Singular behavior of the solution of the Helmholtz equation in weighted L^p -Sobolev spaces

International audience ; We study the Helmholtz equation − ∆u + zu = g in Ω, with Dirichlet boundary conditions in a polygonal domain Ω, where z is a complex number. Here g belongs to L^p_µ (Ω) = {v ∈ L^p_loc (Ω) : r^µ v ∈ L^p (Ω)}, with a real parameter µ and r(x) the distance from x to the set of corners of Ω. We give sufficient conditions on µ, p and Ω that guarantee that the above problem has a unique solution u ∈ H^1_0 (Ω) 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.