Chapter 12  Fourier Series and the Laplace Transform

12.2  The Dirichlet Problem for the Unit Disk

    In Section 12.1 we introduced Fourier Series. The Dirichlet problem for the closed unit disk is to find a real-valued function that is harmonic in the unit disk D and that takes on the boundary values  

(12.10)            ,  

at points    on the unit circle  .  

Theorem 12.7 (The Dirichlet problem in  D). If has period and has the Fourier series representation

            ,  

then the solution to the Dirichlet problem in  the unit disk D is  

(12.11)            ,  

where denotes a complex number in the closed disk .  

Proof.

    The series representation in Equation (12.11) for u takes on the prescribed boundary values in Equation (12.10) at points on the unit circle .  Each term, and , in the series in Equation (12.11) is harmonic, so it is reasonable to conclude that the infinite series representing u will also be harmonic.  The proof follows the proof of Theorem 12.8.

    Theorem 12.8, which is attributed to the French mathematician Siméon Poisson, gives an integral representation for a function that is harmonic in a domain containing the closed unit disk.  The result is the analog to Poisson's integral formula for the upper half-plane.

Theorem 12.8 (Poisson Integral Formula for the Unit Disk). Let be a function that is harmonic in a simply connected domain that contains the closed unit disk .  If takes on the boundary values  

            ,  

then   has the integral representation  

            ,   

which is valid for  .

Proof.

Example 12.3.  Find the function that is harmonic in the unit disk and takes on the boundary values  .

Solution.

    Using Example 12.1, we write the Fourier series for  :

        .  

Using Equation (12.11) for the solution of the Dirichlet problem, we obtain

        .  

    This series representation of      takes on the prescribed boundary values at points where is continuous.
The boundary function is discontinuous at , which corresponds to ; which are points where was not prescribed.
Graphs of the approximations and  , which involve the first seven terms in the preceding two equations, are shown in Figure 12.13.

        Figure 12.13. The functions and .
Explore Solution 12.3.