1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions

8.12 MAXIMUM AND MINIMUM PRINCIPLES FOR

REAL HARMONIC FUNCTIONS

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In Section 6.6 we have seen that the real and imaginary parts of a single-

valued analytic function in some open set A are both harmonic functions

in A. However, it is not necessarily true that a harmonic function u(x, y)

in an open set A is the real part of a single-valued analytic function in A

unless a certain restriction is imposed on A, namely, that A be a simply

connected region.

Theorem 8.34 Let u(x,y) be a real harmonic function in a region G.

Then u(x,y) does not attain either a maximum or a minimum value

anywhere in G, unless u(x,y) is a constant function in G.

Proof If u = Ref, where f is a nonconstant analytic function in G,

consider the function F(z) = ef(z), which is also analytic in G. Since

F(z) 'f: 0 everywhere in G, both Theorems 8.29 and 8.31 apply. Hence

IF(z)I = eu(x,y) does not attain either a maximum or a minimum anywhere

in G, which implies the same property for u(x, y) since the exponential is

a strictly increasing function of u.

If u is not the real part of a nonconstant analytic function in G (which

may happen if G is not simply connected), the conclusion of the theorem

still holds, but this case requires a proof independent of the corresponding

property for analytic functions.

Suppose that M = max(x,y)EG u(x, y) is attained at some point of G.

Then we shall prove that u(x, y) = M everywhere in G. Assuming that

this were not the case, let S be the set of points of G where u takes the

value M. Let z 0 = (xo,Yo) be a point of Sand z1 = (x1,Y1) be a point

of G ~ S, so that u(x 0 ,y 0 ) = M while u(xi,y 1 ) < M. Let P be a simple

oriented polygonal line in G with the initial point z 0 and terminal point

z 1 , and let z* = ( x*, y*) be the last point of P n S. Since z* E S we have

u(x*, y*) = M. Let 0 < r < lz1 - z*I and choose r small enough so that

Nr(z) CG. The circle C(z,r) = {z: jz-z*I = r} will have some points

in common with that part of Plying between z* and z1. At those points

Pi, as well as in some neighborhood of the Pi we have u < M (on account

of the continuity of u ).

By regarding the point z* as the origin we have, by formula (7.28-11),

1 [2"

u(x*,y*) = 2 71" Jo u(r,'l/;)d'lj; (8.12-1)

a formula expressing that the value of u at the center of the circle C is the
integral mean of the values of u on C (mean value property for harmonic
functions). But here we find a contradiction since u( x, y) = M while