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The Poisson problem with mixed boundary conditions in Sobolev and Besov spaces in non-smooth domains

Authors: Irina Mitrea and Marius Mitrea
Journal: Trans. Amer. Math. Soc. 359 (2007), 4143-4182
MSC (2000): Primary 45E05, 47A05; Secondary 35J25, 42B20
Published electronically: April 11, 2007
MathSciNet review: 2309180
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Abstract: We introduce certain Sobolev-Besov spaces which are particularly well adapted for measuring the smoothness of data and solutions of mixed boundary value problems in Lipschitz domains. In particular, these are used to obtain sharp well-posedness results for the Poisson problem for the Laplacian with mixed boundary conditions on bounded Lipschitz domains which satisfy a suitable geometric condition introduced by R.Brown in (1994). In this context, we obtain results which generalize those by D.Jerison and C.Kenig (1995) as well as E.Fabes, O.Mendez and M.Mitrea (1998). Applications to Hodge theory and the regularity of Green operators are also presented.

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Additional Information

Irina Mitrea
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904

Marius Mitrea
Affiliation: Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri 65211

Keywords: Lipschitz domain, Laplacian, mixed boundary conditions, Hardy spaces, Sobolev spaces, Besov spaces, Hodge decompositions, Green operator
Received by editor(s): May 3, 2005
Published electronically: April 11, 2007
Additional Notes: The first author was supported in part by NSF grant DMS - 0547944 and a FEST grant from the University of Virginia
The second author was supported in part by the NSF grants DMS - 0400639 and DMS FRG - 0456306.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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