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.
