Regularity of the solutions for elliptic problems on nonsmooth domains in ℝ³, Part I: countably normed spaces on polyhedral domains

Abstract

This is the first of a series of three papers devoted to the regularity of solutions of elliptic problems on nonsmooth domains in ℝ 3 . The present paper introduces various weighted spaces and countably weighted spaces in the neighbourhood of edges and vertices of polyhedral domains, and it concentrates on exploring the structure of these spaces, such as the embeddings of weighted Sobolev spaces, the relation between weighted Sobolev spaces and weighted continuous function spaces, and the relations between the weighted Sobolev spaces and countably weighted Sobolev spaces in Cartesian coordinates and in the spherical and cylindrical coordinates. These well-defined spaces are the foundation for the comprehensive study of the regularity theory of elliptic problems with piecewise analytic data in ℝ 3 , which are essential for the design of effective computation and the analysis of the h – p version of the finite element method for solving elliptic problems in three-dimensional nonsmooth domains arising from mechanics and engineering.