Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Variational problems in weighted Sobolev spaces on non-smooth domains


Authors: Ana Maria Soane and Rouben Rostamian
Journal: Quart. Appl. Math. 68 (2010), 439-458
MSC (2000): Primary 35J20; Secondary 35Q30, 46E35, 76D05, 65N30
Published electronically: May 19, 2010
MathSciNet review: 2676970
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Abstract: We study the Poisson problem $ -\Delta u = f$ and the Helmholtz problem $ -\Delta u + \lambda 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 $ H^1$ but not in $ H^2$ in general, 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 $ H^2 \cap H^1_0$.

The specific forms of our variational formulations are motivated by, and are particularly suited to, applying a finite element scheme for solving the time-dependent Navier-Stokes equations of fluid mechanics.


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Additional Information

Ana Maria Soane
Affiliation: Department of Mathematics and Statistics, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore 21250, Maryland
Address at time of publication: MOX, Dipartimento di Matematica, Politecnico di Milano, Piazza L. Da Vinci 32, 20133 Milano, Italy
Email: asoane@umbc.edu

Rouben Rostamian
Affiliation: Department of Mathematics and Statistics, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore 21250, Maryland
Email: rostamian@umbc.edu

DOI: http://dx.doi.org/10.1090/S0033-569X-2010-01212-7
Keywords: Poisson problem, Helmholtz problem, corner singularities, weighted Sobolev spaces, finite elements, Navier-Stokes equations
Received by editor(s): August 12, 2008
Published electronically: May 19, 2010
Article copyright: © Copyright 2010 Brown University
The copyright for this article reverts to public domain 28 years after publication.


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