580 Chapters
the integrand on the right is less than or equal to M, in fact, strictly less
than M on some arcs of C containing the points Pi. This contradiction
shows that maxu(x,y) cannot be attained anywhere in G unless u(x,y) is
a constant in that region.
The minimum property follows by applying the preceding result to the
function -u( x, y ).
Theorem 8.35 If u(x,y) is a real harmonic function in a bounded
region G and continuous on 8G, then both max(x,y)EGu(x,y) and
mi~(x y)EG u( x, y) are attained on 8G, unless u( x, y) is a constant function
in G (maximum and minimum principle for harmonic functions).
Proof As before, the theorem follows from the fact that a real continuous
function on a compact set must attain an absolute maximum as well as an
absolute minimum somewhere in the set. Because of Theorem 8.34, such
extrema cannot be attained in G unless u is a constant function in G.
Example Consider the harmonic function u = x^2 - y^2 + 4 on D =
{z: lzl ~ 1}. To find maxu(x,y) and minu(x,y) on that disk it suffices
to look for the extrema of u on lzl = 1. Letting x = cost, y = sin t we have
u = cos2t+4
and it is clear that max u is attained for cos 2t = 1, or t = mr ( n an
integer), while min u is attained for cos 2t = -1, or t =^1 / 2 (2n + l)7r. Hence
max(x,y)EDu = 5 and min(x,y)EDu = 3.
Corollary 8.16 Let u(x, y) be a nonconstant real harmonic function in
the disk lzl < R. Then maxu(x,y) = A(r), 0 ~ r < R, is a strictly
increasing function of r.
Proof Similar to that of Corollary 8.15.
Theorem 8.36 If u(z, y) and U(x, y) are real harmonic functions in a
bounded region G, and continuous on 8G, and if u( x, y) = U ( x, y) for all
(x, y) E 8G, then u(x, y) = U(x, y) everywhere in G. In other terms, the
values of a harmonic function on a bounded region are determined by its
values on the boundary of that region.
Proof Since F( x, y) = u( x, y) - U( x, y) 1s again harmonic m G and
continuous on 8G, we have
max_[u(x, y) - U(x, y)] = max [u(x, y) - U(x, y)] = 0
(x,y)EG (x,y)E8Gand