Elliptic and Parabolic Equations - Abstract

Dauge, Monique

Weighted analytic regularity in corner domains: A long march to 3D polyhedra

Weighted spaces with analytic type control of derivatives were first introduced by Ivo Babuska and Benqi Guo for 2D polygonal domains in the late 80'ies. They have proved that such a regularity holds for several model problems. These spaces allow exponentially fast approximation by piecewise polynomials in the framework of the hp-version of finite elements. Benqi Guo has introduced the corresponding relevant spaces in 3D polyhedra in 1993. Their special feature (absent in 2D) is their anisotropy along edges. The proof that solutions of the Laplace equation with analytic right hand sides and Dirichlet or Neumann boundary condition, is still pending. The hp-investigators take such a regularity as an assumption.
In this talk, we first give a simple proof of the 2D result, for Dirichlet and Neumann case, using a dyadic partition technique. Then we sketch the proof of the anisotropic regularity along edges. The conclusion will be that, combining the two previous steps, one obtains the wanted regularity in a 3D polyhedron.
This result is very recent and is developped in the book:
M. Costabel, M. Dauge, S. Nicaise Corner Singularities and Analytic Regularity for Linear Elliptic Systems, in preparation