Ana Soane, University of Maryland, Baltimore County
Variational problems in weighted Sobolev spaces on non-smooth domains
We study the Poisson problem -Δ u = f and Helmholtz problem -Δ u + λ 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 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 ∩ H10. The specific forms of our variational formulations are motivated by, and applied to, a finite element scheme for the time-dependent Navier-Stokes equations.
TUESDAY, January 22, 2008