Partial Differential Equations, Autumn 2016 Exercise sheet 4 The starred problems are for credit, and should be handed in by Wednesday 07/12/16, 2pm to the MPS School office. 1. ∗ Suppose that v(x, t; s) is the solution of  v vtt (x, s − v ) = 0 xx = 0 , vt(x, s) = f(x, s) x x ∈ ∈ R R, t > s . (a) Show that u(x, t) = R0tv(x, t; s) ds is a solution of  u utt (x, −0) = 0 uxx = , u f(x, t t(x, ) 0) = 0 x x ∈ ∈ R R, t > . 0 (b) Suppose that v, vx, vt ∈ L2(R) for all t ≥ s. Let E(t) = RR(vt 2 + vx 2) dx. Show that dE/dt = 0. 2. Consider the eigenvalue problem for the Laplacian with Neumann boundary conditions: −∆u = λu, x ∈ U ∂u ∂ν = 0, x ∈ ∂U, where ν is the outward unit normal to the boundary. Let λ = RU |∇u|2 dx RU u2 dx = min w∈YR RU RU |∇ w w 2|d 2x dx where YR = {w ∈ C2(U) : w 6≡ 0}, and assuming that a minimiser function u exists. Show that λ is an eigenvalue of the above system with the corresponding eigenfunction u. 3. ∗ Let f(x) be a smooth function satisfying f(0) = f(2) = 0, R0 2(f(x))2dx = 1, and R0 2(f 0(x))2 dx = 1. Does such an f exist? Why? (Hint: Use the minimum principle for the first eigenvalue of d2/dx2 with boundary conditions as for f.) 4. Consider the wave equation utt − ∆u + u = 0, x ∈ U ⊂ Rn, t ∈ R u = 0, x ∈ ∂U, Using the completeness of the eigenfunctions of the Dirichlet problem, show that the solution of the above wave equation satisfies the expansion u(x, t) = ∞ X n =1 An cos(p1 + λn t) + Bn sin(p1 + λn t) vn(x).