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.