Mathematics of Physics and Engineering

Laplace's and Poisson's Equations 161

cian J.P.G.L. Dirichlet, who did pioneering work in potential theory, the
origins of the other two names, Neumann and Robin, are less clear. Most
probably, the Neumann problem is named after the German mathematician
CARL GOTTFRIED NEUMANN (1832-1925) for his contributions to poten-
tial theory. His father FRANZ ERNST NEUMANN (1798-1895) worked in
mathematical physics and could have contributed to the subject as well.
We conclude this section by establishing the Dirichlet principle for the
Poisson equation. As before, let G be a bounded domain whose boundary
dG is a closed orientable piece-wise smooth surface. Let g be a continuous
function on G, and h, a function continuous on some domain containing
dG. We assume that all functions are real-valued.
Denote by U the set of functions so that every function
in U is twice continuously differentiable in an open set
containing G and is equal to h on dG.

For every ligM, define the number

I{u) = Jjj (±\Vu\

2

+g^dV.

In mathematics, a rule that assigns a number to a function is called a
functional; thus, 7 is a functional defined on the set U.

Theorem 3.2.7 (Dirichlet principle) If f is a function in U and

V^2 / = g in G, then

I{f)= mm I(u). (3.2.22)

Conversely, if the function f in U satisfies (3.2.22), then V^2 / = g in G.

Proof. Assume that / is a function in U and V^2 / = g in G. Then

o = ///(v^2 / -g)(f- u)dv = fJJ v^2 / (/ - u)dv - JJfgU ~ u)dv

G G G

(3.2.23)

for every u £U. Being elements of the set U, both / and u are equal to h

on dG. Therefore, / - u = 0 on dG, and we find from (3.2.19) that

HI V

2

/ (/ - u)dV = - HI || V/||

2

dV + JJIV/ • VudV. (3.2.24)