3.6.1 Solution to the Dirichlet Problem on a Disk
A fundamental problem in the theory of partial differential equations is to find a function that is continuous on the closed disk , harmonic on the open disk , and identically equal to a given boundary function on . This is known as the Dirichlet problem (for Laplace’s equation) on a disk.
Theorem 3.6.1.
For a continuous function , the unique real-valued solution that solves
is given by the Poisson integral formula:
where .
Proof.
Since
from (3.6.7), we have that (since each term is independent of either or ). Moreover, by Theorem 1.2.6, (3.6.10) gives that
Our goal is to show that for fixed ,
Let and . Then with ,
For a constant harmonic function identically equal to , we get from (3.6.5). Hence,
By the continuity of , , such that , we have that . Therefore,
Since the Poisson kernel is non-negative,
By continuity of on the compact set , by Theorem 1.2.15, it is bounded and is finite. The Poisson kernel can be rewritten as
where and , with . Then such that with (small enough so that ),
and
as in Figure 5. Then,
We aim to prove that . Since , the condition is satisfied if
and from rearrangement, we can tighten the constraint with:
which follows in particular from
From Figure 5, it is evident that . For (3.6.12) to be true, we previously had that . In other words
Obviously, this is satisfied if . This can be rearranged into
Therefore, we can choose
under which (3.6.12), (3.6.13), and (3.6.14) are satisfied.
Hence, , such that with , we have . Then (3.6.11) follows.
We will now show that is unique. Assume that on also solves the problem. Then is harmonic and vanishes on . By the Poisson Integral Formula ((3.6.5)),
for all . Hence , a contradiction.