Exercise 9.  Establish the following maximum principle for harmonic functions.  

Let    be harmonic and nonconstant in the simply connected domain  D.

Then    does not have a maximum value at any point    in  D.

Solution 9.

See text and/or instructor's solution manual.

Solution.   Let  ,  where    is a harmonic conjugate of  ,  so that    is analytic in  D.

The function    is also analytic in  D,  so that    does not take on a maximum in  D  by the Maximum Modulus Theorem.  

But  .  

Thus the function    does not take on a maximum in  D.

Since the real valued function    is an increasing function       iff    .    

This leads to the conclusion since u is a real valued function, and the real valued function exp is an increasing function.  

Therefore the function    does not take on a maximum in  D.

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell