90 2. FOUR IMPORTANT LINEAR PDE

23. Let S denote the square lying in R × (0, ∞) with corners at the points

(0, 1), (1, 2), (0, 3), (−1, 2). Deﬁne

f(x, t) :=

⎧

⎨

⎩

−1 for (x, t) ∈ S ∩ {t x + 2}

1 for (x, t) ∈ S ∩ {t x + 2}

0 otherwise.

Assume u solves

utt − uxx = f in R × (0, ∞)

u = 0,ut = 0 on R × {t = 0}.

Describe the shape of u for times t 3.

(J. G. Kingston, SIAM Review 30 (1988), 645–649)

24. (Equipartition of energy) Let u solve the initial-value problem for the

wave equation in one dimension:

utt − uxx = 0 in R × (0, ∞)

u = g, ut = h on R × {t = 0}.

Suppose g, h have compact support. The kinetic energy is k(t) :=

1

2

∞

−∞

ut

2(x,

t) dx and the potential energy is p(t) :=

1

2

∞

−∞

ux(x,

2

t) dx.

Prove

(a) k(t) + p(t) is constant in t,

(b) k(t) = p(t) for all large enough times t.

2.6. REFERENCES

Section 2.2 A good source for more on Laplace’s and Poisson’s equations

is Gilbarg–Trudinger [G-T, Chapters 2-4]. The proof of an-

alyticity is from Mikhailov [M]. J. Cooper helped me with

Green’s functions.

Section 2.3 See John [J2, Chapter 7] or Friedman [Fr1] for further in-

formation concerning the heat equation. Theorem 3 is due

to N. Watson (Proc. London Math. Society 26 (1973), 385–

417), as is the proof of Theorem 4. Theorem 6 is taken from

John [J2], and Theorem 8 follows Mikhailov [M]. Theorem 11

is from Payne [Pa, §2.3].

Section 2.4 See Antman (Amer. Math. Monthly 87 (1980), 359–370) for

a careful derivation of the one-dimensional wave equation as

a model for a vibrating string. The solution of the wave

equation presented here follows Folland [F1], Strauss [St2].

Section 2.5 J. Goldstein contributed Problem 24.