Math 220B - Summer 2003Homework 5Due Thursday, July 31, 20031. Let Ω = (0, k)×(0, l). Use separation of variables to solve the following boundary-valueproblem for Laplace’s equation on a square,∆u = 0 (x, y) ∈ Ωu(0, y) = 0, ux(k, y) = φ(y) 0 < y < luy(x, 0) = 0, u(x, l) = 0 0 < x < k.2. Let Ω be an open, bounded subset of Rn. Prove uniqueness of solutions of½∆u = f x ∈ Ω∂u∂ν+ αu = g x ∈ ∂Ωfor α > 0.3. Let Ω ≡ {(x, y) : a2< x2+ y2< b2} be an annular region in R2. Consideruxx+ uyy= 0 (x, y) ∈ Ωdudν+ αu = g(θ) x2+ y2= a2dudν+ βu = h(θ) x2+ y2= b2where ν is the outer unit normal to Ω.(a) Solve this b oundary-value problem in the case when α = β = 1, a = 1, b = 2,h(θ) = 0 and g(θ) is an arbitrary function.(b) From the result from the previous problem, we know the solution to part (a) isunique. Prove that uniqueness may fail if either α or β are negative, by findingtwo solutions ofuxx+ uyy= 0 (x, y) ∈ Ωdudν+ 2u = 0 x2+ y2= 1dudν− u = 0 x2+ y2= 4.4. (a) Find the one-dimensional Green’s function for Ω = (0, l), That is, find the functionG(x, y) such that for each x ∈ Ω,½−∆yG(x, y) = δxy ∈ ΩG(x, y) = 0 y ∈ ∂Ω.You may use the fact that the fundamental solution of Laplace’s equation in onedimension is Φ(x) = −12|x|.(b) Use the Green’s function above to solve the ODEu00(x) = 1 x ∈ (0, 1)u(0) = 3u(1) = 2.15. Find the Green’s function for Laplace’s equation on the half-ball Ω ≡ {(x, y, z) ∈ R3:x2+ y2+ z2< 1, z > 0}.6. Find the Green’s function for Laplace’s equation in the wedge Ω = {(x1, x2) ∈ R2, x2>0, x1>