11.2  Invariance of Laplace's Equation and the Dirichlet Problem

    Let D be a domain whose boundary is made up of piecewise smooth contours joined end to end.  The Dirichlet problem is to find a function that is harmonic in D such that takes on prescribed values at points on the boundary.  Let us first study the problem in the upper half plane.

Theorem 11.1 (Invariance of Laplace's Equation).  Let be harmonic in a domain G in the w-plane. Then satisfies Laplace's equation  

(11-1)            ,   

at each point    in  G.   If    is a conformal mapping from a domain D in the z-plane onto  G, then the composition  

(11-2)            ,  

is harmonic in D, and    satisfies Laplace's equation  

(11-3)            ,  

at each point    in D.

Proof.

Proof of Theorem 11.1 is in the book.
Complex Analysis for Mathematics and Engineering

Example 11.4.  Show that    is harmonic in the disk  .  

Solution.  The results of Exercise 7(b), of Section 10.2, show that the function

            

is a conformal mapping of the disk    onto the right half-plane  ,  it can be rewritten as  

               

The results from Exercise 7(b), Section 5.2, show that the function  

            

is harmonic in the right half-plane  .  Taking the real and imaginary parts of  ,  we have  

                and    .

Substituting these equations into the formula for    and using (11-2), we find that

               is harmonic for  .  

Explore Solution 11.4.

    Let D be a domain whose boundary is made up of piecewise smooth contours joined end to end.  The Dirichlet problem is to find a function that is harmonic in D such that  takes on prescribed values at points on the boundary.  Let's first look at this problem in the upper half-plane.

Example 11.5.  Show that the function  

(11-4)              

is harmonic in the upper half plane    and has boundary values  

Solution.  The function  

is analytic in the upper half-plane  ,  and its imaginary part is the harmonic function  .  

Explore Solution 11.5.

Remark 11.1  Let t be a real number.  We shall use the convention    so that the function    denotes the branch of the inverse tangent that lies in the range  .  Doing so permits us to write the solution in equation (11-4) as  .

Theorem 11.2 (N-Value Dirichlet Problem for the Upper Half Plane).  Let    denote real constants.  The function  

(11-5)              

is harmonic in the upper half plane    and takes on the boundary values

              for    

The situation is illustrated in Figure 11.4.

Figure 11.4  The boundary conditions for the harmonic function  .

Proof.

Proof of Theorem 11.2 is in the book.
Complex Analysis for Mathematics and Engineering

Example 11.6.  Find a function    that is harmonic in the upper half-plane    and takes on the boundary values indicated in Figure 11.5.  That is

              

Solution. This is a four-value Dirichlet problem in the upper half-plane defined by  .  For the z plane, the solution in Equation (11-5) becomes  

            

Here we have    and  ,  which we substitute into equation for    to obtain  

              

Figure 11.5  The boundary values for the Dirichlet problem.

Explore Solution 11.6.

Extra Examples1 (a-d).  Find a function    that is harmonic in the upper half-plane    and takes on the indicated boundary values.

Explore Solution Extra Example 1 (a).

Explore Solution Extra Example 1 (b).

Explore Solution Extra Example 1 (c).

Explore Solution Extra Example 1 (d).

Example 11.7.  Find a function that is harmonic in the upper half-plane  ,  which takes on the boundary values  

              

Solution.  This is a three-value Dirichlet problem in the upper half-plane defined by  .  For the z plane, the solution in Equation (11-5) becomes  

            

Here we have    and  ,  which we substitute into the above equation for    to obtain  

               

A three-dimensional graph of is shown in Figure 11.6.

Figure 11.6  The graph of    

            with the boundary values  , for , and , for .

Explore Solution 11.7.

    We now state the N-value Dirichlet problem for a simply connected domain.  We let D be a simply connected domain bounded by the simple closed contour C and let    denote N points that lie along C in this specified order as C is traversed in the positive direction (counterclockwise).  Then we let denote the portion of C that lies strictly between , for , and let denote the portion that lies strictly between .  Finally, we let    be real constants.  

    We want to find a function that is harmonic in D and continuous on    that takes on the boundary values

        
(11-6)              for    
        

The situation is illustrated in Figure 11.7.

Figure 11.7  The boundary values for for the Dirichlet problem in the simply connected domain D.

    One method for finding is to find a conformal mapping  

(11-7)            

of D onto the upper half-plane , such that the N points    are mapped onto the points  ,  for  ,  and    is mapped onto    along the u axis in the w -plane.

    When we use Theorem 11.1, the mapping in Equation (11-7) gives rise to a new N-value Dirichlet problem in the upper half-plane for which the solution is given by Theorem 11.2.  If we set  ,  then the solution to the Dirichlet problem in D with the boundary values from Equation (11-6) is

This method relies on our ability to construct a conformal mapping from D onto the upper half-plane .  Theorem 10.4 guarantees the existence of such a conformal mapping.

Example 11.8.  Find a function    that is harmonic in the unit disk  ,  which takes on the boundary values

(11-8)              

Solution.  Example 10.3 showed that the transformation  

is a one-to-one conformal mapping of the unit disk onto the upper half-plane , which can be written as

(11-9)               

Equation (11-9) reveals that the points lying on the upper semicircle    are mapped onto the positive u axis.  Similarly, the lower semicircle is mapped onto the negative u axis, as shown in Figure 11.8.  The mapping given by Equation (11-9) gives rise to a new Dirichlet problem of finding a harmonic function    that has the boundary values  

            ,   for  ,   and      
            
            ,   for  ,       

as shown in Figure 11.8.  Using the result of Example 11.5 and the functions and from Equation (11-9), we get the solution to Equations (11-9):  

            .  

Figure 11.8  The Dirichlet problems for and .  

Explore Solution 11.8.

Example 11.9.  Find a function    that is harmonic in the upper half disk  ,  which takes on the boundary values

              

Solution.  When we use the result of Exercise 4, Section 10.2, the transformation in Equation (11-9) maps the upper half-disk H onto the first quadrant .  The conformal mapping given in Equation (11-9) maps the points that lie on the segment    onto the positive v axis.

    Equation (11-9) gives rise to a new Dirichlet problem of finding a harmonic function in Q that has the boundary values

            ,   for  ,   and      
            
            ,   for  ,       

as shown in Figure 11.9.  In this case, the method in Example 11.2 can be used to show that is given by

            

Using the functions and in Equation (11-9) in the preceding equation, we find the solution of the Dirichlet problem in H:  

            .  

A three-dimensional graph of in cylindrical coordinates is shown in Figure 11.10.

Figure 11.9  The Dirichlet problems for the domains H and Q.  

Figure 11.10  The graph .

Explore Solution 11.9.

Example 11.10.  Find a function    that is harmonic in the quarter disk  ,  which takes on the boundary values

Solution.  The function  

(11-10)          

maps the quarter-disk onto the upper half-disk  .  The new Dirichlet problem in D is shown in Figure 11.11.  From the result of Example 11.9 the solution in H is  

(11-11)        .  

Using Equation (11-10), and    and    we obtain  

            

which we use in Equation (11-11) to construct the solution  in G:  

            .

A three-dimensional graph of in cylindrical coordinates is shown in Figure 11.12.

            Figure 11.11  The Dirichlet problems for the domains G and H.  

Figure 11.12  The graph  .

Explore Solution 11.10.

Exercises for Section 11.2.  Invariance of Laplace's Equation and the Dirichlet Problem