2.5. PROBLEMS 87 (b) Now assume only that u ∈ C2(U +) ∩ C(U +). Show that v is harmonic within U. (Hint: Use Poisson’s formula for the ball.) 11. (Kelvin transform for Laplace’s equation) The Kelvin transform Ku = ¯ u of a function u : Rn → R is ¯(x) u := u(¯)|¯|n−2 x x = u(x/|x|2)|x|2−n (x = 0), where ¯ x = x/|x|2. Show that if u is harmonic, then so is ¯. u (Hint: First show that Dx¯(Dx¯)T x x = |¯|4I. x The mapping x → ¯ x is conformal, meaning angle preserving.) 12. Suppose u is smooth and solves ut − Δu = 0 in Rn × (0, ∞). (a) Show uλ(x, t) := u(λx, λ2t) also solves the heat equation for each λ ∈ R. (b) Use (a) to show v(x, t) := x·Du(x, t)+2tut(x, t) solves the heat equation as well. 13. Assume n = 1 and u(x, t) = v( x √ t ). (a) Show ut = uxx if and only if (∗) v + z 2 v = 0. Show that the general solution of (∗) is v(z) = c z 0 e−s2/4 ds + d. (b) Differentiate u(x, t) = v( x √ t ) with respect to x and select the constant c properly, to obtain the fundamental solution Φ for n = 1. Explain why this procedure produces the fundamental solution. (Hint: What is the initial condition for u?) 14. Write down an explicit formula for a solution of ut − Δu + cu = f in Rn × (0, ∞) u = g on Rn × {t = 0}, where c ∈ R. 15. Given g : [0, ∞) → R, with g(0) = 0, derive the formula u(x, t) = x √ 4π t 0 1 (t − s)3/2 e −x 2 4(t−s) g(s) ds
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