Helms L.L. Potential Theory

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x Contents

3 Green Functions .......................................... 107

3.1 Introduction ........................................... 107

3.2 GreenFunctions........................................ 107

3.3 Symmetryof the GreenFunction ......................... 115

3.4 GreenPotentials ....................................... 122

3.5 RieszDecomposition.................................... 135

3.6 PropertiesofPotentials ................................. 144

4 Negligible Sets............................................ 149

4.1 Introduction ........................................... 149

4.2 SuperharmonicExtensions............................... 149

4.3 ReductionofSuperharmonicFunctions.................... 158

4.4 Capacity .............................................. 163

4.5 Boundary Behavior of the Green Function . . . . . . . . . . . . . . . . . 178

4.6 Applications ........................................... 182

4.7 Sweeping.............................................. 189

5 Dirichlet Problem for Unbounded Regions ................ 197

5.1 Introduction ........................................... 197

5.2 ExteriorDirichletProblem............................... 197

5.3 PWB Method for Unbounded Regions . . . . . . . . . . . . . . . . . . . . 204

5.4 Boundary Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

5.5 Intrinsic Topology ...................................... 223

5.6 ThinSets.............................................. 224

5.7 ThinnessandRegularity ................................ 228

6Energy................................................... 241

6.1 Introduction ........................................... 241

6.2 EnergyPrinciple ....................................... 241

6.3 MutualEnergy......................................... 249

6.4 ProjectionsofMeasures ................................. 257

6.5 Wiener’sTest.......................................... 260

7 Interpolation and Monotonicity ........................... 267

7.1 Introduction ........................................... 267

7.2 H¨older Spaces.......................................... 268

7.3 GlobalInterpolation .................................... 273

7.4 Interpolationof WeightedNorms ......................... 280

7.5 Inner Norms ........................................... 284

7.6 Monotonicity .......................................... 288

8 Newtonian Potential ...................................... 303

8.1 Introduction ........................................... 303

8.2 Subnewtonian Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

8.3 Poisson’sEquation ..................................... 316

8.4 H¨older ContinuityofSecond Derivatives................... 319

8.5 TheReﬂection Principle................................. 326

Contents xi

9 Elliptic Operators ........................................ 333

9.1 Introduction ........................................... 333

9.2 LinearSpaces.......................................... 334

9.3 Constant Coeﬃcients ................................... 335

9.4 SchauderInteriorEstimates.............................. 338

9.5 MaximumPrinciples.................................... 343

9.6 TheDirichletProblemfor a Ball ......................... 347

9.7 Dirichlet Problem for Bounded Domains . . . . . . . . . . . . . . . . . . . 353

9.8 Barriers............................................... 358

10 Apriori Bounds ........................................... 371

10.1 Introduction ........................................... 371

10.2 Green Function foraHalf-space .......................... 372

10.3 Mixed Boundary Conditions for Laplacian . . . . . . . . . . . . . . . . . 376

10.4 NonconstantCoeﬃcients ................................ 385

11 Oblique Derivative Problem .............................. 391

11.1 Introduction ........................................... 391

11.2 Boundary Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 391

11.3 Curved Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405

11.4 Superfunctions for Elliptic Operators . . . . . . . . . . . . . . . . . . . . . 410

11.5 Regularity of Boundary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

References .................................................... 431

Index ......................................................... 435

Notation ...................................................... 439

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Chapter 0

Preliminaries

0.1 Notation

An element of the n-dimensional Euclidean space R

,n≥ 2, will be denoted

by x =(x

,...,x

)anditslength|x| is deﬁned by |x| =(



i=1

)

1/2

.If

x =(x

,...,x

)andy =(y

,...,y

) ∈ R

, the inner product x ·y is deﬁned

by x·y =



i=1

, and the distance between x and y is deﬁned to be |x−y|.

The angle between two nonzero vectors x and y is deﬁned to be the angle φ

such that 0 ≤ φ ≤ π and

cos φ =

x · y

|x||y|

The zero vector of R

is denoted by 0 = (0,...,0). The ball B

x,ρ

with center

x and radius ρ is deﬁned by B

x,ρ

= {y; |x −y| <ρ}; the sphere with center x

and radius ρ is deﬁned by S

x,ρ

= ∂B

x,ρ

= {y; |x−y| = ρ}. The closure of a set

F ⊂ R

is denoted by F

−

or by cl F , its interior by int F , and its complement

by ∼ F . The standard basis for R

will be denoted by {ε

,...,ε

} where ε

has a 1 in the ith position and 0 in all other positions.

The spherical coordinates of a point x =(x

,...,x

) =0inR

are deﬁned

as follows: if r = |x|,then

θ =



,...,



is a point of S

0,1

= ∂B

0,1

, the unit sphere with center at 0. The r and θ

of the ordered pair (r, θ) are called the spherical coordinates of the point x.

This transformation from rectangular coordinates to spherical coordinates is

essentially the mapping (x

,...,x

) → (θ

,...,θ

n−1

,r)where

L.L. Helms, Potential Theory, Universitext, 1

 Springer-Verlag London Limited 2009

2 0 Preliminaries

n−1

r =(x

+ ···+ x

)

1/2

Letting θ

= x

/r, θ

is the cosine of the angle between x and the vector

(0,...,0, 1); that is, the angle between x and the x

-axis. The Jacobian of

the map is easily calculated and its absolute value is given by



∂(x

,...,x

)

∂(θ

,...,θ

n−1

,r)



n−1

(1 − θ

−···−θ

n−1

)

1/2

n−1

|θ

Surface area on a sphere S

x,ρ

is a ﬁnite measure on the Borel subsets of

x,ρ

. The integral of a Borel measurable function f on S

x,ρ

relative to surface

area will be denoted by



x,ρ

fdσ or



x,ρ

f(x) dσ(x).

The volume of B

x,ρ

⊂ R

and surface area of S

x,ρ

will be denoted by ν

(ρ)

and σ

(ρ), respectively; when ρ = 1 these quantities will be denoted by

and σ

, respectively. The ν

(ρ)andσ

(ρ) are related by the equations

(ρ)=ρ

,σ

(ρ)=ρ

n−1

,andν

= σ

/n. Derivation of these equations

can be found in Helms [28]. The σ

are given by

⎧

⎪

⎨

⎪

⎩

n/2

(n/2)!

if n is even ,

(n+1)/2

(n−1)/2

1·3·5···(n−2)

if n is odd and > 1.

(0.1)

Consider an extended real-valued function f deﬁned on a subset of R

When f is considered as a function of the spherical coordinates (r, θ)ofx,

f(x) will be denoted by f(r, θ). Suppose f is integrable on a ball B

0,ρ

.Then

the integral of f over B

0,ρ

can be evaluated using spherical coordinates as in



0,ρ

f(x) dx =



n−1





|θ|=1

f(r, θ) dσ(θ)



dr.

If β

,...,β

are nonnegative integers, β =(β

,...,β

) is called a multi-

index, |β| is deﬁned by |β| =



i=1

,andβ! is deﬁned by β!=β

×β

×...×

. If the real-valued function u = u(x

,...,x

) is deﬁned on a neighborhood

of a point x ∈ R

, put D

u = D

u = ∂u/∂x

u = D

u = ∂

u/∂x

∂x

;

more generally, if β =(β

,...,β

) is a multi-index, put

0.1 Notation 3

u(x)=

∂

|β|

∂x

···∂x

(x)

provided the indicated partial derivatives are deﬁned. The gradient ∇u of u

at x is deﬁned by

∇u(x)=(D

u(x),...,D

u(x)).

If the function u is a function of x, y ∈ R

, then derivatives with respect to

just one of the variables, say x, will be denoted by D

(x)

If Ω is an open subset of R

and f is a bounded function on Ω, the norm

of f is deﬁned by  f 

0,Ω

=sup

x∈Ω

|f(x)|. The set of continuous functions on

Ω will be denoted by C

(Ω), the set of bounded continuous functions on Ω

will be denoted by C

(Ω), and C

(Ω

−

) will denote the set of functions on Ω

having continuous extensions to Ω

−

.Iff ∈ C

(Ω) has a continuous extension

to Ω

−

, the same symbol f will be used for its extension. If f ∈ C

(Ω)and

lim

y→x,y∈Ω

f(y)existsinR for all x ∈ ∂Ω,thenf has a continuous extension

to Ω

−

.Ifk is a positive integer or ∞,C

(Ω) will denote the set functions

having continuous derivatives of all orders less than or equal to k on Ω;

(Ω

−

) will denote the set of functions on Ω with derivatives of order less

than or equal to k having continuous extensions to Ω

−

.Ifk is a positive

integer,  D

u 

0,Ω

will denote sup

|β|=k

 D

u 

0,Ω

. A subscript 0 on C

(Ω)

will signify the subset C

(Ω) of functions in C

(Ω) having compact support

in Ω. A measure on the Borel subsets of Ω which is ﬁnite on compact subsets

of Ω is called a Borel measure and is determined by its values on the compact

subsets of Ω.

If u is an extended real-valued function with domain Ω ⊂ R

,x∈ Ω,and

u(x)=+∞ (u(x)=−∞), then u is continuous in the extended sense

at x if lim

y→x,y∈Ω

u(y)=+∞ (lim

y→x,y∈Ω

(y)=−∞); u is continuous in the

extended sense on Ω if u is either continuous or continuous in the extended

sense at each point of Ω.

Let u be an extended real-valued function with domain Ω ⊂ R

.Foreach

y ∈ R

,letN(y) be the collection of neighborhoods of y.IfΩ

⊂ Ω and x is

any point of Ω

−

, deﬁne

lim inf

y→x,y∈Ω

u(y)= sup

N∈N (x)



inf

y∈(N∩Ω

)∼{x}

u(y)



lim sup

y→x,y∈Ω

u(y)= inf

N∈N (x)



sup

y∈(N∩Ω

)∼{x}

u(y)



;

if Ω

= Ω, the notation lim inf

y→x

u(y) and lim sup

y→x

u(y) will be used.

The function u is lower semicontinuous (l.s.c.) at x ∈ Ω if u(x) ≤

lim inf

y→x

u(y); upper semicontinuous (u.s.c.) at x ∈ Ω if u(x) ≥

lim sup

y→x

u(y). If u is l.s.c. on Ω,thencu is l.s.c. or u.s.c. on Ω accord-

ing as c ≥ 0orc ≤ 0. If u is l.s.c. on Ω,then−u is u.s.c. on Ω. The function

u is l.s.c. on Ω if and only if the set {x ∈ Ω; u(x) >c} is relatively open

4 0 Preliminaries

in Ω for every real number c.If{u

; a ∈ A} is a family of l.s.c. functions

with common domain Ω,thenu =sup

a∈A

is l.s.c. on Ω.Ifu and v are

l.s.c. functions with a common domain Ω,thenmin(u, v) is l.s.c. as is u + v if

deﬁned on Ω.Ifu is any extended real-valued function on Ω ⊂ R

,thelower

regularization ˆu of u deﬁned for x ∈ Ω

−

by ˆu(x) = lim inf

y→x

u(y) is l.s.c.

on Ω

−

.Ifu is l.s.c. on the compact set Ω ⊂ R

, then it attains its minimum

value on Ω.Moreover,ifu is l.s.c. on Ω ⊂ R

and there is a continuous

function f on R

such that u ≥ f on Ω, then there is an increasing sequence

of continuous functions {f

} on R

such that lim

j→∞

= u on Ω.

0.2 Useful Theorems

In this section, several results used repeatedly in subsequent chapters will be

stated without proofs. Proofs can be found in many intermediate real analysis

texts; e.g., Apostol [1]. The following theorem is easily proved using the one-

dimensional version by introducing a function g(t)=u(tx +(1− t)y), 0 ≤

t ≤ 1.

Theorem 0.2.1 (Mean Value Theorem) If Ω is an open subset of R

u has continuous ﬁrst partial derivatives on Ω, x and y are two points of Ω

suchthatthelinesegment{tx +(1−t)y;0≤ t ≤ 1}⊂Ω, then there is point

z on the line segment joining x and y such that

u(y) − u(x)=∇u(z) · (y − x).

The relationship of the limit of a convergent sequence of functions to the

limit of the corresponding sequence of derivatives is the content of the next

theorem.

Theorem 0.2.2 Let {f

} be a sequence of real-valued functions on an open

interval (a,b) having real-valued derivatives at each point of (a, b) such that

asequence{f

)} converges for some x

∈ (a, b). If the sequence {f



}

converges uniformly to a function g on (a, b),then

(i) the sequence {f

} converges uniformly to a function f on (a, b),and

(ii) for each x ∈ (a, b), f



(x) exists and f



(x)=g(x).

Recall that a family F of functions f on Ω ⊂ R

is equicontinuous at

x ∈ Ω if for each >0thereisaδ>0 such that |f (x) − f(y)| <for all

f ∈Fand all y ∈ B

x,δ

∩Ω, equicontinuous on Ω if equicontinuous at each

point of Ω,anduniformly equicontinuous if the above δ does not depend

upon x.

Theorem 0.2.3 (Arzel`a-Ascoli Theorem) Let {f

} be a sequence of uni-

formly bounded and equicontinuous functions on a compact set K ⊂ R

.Then

there is a subsequence {f

} which converges uniformly on K to a continuous

function f.

0.2 Useful Theorems 5

The Arzel`a-Ascoli theorem is valid for generalized sequences that are deﬁned

as follows. Recall that a nonempty set A and a relation  constitute a di-

rected set if  is reﬂexive and transitive, and for each pair α, β ∈Athere

is a γ ∈Asuch that γ  α and γ  β.Anet {x

; α ∈A}is a function from

the directed set A into a set X.IfX is a metric space with metric d,the

net {x

; α ∈A}converges to x ∈ X, written lim

= x, if for every >0

there is an α



∈Asuch that d(x

,x) <for all α  α



Theorem 0.2.4 (Arzel`a-Ascoli Theorem) Let {f

; α ∈A}be a net of

uniformly bounded and equicontinuous functions on a compact set K ⊂ R

Then there is a subnet {f

} which converges uniformly on K to a continuous

function f.

If (X, M) is a measurable space, a signed measure μ on M is an extended

real-valued, countably additive set function on M with μ(∅) = 0 and taking

on only one of the two values +∞, −∞. A signed measure μ on a measurable

space (X, M ) has a decomposition μ = μ

−μ

−

into a positive part μ

and

a negative part μ

−

.Thetotal variation |μ| of μ is deﬁned by |μ| = μ

+μ

−

If both μ

and μ

−

are ﬁnite measures, μ is said to be of bounded variation

on X,inwhichcasethenorm of μ is deﬁned by μ = |μ|(X).

If X is a locally compact Hausdorﬀ space, the smallest σ-algebra of subsets

of X containing the closed subsets of X is called the class of Borel sets of

X and is denoted by B(X). A measure μ on B(X)isaBorel measure if

ﬁnite on each compact set. All Borel measures will be assumed to be regular

Borel measures;thatis,μ(Λ)=sup{μ(Γ ); Γ ⊂ Λ, Γ compact },Λ∈B(X).

If {μ

} is a sequence of signed measures on B(X) of bounded variation, the

sequence converges in the w

∗

- topology to a signed measure μ of bounded

variation if

lim

j→∞



fdμ



fdμ

for all f ∈ C

(X). Lebesgue measure on R

will be denoted by μ

The function u on the open set Ω ⊂ R

is locally integrable on Ω if

integrable with respect to Lebesgue measure on each compact subset of Ω.

The following theorem is a useful tool for extracting a convergent subsequence

from a sequence of signed measures.

Theorem 0.2.5 Let {μ

} be a sequence of signed measures on the Borel

subsets of a compact metric space X with μ

≤M for all j ≥ 1.Then

there is a signed measure μ on the Borel subsets of X with u≤M and a

subsequence {μ

} that converges to μ in the w

∗

-topology ; that is,

lim

i→∞



fdμ



fdμ for all f ∈ C

(X).

Sketch of Proof: Since X is a compact metric space, C

(X) is separa-

ble. Let {f

} be countable dense subset of C

(X). For each k ≥ 1, the

6 0 Preliminaries

sequence



dμ

is bounded, and there is a subsequence {μ

(k)

} of the se-

quences {μ

(1)

},...{μ

(k−1)

} such that lim

j→∞



dμ

(k)

exists. The Cantor

diagonalization procedure then can be invoked to assert the existence of a

subsequence that works for all f

Corollary 0.2.6 If {f

} is a sequence in C

(X) that converges uniformly to

f ∈ C

(X) and {μ

} is a sequence of signed measures with μ

≤M<+∞

for all j ≥ 1 that converges to μ in the w

∗

- topology, then

lim

j→∞



dμ



fdμ.

Sketch of Proof: Given >0, there is an j() such that |



fdμ

−



fdμ| <

/2and|f

− f | </2M for all j>j(). Apply triangle inequality.

A linear space X is a normed linear space if to each x ∈ X there is

associated a real number x with the following properties:

(i) x≥0and x =0ifandonlyifx =0,

(ii) αx = |α|x for all α ∈ R, x ∈ X,and

(iii)x + y≤x + y for all x, y ∈ X.

The function ·is called a norm on X.If,inaddition,(X,d)isacomplete

metric space where d(x, y)=x−y,thenX is called a Banach space under

the norm ·. If it is necessary to consider more than one Banach space, the

norm will be denoted by ·

. If only the ﬁrst part of (i) holds, then ·

is called a seminorm on X.Alinear operator on X is a linear map whose

domain D(A) is a subspace of X and range R(A) is a subset of X.IfA is a

linear operator on the Banach space X with domain D(A)=X and there is

a constant M such that  Ax ≤ M  x  for all x ∈ X,thenA is called a

bounded linear operator,inwhichcasethenormofA is deﬁned by

 A =sup

x=0

 Ax 

 x 

A linear operator need not be bounded.

Chapter 1

Laplace’s Equation

1.1 Introduction

Potential theory has its origins in gravitational theory and electromagnetic

theory. The common element of these two is the inverse square law govern-

ing the interaction of two bodies. The concept of potential function arose

as a result oﬀ the work done in moving a unit charge from one point of

space to another in the presence of another charged body. A basic potential

function 1/r, the reciprocal distance function, has the important property

that it satisﬁes Laplace’s equation except when r = 0. Such a function is

called a harmonic function. Since the potential energy at a point due to a

distribution of charge can be regarded as the sum of a large number of po-

tential energies due to point charges, the corresponding potential function

should also satisfy Laplace’s equation. Applications to electromagnetic the-

ory led to the problem of determining a harmonic function on a region having

prescribed values on the boundary of the region. This problem came to be

known as the Dirichlet problem. A similar problem, connected with steady-

state heat distribution, asks for a harmonic function with prescribed ﬂux or

normal derivative at each point of the boundary. This problem is known as

the Neumann problem. Another problem, Robin’s problem, asks for a har-

monic function satisfying a condition at points of the boundary which is a

linear combination of prescribed values and prescribed ﬂux.

In this chapter, explicit formulas will be developed for solving the Dirichlet

problem for a ball in n-space, uniqueness of the solution will be demonstrated,

and the solution will be proved to have the right “boundary values.” Not

nearly as much is possible for the Neumann problem. Explicit formulas for

solving the Neumann problem for a ball are known only for the n =2and

n = 3 cases.

L.L. Helms, Potential Theory, Universitext, 7

 Springer-Verlag London Limited 2009