Math 295a Fall Term 2001

Assignment 4

1. Compute a fundamental solution G_{2} for the wave operator ∂_{t}^{2} − △ on ℝ × ℝ^{2}. [Hint: Let G_{3} be the fundamental solution for the wave operator on ℝ × ℝ^{3} introduced in class and show that

⟨G_{3}(t, x, x_{3}), φ(t, x) 1(x_{3})⟩

can be made sense of for φ ∈ S(ℝ × ℝ^{2}). Define G_{2} through

⟨G_{2}, φ⟩ = ⟨G_{3}, φ 1(x_{3})⟩ = \frac{1}{4π} ∫_{0}^{∞} \frac{1}{t} ∫_{𝕊_{t}^{2}} φ(t, x) dσ_{𝕊_{t}^{2}}(x)dt

and use the fact that φ is independent of x_{3} to simplify the integral.]

2. A fundamental solution G(·, y) for the Laplacian −△ on a domain Ω ⊂ ℝ^{n} with pole at x = y and satisfying G(·, y)|_{∂Ω} = 0 is called Green’s function for the Dirichlet problem in Ω (considered as a function of (x, y) ∈ Ω×Ω). Compute a Green’s function for the Dirichlet problem in the half-space ℍ^{n} = ℝ^{n−1} × (0, ∞) , n ≥ 2.

3. Let G be the fundamental solution for the heat operator ∂_{t} − △ on ℝ × ℝ^{n} introduced in class. Show that

G(t, ·) → δ , t → 0+ , in S′(ℝ^{n})

and find a representation for the solution of the Cauchy problem

∂_{t}u − △u = f(t, x) , u(0, ·) = u_{0}

on (0, ∞) × ℝ^{n} in terms of G.

4. Compute the general solution to the following initial value problem for the wave equation

∂_{t}^{2}u − ∂_{x}^{2}u = f(t, x) , u(0, ·) = u_{0} , ∂_{t}u(0, ·) = u_{1} ,

in (0, ∞) × ℝ in terms of a fundamental solution for the wave operator.

5. Let u : Ω → ℝ be a harmonic function on the domain Ω. Prove Gauss’s law of the arithmetic mean

u(x) = \frac{1}{ω_{n} r^{n−1}} ∫_{|x−y|=r} u(y) dσ_{r}(y)

valid for all x and r such that 𝔹(x, r) ⊂ Ω. [Hint: Use Green’s formula with u and a Green’s function for the Dirichlet problem on 𝔹(x, r).

The Homework is due by November 16 2001