Properties of solutions to potential equations
*©
Copyright 19942000 by Evans M. Harrell
II and James V. Herod. All rights reserved.
version of 30 August 2000.
Properties of potential equations
Even if there is an explicit formula for the solution of an equation,
you may not have enough information about the boundary or intiial
conditions to calculate that solution. Also, in its
complexity, an explicit solution may
hide important physical properties.
For these among other reasons it is often
useful analyze the features which will be
shared by all solutions to a
particular partial differential equation.
If there are some such general features, we can hope to find them
either by analyzing the equation itself or by analyzing a formula
for the solution. For instance, the
Green function for a partial
differential equation must contain all the physical
properties shared by the solutions it is capable of representing.
We begin by discussing the
potential equations
of
Laplace
and
Poisson,
which are discussed in Chapters
VI,
IX, and
XIX.
Our main tool will be integration by parts, as
encapsulated by
Green's identities.
The question we ask is the relationship between a function
u(x) and its average value over a small sphere centered
at x. The average would be expected to depend on the
radius of the ball, which we'll call r. Thus:
where _{n}
is the area of a sphere in n dimensions, of radius 1. Differentiating
with the vector chain rule, we get
This is now in a form where we can apply
Green's first identity
with the simplification that the function f is 1 (while g is u(x+ry)). The result is that
(pr1)
(the factor r arises because of the chain rule:
the gradient in Green's identity is
with respect to y rather than x).
This formula has several interesting consequences:
Definition PR.1.
A function is harmonic if it satisfies
Laplace's equation,
.
It is subharmonic if
^{2}u
0.
Finally, it is superharmonic if
^{2}u
0.
In these definitions, we assume that u is a smooth function and satisfies
the differential equation or inequality on an open region. Note
that any superharmonic function is the negative of a
subharmonic function and vice versa, so they are
not really distinct notions.
Some sources define subharmonic
and superharmonic functions by the property mentioned in
Theorem PR2, below, rather than by
a differential inequality.
From (pr1) we get the following theorem:
Theorem PR.2.

Suppose that u is harmonic on an open region D
and that
r is less than the
distance from x to the boundary of D
Then u = U(r;x). Moreover,
u is also equal to its average over the ball
{y: yx < r}.

Suppose that u is subharmonic (resp. superharmonic)
on an open region D
as in part 1,
Then u < U(r;x) (resp.
u > U(r;x)). Moreover, in this statement
the average over the sphere may be replaced
by the average over the ball
{y: yx < r}.

If the derivatives of second order of a function u
are continuous, the meanvalue properties stated in
parts 1 and 2 imply that u is
harmonic, subharmonic, or superharmonic,
accordingly.
Remarks about the proof.
Replacing the spheres by the balls in parts 1 and 2
is merely a matter
of performing an additional integration over r.
Other than that these statements are immediate from
(pr1). The converse, part 3,
also follows from that formula, with
contrapositive logic:
We consider the converse of part 1, as representative. Suppose that
^{2}u is continuous and not
equal to 0 at some point x. Then either there is some ball
centered at x where the right side of
(pr1) is positive, or else a ball
centered at x where the right side of
(pr1) is negative.
Whichever statement applies, U(r,x) changes as r varies,
which contradicts the meanvalue property.
As an example of the physical consequences of
this, consider
heat flow, as in
Chapter VIII.
If the temperature distribution of a
homogeneous
body does
not have the meanvalue property guaranteed
by
Theorem PR.2 for
harmonic functions, then it is not in equilibrium,
and the temperature will redistribute itself.
The meanvalue property is also related to the
maximum principle, which was stated and
proved
for the heat equation in
Chapter VIII. In
the case of harmonic functions it reads as follows.
Theorem PR.3.
Suppose that u is harmonic on an open region D.
If u attains its maximum or minimum value
within D
then u must be a constant throughout D.
Another stetment of the maximum principle
is that: for any closed region on which u
is harmonic, its maximum and its minimum are
attained on the boundary.
Finally, the maximum principle is an important tool for knowing when
a problem is wellposed, for it implies a uniqueness theorem
for
Poisson's equation
(see Chapter 19):
Theorem PR.4. Suppose that
^{2}u = f on D,
u = g on the boundary of D.
has two solutions, u_{1} and u_{2}.
Then u_{1} = u_{2}.
Proof.
Let w(x) := u_{1}(x)  u_{2}(x).
Then we know that w solves Laplace's equation and is
identically 0 on the boundary of D. As a consequence of the
maximum principle, both the maximum and the minimum of w in D
must equal 0, which means that u_{1} = u_{2}.
QED
Exercises PR

Provide two proofs for the maximum principle
for harmonic functions:

State and prove maximum or minimum principles for
subharmonic and superharmonic functions.
Link to
chapter VI
chapter IX
chapter XIX
Table of Contents
Evans Harrell's home page
Jim Herod's home page