Jost J. Partial Differential Equations

Подождите немного. Документ загружается.

11.2 The Schauder Estimates 297

As in Section 9.3, let

(0,R)=



x =(x

,...,x

) ∈ R

; |x| <R,x

> 0



Moreover, let

∂

(0,R):=∂B

(0,R) ∩





∂

(0,R):=∂B

(0,R) \ ∂

(0,R).

We consider f ∈ C



(0,R)



with

f =0 on∂

(0,R).

In contrast to the situation considered in Theorem 11.1.1(b), f no longer must

vanish on all of the boundary of our domain Ω = B

(0,R), but only on a

certain portion of it. Again, we consider the corresponding Newton potential

u(x):=



(0,R)

Γ (x, y)f (y)dy. (11.2.22)

Up to a constant factor, the ﬁrst derivatives of u are given by

(x)=



(0,R)

− y

|x − y|

f(y)dy (i =1,...,d), (11.2.23)

and they can be estimated as in the proof of Theorem 11.1.1(a), since there,

we did not need any assumption on the boundary values.

Up to a constant factor, the second derivatives are given by

(x)=



(0,R)

∂

∂x



− y

|x − y|



f(y)dy



= w

(x)



. (11.2.24)

For K(x − y)=

∂

∂x



−y

|x−y|



,andi = d or j = d,



<|y|<R

K(y)dy = 0 (11.2.25)

by homogeneity as in (11.1.12). Thus, for i = d or j = d,theα-H¨older

norm of the second derivative

∂

∂x

u canbeestimatedasintheproofof

Theorem 11.1.1(b). The diﬀerential equation Δu = f implies

∂

(∂x

)

u = f −

d−1



i=1

∂

(∂x

)

u, (11.2.26)

and so we obtain estimates for the α-H¨older norm of

∂

(∂x

)

u as well. We can

thus estimate all second derivatives of u.

298 11. Existence Techniques IV

As in the proof of Theorem 11.1.2, we then obtain C

2,α

-estimates in

(0,R) for solutions of

Δu = f in B

(0,R)withf ∈ C



(0,R)



u =0 on∂

(0,R),

(11.2.27)

for 0 <r<R:

u

2,α

(0,r))

≤ c



f

(0,R))

+ u

(0,R))



. (11.2.28)

Namely, putting

ϕ := ηu

as in (11.1.23) with the same cutoﬀ function as in (11.1.22), we have ϕ =0

on ∂B

(0,R

)(0<R

<R), since η vanishes on ∂

(0,R

), and u

on ∂

(0,R

). Thus, again

ϕ(x)=



(0,R)

Γ (x, y)Δϕ(y)dy

is a Newton potential, and the preceding estimates can be used to deduce

the same result as in Theorem 11.1.2: For 0 <r<R,

u

2,α

(0,r))

≤ c



f

(0,R))

+ u

(0,R))



. (11.2.29)

We next consider a solution of

Δu = f in B

(0,R)withf ∈ C



(0,R)



, (11.2.30)

u = g on ∂

(0,R)withg ∈ C

2,α



(0,R)



. (11.2.31)

As in Section 9.3, we put ¯u := u − g. We see that ¯u satisﬁes

Δ¯u = f − Δg =:

f ∈ C



(0,R)



in B

(0,R),

¯u =0on∂

(0,R).

We have thus reduced our considerations to the above case (11.2.27), and so,

from (11.2.29), we obtain

u

2,α

(0,r))

≤¯u

2,α

(0,r))

+ g

2,α

(0,r))

≤ c

)

(0,R))

+ ¯u

(0,R))

+ g

2,α

(0,R))

≤ c

f

(0,R))

+ g

2,α

(0,r))

+ u

(0,R))

(11.2.32)

11.3 Existence Techniques IV: The Continuity Method 299

In order to ﬁnally treat the situation of Theorem 11.2.2, as in Section 9.3,

we transform a neighborhood U of a boundary point x

∈ ∂Ω with a C

2,α

diﬀeomorphism φ to the ball

B(0,R), such that the portion of u that is

contained in Ω is mapped to B

(0,R), and the intersection of U with ∂Ω

is mapped to ∂

(0,R). Again, ˜u := u ◦ φ

−1

on B

(0,R) satisﬁes a dif-

ferential equation of the same type as Lu = f,

L˜u =

f, again with diﬀerent

constants λ, K in (A) and (B). By the preceding considerations, we obtain a

2,α

-estimate for ˜u in B

(0,R/2). Again φ transforms this estimate into one

for u on a subset U



of U.SinceΩ is bounded, ∂Ω is compact and can thus

be covered by ﬁnitely many such neighborhoods U



. The resulting estimates,

together with the interior estimate of Theorem 11.2.1, applied to the comple-

ment Ω

of those neighborhoods in Ω, yield the claim of Theorem 11.2.2. 

Corollary 11.2.1: In addition to the assumptions of Theorem 11.2.2, sup-

pose that c(x) ≤ 0 in Ω. Then

u

2,α

(Ω)

≤ c



f

(Ω)

+ g

2,α

(Ω)



. (11.2.33)

Proof: Because of c ≤ 0, the maximum principle (see, e.g., Theorem 2.3.2)

implies

sup

|u|≤max

∂Ω

|u| + c

sup

|f| =max

∂Ω

|g| + c

sup

|f|.

Therefore, the L

-norm of u can be estimated in terms of the C

-norms of f

and g, and the claim follows from (11.2.21). 

11.3 Existence Techniques IV: The Continuity Method

In this section, we wish to study the existence problem

Lu = f in Ω,

u = g on ∂Ω,

in a C

2,α

-region Ω with f ∈ C

(

Ω), g ∈ C

2,α

(

Ω). The starting point for

our considerations will be the corresponding result for the Poisson equation,

Kellogg’s theorem:

Theorem 11.3.1: Let Ω be a bounded domain of class C

∞

in R

, f ∈

(

Ω), g ∈ C

2,α

(

Ω). The Dirichlet problem

Δu = f in Ω,

u = g on ∂Ω,

(11.3.1)

then possesses a unique solution u of class C

2,α

(

Ω).

300 11. Existence Techniques IV

Proof: Uniqueness follows from the maximum principle (see Corollary 2.1.1).

For the existence proof, we ﬁrst assume that f and g are of class C

∞

.The

variational methods of Section 8.3 yield a weak solution, which then is of

class C

∞

(Ω) by Theorem 9.3.1. Moreover, by Corollary 11.2.1,

u

2,α

(Ω)

≤ c



f

(Ω)

+ g

2,α

(Ω)



. (11.3.2)

We now return to the C

2,α

-case. We approximate f and g by C

∞

-functions

and g

that are deﬁned on Ω.Letu

be the solution of the corresponding

Dirichlet problem

Δu

= f

in Ω,

= g

on ∂Ω.

For n ≥ m, u

− u

then satisﬁes (11.3.2) on Ω, i.e.,

u

− u



2,α

(Ω)

≤ c



f

− f



(Ω)

+ g

− g



2,α

(Ω)



. (11.3.3)

Here, the constant c

does not depend on the solutions; it depends only

on the C

2,α

-geometry of the domain. We assume that f

converges to f in

(Ω), and g

to g in C

2,α

(Ω), and so the u

constitute a Cauchy sequence

in C

2,α

(Ω) and therefore converge towards some u ∈ C

2,α

(Ω)thatsatisﬁes

Δu = f in Ω,

u = g on ∂Ω,

and the estimate (11.3.2). 

We now state the main existence result of this chapter:

Theorem 11.3.2: Let Ω be a bounded domain of class C

∞

in R

. Let the

diﬀerential operator

L =



i,j=1

(x)

∂

∂x



i=1

(x)

∂

∂x

+ c(x) (11.3.4)

satisfy (A) and (B) from Section 11.2, and in addition,

c(x) ≤ 0 in Ω. (11.3.5)

For any f ∈ C

(

Ω), g ∈ C

2,α

(

Ω) there then exists a unique solution u ∈

2,α

(

Ω) of the Dirichlet problem

Lu = f in Ω,

u = g on ∂Ω.

(11.3.6)

11.3 Existence Techniques IV: The Continuity Method 301

Remark: It is quite instructive to compare this result and its assumptions

with Theorem 9.4.4 above.

Proof: Considering, as usual, ¯u = u −g in place of u, we may assume g =0,

as our problem is equivalent to

L¯u =

f := f − Lg ∈ C

(Ω),

¯u =0 on∂Ω.

We thus assume g = 0 (and write u in place of ¯u). We consider the family of

equations

u = f for 0 ≤ t ≤ 1,

u =0 on∂Ω,

(11.3.7)

with

= tL +(1− t)Δ. (11.3.8)

The diﬀerential operators L

satisfy the structural conditions (A) and (B)

with

=min(1,λ),K

= max(1,K). (11.3.9)

We have L

= Δ, L

= L. By Theorem 11.3.1, we can solve (11.3.7) for t =0.

We intend to show that we may then also solve this equation for all t ∈ [0, 1],

in particular for t = 1. The latter is what is claimed in the theorem.

The operator

: B

:= C

2,α

(

Ω) ∩{u : u =0 on∂Ω}→C

(

Ω)=:B

is a bounded linear operator between the Banach spaces B

and B

.Letu

be a solution of L

= f, u

=0on∂Ω. By Corollary 11.2.1,

u



2,α

(Ω)

≤ c

f

(Ω)

i.e.,

u

≤ c

L

u

, (11.3.10)

for all u ∈ B

. Here, the constant c

does not depend on t, because by

(11.3.9), the structure constants λ

, K

of the operators L

can be controlled

independently of t.

We want to show that for any f ∈ B

there exists a solution u

of (11.3.7),

i.e., of L

= f,inB

. In other words, we want to show that the operators

: B

→ B

are surjective for 0 ≤ t ≤ 1. This, however, follows from the

general result stated as the next theorem. With that result, we then conclude

the proof of Theorem 11.3.2. 

302 11. Existence Techniques IV

Theorem 11.3.3: Let L

: B

→ B

be bounded linear operators between

the Banach spaces B

.Weput

:= (1 −t)L

+ tL

for 0 ≤ t ≤ 1.

We assume that there exists a constant c that does not depend on t,with

u

≤ c L

u

for all u ∈ B

. (11.3.11)

If then L

is surjective, so is L

Proof: Let L

be surjective for some τ ∈ [0, 1]. By (11.3.11), L

then is

injective as well, and thus bijective. We therefore have an inverse operator

−1

: B

→ B

For t ∈ [0, 1], we rewrite the equation

u = f for f ∈ B

(11.3.12)

u = f +(L

− L

)u = f +(t − τ )(L

u − L

u),

u = L

−1

f +(t − τ)L

−1

− L

)u =: Λu.

Thus, for solving (11.3.12), we need to ﬁnd a ﬁxed point of the operator

Λ : B

→ B

. By the Banach ﬁxed point theorem, such a ﬁxed point exists

if we can ﬁnd some q<1with

Λu − Λv

≤ q u − v

We have

Λu − Λv≤

)

−1

)

(L

 + L

) |t − τ |u − v.

By (11.3.11),

)

−1

)

≤ c. Therefore, it suﬃces to choose

|t − τ|≤

(c (L

 + L

))

−1

=: η

for obtaining the desired ﬁxed point. This means that if L

u = f is solvable,

so is L

u = f for all t with |t − τ|≤η.SinceL

is surjective by assumption,

then is surjective for 0 ≤ t ≤ η. Repeating the preceding argument, this

time for τ = η, we obtain surjectivity for η ≤ t ≤ 2η. Iteratively, all L

for

t ∈ [0, 1], and in particular L

, are surjective. 

Basic references about Schauder’s approach are [1, 9]. Our treatment of the

fundamental C

-estimate for the Poisson equation uses scaling relations in

place of the usual weighted H¨older spaces and is hopefully a little simpler.

Exercises 303

Summary

A solution of the Poisson equation

Δu = f

with α-H¨older continuous f is contained in the space C

2,α

; i.e., it possesses

α-H¨older continuous second derivatives for 0 <α<1. (This is no longer true

for α =0orα = 1. For example, if f is only continuous, a solution need not

be twice continuously diﬀerentiable.) By linear coordinate transformations

this result can be easily extended to linear elliptic diﬀerential equations with

constant coeﬃcients. Schauder then succeeded in extending these results to

solutions of elliptic equations

Lu(x):=



i,j

(x)

∂

u(x)

∂x



(x)

∂u

∂x

+ c(x)u(x)=f(x)

with α-H¨older continuous coeﬃcients, by considering such an operator L as

a local perturbation of an operator with constant coeﬃcients a

,c.

The continuity method reduces the solution of

Lu = f

to that of the Poisson equation

Δu = f

by considering the operators

:= tL +(1− t)Δ

for 0 ≤ t ≤ 1, and showing that the set of t ∈ [0, 1] for which

u = f

can be solved is open and closed (and nonempty, because the Poisson equation

can be solved). The proof of closedness rests on Schauder’s estimates.

Exercises

11.1 Let K ⊂ R

be bounded, f

: K → R (n ∈ N) a sequence of functions

with

f



(K)

≤ const (independent of n),

for some 0 <α≤ 1. (Here and in the next exercise, in the case α =1we

consider the space C

0,1

of Lipschitz continuous functions.) Show that

then (f

)

n∈N

has to contain a uniformly convergent subsequence.

304 11. Existence Techniques IV

11.2 Is it true that for all domains Ω ⊂ R

, 0 <α<β≤ 1,

(Ω) ⊂ C

(Ω)?

11.3 Let u ∈ C

k,α

(Ω) satisfy

Lu = f

for some f ∈ C

k,α

(Ω)(k ∈ N, 0 <α<1). Here, we assume that the

operator L from (11.2.1) satisﬁes the ellipticity condition (A) as well as

)

k,α

(Ω)

)

k,α

(Ω)

, c

k,α

(Ω)

≤ K

for all i, j. Show that u ∈ C

k+2,α

(Ω

) for any Ω

⊂⊂ Ω,and

u

k+2,α

(Ω)

≤ c(f

k,α

(Ω)

+ u

(Ω)

with a constant c depending on K and the quantities of Theorem 11.2.1.

12. The Moser Iteration Method and the

Regularity Theorem of de Giorgi and Nash

12.1 The Moser–Harnack Inequality

In this chapter, as in Chapter 9, we shall consider elliptic diﬀerential operators

of divergence type. In order to concentrate on the essential aspects and not

to burden the proofs with too many technical details, in this chapter we shall

omit all lower-order terms and consider only solutions of the homogeneous

equation. Thus, we shall investigate (weak) solutions of

Lu =



i,j=1

∂

∂x



(x)

∂

∂x

u(x)



=0,

where the coeﬃcients a

are (measurable and) bounded and satisfy an ellip-

ticity condition. We thus assume that there exist constants 0 <λ≤ Λ<∞

with

λ |ξ|

≤



i,j=1

(x)ξ

(12.1.1)

for all x in the domain of deﬁnition Ω of u and all ξ ∈ R

,and

sup

i,j,x



(x)



≤ Λ.

Deﬁnition 12.1.1: A function u ∈ W

1,2

(Ω) is called a weak subsolution of

L, and we write this as Lu ≥ 0, if for all ϕ ∈ H

1,2

(Ω), ϕ ≥ 0 in Ω,





i,j

(x)D

ϕdx ≤ 0. (12.1.2)

Similarly, it is called a weak supersolution (Lu ≤ 0), if we have ≥ in (11.1.2).

Inequalities like ϕ ≥ 0 are assumed to hold pointwise almost everywhere, here

and in the sequel. Likewise, sup and inf will denote the essential supremum

and inﬁmum, respectively. Finally, as always, −



will denote the average mean

integral: