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Mixed boundary value problems for the Stokes system


Authors: R. Brown, I. Mitrea, M. Mitrea and M. Wright
Journal: Trans. Amer. Math. Soc. 362 (2010), 1211-1230
MSC (2000): Primary 35J25, 42B20; Secondary 35J05, 45B05, 31B10
DOI: https://doi.org/10.1090/S0002-9947-09-04774-6
Published electronically: October 9, 2009
MathSciNet review: 2563727
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Abstract: We prove the well-posedness of the mixed problem for the Stokes system in a class of Lipschitz domains in $ {\mathbb{R}}^n$, $ n\geq 3$. The strategy is to reduce the original problem to a boundary integral equation, and we establish certain new Rellich-type estimates which imply that the intervening boundary integral operator is semi-Fredholm. We then prove that its index is zero by performing a homotopic deformation of it onto an operator related to the Lamé system, which has recently been shown to be invertible.


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Additional Information

R. Brown
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
Email: russell.brown@uky.edu

I. Mitrea
Affiliation: Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, Massachusetts 01609
Email: imitrea@wpi.edu

M. Mitrea
Affiliation: Department of Mathematics, University of Missouri at Columbia, Columbia, Missouri 65211
Email: marius@math.missouri.edu

M. Wright
Affiliation: Department of Mathematics, University of Missouri at Columbia, Columbia, Missouri 65211
Email: wrightm@math.missouri.edu

DOI: https://doi.org/10.1090/S0002-9947-09-04774-6
Keywords: Stokes system, Lipschitz domains, mixed boundary value problems, layer potentials, well-posedness.
Received by editor(s): June 25, 2007
Published electronically: October 9, 2009
Additional Notes: The research of the authors was supported in part by the NSF
Dedicated: Dedicated to the memory of Misha Cotlar
Article copyright: © Copyright 2009 American Mathematical Society

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