UMD PDE Qualifying Exams/Jan2007PDE

Problem 1 a) Show that the function is a solution in the distribution sense of the equation . b) Use part (a) to write a solution of Solution (a) We want to show for every test function . One can compute and . Therefore, away from 0, we have , that is, a.e. and . We now compute by an integration by parts: A similar calculation gives So we have shown that for all which gives the desired result. (b) We guess . Then by part (a), . Problem 6 Let be the unit ball in . Consider the eigenvalue problem, where denotes the normal derivative on the boundary . Show that all eigenvalues are positive and the eigenfunctions corresponding to different eigenvalues are orthogonal to each other. Solution Multiply the PDE by and integrate: . Of course we know that is an eigenvalue of corresponding to a constant eigenfunction. But a constant function has which implies by the boundary condition. Hence is no longer an eigenvalue. This forces . To see orthogonality of the eigenfunctions, let be two eigenfunctions corresponding to distinct eigenvalues , respectively. Then by an integration of parts, So by the PDE, . Since this implies that are pairwise orthogonal in .