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.

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