2.2. LAPLACE’S EQUATION 21
Substituting into (3), we obtain Laplace’s equation
div(Du) = Δu = 0.
If u denotes the
equation (4) is
Fick’s law of diﬀusion
Fourier’s law of heat conduction
Ohm’s law of electrical conduction.
See Feynman–Leighton–Sands [F-L-S, Chapter 12] for a discussion of the
ubiquity of Laplace’s equation in mathematical physics. Laplace’s equa-
tion arises as well in the study of analytic functions and the probabilistic
investigation of Brownian motion.
2.2.1. Fundamental solution.
a. Derivation of fundamental solution. One good strategy for inves-
tigating any partial diﬀerential equation is ﬁrst to identify some explicit
solutions and then, provided the PDE is linear, to assemble more compli-
cated solutions out of the speciﬁc ones previously noted. Furthermore, in
looking for explicit solutions, it is often wise to restrict attention to classes
of functions with certain symmetry properties. Since Laplace’s equation is
invariant under rotations (Problem 2), it consequently seems advisable to
search ﬁrst for radial solutions, that is, functions of r = |x|.
Let us therefore attempt to ﬁnd a solution u of Laplace’s equation (1)
in U =
having the form
u(x) = v(r),
where r = |x| = (x1
+ · · · +
and v is to be selected (if possible) so
that Δu = 0 holds. First note for i = 1,...,n that
+ · · · + xn
(x = 0).
We thus have
uxi = v (r)
, uxixi = v (r)
+ v (r)