regularity theorem for the Laplace equation

regularity theorem for the Laplace equation

Warning: This entry is still in the process of being written, hence is not yet .

Let

D

be an open subset of

ℝ

n

. Suppose that

f

:

D

→

ℝ

is twice

differentiable

and satisfies Laplace’s equation. Then

f

has

derivatives

of all orders and is, in fact

analytic

.

Proof:

Let

𝐩

be any point of

D

. We shall show that

f

is analytic at

𝐩

. Since

D

is an open set, there must exist a

real number

r

>

0

such that the

closed ball

of radius

r

about

𝐩

lies inside of

D

.

Since

f

satisfies Laplace’s equation, we can express the value of

f

inside this ball in terms of its values on the boundary of the ball by using Poisson’s formula:

f

⁢

(

𝐱

)

=

1

r

n

-

1

⁢

A

⁢

(

n

-

1

)

⁢

∫

|

𝐲

-

p

|

=

r

f

⁢

(

𝐲

)

⁢

r

2

-

|

𝐱

-

𝐩

|

2

|

𝐱

-

𝐲

|

n

⁢

𝑑

Ω

⁢

(

𝐲

)

Here,

A

⁢

(

k

)

denotes the

http://planetmath.org/node/4495

area of the

k

-dimensional sphere and

d

⁢

Ω

denotes the measure on the sphere of radius

r

about

𝐩

.

We shall show that

f

is analytic by deriving a

convergent

power series for

f

. From this, it will automatically follow that

f

has derivatives of all orders, so a separate proof of this fact will not be necessary.

Since this involves manipulating power series in several variables, we shall make use of

multi-index notation

to keep the equations from becoming unnecessarily complicated and drowning in a plethora of indices.

First, note that since

f

is assumed to be twice differentiable in

D

, it is

continuous

in

D

and, hence, since the sphere of radius

r

about

s

is

compact

, it attains a

maximum

on this sphere. Let us denote this maxmum by

M

. Next, let us consider the quantity

1

|

𝐱

-

𝐲

|

n

which appears in the

integral

. We may write this quantity more explicitly as

(

|

𝐲

-

𝐩

|

2

-

2

⁢

(

𝐱

-

𝐩

)

⋅

(

𝐲

-

𝐩

)

+

|

𝐱

-

𝐩

|

2

)

-

n

2

.

Since the values of the variable

y

has been restricted by the condition

|

𝐲

-

𝐩

|

=

r

, we may rewrite this as

1

r

n

⁢

(

1

+

-

2

⁢

(

𝐱

-

𝐩

)

⋅

(

𝐲

-

𝐩

)

+

|

𝐱

-

𝐩

|

2

r

2

)

-

n

2

.

Assume that

|

𝐱

-

𝐩

|

<

r

/

4

. Then we have

|

-

2

⁢

(

𝐱

-

𝐩

)

⋅

(

𝐲

-

𝐩

)

+

|

𝐱

-

𝐩

|

2

r

2

|

≤

2

⁢

|

(

𝐱

-

𝐩

)

⋅

(

𝐲

-

𝐩

)

|

r

2

+

|

𝐱

-

𝐩

|

2

r

2

≤

2

⁢

|

𝐱

-

𝐩

|

⁢

|

𝐲

-

𝐩

|

r

2

+

(

|

𝐱

-

𝐩

|

r

)

2

≤

2

⋅

1

4

+

(

1

4

)

2

=

9

16

<

1

.

Since this

absolute value

is less than one, we may apply the

binomial theorem

to obtain the series

1

|

𝐱

-

𝐲

|

n

=

1

r

n

⁢

(

1

+

-

2

⁢

(

𝐱

-

𝐩

)

⋅

(

𝐲

-

𝐩

)

+

|

𝐱

-

𝐩

|

2

r

2

)

n

2

=

∑

m

=

0

∞

(

n

2

)

m

¯

m

!

⁢

(

-

2

⁢

(

𝐱

-

𝐩

)

⋅

(

𝐲

-

𝐩

)

+

|

𝐱

-

𝐩

|

2

r

2

)

m

Note that each term in this sum is a

polynomial

in

x

-

p

. The powers of the various

components

of

x

-

p

that appear in the

m

-th term range between

m

and

2

⁢

m

. Moreover, let us note that we can strengthen the assertion used to show that the

binomial series

converged by inserting absolute value bars. If we write

-

2

⁢

(

x

-

p

)

⋅

(

y

-

p

)

+

|

x

-

p

|

2

r

2

=

∑

k

=

0

n

c

k

⁢

(

y

)

⁢

(

x

-

p

)

k

+

∑

k

1

,

k

2

=

0

n

c

k

1

⁢

k

2

⁢

(

y

)

⁢

(

x

-

p

)

k

1

⁢

(

x

-

p

)

k

2

,

(actually, the coefficients

c

k

1

⁢

k

2

depend on

y

trivially, but the dependence on

y

has been indicated for the sake of uniformity) then

∑

k

=

0

n

|

c

k

⁢

(

y

)

|

⁢

|

(

x

-

p

)

k

|

+

∑

k

1

,

k

2

=

0

n

|

c

k

1

⁢

k

2

⁢

(

y

)

|

⁢

|

(

x

-

p

)

k

1

|

⁢

|

(

x

-

p

)

k

2

|

≤

9

16

.

Raising this to the

m

-th power, we see that, if we define

(

-

2

⁢

(

x

-

p

)

⋅

(

y

-

p

)

+

|

x

-

p

|

2

r

2

)

m

=

∑

k

1

,

k

2

,

…

,

k

m

=

0

n

c

k

1

⁢

k

2

,

⋯

⁢

k

m

⁢

(

y

)

⁢

(

x

-

p

)

k

1

⁢

(

x

-

p

)

k

2

⁢

⋯

⁢

(

x

-

p

)

k

m

,

then we have

∑

k

1

,

k

2

,

…

,

k

m

=

0

n

|

c

k

1

⁢

k

2

,

⋯

⁢

k

m

⁢

(

y

)

|

⁢

|

(

x

-

p

)

k

1

|

⁢

|

(

x

-

p

)

k

2

|

⁢

⋯

⁢

|

(

x

-

p

)

k

m

|

≤

(

9

16

)

m

Because of the fact that one may freely rearrange and regroup the terms in an absolutely convegent series, we may conclude that the expansion of

|

x

-

y

|

-

n

in powers of

x

-

p

converges absolutely. Furthermore, there exist constants

b

k

1

⁢

k

2

,

⋯

⁢

k

m

such that the term involving

|

(

x

-

p

)

k

1

|

⁢

|

(

x

-

p

)

k

2

|

⁢

⋯

⁢

|

(

x

-

p

)

k

m

|

in the power series is

bounded

by

b

k

1

⁢

k

2

,

⋯

⁢

k

m

.

Title

regularity theorem for the

Laplace equation

Canonical name

RegularityTheoremForTheLaplaceEquation

Date of creation

2013-03-22 14:57:29

Last modified on

2013-03-22 14:57:29

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

17

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 26B12

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