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

t
0 ∂B(x,t−s)
f(y, s)
(t s)
dSds
=
1

t
0 ∂B(x,r)
f(y, t r)
r
dSdr.
Therefore
(44) u(x, t) =
1

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).
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