2.4. WAVE EQUATION 83
Cone of dependence
Proof. If ˜ u is another such solution, then w := u − ˜ u solves
⎧
⎨
⎩
wtt − Δw = 0 in UT
w = 0 on ΓT
wt = 0 on U × {t = 0}.
Define the “energy”
E(t) :=
1
2
U
wt
2(x,
t) + |Dw(x,
t)|2
dx (0 ≤ t ≤ T ).
We compute
˙
E (t) =
U
wtwtt + Dw · Dwt dx ˙=
d
dt
=
U
wt(wtt − Δw) dx = 0.
There is no boundary term since w = 0, and hence wt = 0, on ∂U × [0,T ].
Thus for all 0 ≤ t ≤ T , E(t) = E(0) = 0, and so wt,Dw ≡ 0 within UT .
Since w ≡ 0 on U × {t = 0}, we conclude w = u − ˜ u ≡ 0 in UT .
b. Domain of dependence. As another illustration of energy methods,
let us examine again the domain of dependence of solutions to the wave
equation in all of space. For this, suppose u ∈
C2
solves
utt − Δu = 0 in
Rn
× (0, ∞).
Fix x0 ∈
Rn,
t0 0 and consider the backwards wave cone with apex (x0,t0)
K(x0,t0) := {(x, t) | 0 ≤ t ≤ t0, |x − x0| ≤ t0 − t}.
