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{\it Variational Problems in Weighted Sobolev Spaces}

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{\bf Ana Maria Soane}\\
University of Maryland, Baltimore County\\
{\tt asoane@math.umbc.edu} 
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\noindent {\bf 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. 

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