Chapter 12  Fourier Series and the Laplace Transform

12.2  The Dirichlet Problem for the Unit Disk

        We have emphasized practical applications of harmonic functions to steady state temperatures, electrostatics, and fluid flow in Chapter 11.  

In Section 11.2 we introduced the N-Value Dirichlet problem, and showed how to solve Laplace's equation  ,  

in the upper-half plane, for a harmonic function   that has certain specified boundary values on the real axis.  

Now we will develop the solution to the Dirichlet problem in the closed unit disk  .  

Dirichlet Problem for the Unit Disk

        Given a real valued function    that is both piecewise continuous and a bounded function.  
        
Let    be considered as boundary values on the unit circle ,  in the sense that  

(12.10)           ,      at points    on the unit circle.

The Dirichlet problem for the closed unit disk    is to extend    to be

                           for     ,   

where    is harmonic in the unit disk and take on the boundary values (12.10)   at points where    is continuous.   

        Our first method of solution uses the Fourier Series representation for    that was developed in Section 12.1.  

Theorem 12.7 (Extended Fourier Series in the unit disk).   Let    be the boundary values on the unit circle ,  

(12.10.1)      .  

If    has the Fourier series representation  

                     ,  

then

(12.11.1)      ,  

solves the Dirichlet problem and is a harmonic function in the unit disk  .  

        We can motivate Theorem 12.7 if we are allowed to make the following claim:  

The series in (12.11.1) takes on the boundary values in Equation (12.10.1) at points on    where the radial limits exist, that is

                       

Furthermore, Exercise 12 in Section 3.3 shows that each of the terms,     and      are harmonic,

and so it is reasonable to conclude that the infinite series representing    in Equation (12.11.1) will be harmonic.  

Remark.  The radial limits will exist except at the finite number of points where    is discontinuous.  

There are details regarding the convergence of the series and the existence of radial limits that are left for advanced study.

Proof.

Example 12.3.  Find the function    that is harmonic in the unit disk    

and takes on the boundary values   .  

Solution.

        Using the result of Example 12.1 in Section 12.1, we write the Fourier series for  ,   

                    .  

Using Equation (12.11.1) for the extended Fourier series solution of the Dirichlet problem, we obtain

                    .  

    The series representations of      take 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.

The approximations     and   ,  

and the true solutions      and      are shown in the figures below.  

                    Figure 12.13.A. The functions    and   .  

                    Figure 12.13.B. The functions    and   .  

Explore Solution 12.3.

    Our next result, Theorem 12.8, is the analog to Theorem 11.3, Poisson's integral formula for the upper half-plane, that was introduced in Section 11.3.

It shows that the value of the harmonic function    inside the unit disk is a special "average of the values    on the boundary."  

In the integrand the function is multiplied by the Poisson kernel    which includes the variables  .

Theorem 12.8 (Poisson Integral Formula in the unit disk).   Let    be the boundary values on the unit circle ,  

(12.10.2)        .  

If    is both piecewise continuous and bounded, then  

(12.11.2)        ,      ,    

solves the Dirichlet problem and is a harmonic function in the unit disk  .  

Observation.  The integrand is taken over the unit circle where the parameter of integration is  ,  and the numerator includes the function  .

Proof.

Corollary 12.1  (N-Value Dirichlet solution for the unit disk).   Assume that  .  

If    is the boundary value function on the unit circle  ,  

(12.10.3)          

then   

(12.11.3)        ,  

solves the Dirichlet problem and is a harmonic function in the unit disk  .  

Proof.  Use the indefinite integral  

                    .

Then

                    

Remark.  When applying formula (12.11.3) it is necessary to pay attention and to use appropriate branches and to beware of branch cuts.

Extra Example 1.  Find the function    that is harmonic in the unit disk  ,  

and takes on the boundary values     

                    Figure 1.  The graphs of        and    .    

Solution using Fourier series.

In Exercise 9 in Section 12.1, we showed that the Fourier series for   can be written as  

                    .  

Use Equation (12.11.3) to construct the extended Fourier series solution of the Dirichlet problem.

Therefore, the Fourier series solution is

                    .  

Solution using Poisson's integral.

Use the known anti-derivative      and get  


                              
Therefore, the Poisson integral solution is

                    .

Solution using N-Value Dirichlet formula.

        Set    and    and then use the formula  (12.11.3)  and get

                    

Therefore, the N-Value Dirichlet solution is

                    .
         
Remark.  This is just a straightforward way of calculating the Poisson integral solution, and is recommended for working some of the exercises.

Explore Extra Solution 1.

Extra Example 2.  Find the function    that is harmonic in the unit disk   ,  

and takes on the boundary values   .  

                    Figure 2.  The graphs of        and    .    

Explore Extra Solution 2.

Remarks.  Some integrals of the form     are readily available.  

For      the computer algebra systems Mathematica and Maple can be used to obtain the following formulae.


                    
                    
                    
                      
                    
                    
                      

The software Maple will also compute the above integrals, but the Maple syntax will use the notation    and  .  

The first integral      can easily be verified using techniques from calculus.

Caveat.  It is beyond the scope of this book to use the other two integrals involving  .  

Exercises for Section 12.2.  The Dirichlet Problem for the Unit Disk