Variational Problems in Weighted Sobolev Spaces

Ana Maria Soane
University of Maryland, Baltimore County

Abstract: We study the Poisson problem $ -\Delta u = f$ and Helmholtz problem $ -\Delta u + \lambda u = 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 \cap 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.