160 The Three Theorems
(b) If dfi/dn = dfi/dn on S then there exists a real number c so that
/i — fi = c everywhere in Gs-
EXERCISE 3.2.13.c Prove Theorem 3.2.6. Hint: consider the function f =
/i — fi and apply (3.2.19) with g = f to conclude that V/ = 0 everywhere in Gs-
Therefore f must be constant (Exercise 3.1.3 on page 125); in part (a), f = 0 on
S, so the constant must be zero.
Let G be a bounded domain whose boundary dG is a closed orientable
piece-wise smooth surface. Let g be a continuous function in G, and h,
a function continuous in some domain containing dG. The Dirichlet, or
first, boundary value problem for the Poisson equation (3.2.15) in G is
to find a function / that is twice continuously differentiable in an open
set containing G, satisfies equation (3.2.15) everywhere in G, and / = h
on dG. The Neumann, or second, boundary value problem for the Poisson
equation is to find a function / that is twice continuously differentiable in
an open set containing G, satisfies equation (3.2.15) everywhere in G, and
df/dn — h on dG. With some additional constructions, a formulation of
both problems is possible by considering the function / only in the closure
of G, that is, without using an open set containing G
EXERCISE 3.2.14. c (a) Show that the Dirichlet problem for the Poisson
equation can have at most one solution. Hint: use the first part of Theorem
3.2.6; the difference of two solutions is a harmonic function, (b) Show that
every two solutions of the Neumann problem differ by a constant. Hint: use
the second part of Theorem 3.2.6. (c) Show that the Neumann problem for the
Laplace equation has no solutions unless J J hda = 0. Hint: use (3.2.18).
dG
The difference between the Dirichlet and Neumann problems is the
boundary conditions imposed on the function /: in the first case, we
prescribe the values on the boundary of G to the function itself, and in the
second case, to the normal derivative of the function. A combination of the
two is possible and is called the Robin boundary value problem, when we
prescribe the values on dG to the expression af + (df/dn), where a is a
known function defined on dG. Finally, there is an oblique derivative
problem, when the values on dG are prescribed to the expression V/ • u,
where ix is a continuous unit vector field on dG. We only mention these
boundary conditions to illustrate the numerous problems that can be stud-
ied for the Poisson and Laplace equations.
While the Dirichlet problem is named after the German mathemati-