Helms L.L. Potential Theory

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338 9 Elliptic Operators

The following lemma is an immediate consequence of Theorem 8.4.5 and

Inequalities (9.5) and (9.6).

Lemma 9.3.1 If u ∈ C

(Ω),f ∈ H

(2)

(Ω; d),α ∈ (0, 1),andL

u = f on Ω,

then

|u; d|

2+α,Ω

≤ C(n, α, m, M)(u

0,Ω

+ |f ; d|

(2)

α,Ω

9.4 Schauder Interior Estimates

The following terminology regarding the operator L will be used. If there are

positive functions m and M on the open set Ω ⊂ R

such that

m|λ|

≤



i,j=1

(x)λ

≤ M |λ|

for all x ∈ Ω, λ ∈ R

, (9.9)

the operator L is said to be elliptic on Ω;ifm is a positive constant, L is said

to be strictly elliptic on Ω. Granted that the solution of the equation Lu =

f is bounded, a bound for the weighted norm |u; d|

2+α,Ω

can be obtained.

Such estimates are called Schauder interior estimates. All of the results

in this section can be found in [25].

Lemma 9.4.1 If α ∈ (0, 1),a + α ≥ 0,b + α ≥ 0,a + b + α ≥ 0,f ∈

(a)

(Ω; φ),g ∈ H

(b)

(Ω; φ),thenfg ∈ H

(a+b)

(Ω; φ) and

|fg; φ|

(a+b)

α,Ω

≤|f; φ|

(a)

α,Ω

|g; φ|

(b)

α,Ω

. (9.10)

Proof: By deﬁnition,

|fg; φ|

(a+b)

α,Ω

=sup

x∈Ω

a+b

(x)|f(x)g(x)|

+sup

x,y∈Ω

a+b+α

(x, y)

|f(x)g(x) − f (y)g(y)|

|x − y|

≤|f; φ|

(a)

0,Ω

|g; φ|

(b)

0,Ω

+sup

x,y∈Ω

a+b+α

(x, y)

|f(x)||g(x) − g(y)| + |g(y)||f(x) − f(y)|

|x − y|

≤|f; φ|

(a)

0,Ω

|g; φ|

(b)

0,Ω

+[f; φ]

(a)

0,Ω

[g; φ]

(b)

α,Ω

+|g; φ|

(b)

0,Ω

[f; φ]

(a)

α,Ω

≤



|f; φ|

(a)

0,Ω

+[f; φ]

(a)

α,Ω



|g; φ|

(b)

0,Ω

+[g; φ]

(b)

α,Ω



= |f; φ|

(a)

α,Ω

|g; φ|

(b)

α,Ω

9.4 Schauder Interior Estimates 339

Lemma 9.4.2 If α ∈ (0, 1),g ∈ H

(2)

(Ω,d),μ ≤ 1/2,x

∈ Ω, δ = μd(x

and B = B

,δ

,then

|g; d

(2)

α,B

≤ 4μ

[g; d]

(2)

0,Ω

+8μ

2+α

[g; d]

(2)

α,Ω

≤ 8μ

|g; d|

(2)

α,Ω

. (9.11)

Proof: For x ∈ B, d

(x)=d(x, ∂B) <δand d(x) ≥ (1 − μ)d(x

)sothat

|g; d

(2)

α,B

=sup

x∈B

(x)|g(x)| +sup

x,y∈B

2+α

(x, y)

|g(x) − g(y)|

|x − y|

≤ δ

sup

x∈B

−2

(x)d

(x)|g(x)|

+ δ

2+α

sup

x,y∈B

−2−α

(x, y)d

2+α

(x, y)

|g(x) − g(y)|

|x − y|

≤

(1 − μ)

)

sup

x∈Ω

(x)|g(x)|

2+α

(1 − μ)

2+α

)

sup

x,y∈Ω

2+α

(x, y)

|g(x) − g(y)|

|x − y|

≤

(1 − μ)

[g; d]

(2)

0,Ω

2+α

(1 − μ)

2+α

[g; d]

(2)

α,Ω

≤ 4μ

[g; d]

(2)

0,Ω

+8μ

2+α

[g; d]

(2)

α,Ω

≤ 8μ

|g; d|

(2)

α,Ω

Lemma 9.4.3 Let Ω be a bounded open subset of R

and let L be an elliptic

operator on Ω satisfying

; d|

(0)

α,Ω

, |b

; d|

(1)

α,Ω

, |c; d|

(2)

α,Ω

≤ K, α ∈ (0, 1),i,j =1,...,n. (9.12)

If u ∈ H

2+α

(Ω; d),f ∈ H

(2)

(Ω; d),α ∈ (0, 1),andLu = f ,then

|u; d|

2+α,Ω

≤ C(u

0,Ω

+ |f ; d|

(2)

α,Ω

). (9.13)

where C = C(m, n, α, K).

Proof: By Inequality (7.10), it suﬃces to prove the result for [u; d]

2+α,Ω

.Let

be two points of Ω with x

= y

and labeled so that d(x

) ≤ d(y

), let

μ ≤ 1/2 be a constant that will be further speciﬁed later, let δ = μd(x

), and

let B = B

,δ

. The equation Lu = f can be related to an elliptic operator

with constant coeﬃcients by writing



i,j=1

u(x)=



i,j=1

) − a

(x))D

u(x)

−



i=1

(x)D

u(x) − c(x)u(x)+f(x).

340 9 Elliptic Operators

Denote the left side of this equation by L

u and the right side by f

.Note

that for y

∈ B

,δ/2

,d(y

,∂B) ≥ δ/2 so that by Lemma 9.3.1,





2+α

u(x

) − D

u(y

− y

≤ C



u

0,B

+ |f

; d

(2)

α,B



(9.14)

where C = C(m, n, α, K). Since δ = μd(x

)=μd(x

2+α

)

u(x

) − D

u(y

− y

≤

2+α



u

0,B

+ |f

; d

(2)

α,B



. (9.15)

Each of the terms on the right side of the inequality

; d

(2)

α,B

≤



i,j=1

|(a

) − a

u; d

(2)

α,B



i=1

u; d

(2)

α,B

+ |cu; d

(2)

α,B

+ |f ; d

(2)

α,B

(9.16)

can be estimated as follows. The subscripts on a

will be omitted temporarily.

Keep in mind that D

is a generic symbol for a second-order partial derivative.

By Inequalities (9.10) and (9.11),

|(a(x

) − a)D

u; d

(2)

α,B

≤|a(x

) − a; d

(0)

α,B

u; d

(2)

α,B

≤|a(x

) − a; d

α,B



4μ

u; d]

(2)

0,Ω

+8μ

2+α

u; d]

(2)

α,Ω



≤|a(x

) − a; d

α,B



4μ

[u; d]

2+0,Ω

+8μ

2+α

[u; d]

2+α,Ω



Using the fact that (1 − μ)d(x

) ≤ d(x), (1 − μ)d(x

) ≤ min (d(x

),d(x)) =

d(x

,x)forx ∈ B,

|a(x

) − a(x)| = |x − x

|a(x

) − a(x)|

|x − x

≤ μ

d(x

)

|a(x

) − a(x)|

|x − x

≤

(1 − μ)

,x)

|a(x

) − a(x)|

|x − x

≤

(1 − μ)

[a; d]

α,Ω

≤ 2

[a; d]

α,Ω

so that a(x

) − a

0,B

≤ 2

[a; d]

α,Ω

.Moreover,

[a(x

) − a; d

]

α,B

=sup

x,y∈B

(x, y)

|a(x) − a(y)|

|x − y|

≤ μ

d(x

)

sup

x,y∈B

d(x, y)

−α

d(x, y)

|a(x) − a(y)|

|x − y|

≤

d(x

)

(1 − μ)

d(x

)

[a; d]

α,Ω

≤ 2

[a; d]

α,Ω

9.4 Schauder Interior Estimates 341

Combining these two inequalities,

|a(x

) − a; d

α,B

≤ 2

α+1

[a; d]

α,Ω

Since |a; d|

α,Ω

≤ K by Inequality (9.12),

|(a(x

)−a)D

u; d

(2)

α,B

≤ 2

α+1

[a; d]

α,Ω



4μ

[u; d]

2+0,Ω

+8μ

2+α

[u; d]

2+α,Ω



≤ 32Kμ

2+α



[u; d]

2+0,Ω

+μ

[u; d]

2+α,Ω



Applying Inequality (7.10) to [u; d]

2+0,Ω

with  = μ

, there is a constant

C = C(μ, α) such that

|(a(x

) − a)D

u; d

(2)

α,B

≤ 64Kμ

2+α

(Cu

0,Ω

+ μ

[u; d]

2+α,Ω

). (9.17)

By Inequalities (9.11) and (9.10),

u; d

(2)

α,B

≤ 8μ

u; d|

(2)

α,Ω

≤ 8μ

; d|

(1)

α,Ω

u; d|

(1)

α,Ω

≤ 8Kμ

|u; d|

1+α,Ω

Applying Inequality (7.10) to |u; d|

1+α,Ω

with  = μ

2α

, there is a constant

C = C(μ, α) such that

u; d

(2)

α,B

≤ 8Kμ



Cu

0,Ω

+ μ

2α

|u; d|

2+α,Ω



. (9.18)

Similarly, thereisaconstantC = C(μ, α) such that

|cu; d

(2)

α,B

≤ 8μ

|c; d|

(2)

α,Ω

|u; d|

(0)

α,Ω

≤ 8Kμ

(Cu

0,Ω

+ μ

2α

|u; d|

2+α,Ω

). (9.19)

By Inequality (9.11),

|f; d

(2)

α,B

≤ 8μ

|f; d|

(2)

α,Ω

. (9.20)

In combining the above inequalities to estimate |f

; d

(2)

α,B

, it is essential to

keep track of the dependence of constants on the data; in particular, it is

essential that the coeﬃcient of μ

2+2α

[u; d]

2+α,Ω

does not depend upon μ.As

usual, C will be a generic symbol for constants depending upon the data but

the symbol for the coeﬃcient of μ

2+2α

[u; d]

2+α,Ω

will exhibit the dependence

explicitly. Returning to Inequality (9.16) and using Inequalities (9.17), (9.18),

(9.19), and (9.20),

; d|

(2)

α,B

≤ Cμ

u

0,Ω

+ C(K)μ

2+2α

[u]

2+α,Ω

+ Cμ

[f; d]

(2)

α,Ω

342 9 Elliptic Operators

Returning to Inequality (9.15), for y

∈ B

,δ/2

d(x

)

2+α

u(x

) − D

u(y

− y

≤ C

u

0,Ω

2+α

+ C

u

0,Ω

+ C(K)μ

|u; d|

2+α,Ω

+ C

|f; d|

(2)

α,Ω

(9.21)

Using Inequality (7.10) with  = μ

2α

α+1

,fory

∈ B

,δ/2

d(x

)

2+α

u(x

) − D

u(y

− y

≤

α+1

d(x

)

[u; d]

2+0,Ω

≤

α+1

(Cu

0,Ω

2α

α+1

[u; d]

2+α,Ω

)

≤ Cu

0,Ω

+ μ

[u; d]

2+α,Ω

Combining this result with Inequality (9.21),

[u; d]

2+α,Ω

≤ Cu

0,Ω

+ C(K)μ

|u; d|

2+α,Ω

+ C

|f; d|

(2)

α,Ω

Adding |u; d|

2+0,Ω

to both sides and applying Inequality (7.10) with  = μ

and choosing μ<1/2sothat(C(K)+1)μ

< 1/2, there is a constant

C = C(K) such that

|u; d|

2+α,Ω

≤ C



u

0,Ω

+ |f ; d|

(2)

α,Ω



Theorem 9.4.4 Let L be an elliptic operator satisfying

; d|

(0)

α,Ω

, |b

; d|

(1)

α,Ω

, |c; d|

(2)

α,Ω

≤ K, i, j =1,...,n. (9.22)

If f ∈ H

(2)

(Ω; d),α ∈ (0, 1),b∈ (−1, 0],and{u

} is a sequence of uniformly

bounded functions in H

(b)

2+α

(Ω; d) satisfying the equation Lu

= f on Ω,then

there is a subsequence that converges uniformly on compact subsets of Ω to

afunctionu ∈ H

2+α

(Ω; d) ⊂ H

2+α,loc

(Ω) for which Lu = f on Ω.

Proof: Suppose ﬁrst that b ∈ (−1, 0). Since d(x) ≤ d(Ω),d(x)

≥ d(Ω)

,x∈

Ω,andd(Ω)

; d|

2+α,Ω

≤|u

; d|

(b)

2+α,Ω

< +∞,i≥ 1. By Lemma 9.4.3,

; d|

2+α,Ω

≤ C



u



0,Ω

+ |f ; d|

(2)

α,Ω



≤ C





+ |f ; d|

(2)

α,Ω



< +∞ (9.23)

where K



=sup

i≥1

u



0,Ω

.Ifb = 0, this inequality follows from the same

lemma. Now let Γ be any compact, convex subset of Ω and choose 0 <

δ<min (1,d(Γ, ∂Ω)sothatΓ ⊂ Ω

.Since|u

; d|

2+α,

≤|u; d|

2+α,Ω

≤

C(K



+ |f; d|

(2)

α,Ω

< +∞, it follows from the subsequence selection principle,

9.5 Maximum Principles 343

Theorem 7.2.4, that the sequences {u

}, {Du

},and{D

} are uniformly

bounded and equicontinuous on Γ .Let{B

} be a sequence of balls with

closures in Ω which exhausts Ω. Applying the above argument to each B

−

there is a function u

(j)

on each B

and a subsequence {u

(j)

} of the {u

}

sequence such that the sequences {u

(j)

}, {Du

(j)

},and{D

(j)

} converge to

(j)

,Du

(j)

,andD

(j)

, respectively, on B

−

. Note that the u

(j)

agree on

overlapping parts of the B

. The sequence {u

(i)

} is a subsequence of each of

the above sequences and there is a function u on Ω such that the sequences

(i)

}, {Du

(i)

},and{D

(i)

} converge to u, Du,andD

u, respectively, on Ω.

Since Inequality (9.23) holds for each u

(i)

,eachtermof|u

(i)

2+α,Ω

is bounded

by C(K



+ |f ; d|

(2)

α,Ω

); for example,

d(x, y)

2+α

(i)

(x) − D

(i)

(y)|

|x − y|

≤ C





+ |f ; d|

(2)

α,Ω



,i≥ 1,x,y ∈ Ω.

Letting i →∞and taking the supremum over x, y ∈ Ω and D

, [u; d]

2+α,Ω

+∞. Similarly, for the other terms of |u; d|

2+α,Ω

.Thus,u ∈ H

2+α

(Ω; d) ⊂

2+α,loc

(Ω).

9.5 Maximum Principles

Although an existence theorem for the Dirichlet problem corresponding to an

elliptic operator L has not been proven, a uniqueness result can be proved us-

ing a weak maximum principle. Before stating the principle, some results

from basic analysis will be reviewed. Consider the quadratic function

Q(y)=



p,q=1

where the n ×n matrix [c

] is a real symmetric matrix. The quadratic form

Q is said to be nonnegative deﬁnite if Q(y) ≥ 0 for all y ∈ R

;if,in

addition, Q(y) = 0 only when y =0,thenQ is said to be positive deﬁnite.

Lemma 9.5.1 If u ∈ C

(Ω),x

∈ Ω,andu(x

)=sup

x∈Ω

(x), then the

matrix [−D

)] is nonnegative deﬁnite.

Proof: Let B = B

,δ

⊂ Ω and let x ∈ R

.Sinceu attains its maximum

value at x

u(x

)=0, 1 ≤ i ≤ n. By Taylor’s formula with remainder

(c.f. [1]), for 0 <t|x| <δ,

0 ≥ u(x

+ tx) − u(x



i,j=1

u(x

+ t



x)tx

344 9 Elliptic Operators

for some 0 <t



<t.Thus,



i,j=1

u(x

+ t



x)x

≤ 0.

Letting δ → 0,



i.j=1

u(x

≤ 0.

Lemma 9.5.2 Let Ω be a bounded open subset of R

and let L be an elliptic

operator with c =0and sup

x∈Ω,1≤i≤n

(|b

(x)|/m(x)| < +∞.Ifu ∈ C

(Ω

−

)∩

(Ω) and Lu ≥ 0 on Ω, then u attains its maximum value on ∂Ω; moreover,

if Lu>0 on Ω,thenu cannot attain its maximum value at an interior point

of Ω.

Proof: ItwillbeshownﬁrstthatifLu>0, then u cannot attain its maxi-

mum value at an interior point of Ω. Suppose u attains its maximum value

at an interior point x

of Ω. At such a point, the matrix [−D

u(x

)] is real,

symmetric, and nonnegative deﬁnite. Since the matrix A =[a

)] of coef-

ﬁcients is positive deﬁnite, there is a nonsingular matrix S =[s

] such that

A = S

S (c.f.,[31]). Since a



k=1

Lu(x

)=−



i,j=1

)(−D

u(x

))

= −



i,j=1





k=1



(−D

u(x

))

= −



k=1





i,j=1

(−D

u(x

))s



Since the double sum is nonnegative for each k =1,...,n,Lu(x

) ≤ 0, a

contradiction. Therefore, u cannot attain its maximum value at an interior

point of Ω if Lu>0onΩ. Suppose now that Lu ≥ 0onΩ. By hypothesis,

there is a constant M

such that |b

|/m ≤ M

,i =1,...,n.Sincea

≥ m,

for any constant γ>M

γx

=(γ

+ γb

γx

≥ m(x)(γ

− γM

γx

> 0.

For every >0, L(u + e

γx

) ≥ Le

γx

> 0onΩ. By the ﬁrst part of the

proof, there is a z = z() ∈ ∂Ω such that u(z)+e

γz

≥ u(x)+e

γx

for

all x ∈ Ω. Thus, sup

z∈∂Ω

u(z)+ sup

z∈∂Ω

γz

≥ sup

z∈∂Ω

(u(z)+e

γz

) ≥

u(x)+e

γx

≥ u(x) for all x ∈ Ω. Letting  → 0, sup

z∈∂Ω

u(z) ≥ u(x) for all

x ∈ Ω. Thus, sup

z∈∂Ω

u(z) ≥ sup

x∈Ω

u(x).

Recall that u

=max(u, 0) and u

−

=max(−u, 0).

9.5 Maximum Principles 345

Corollary 9.5.3 (Weak Maximum Principle) Let Ω be a bounded open

subset of R

and let L be an elliptic operator on Ω with c ≤ 0 and

sup

x∈Ω,1≤i≤n

(|b

(x)|/m(x)) < +∞.

If u ∈ C

(Ω

−

) ∩ C

(Ω) and Lu ≥ 0 or Lu ≤ 0,then

sup

u ≤ sup

∂Ω

or inf

∂Ω

−

≤ inf

respectively; if Lu =0,then

sup

|u|≤sup

∂Ω

|u|.

Proof: Suppose Lu ≥ 0onΩ and let L

u = Lu − cu.SinceLu ≥ 0onΩ,

u ≥−cu on Ω. Letting Ω

= {x ∈ Ω; u(x) > 0}, L

u ≥−cu ≥ 0onΩ

If Ω

= ∅,thenu ≤ 0onΩ, u

=0,andsup

u ≤ sup

∂Ω

;ifΩ

= ∅,

applying the previous lemma to L

on Ω

,u attains its maximum value on

∂Ω

. But since u =0on∂Ω

∩Ω, u attains its maximum value on ∂Ω where

u = u

. Thus, sup

u ≤ sup

∂Ω

. Suppose now that Lu =0onΩ. For any

x ∈ Ω,u(x) ≤ sup

∂Ω

≤ sup

∂Ω

|u|;sinceL(−u)=0, −u(x) ≤ sup

∂Ω

|u|.

Therefore, |u(x)|≤sup

∂Ω

|u| for all x ∈ Ω.

A uniqueness theorem and comparison principle follow easily from the

weak maximum principle.

Theorem 9.5.4 Let Ω be a bounded open subset of R

and let L be an

elliptic operator with c ≤ 0 and sup

x∈Ω,1≤i≤n

(|b

(x)|/m(x)) < +∞.Ifu, v ∈

(Ω

−

) ∩C

(Ω) satisfy Lu = Lv on Ω and u = v on ∂Ω,thenu = v on Ω.

If Lu ≥ Lv on Ω and u ≤ v on ∂Ω,thenu ≤ v on Ω.

The weak maximum principle does not exclude the possibility that a func-

tion u satisfying Lu ≥ 0 attains a maximum value at an interior point of Ω as

was the case with the Laplacian for nontrivial functions. A stronger form of

the weak maximum principle is needed to exclude this possibility. The open

set Ω ⊂ R

is said to satisfy an interior sphere condition at x

∈ ∂Ω if

there is a ball B ⊂ Ω with x

∈ ∂B. The upper inner normal derivate of u

at x

∈ ∂Ω is deﬁned by

u(x

) = lim sup

t→0

u(x

+ tν(x

)) − u(x

)

where ν = −n.

Lemma 9.5.5 (Hopf [33], Boundary Point Lemma) Let L be a strictly

elliptic operator on the open set Ω ⊂ R

with c ≤ 0 and with |a

|, |b

|, |c|≤

K, i, j =1,...,n and let u ∈ C

(Ω) satisfy Lu ≥ 0 on Ω.Ifforx

∈ ∂Ω,

346 9 Elliptic Operators

(i) u can be extended continuously to x

(ii) u(x

) >u(x) for all x ∈ Ω,and

(iii) the interior sphere condition is satisﬁed at x

then the upper inner normal derivative at x

satisﬁes D

u(x

) < 0 whenever

u(x

) ≥ 0; if, in addition, c =0,thenD

) < 0 regardless of the sign of

u(x

Proof: Let B = B

y,ρ

be a ball in Ω with x

∈ ∂B.Fix0<ρ

<ρ

and

deﬁne a function w on the closed annulus A = B

−

y,ρ

∼ B

y,ρ

by putting

w(x)=e

−γr

− e

−γρ

where ρ

<r= |x − y| <ρ

and γ is a constant to be speciﬁed below. By

Schwarz’s inequality,

Lw(x)=e

−γr



4γ



i,j=1

− y

)(x

− y

)

− 2γ



i=1

+ b

− y

)



+ cw

≥ e

−γr



4γ

− 2γ



i=1

− 2γ|b|r + c



≥ e

−γr

(4γ

mρ

− 2γnK − 2γ

√

nKρ

− K).

It follows that γ can be chosen so that Lw ≥ 0onA.Sinceu −u(x

) < 0on

∂B

y,ρ

,thereisan>0 such that

u − u(x

)+w ≤ 0on∂B

y,ρ

Since w =0on∂B

y,ρ

,u− u(x

)+w ≤ 0on∂B

y,ρ

and therefore on ∂A.

Since L(u − u(x

)+w)=Lu − cu(x

)+Lw ≥−cu(x

) ≥ 0 whenever

u(x

) ≥ 0, u − u(x

)+w ≤ 0onA by Corollary 9.5.3. For x

+ tν ∈ A,

u(x

+ tν(x

)) − u(x

)

≤−

w(x

+ tν(x

))

= −

w(x

+ tν(x

)) − w(x

)

so that D

u(x

) ≤−D

w(x

)=D

r=ρ

= −2γρ

−γρ

< 0. If c =0,

then the preceding steps are valid without any condition on u(x

Theorem 9.5.6 (Hopf [32], Strong Maximum Principle) Let L be a

strictly elliptic operator on the connected open set Ω ⊂ R

with c ≤ 0 and

|, |b

|, |c|≤K, i, j =1,...,n,andletu ∈ C

(Ω

−

)∩C

(Ω) satisfy Lu ≥ 0

on Ω.Thenu cannot attain a nonnegative maximum value at a point of Ω

unless it is constant; if, in addition, c =0and u attains its maximum value

at a point of Ω,thenu is a constant function.

9.6 The Dirichlet Problem for a Ball 347

Proof: Assume that u is nonconstant and attains a nonnegative maximum

value M at an interior point of Ω.LetΩ

= {x ∈ Ω; u(x) <M} = ∅.

Under these assumptions, ∂Ω

∩ Ω = ∅, for if not, then ∂Ω

⊂ ∂Ω which

implies that Ω = Ω

∪ (Ω ∼ Ω

)=Ω

∪ (Ω ∼ Ω

−

), contradicting

the fact that Ω is connected. Thus, there is a point x

∈ Ω

such that

d(x

,∂Ω

) <d(x

,∂Ω). Let B be the largest ball in Ω

having x

as its

center. Then u(y)=M for some y ∈ ∂B and u<Mon B.SinceΩ

satisﬁes the interior sphere condition at y, the inner normal derivative of u

on B at y is strictly negative according to the preceding lemma. But this is

a contradiction since any directional derivative of u at y must be zero. Thus,

if u attains a nonnegative maximum value at an interior point of Ω,itmust

be a constant function. If, in addition, c = 0, the preceding lemma can be

applied in the same way.

Theorem 9.5.7 (Hopf [32]) Let Ω be a connected open subset of R

,letL

be a strictly elliptic operator with c ≤ 0 and locally bounded coeﬃcients. If u ∈

(Ω

−

) ∩ C

(Ω) satisﬁes Lu ≥ 0 on Ω,thenu cannot attain a nonnegative

maximum value on Ω unless it is constant on Ω. If, in addition, c =0,then

u cannot attain its maximum value on Ω unless it is constant on Ω.

Proof: Suppose M =sup

x∈Ω

−

u(x) ≥ 0andletΩ

= {y ∈ Ω; u(y)=

M}.ThenΩ

is a relatively closed subset of Ω. Suppose x

∈ Ω

and

B = B

,δ

⊂ B

−

,δ

⊂ Ω. By the preceding theorem, u = M on B since

u(x

)=M. Therefore, B ⊂ Ω

;thatis,Ω

is relatively closed and open in

Ω.SinceΩ is connected, Ω

= ∅ or Ω

= Ω.

9.6 The Dirichlet Problem for a Ball

It will be assumed in this section that the coeﬃcients of L belong to

α,loc

(Ω); in particular, the coeﬃcients belong to H

(B) when restricted

toaballB with closure in Ω.

If it were possible to solve the Dirichlet problem corresponding to an ellip-

tic operator L and a ball B, then the Perron-Wiener-Brelot method could be

used to solve the Dirichlet problem for more general regions. The following

lemma reduces to Theorem 9.4.3 when b =0.

Lemma 9.6.1 Let L be a strictly elliptic operator on the bounded open set

Ω ⊂ R

satisfying Inequalities (9.12). If u ∈ H

2+α

(Ω; d),α ∈ (0, 1),satisﬁes

the equation Lu = f on Ω, |u; d|

(b)

0,Ω

< +∞,and|f; d|

(2+b)

α,Ω

< +∞ for b ∈

(−1, 0),then

|u; d|

(b)

2+α,Ω

≤ C



|u; d|

(b)

0,Ω

+ |f ; d|

(2+b)

α,Ω



where C = C(n, α, b).