Introduction Program Talks & posters Participants Practical Info
Young Women in Harmonic Analysis and PDE
December 2-4, 2016
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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