Abstract: We study the Poisson problem and Helmholtz problem in bounded domains with corners in the plane and on the boundary. On non-convex domains of this type, the solutions are in the Sobolev space but not in even though may be very regular. We formulate these as variational problems in weighted Sobolev spaces and prove existence and uniqueness of solutions in what would be weighted counterparts of . The specific forms of our variational formulations are motivated by, and applied to, a finite element scheme for the time-dependent Navier-Stokes equations.