Mixed boundary value problems for the Stokes system

Brown, R.; Mitrea, I.; Mitrea, M.; Wright, M.

doi:10.1090/S0002-9947-09-04774-6

Mixed boundary value problems for the Stokes system
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by R. Brown, I. Mitrea, M. Mitrea and M. Wright

Trans. Amer. Math. Soc. 362 (2010), 1211-1230

DOI: https://doi.org/10.1090/S0002-9947-09-04774-6

Published electronically: October 9, 2009

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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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