Analytic Regularity for Linear Elliptic Systems in Polygons and Polyhedra

We prove weighted anisotropic analytic estimates for solutions of second order elliptic boundary value problems in polyhedra. The weighted analytic classes which we use are the same as those introduced by Guo in 1993 in view of establishing exponential convergence for hp finite element methods in polyhedra. We first give a simple proof of the known weighted analytic regularity in a polygon, relying on a new formulation of elliptic a priori estimates in smooth domains with analytic control of derivatives. The technique is based on dyadic partitions near the corners. This technique can successfully be extended to polyhedra, providing isotropic analytic regularity. This is not optimal, because it does not take advantage of the full regularity along the edges. We combine it with a nested open set technique to obtain the desired three-dimensional anisotropic analytic regularity result. Our proofs are global and do not require the analysis of singular functions. CONTENTS Introduction 2 Motivation 2 Organization of the paper 3 1. Illustration 5 1.1. Dirichlet conditions 5 1.2. Neumann conditions 9 1.3. Bibliographical comments 12 2. Local Cauchy-type estimates in smooth domains 13 3. Weighted Cauchy-type estimates in plane sectors 14 3.1. Weighted spaces with homogeneous norms 15 3.2. Weighted spaces with non-homogeneous norms 17 4. Natural weighted regularity shift in polygons 19 5. Local anisotropic Cauchy-type estimates in dihedral domains 22 5.1. Isotropic estimates: natural regularity shift 22 5.2. Tangential regularity along the edge (homogeneous norms) 23 5.3. Anisotropic estimates in dihedral domains (homogeneous norms) 28 5.4. Anisotropic estimates in dihedral domains (non-homogeneous norms) 30 6. Natural anisotropic weighted regularity shift in polyhedra 33 6.1. Edge and corner neighborhoods 33 6.2. Anisotropic weighted spaces with homogeneous norms 35 Date: October 28, 2011. 2000 Mathematics Subject Classification. 35B65, 35J25, 65N30 .