78 2. FOUR IMPORTANT LINEAR PDE
Remarks. (i) Notice that to compute u(x, t) we need only have information
on g, h and their derivatives on the sphere ∂B(x, t), and not on the entire
ball B(x, t).
(ii) Comparing formula (31) with d’Alembert’s formula (8) (n = 1), we
observe that the latter does not involve the derivatives of g. This suggests
that for n 1, a solution of the wave equation (11) need not for times t 0
be as smooth as its initial value g: irregularities in g may focus at times
t 0, thereby causing u to be less regular. (We will see later in §2.4.3 that
the “energy norm” of u does not deteriorate for t 0.)
(iii) Once again (as in the case n = 1) we see the phenomenon of finite
propagation speed of the initial disturbance.
(iv) A completely different derivation of formula (31) (using the heat
equation!) is in §4.3.3.
e. Solution for even n. Assume now
n is an even integer.
Suppose u is a
Cm
solution of (11), m =
n+2
2
. We want to fashion a repre-
sentation formula like (31) for u. The trick, as above for n = 2, is to note
that
(33) ¯(x1,...,xn+1,t) u := u(x1,...,xn,t)
solves the wave equation in
Rn+1
× (0, ∞), with the initial conditions
¯ u = ¯ g, ¯t u =
¯
h on
Rn+1
× {t = 0},
where
(34)
¯(x1,...,xn+1) g := g(x1,...,xn)
¯(x1,...,xn+1)
h := h(x1,...,xn).
As n + 1 is odd, we may employ (31) (with n + 1 replacing n) to secure
a representation formula for ¯ u in terms of ¯ g,
¯.
h But then (33) and (34) yield
at once a formula for u in terms of g, h. This is again the method of descent.
To carry out the details, let us fix x
Rn,
t 0, and write ¯ x =
(x1,... , xn, 0)
Rn+1.
Then (31), with n + 1 replacing n, gives
(35)
u(x, t) =
1
γn+1

∂t
1
t

∂t
n−2
2
tn−1


¯(¯
B x,t )
¯ g

S
+
1
t

∂t
n−2
2
tn−1


¯(¯
B x,t )
¯
h

S ,
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