Real and Complex Analysis (Rudin)

HARMONIC FUNCTIONS 235
Note: This theorem provides the solution of a boundary value problem (the
Dirichlet problem): A continuous functionfis given on T and it is required to find
a harmonic function F in U "whose boundary values are f." The theorem exhibits
a solution, by means of the Poisson integral of f, and it states the relation
between f and F more precisely. The uniqueness theorem which corresponds to
this existence theorem is contained in the following result.
11.9 Theorem Suppose u is a continuous real function on the closed unit disc 0,
and suppose u is harmonic in U. Then (in U) u is the Poisson integral of its
restriction to T, and u is the real part of the holomorphic function
1 I" e il + z .
f(z) = - -.- u(e") dt
2n -" e" - z
(z E U). (1)
PROOF Theorem 10.7 shows that f E H(U). If Ul = Re f, then (1) shows that
Ul is the Poisson integral of the boundary values of u, and the theorem will
be proved as soon as we show that u = u 1•
Put h = u - Ul. Then h is continuous on 0 (apply Theorem 11.8 to u 1), h
is harmonic in U, and h = 0 at all points of T. Assume (this will lead to a
contradiction) that h(zo) > 0 for some Zo E U. Fix E so that 0 < E < h(zo), and
define
g(z) = h(z) + Ei Z 12 (z EO). (2)
Then g(zo) ~ h(zo) > E. Since g E qO) and since g = E at all points of T,
there exists a point z 1 E U at which g has a local maximum. This implies that
gxx ~ 0 and gyy ~ 0 at Zl. But (2) shows that the Laplacian of g is 4E > 0, and
we have a contradiction.
Thus u - Ul ~ O. The same argument shows that U 1 - u ~ O. Hence u =
u 1, and the proof is complete. / / / /
11.10 So far we have considered only the unit disc U = D(O; 1). It is clear that
the preceding work can be carried over to arbitrary circular discs, by a simple
change of variables. Hence we shall merely summarize some of the results:
If u is a continuous real function on the boundary of the disc D(a; R) and if u
is defined in D(a; R) by the Poisson integral
·9 1 I" R2 - r2 .
u(a + re' ) = -2 2 2R (0 ) 2 u(a + Re'~ dt
n -" R - r cos - t + r
then u is continuous on D(a; R) and harmonic in D(a; R).
If u is harmonic (and real) in an open set n and if D(a; R) c n, then u
satisfies (1) in D(a; R) and there is a hoi om orphic function f defined in D(a; R)
whose real part is u. This f is uniquely defined, up to a pure imaginary additive
constant. For if two functions, holomorphic in the same region, have the same
real part, their difference must be constant (a corollary of the open mapping
theorem, or the Cauchy-Riemann equations).
(1)