Math 220B - Summer 2003Homework 6Due Thursday, August 7, 20031. Consider the Neumann problem,½−∆u = f x ∈ Ω∂u∂ν= g x ∈ ∂ΩAssume the compatibility condition holds. That is,−ZΩf(x) dx =Z∂Ωg(x) dS(x).Just as the Green’s function allowed us to find a representation formula for solutionsto Poisson’s equation on a bounded domain Ω, here we construct a Neumann functionto derive a representation formula for the Neumann problem. Let N(x, y) be definedas follows. LetN(x, y) = Φ(y − x) −ehx(y) ∀y ∈ Ωwhereehx(y) is a solution of(∆yehx(y) = 0 ∀y ∈ Ω∂ehx∂ν(y) =∂Φ∂ν(y − x) − C ∀y ∈ ∂Ωfor some appropriately chosen constant C. (In part (b), you will determine the neces-sary constant for a given region Ω. For now, you may assume C is arbitrary.)(a) Use N(x, y) to write a solution formula for½−∆u = f x ∈ Ω∂u∂ν= g x ∈ ∂Ωin terms of f, g, and N. (Note: As we know, Poisson’s equation with Neumannboundary conditions is only unique up to constants. Therefore, adding any con-stant to your solution formula will also give you a solution.)(b) In the definition ofehx, what must the constant C be? Explain.2. (a) Find the Neumann function for Rn+.(b) Use the Neumann function for Rn+to find the solution formula for½∆u = 0 x ∈ Rn+∂u∂ν= g x ∈ ∂Rn+.13. Let Ω be an open, bounded subset of Rnwith C2boundary. Let h be a continuousfunction on ∂Ω. Let Φ be the fundamental solution of Laplace’s equation on Rn. Definethe single-layer potential with moment h asu(x) = −Z∂Ωh(y)Φ(y − x) dS(y).(a) Show that u is defined and continuous for all x ∈ Rn.(b) Show that ∆u(x) = 0 for x /∈ ∂Ω.4. Let Ω be an open, bounded set in Rnwith smooth boundary. Let Ωc≡ Rn\Ω. Considerthe exterior Neumann problem,(∗)½∆u = 0 x ∈ Ωc∂u∂ν= g x ∈ ∂Ωc.Assume g satisfies the condition,Z∂Ωg(x) dS(x) = 0. (∗∗)(Note: Recall: This is not a necessary condition for solvability of the exterior Neumannproblem.) Suppose a solution u of (*) is given by the single-layer potential,u(x) ≡ −Z∂Ωh(y)Φ(x − y) dS(y)where h satisfies the integral equationg(x) =12h(x) −Z∂Ωh(y)∂Φ(x − y)∂νxdS(y).(a) Show that if g satisfies the condition (**), thenZ∂Ωh(y) dS(y) = 0.(b) Show that the solution u will have decay rate O(|x|1−n) In particular, show|u(x)| ≤ C|x|1−n. Hint: By (a), write u(x) = −R∂Ωh(y)[Φ(x − y) − Φ(x)] dS(y).5. Let Ω be an open, bounded subset of Rn. Let Ωc≡ Rn\Ω. Prove there exists at mostone solution u which decays to 0 as |x| → +∞ of the following½∆u = f x ∈ Ωcu = g x ∈