2.2. LAPLACE’S EQUATION 21

Substituting into (3), we obtain Laplace’s equation

div(Du) = Δu = 0.

If u denotes the

⎧

⎨

⎩

chemical concentration

temperature

electrostatic potential,

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 =

Rn,

having the form

u(x) = v(r),

where r = |x| = (x1

2

+ · · · +

xn)1/2 2

and v is to be selected (if possible) so

that Δu = 0 holds. First note for i = 1,...,n that

∂r

∂xi

=

1

2

(

x1

2

+ · · · + xn

2

)−1/2

2xi =

xi

r

(x = 0).

We thus have

uxi = v (r)

xi

r

, uxixi = v (r)

xi

2

r2

+ v (r)

1

r

−

xi2

r3