Variational Problems in Weighted Sobolev Spaces

Ana Maria Soane University of Maryland, Baltimore County

                     asoane@math.umbc.edu

Abstract: We study the Poisson problem −Δu = f and Helmholtz problem −Δu + λu = f in bounded domains with corners in the plane and u = 0 on the boundary. On non-convex domains of this type, the solutions are in the Sobolev space H1 but not in H2 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 H2∩H1_{0}. The specific forms of our variational formulations are motivated by, and applied to, a finite element scheme for the time-dependent Navier-Stokes equations.