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Young Women in Harmonic Analysis and PDE

December 2-4, 2016

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Laura Westermann (University of Düsseldorf)

Optimal Sobolev Regularity for Laplace and Stokes Operator on Wegde Type Domains subject to Navier Slip Boundary Conditions

The aim of this talk is to show for $1 < p < \infty$ the $W^{2,p}$-Sobolev regularity for Laplace and Stokes operator in the $L^p$-space on two-dimensional wedge type domains subject to Navier slip boundary conditions. Introducing polar coordinates and applying the Euler transformation, we transform the elliptic problem of the Laplace operator from wedge type domains to a layer domain. The resulting operator on a layer is described by the sum of two linear closed operators. The proof of the $W^{2,p}$-Sobolev regularity requires a subtle spectral analysis of these two operators. This then allows for applying elements of $H^\infty$-calculus and a consequence of the Kalton-Weis theorem to prove invertibility of the full operator on the layer domain. Thanks to this result and after back transformation to wedge type domains, we are able to prove the well-posedness of the Stokes equations. The topic of the talk is content of [1], [2].

References

[1] J. Saal, L. Westermann. Optimal Sobolev Regularity for the Stokes Operator subject to Navier Slip on two dimensional Wedges. In preparation.
[2] L. Westermann. Optimal Regularity for Laplace and Stokes Equation on Polygonal Domains. Master’s thesis, Heinrich-Heine-Universität Düsseldorf, 2016.

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