ζ + 2β
aij Di wm Dj wm ζ 2 |wm |2β−1
B1
B1
B1
Z
Z
1
)|wm |2β + (2β)2β−1
≤2
aij Di wDj ζ|wm |2β ζ +
aij Di wm Dj wm ζ 2 (1 −
2β
B1
B1
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
Z
Z
aij Di wDj wζ 2 |wm |2β +
aij Di wDj wζ 2 |wm |2β
{Dwm 6=0}
{Dwm =0}
Z
Z
ij
2β
≤ 4β
a Di wDj ζ|wm | ζ + (2β)2β
aij Di wm Dj wm ζ 2 .
B1
B1
Using ellipticity again, this gives
Z
Z
|Dw|2 ζ 2 |wm |2β . β
B1
|wm |2β ζ|Dw||Dζ| + (2β)2β
Z
B1
ζ 2 |Dwm |2
B1
The first term on the RHS can be dealt with using Cauchy’s inequality, and we obtain
Z
Z
Z
Z
|Dwm |2 ζ 2 |wm |2β ≤
|Dw|2 ζ 2 |wm |2β . β 2
|wm |2β |Dζ|2 + (2β)2β
ζ 2 |Dwm |2 .
B1
B1
B1
(6.17)
B1
In what follows, we drop the m in wm and take the limit m → ∞ at the end, so the final result holds
indeed true for w. We have
Z
Z
Z
β 2
2
2β
2
|D(ζ|w| )| ≤ 2
|Dζ| |w| + 2β
ζ 2 |w|2β−2 |Dw|2
B1
B1
B1
Z
Z
Z
β−1
2β 2
2
2β−1
2
2β
|w| ζ |Dw| + 2β
ζ 2 |Dw|2
≤2
|Dζ| |w| + 2
β
B1
B1
B1
Z
Z
Z
Z
β−1
β−1 2
2
2β
2
2
2β
2
2β−1
2β
|Dζ| |w| +
ζ 2 |Dw|2
.
ζ |Dw| +
|w| |Dζ| + β
(2β)
β
β
β
B1
B1
B1
B1
Z
Z
2
2
2
2β
2β
2
ζ |Dw|
|Dζ| |w| + (2β)
.β
B1
B1
where we used Young’s inequality in the second line and (6.17) in the third line. Then by the Sobolev
inequality (6.3) applied to ζ|w|β ∈ W01,2 (Rn ) and (6.15), we obtain
Z
χ1
Z
2χ
2βχ
ζ |w|
.
|D(ζ|w|β )|2
B1
B1
Z
Z
2
2
2β
2β
ζ 2 |Dw|2
.β
|Dζ| |w| + (2β)
B1
B1
Z
Z
2
2
2β
2β
.β
|Dζ| |w| + (2β)
|Dζ|2
B1
B1
where χ = n/(n − 2) > 1 if n > 2 and χ > 1 arbitrary if n = 2. Now, we can choose a suitable cut-off
function ζ ∈ C01 (B1 ) as follows: for τ ≤ r < R ≤ 1, ζ ∈ C01 (BR ) with ζ ≡ 1 in Br and |Dζ| ≤ 2/(R − r).
Then
Z
χ1
Z
1
2β
2
2β
|w|2βχ
.
(2β)
+
β
|w|
(6.18)
(R − r)2
Br
BR
R
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 0 < 1 and i ∈ Z≥0 , let us define
βi = χ i
ri = τ +
1 0
(τ − τ ),
2i
30
so βi ↑ ∞ and ri ↓ τ as i → ∞. Note that
βi = χβi−1 ,
1 0
(τ − τ ).
2i
ri−1 − ri =
Then using (6.18), we obtain
i−1
i−1 1
Ii ≤ C χi−1 2χi−1 + (χi−1 ) χi−1 Ii−1 ≤ (Cχ) χi−1 2χi−1 + Ii−1 ,
where C = C(n, λ, Λ, τ, τ 0 ) > 0 is a positive constant and where we define
Ii = kwkL2χi (Br ) .
i
Iterating the above inequality we obtain
i−1
Ii ≤ (Cχ) χi−1 2χi−1 + Ii−1
≤
i Y
i
X
k−1
(Cχ) χ−1 2χs−1 +
s=1 k=s
.
i
X
i
Y
k−1
(Cχ) χ−1 I0
k=1
χs−1 + I0
s=1
i
. χ + I0
P∞
since k=0 k/χk < ∞ and χ > 1. Now, for an arbitrary β ∈ N, β ≥ 2, there exists j ∈ N such that
2χj−1 ≤ β ≤ 2χj . It follows, using the above estimate and boundedness of I0 given by (6.16), that
Z
β
β1
. Ij . χj + I0 . β + I0 ≤ Cβ,
|w|
Bτ
and by Stirling’s formula,
Z
|w|β ≤ C β β β ≤ C β eβ β!,
Bτ
where C = C(n, λ, Λ, τ, τ 0 ) is a positive constant. Choose
p0 ≤
1
.
2Ce
Then we arrrive at the desired bound
Z
Bτ
and hence we conclude
Z
Bτ
(p0 |w|)β
1
≤ β,
β!
2
ep0 |w| ≤ 1 +
1
1
+
+ ... = 2,
2 22
which proves the claim.
Remark 6.11. The above proof of claim (6.13) is long but very elementary in nature. There is a
shorter but more indirect proof using the John-Nirenberg Lemma by showing that w has bounded mean
oscillations.
31
Step 2: Let us now generalise the result from a specific p0 to any 0 < p < n/(n − 2). Our aim is to
show that for any 0 < r1 < r2 < 1 and for any 0 < p2 < p1 < n/(n − 2), we have
kūkLp1 (Br1 ) ≤ CkūkLp2 (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 with
0 < p0 ≤ 1 < n/(n − 2) and for any radius τ sucht that 0 < θ < τ < 1, and since p0 can be made
arbitrary small, equation (6.19) implies the more general result (6.9).
Recall that ū = u + k for some k > 0. Take ϕ = ū−β ζ 2 , β ∈ (0, 1), as a test function in theorem 6.9.
Then
Z
Z
0 ≤ −β
aij Di ūDj ū ζ 2 ū−β−1 + 2
aij Di ūDj ζ ū−β ζ
B1
B1
Z
Z
. −β
|Dū|2 ū−β−1 ζ 2 +
|Dū||Dζ|ū−β ζ
B1
B1
Z
Z
1
. −β
|Dū|2 ū−β−1 ζ 2 +
|Dζ|2 ū1−β ,
β B1
B1
and hence for γ = 1 − β ∈ (0, 1)
Z
|Dū|2 ūγ−2 ζ 2 .
B1
Z
1
(1 − γ)2
|Dζ|2 ūγ .
B1
Setting w = ūγ/2 , we arrive at the energy estimate
Z
Z
Z
γ2
1
2 γ
|Dw|2 ζ 2 .
|Dζ|
ū
.
|Dζ|2 w2
2
2
(1
−
γ)
(1
−
γ)
B1
B1
B1
and therefore
Z
|D(wζ)|2 .
B1
1
(1 − γ)2
Z
|Dζ|2 w2 .
B1
Applying Sobolov inequality (6.3) with χ = n/(n − 2) if n > 2 and χ > 1 arbitrary if n = 2, and
choosing the cut-off function ζ ∈ C01 (B1 ) such that ζ ≡ 1 in Br , ζ ∈ C01 (BR ) and |Dζ| ≤ 2/(R − r) for
0 < r < R < 1, we obtain
Z
χ1
Z
Z
1
1
w2χ
≤
|D(wζ)|2 .
w2 .
2 (R − r)2
(1
−
γ)
Br
B1
BR
So we conclude for ū:
Z
χ1
Z
1
1
ūγ .
(6.20)
(1 − γ)2 (R − r)2 BR
Br
Using this inequality, we can do an iterative procedure similar to the approach used in step 1. Given
0 < p2 < p1 < n/(n − 2), let us assume there exists i0 ∈ Q such that p1 = p2 χi0 . If not, the result still
follows by density of Q in R. For 0 < r1 < r2 < 1 and i ∈ Q≥0 , we define
ūγχ
.
pi = p2 χi
1
2i0 r1 − r2
+ i
ri =
2i0 − 1
2
r2 − r1
1 − 21i0
!
,
so r0 = r2 , ri0 = r1 and pi ↑ ∞. From (6.20) it follows that
! χ1
Z
Z
2i (1 − 21i0 )
1
pi
ū
.
ūpi−1
(1 − pi−1 )2 (r2 − r1 )2 Bri−1
Bri
and so after i0 iterations,
kūkLp1 (Br1 ) ≤ C(n, λ, Λ, p1 , p2 , r1 , r2 )kūkLp2 (Br2 )
as claimed. Taking k → ∞, the above statements hold true for any non-negative supersolution u ∈
H 1 (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 which
in term yields Theorem 6.1. Theorem 6.10 also implies a number of other well-known results such as
Liouville’s theorem and a weak maximum principle.
6.3
Global Hölder Estimates
In order to apply the De Giorgi–Nash–Moser theory to the minimal surface equation and obtain a
bound satisfying the conditions of the Leray–Schauder Existence Theorem (2.8), we require a global
Hölder estimate, extending Theorem 6.1 up to the boundary ∂Ω. We will here state, but not prove
the necessary results (for more details, see [2]). Consider the following variation of the interior Hölder
estimate 6.1:
Theorem 6.12 (Interior Hölder estimate I). Suppose aij ∈ L∞ (Ω) is such that
λ|ξ|2 ≤ aij (x)ξi ξj ≤ Λ|ξ|2 ,
∀x ∈ Ω, ξ ∈ Rn
for some positive constants λ, Λ > 0. If u ∈ H 1 (Ω) is a weak solution in Ω,
Z
aij Di uDj ϕ = 0, ∀ϕ ∈ H01 (Ω),
Ω
then we have for any Ω0 ⊂⊂ Ω the estimate
kukC 0,α (Ω̄0 ) ≤ CkukL2 (Ω)
λ
, d0 ), d0 = dist (Ω0 , ∂Ω) and α = α n, Λ
where C = C(n, Λ
λ > 0.
The following theorem combines seperate Hölder estimates in the interior (Theorem 6.12) and on the
boundary into a global Hölder estimate:
Theorem 6.13 (Global Hölder estimate I). Suppose aij ∈ L∞ (Ω) is such that
λ|ξ|2 ≤ aij (x)ξi ξj ≤ Λ|ξ|2 ,
∀x ∈ Ω, ξ ∈ Rn
for some positive constants λ, Λ > 0. Suppose Ω satisfies a uniform exterior cone condition on the
boundary ∂Ω. Then if u ∈ H 1 (Ω) is a weak solution in Ω,
Z
aij Di uDj ϕ = 0, ∀ϕ ∈ H01 (Ω),
Ω
and if there exist K, α0 > 0 such that
osc∂Ω∩BR (x0 ) u ≤ KRα0 ,
∀x0 ∈ ∂Ω, ∀R > 0,
then u ∈ C α (Ω̄), and for any Ω0 ⊂⊂ Ω we have the estimate
kukC α (Ω) ≤ C sup |u| + K,
Ω
λ
where C = C(n, Λ
, α0 , diam Ω), α = α n, Λ
λ , α0 > 0.
In particular, let us suppose the quasililinear elliptic operator Q with smallest and largest eigenvalues
λ, Λ > 0 respectively is of the special divergence form (4.1),
Qu = div A(Du) = 0,
where A ∈ C 1 (Rn ). Then we have shown in Section 4 that if u ∈ C 2 (Ω) satisfies Qu = 0 in Ω, then for
k ∈ {1, ..., n}), w = Dk u ∈ C 1 (Ω) is a weak solution of the linear elliptic equation
Di (ãij (x)Dj w) = 0,
ãij (x) = aij (Du(x)).
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By replacing Ω if necessary by a strictly contained subdomain, we can assume ãij bounded and strictly
elliptic in Ω. The assumptions of Theorem 6.12 are therefore satisfied. Hence, choosing λK , ΛK such
that
0 < λk ≤ λ(Du), |ãij (Du)| ≤ ΛK
for K := supΩ |Du|, we have the following interior and global Hölder estimates:
Theorem 6.14 (Interior Hölder estimate II). Let u ∈ C 2 (Ω) satisfy Qu = 0 in Ω where Q is elliptic
and of the form (4.1) with A ∈ C 1 (Rn ). Then for any Ω0 ⊂⊂ Ω we have the estimate
[Du]α,Ω0 ≤ Cd−α ,
where C = C(n, K, ΛK /λK , diam Ω), d = dist (Ω0 , ∂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 Hölder estimate II). Let u ∈ C 2 (Ω̄) satisfy Qu = 0 in Ω where Q is elliptic in
Ω̄ and of the form (4.1) with A ∈ C 1 (Rn ). Then if ϕ ∈ C 2 (∂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 minimal
surface equation. We will now show how to apply the developed theory to this historical example. Let
Ω be a sufficiently smooth bounded domain in Rn , and u ∈ C 2 (Ω) ∩ C 0 (Ω̄) satisfy M(u) = 0 in Ω and
u = ϕ on ∂Ω, ϕ ∈ C 0 (∂Ω). Our goal is to show that there exists some β ∈ (0, 1) and M > 0 such that
(1.4) is bounded by M ,
(7.1)
kukC 1,β (Ω) ≤ sup |u| + sup |Du| + [Du]β,Ω ≤ M.
Ω
Ω
Let us refer to the strategy outlined in Section 1. Step 1 has been completed in Example 3.3, where we
obtained for the minimal surface equation that
sup |u| ≤ sup |u|,
Ω
∂Ω
which is bounded in terms of the boundary data ϕ. Concerning Step 2, we recall that we established the
equality
sup |Du| = sup |Du|
Ω
∂Ω
in Section 4. If we make some further assumptions, in particular ϕ ∈ C 2 (Ω̄) and ∂Ω ∈ C 2 is mean
convex, i.e. its mean curvature is non-negative at all x0 ∈ ∂Ω, then it follows from Theorem 5.11 that
sup |Du| ≤ C.
∂Ω
Hence the first and second terms on the LHS of (7.1) can be controlled using the theory developped in
Sections 3 and 5. In order to bound the Hölder Semi-norm, we apply the De Girogi–Nash–Moser theory
from Section 6 to |Du|. To do so, we use the gradient estimates in Section 4 to justify that we indeed
satisfy the conditions of the Hölder estimate 6.1. In fact, we require here a variation of theorem 6.1 that
includes the boundary ∂Ω. Note that the minimal surface equation is of divergence form (4.1) with
Du
.
A(Du) = p
1 + |Du|2
We showed in Section 1.2 that aij (x, z, p) = δij −
pi pj
(1+|p|2 )
satisfies the ellipticity condition
λ(Du(x))|ξ|2 ≤ aij (Du(x))ξi ξj ≤ Λ,
34
∀x ∈ Ω, ξ ∈ Rn
with λ(Du(x)) = 1/(1 + |Du|2 ), Λ = 1. By equation (4.3) in Section 4, we conclude
λ(Du(x)) ≥
1
1
=
≥ C > 0,
(1 + supΩ |Du|2 )
(1 + sup∂Ω |Du|2 )
where we used again the boundary gradient estimate provided by Theorem 5.11. Hence, the coefficient
matrix ãij ∈ L∞ (Ω) defined in (6.3) is uniformly elliptic, and w = Dk u, k ∈ {1, .., n} satisfies
Z
ãij Di wDj ψ = 0, ∀ψ ∈ H01 (Ω).
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 the
Leray-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 differenziabilità e l’analiticità 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. Government Printing Office, 2001
[3] Guisti, E., Minimal Surfaces and Functions of Bounded Variation, Birkhäuser Boston, 1984
[4] Han, Q. and Lin, F., Elliptic Partial Differential Equations, Courant Institute of Mathematical
Sciences, New York University, 2011
[5] Moser, J., A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math., 1960
[6] Nash, J., Parabolic Equations, Proceedings of the National Academy of Sciences, 1957
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