82 2. FOUR IMPORTANT LINEAR PDE

(ii) For n = 3, Kirchhoﬀ’s formula (22) implies

u(x, t; s) = (t − s)−

∂B(x,t−s)

f(y, s) dS,

so that

u(x, t) =

t

0

(t − s) −

∂B(x,t−s)

f(y, s) dS ds

=

1

4π

t

0 ∂B(x,t−s)

f(y, s)

(t − s)

dSds

=

1

4π

t

0 ∂B(x,r)

f(y, t − r)

r

dSdr.

Therefore

(44) u(x, t) =

1

4π

B(x,t)

f(y, t − |y − x|)

|y − x|

dy (x ∈

R3,

t ≥ 0)

solves (42) for n = 3. The integrand on the right is called a retarded potential.

2.4.3. Energy methods.

The explicit formulas (31) and (38) demonstrate the necessity of making

more and more smoothness assumptions upon the data g and h to ensure

the existence of a

C2

solution of the wave equation for larger and larger

n. This suggests that perhaps some other way of measuring the size and

smoothness of functions may be more appropriate. Indeed we will see in this

subsection that the wave equation is nicely behaved (for all n) with respect

to certain integral “energy” norms.

a. Uniqueness. Let U ⊂

Rn

be a bounded, open set with a smooth

boundary ∂U, and as usual set UT = U × (0,T ], ΓT =

¯

U

T

− UT , where

T 0.

We are interested in the initial/boundary-value problem

(45)

⎧

⎨

⎩

utt − Δu = f in UT

u = g on ΓT

ut = h on U × {t = 0}.

THEOREM 5 (Uniqueness for wave equation). There exists at most one

function u ∈

C2(

¯

U

T

) solving (45).