Partial Differential Equations

1.2 The Laplace Equation 5
Fig. 1.2. Transport equation
Exercise 1.1.2. Provide an explicit solution of the following initial value problem:
{
ut + b · Du + cu = 0 on R n × ]0, ∞[,
u = g on R n × {0}.
Here, c ∈ R and b ∈ R n are constants.
1.2 The Laplace Equation
The Laplace equation is the partial differential equation
∆u = 0,
where u: R n → R and ∆u = ∑ n
j=1 ∂2 u
∂x j
2 . The solutions of the Laplace equation are called harmonic
functions. The inhomogeneous version
−∆u = f
of the Laplace equation, where f : R n → R is a given function, and again u: R n → R is the unknown
function, is also called Poisson equation.
The Laplace equation describes an equilibrium state of densities u whose flux F is described by
F = −aDu. By the divergence theorem, equilibrium then means that
∫ ∫
div F = F · ν dS ∂V = 0,
V
hence, div F = 0 (with infinitesimal V ), and therefore, ∆u = div Du = − 1 adiv F = 0.
∂V
First, we look for radial solutions of the Laplace, i.e. solutions u of the form u(x) = v(r), where
r = √ x 2 1 + . . . + x2 n is the Euclidean norm |x| of x. Because of
for x ≠ 0, we have, for radial u, that
∂r x
= √ j
= x j
∂x j x
2
1 + . . . + x 2 r
n
u xj (x) = v ′ (r) x j
r ,
u xjx j
(x) = v ′′ (r) x2 j
r 2 + v′ (r) 1 r
(
1 − x2 j
r 2 )
for all j = 1, . . . , n, and thus,
∆u(x) = v ′′ (r) + n − 1 v ′ (r).
r
Therefore, ∆u = 0 is equivalent to the ordinary differential equation