Geometric Singularities of the Poisson’s Equation in a Non-Smooth Domain with Applications of Weighted Sobolev Spaces

Abstract

The solution fields of the elliptic boundary value problems may exhibit singularities near the corners, edges, crack tips, and so forth of the physical domain. This paper deals with the boundary singularities of weak solutions of boundary value problems governed by the Poisson equation in a two-dimensional non-smooth domain with singular points on the boundary. The presence of these points on the boundary, generally, generates local singularities in the solution. The applications of Fourier transform and weighted Sobolev spaces make it possible to describe the qualitative properties of the solution including its regularity. The general theory of V. A. Kondratiev is followed to obtain these results.