Measure, Integration & Real Analysis, 2021a

348 Chapter 11 Fourier Analysis
Solution to Dirichlet Problem on Disk
As a bonus to our investigation into Fourier series, the previous result provides the
solution to the Dirichlet problem on the unit disk. To state the Dirichlet problem, we
first need a few definitions. As usual, we identify C with R 2 . Thus for x, y ∈ R, we
can think of w = x + yi ∈ C or w =(x, y) ∈ R 2 . Hence
D = {w ∈ C : |w| < 1} = {(x, y) ∈ R 2 : x 2 + y 2 < 1}.
For a function u : G → C on an open subset G of C (or an open subset G of
R 2 ), the partial derivatives D 1 u and D 2 u are defined as in 5.46 except that now we
allow u to be a complex-valued function. Clearly D j u = D j (Re u)+iD j (Im u) for
j = 1, 2.
11.19 Definition harmonic function; Laplacian; Δu
A function u : G → C on an open subset G of R 2 is called harmonic if
(
D1 (D 1 u) ) (w)+ ( D 2 (D 2 u) ) (w) =0
for all w ∈ G. The left side of the equation above is called the Laplacian of u at
w and is often denoted by (Δu)(w).
11.20 Example harmonic functions
• If g : G → C is an analytic function on an open set G ⊂ C, then the functions
Re g, Im g, g, and g are all harmonic functions on G, as is usually discussed
near the beginning of a course on complex analysis.
• If ζ ∈ ∂D, then the function
w ↦→ 1 −|w|2
|1 − ζw| 2
is harmonic on C \{ζ} (see Exercise 7).
• The function u : C \{0} → R defined by u(w) = log|w| is harmonic on
C \{0}, as you should verify. However, there does not exist a function g
analytic on C \{0} such that u = Re g.
The Dirichlet problem asks to extend a continuous function on the boundary of an
open subset of R 2 to a function that is harmonic on the open set and continuous on
the closure of the open set. Here is a more formal statement:
11.21
Dirichlet problem on G: Suppose G ⊂ R 2 is an open set and
f : ∂G → C is a continuous function. Find a continuous function
u : G → C such that u| G is harmonic and u| ∂G = f .
For some open sets G ⊂ R 2 , there exist continuous functions f on ∂G whose
Dirichlet problem has no solution. However, the situation on the open unit disk D is
much nicer, as we will soon see.
Measure, Integration & Real Analysis, by Sheldon Axler