Regularity of the solutions for elliptic problems on nonsmooth domains in ℝ3. Part II: Regularity in neighbourhoods of edges

Abstract

This paper is the second in a series of three devoted to the analysis of the regularity of solutions of elliptic problems on nonsmooth domains in ℝ3. The present paper concentrates on the regularity of solutions of the Poisson equation in neighbourhoods of edges of a polyhedral domain in the framework of the weighted Sobolev spaces and countably normed spaces. These results can be generalised to elliptic problems arising from mechanics and engineering, for instance, the elasticity problem on polyhedral domains. Hence, the results are not only important to understand comprehensively the qualitative and quantitative aspects of the behaviours of the solution and its derivatives of all orders in neighbourhoods of edges, but also essential to design an effective computation and analyse the optimal convergence of the finite elements solutions for these problems.