SINGULAR BEHAVIOR OF THE SOLUTION OF THE HELMHOLTZ EQUATION IN WEIGHTED Lp-SOBOLEV SPACES

We study the Helmholtz equation (1) -Delta u + zu = g in Omega, with Dirichlet boundary conditions in a polygonal domain Omega, where z is a complex number. Here g belongs to L-mu(p)(Omega) = {v is an element of L-loc(p)(Omega) : r(mu)v is an element of L-p(Omega)}, with a real parameter mu, and r(x) the distance from x to the set of corners of Omega. We give sufficient conditions on mu, p, and Omega that guarantee that problem (1) has a unique solution u epsilon H-0(1)(Omega) that admits a decomposition into a regular part in weighted L-p-Sobolev spaces and an explicit singular part. We further obtain some estimates where the explicit dependence on |z| is given.