38 1. Simple Examples of Propagation

This suﬃces to prove uniqueness of solutions. A localized argument as in

§1.6.1, shows that signals travel at most at speed one.

Exercise 1.7.1. State and prove this finite speed result.

A function u(t, x) is called piecewise smooth if its restriction to x1 0

(resp., x1 0) has a

C∞

extension to x1 ≤ 0 (resp., x1 ≥ 0). The Cauchy

data of piecewise smooth solutions must be piecewise smooth (with the

analogous definition for functions of x only). They must, in addition, satisfy

conditions analogous to (1.6.3).

Proposition 1.7.1. If u is a piecewise smooth solution u of the transmission

problem, then the partial derivatives satisfy the sequence of compatibility

conditions, for all j ≥ 0,

Δj{u,

ut}(t,

0−,x2,x3)

=

(c2Δ)j{u,

ut}(t,

0+,x2,x3)

,

Δj∂1{u,,ut}(t, 0−,x2,x3)

=

(c2Δ)j∂1{u,

ut}(t,

0+,x2,x3)

.

ii. Conversely, if the piecewise smooth f, g satisfy for all j ≥ 0,

Δj{f, g}(0−,x2,x3)

=

(c2Δ)j{f, g}(0+,x2,x3)

,

Δj∂1{f, g}(0−,x2,x3)

=

(c2Δ)j∂1{f, g}(0+,x2,x3)

,

then there is a piecewise smooth solution with these Cauchy data.

Proof. i. Differentiating (1.7.2) with respect to t yields

(1.7.4) ∂t

ju

= 0 , ∂t

j∂1u

= 0 .

Compute for k ≥ 1,

∂t

2ku

=

Δku when x1 0,

(c2Δ)ku

when x1 0,

∂t

2kut

=

Δkut when x1 0,

(c2Δ)kut

when x1 0.

The transmission conditions (1.7.4) prove i.

The proof of ii is technical, interesting, and omitted. One can construct

solutions using finite differences almost as in §2.2. The shortest existence

proof to state uses the spectral theorem for selfadjoint

operators.4

The

general regularity theory for such transmission problems can be obtained by

4For

those with suﬃcient background, the Hilbert space is H :=

L2(Rd

; γ dx).

D(A) := w ∈

H2(R+) d

∩

H2(R−) d

: [w] = [∂1w] = 0 ,

Aw := Δw in x1 0, Aw :=

c2Δ

in x1 0 .

Then,

(Au, v)H = (u, Av)H = − ∇u · ∇v dx ,

so −A ≥ 0. The Elliptic Regularity Theorem implies that A is selfadjoint. The regularity theorem

is proved, for example, by the methods in [Rauch, 1991, Chapter 10]. The solution of the initial