Partial Differential Equations

1.4 The Wave Equation 29
Lemma 1.4.6. Let u be a solution of the homogeneous initial value problem (1.23), and fix x ∈ R n . If
u ∈ C m (R n × ]0, ∞[), and if all of its derivatives up to order m extend continuously to R n × [0, ∞[, then
the spherical means U(x; r, t) define a m-times differentiable function on ]0, ∞[ × ]0, ∞[ whose derivatives
extend continuously to [0, ∞[ × [0, ∞[. They satisfy the following initial value problem:
⎧
⎨ U tt − U rr − n−1
r
U r = 0 on ]0, ∞[ × ]0, ∞[,
U = G on ]0, ∞[ ×{0},
(1.24)
⎩
U t = H on ]0, ∞[ ×{0}.
Furthermore, lim r↘0 U r (x; r, t) = 0 and lim r↘0 U rr (x; r, t) = 1 n∆u(x, t).
Proof. Idea: Use Lemma 1.4.5 to write U r(x; r, t) as an integral and calculate U rr(x; r, t) integrating by parts.
This yields the limit behavior. Replacing ∆u by u tt in the integral leads to equation (1.24).
The first claim immediately follows from the definition of the spherical means and our first comments
on them. Lemma 1.4.5 gives the identity
∫
r
U r (x; r, t) =
n vol B(x;r)
∆u(y, t) dy (1.25)
B(x;r)
which shows that lim r↘0 U r (x; r, t) = 0. Integration by parts, i.e. formula (1.6), allows to differentiate
(1.25) with respect to r:
(
U rr (x; r, t) = ∂ ∫
)
r
∂r
n vol B(0;1)
∆u(x + ry, t) dy
B(0;1)
∫
∫
1
r
∂
=
n vol B(0;1)
∆u(x + ry, t) dy +
n vol B(0;1)
(∆u(x + ry, t)) dy
∂r
=
1
n vol B(x;r)
=
1
n vol B(x;r)
∫
∫
+
r
n vol B(0;1)
∫
1
=
n vol B(x;r)
+
r
vol ∂B(0;1)
B(0;1)
B(x;r)
B(x;r)
∫
∆u(y, t) dy +
∆u(y, t) dy −
∂B(0;1) j=1
B(x;r)
∫
∂B(0;1)
= ( 1
n − r) 1
vol B(x;r)
∫
r
n vol B(0;1)
r
n vol B(0;1)
∫
B(0;1)
B(0;1) j=1
B(0;1) j=1
n∑
∆u(x + ry, t)yj 2 dS ∂B(0;1) (y)
∆u(y, t)dy −
∫
∫
r
vol B(x;r)
∆u(y, t) dy
B(x;r)
∆u(x + ry, t) dS ∂B(0;1) (y)
B(x;r)
∆u(y, t)dy +
n∑
(D j (∆u)(x + ry, t)) y j dy
n∑
∆u(x + ry, t) dy
∫
r
vol ∂B(x;r)
∆u(y, t) dS ∂B(x;r) (y),
∂B(x;r)
where D j is the partial derivative with respect to the j-th variable in R n . From this, one derives
lim r↘0 U rr (x; r, t) = 1 n ∆u(x, t). Further, u tt = ∆u and (1.25) imply the identity
∫
r n−1 U r (x; r, t) = 1
nα(n)
u tt (y, t) dy
B(x;r)
which, calculating in polar coordinates, leads to