VARIATIONAL PROBLEMS IN WEIGHTED SOBOLEV SPACES ON NON-SMOOTH DOMAINS

We study the Poisson problem -Delta u = f and the Helmholtz problem -Delta u + lambda u = f in bounded domains with angular corners in the plane and u = 0 on the boundary. On non-convex domains of this type, the solutions are in the Sobolev space HI but not in H(2) in general, even though f 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 H(2) boolean AND H(0)(1). The specific forms of our variational formulations are motivated by, and are particularly suited to, applying a finite element scheme for solving the time-dependent Navier-Stokes equations of fluid mechanics.