Module

for

Laplace's Equation and Dirichlet Problem

 

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 [Graphics:Images/DirichletProblemMod_gr_1.gif] that is harmonic in D such that [Graphics:Images/DirichletProblemMod_gr_2.gif] 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 [Graphics:Images/DirichletProblemMod_gr_3.gif] be harmonic in a domain G in the w-plane. Then [Graphics:Images/DirichletProblemMod_gr_4.gif] satisfies Laplace's equation  

(11-1)            [Graphics:Images/DirichletProblemMod_gr_5.gif],   

at each point  [Graphics:Images/DirichletProblemMod_gr_6.gif]  in  G.   If  [Graphics:Images/DirichletProblemMod_gr_7.gif]  is a conformal mapping from a domain D in the z-plane onto  G, then the composition  

(11-2)            [Graphics:Images/DirichletProblemMod_gr_8.gif],  

is harmonic in D, and  [Graphics:Images/DirichletProblemMod_gr_9.gif]  satisfies Laplace's equation  

(11-3)            [Graphics:Images/DirichletProblemMod_gr_10.gif],  

at each point  [Graphics:Images/DirichletProblemMod_gr_11.gif]  in D.

Proof.

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

 

Example 11.4.  Show that  [Graphics:Images/DirichletProblemMod_gr_12.gif]  is harmonic in the disk  [Graphics:Images/DirichletProblemMod_gr_13.gif].  

[Graphics:Images/DirichletProblemMod_gr_14.gif]

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

            [Graphics:Images/DirichletProblemMod_gr_15.gif]

is a conformal mapping of the disk  [Graphics:Images/DirichletProblemMod_gr_16.gif]  onto the right half-plane  [Graphics:Images/DirichletProblemMod_gr_17.gif],  it can be rewritten as  

            [Graphics:Images/DirichletProblemMod_gr_18.gif]   

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

            [Graphics:Images/DirichletProblemMod_gr_19.gif]

is harmonic in the right half-plane  [Graphics:Images/DirichletProblemMod_gr_20.gif].  Taking the real and imaginary parts of  [Graphics:Images/DirichletProblemMod_gr_21.gif],  we have  

            [Graphics:Images/DirichletProblemMod_gr_22.gif]    and    [Graphics:Images/DirichletProblemMod_gr_23.gif].

Substituting these equations into the formula for  [Graphics:Images/DirichletProblemMod_gr_24.gif]  and using (11-2), we find that

            [Graphics:Images/DirichletProblemMod_gr_25.gif]   is harmonic for  [Graphics:Images/DirichletProblemMod_gr_26.gif].  

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 [Graphics:Images/DirichletProblemMod_gr_46.gif] 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)            [Graphics:Images/DirichletProblemMod_gr_47.gif]  

is harmonic in the upper half plane  [Graphics:Images/DirichletProblemMod_gr_48.gif]  and has boundary values  

            [Graphics:Images/DirichletProblemMod_gr_49.gif]  

Solution.  The function  

            [Graphics:Images/DirichletProblemMod_gr_50.gif][Graphics:Images/DirichletProblemMod_gr_51.gif]


is analytic in the upper half-plane  [Graphics:Images/DirichletProblemMod_gr_52.gif],  and its imaginary part is the harmonic function  [Graphics:Images/DirichletProblemMod_gr_53.gif].  

Explore Solution 11.5.

 

Remark 11.1  Let t be a real number.  We shall use the convention  [Graphics:Images/DirichletProblemMod_gr_67.gif]  so that the function  [Graphics:Images/DirichletProblemMod_gr_68.gif]  denotes the branch of the inverse tangent that lies in the range  [Graphics:Images/DirichletProblemMod_gr_69.gif].  Doing so permits us to write the solution in equation (11-4) as  [Graphics:Images/DirichletProblemMod_gr_70.gif].

 

Theorem 11.2 (N-Value Dirichlet Problem for the Upper Half Plane).  Let  [Graphics:Images/DirichletProblemMod_gr_71.gif]  denote [Graphics:Images/DirichletProblemMod_gr_72.gif] real constants.  The function  

(11-5)            [Graphics:Images/DirichletProblemMod_gr_73.gif]  

is harmonic in the upper half plane  [Graphics:Images/DirichletProblemMod_gr_74.gif]  and takes on the boundary values

            [Graphics:Images/DirichletProblemMod_gr_75.gif]   

        

            [Graphics:Images/DirichletProblemMod_gr_76.gif]  for  [Graphics:Images/DirichletProblemMod_gr_77.gif]  

        

            [Graphics:Images/DirichletProblemMod_gr_78.gif]   

 

The situation is illustrated in Figure 11.4.

Figure 11.4  The boundary conditions for the harmonic function  [Graphics:Images/DirichletProblemMod_gr_79.gif].

Proof.

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

 

Example 11.6.  Find a function  [Graphics:Images/DirichletProblemMod_gr_80.gif]  that is harmonic in the upper half-plane  [Graphics:Images/DirichletProblemMod_gr_81.gif]  and takes on the boundary values indicated in Figure 11.5.  That is

            [Graphics:Images/DirichletProblemMod_gr_82.gif]  

[Graphics:Images/DirichletProblemMod_gr_83.gif]

Solution. This is a four-value Dirichlet problem in the upper half-plane defined by  [Graphics:Images/DirichletProblemMod_gr_84.gif].  For the z plane, the solution in Equation (11-5) becomes  

            [Graphics:Images/DirichletProblemMod_gr_85.gif]

Here we have  [Graphics:Images/DirichletProblemMod_gr_86.gif]  and  [Graphics:Images/DirichletProblemMod_gr_87.gif],  which we substitute into equation for  [Graphics:Images/DirichletProblemMod_gr_88.gif]  to obtain  

            [Graphics:Images/DirichletProblemMod_gr_89.gif]  

Figure 11.5  The boundary values for the Dirichlet problem.

Explore Solution 11.6.

 

Extra Examples1 (a-d).  Find a function  [Graphics:Images/DirichletProblemMod_gr_100.gif]  that is harmonic in the upper half-plane  [Graphics:Images/DirichletProblemMod_gr_101.gif]  and takes on the indicated boundary values.

[Graphics:Images/DirichletProblemMod_gr_102.gif]

[Graphics:Images/DirichletProblemMod_gr_103.gif]

[Graphics:Images/DirichletProblemMod_gr_104.gif]

[Graphics:Images/DirichletProblemMod_gr_105.gif]

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 [Graphics:Images/DirichletProblemMod_gr_150.gif] that is harmonic in the upper half-plane  [Graphics:Images/DirichletProblemMod_gr_151.gif],  which takes on the boundary values  

            [Graphics:Images/DirichletProblemMod_gr_152.gif]  

[Graphics:Images/DirichletProblemMod_gr_153.gif]

Solution.  This is a three-value Dirichlet problem in the upper half-plane defined by  [Graphics:Images/DirichletProblemMod_gr_154.gif].  For the z plane, the solution in Equation (11-5) becomes  

            [Graphics:Images/DirichletProblemMod_gr_155.gif]

Here we have  [Graphics:Images/DirichletProblemMod_gr_156.gif]  and  [Graphics:Images/DirichletProblemMod_gr_157.gif],  which we substitute into the above equation for  [Graphics:Images/DirichletProblemMod_gr_158.gif]  to obtain  

            [Graphics:Images/DirichletProblemMod_gr_159.gif]   

A three-dimensional graph of [Graphics:Images/DirichletProblemMod_gr_160.gif] is shown in Figure 11.6.

Figure 11.6  The graph of  [Graphics:Images/DirichletProblemMod_gr_161.gif]  

            with the boundary values  [Graphics:Images/DirichletProblemMod_gr_162.gif], for [Graphics:Images/DirichletProblemMod_gr_163.gif], and [Graphics:Images/DirichletProblemMod_gr_164.gif], for [Graphics:Images/DirichletProblemMod_gr_165.gif].

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  [Graphics:Images/DirichletProblemMod_gr_176.gif]  denote N points that lie along C in this specified order as C is traversed in the positive direction (counterclockwise).  Then we let [Graphics:Images/DirichletProblemMod_gr_177.gif] denote the portion of C that lies strictly between [Graphics:Images/DirichletProblemMod_gr_178.gif], for [Graphics:Images/DirichletProblemMod_gr_179.gif], and let [Graphics:Images/DirichletProblemMod_gr_180.gif] denote the portion that lies strictly between [Graphics:Images/DirichletProblemMod_gr_181.gif].  Finally, we let  [Graphics:Images/DirichletProblemMod_gr_182.gif]  be real constants.  

    We want to find a function [Graphics:Images/DirichletProblemMod_gr_183.gif] that is harmonic in D and continuous on  [Graphics:Images/DirichletProblemMod_gr_184.gif]  that takes on the boundary values

            [Graphics:Images/DirichletProblemMod_gr_185.gif]   

            

            [Graphics:Images/DirichletProblemMod_gr_186.gif]   

        
(11-6)            [Graphics:Images/DirichletProblemMod_gr_187.gif]  for  [Graphics:Images/DirichletProblemMod_gr_188.gif]  
        

            [Graphics:Images/DirichletProblemMod_gr_189.gif]   


The situation is illustrated in Figure 11.7.

Figure 11.7  The boundary values for [Graphics:Images/DirichletProblemMod_gr_190.gif] for the Dirichlet problem in the simply connected domain D.

 

 

    One method for finding [Graphics:Images/DirichletProblemMod_gr_191.gif] is to find a conformal mapping  

(11-7)            [Graphics:Images/DirichletProblemMod_gr_192.gif]

of D onto the upper half-plane [Graphics:Images/DirichletProblemMod_gr_193.gif], such that the N points  [Graphics:Images/DirichletProblemMod_gr_194.gif]  are mapped onto the points  [Graphics:Images/DirichletProblemMod_gr_195.gif],  for  [Graphics:Images/DirichletProblemMod_gr_196.gif],  and  [Graphics:Images/DirichletProblemMod_gr_197.gif]  is mapped onto  [Graphics:Images/DirichletProblemMod_gr_198.gif]  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 [Graphics:Images/DirichletProblemMod_gr_199.gif] for which the solution is given by Theorem 11.2.  If we set  [Graphics:Images/DirichletProblemMod_gr_200.gif],  then the solution to the Dirichlet problem in D with the boundary values from Equation (11-6) is

            [Graphics:Images/DirichletProblemMod_gr_201.gif]  


This method relies on our ability to construct a conformal mapping from D onto the upper half-plane [Graphics:Images/DirichletProblemMod_gr_202.gif].  Theorem 10.4 guarantees the existence of such a conformal mapping.

 

Example 11.8.  Find a function  [Graphics:Images/DirichletProblemMod_gr_203.gif]  that is harmonic in the unit disk  [Graphics:Images/DirichletProblemMod_gr_204.gif],  which takes on the boundary values

(11-8)            [Graphics:Images/DirichletProblemMod_gr_205.gif]  

Solution.  Example 10.3 showed that the transformation  

            [Graphics:Images/DirichletProblemMod_gr_206.gif]


is a one-to-one conformal mapping of the unit disk [Graphics:Images/DirichletProblemMod_gr_207.gif] onto the upper half-plane [Graphics:Images/DirichletProblemMod_gr_208.gif], which can be written as

(11-9)            [Graphics:Images/DirichletProblemMod_gr_209.gif]   

Equation (11-9) reveals that the points [Graphics:Images/DirichletProblemMod_gr_210.gif] lying on the upper semicircle  [Graphics:Images/DirichletProblemMod_gr_211.gif]  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  [Graphics:Images/DirichletProblemMod_gr_212.gif]  that has the boundary values  

            [Graphics:Images/DirichletProblemMod_gr_213.gif],   for  [Graphics:Images/DirichletProblemMod_gr_214.gif],   and      
            
            [Graphics:Images/DirichletProblemMod_gr_215.gif],   for  [Graphics:Images/DirichletProblemMod_gr_216.gif],       

as shown in Figure 11.8.  Using the result of Example 11.5 and the functions [Graphics:Images/DirichletProblemMod_gr_217.gif] and [Graphics:Images/DirichletProblemMod_gr_218.gif] from Equation (11-9), we get the solution to Equations (11-9):  

            [Graphics:Images/DirichletProblemMod_gr_219.gif].  

Figure 11.8  The Dirichlet problems for [Graphics:Images/DirichletProblemMod_gr_220.gif] and [Graphics:Images/DirichletProblemMod_gr_221.gif].  

Explore Solution 11.8.

 

Example 11.9.  Find a function  [Graphics:Images/DirichletProblemMod_gr_233.gif]  that is harmonic in the upper half disk  [Graphics:Images/DirichletProblemMod_gr_234.gif],  which takes on the boundary values

            [Graphics:Images/DirichletProblemMod_gr_235.gif]  

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 [Graphics:Images/DirichletProblemMod_gr_236.gif].  The conformal mapping given in Equation (11-9) maps the points [Graphics:Images/DirichletProblemMod_gr_237.gif] that lie on the segment  [Graphics:Images/DirichletProblemMod_gr_238.gif]  onto the positive v axis.

    Equation (11-9) gives rise to a new Dirichlet problem of finding a harmonic function [Graphics:Images/DirichletProblemMod_gr_239.gif] in Q that has the boundary values

            [Graphics:Images/DirichletProblemMod_gr_240.gif],   for  [Graphics:Images/DirichletProblemMod_gr_241.gif],   and      
            
            [Graphics:Images/DirichletProblemMod_gr_242.gif],   for  [Graphics:Images/DirichletProblemMod_gr_243.gif],       

as shown in Figure 11.9.  In this case, the method in Example 11.2 can be used to show that [Graphics:Images/DirichletProblemMod_gr_244.gif] is given by

            [Graphics:Images/DirichletProblemMod_gr_245.gif]

Using the functions [Graphics:Images/DirichletProblemMod_gr_246.gif] and [Graphics:Images/DirichletProblemMod_gr_247.gif] in Equation (11-9) in the preceding equation, we find the solution of the Dirichlet problem in H:  

            [Graphics:Images/DirichletProblemMod_gr_248.gif].  

A three-dimensional graph of [Graphics:Images/DirichletProblemMod_gr_249.gif] 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 [Graphics:Images/DirichletProblemMod_gr_250.gif].

Explore Solution 11.9.

 

Example 11.10.  Find a function  [Graphics:Images/DirichletProblemMod_gr_262.gif]  that is harmonic in the quarter disk  [Graphics:Images/DirichletProblemMod_gr_263.gif],  which takes on the boundary values

            [Graphics:Images/DirichletProblemMod_gr_264.gif]    

Solution.  The function  

(11-10)        [Graphics:Images/DirichletProblemMod_gr_265.gif]  

maps the quarter-disk onto the upper half-disk  [Graphics:Images/DirichletProblemMod_gr_266.gif][Graphics:Images/DirichletProblemMod_gr_267.gif].  The new Dirichlet problem in D is shown in Figure 11.11.  From the result of Example 11.9 the solution [Graphics:Images/DirichletProblemMod_gr_268.gif] in H is  

(11-11)        [Graphics:Images/DirichletProblemMod_gr_269.gif].  

Using Equation (11-10), and  [Graphics:Images/DirichletProblemMod_gr_270.gif]  and  [Graphics:Images/DirichletProblemMod_gr_271.gif]  we obtain  

            [Graphics:Images/DirichletProblemMod_gr_272.gif]

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

            [Graphics:Images/DirichletProblemMod_gr_273.gif].

A three-dimensional graph of [Graphics:Images/DirichletProblemMod_gr_274.gif] 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  [Graphics:Images/DirichletProblemMod_gr_275.gif].

Explore Solution 11.10.

 

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

 

Library Research Experience for Undergraduates

Dirichlet Problem

Neumann Problem

Poisson Integral

 

 

 

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Poisson's Integral Formula for the Upper Half-Plane

 

 

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This material is coordinated with our book Complex Analysis for Mathematics and Engineering.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) 2012 John H. Mathews, Russell W. Howell