Regularity of the Solution of Elliptic Problems with Piecewise Analytic Data. Part I. Boundary Value Problems for Linear Elliptic Equation of Second Order

This paper is the first in a series devoted to the analysis of the regularity of the solution of elliptic partial differential equations with piecewise analytic data. The present paper analyzes the case of linear, second order partial differential equation of elliptic type. It concentrates on the case when the domain Ω ⊂ R² is a polygon, boundary conditions are of changing type and coefficients are analytic on Ω. The main result states that the solution belongs to a countably normed space based on weighted Sobolev spaces of all orders with weights located in the vertices of the domain and at the points where the type of boundary conditions changes. These results are essential for the design and the analysis of the h-p version of the finite element method for solving the elliptic differential equations of structural engineering (see [6], [11], [12]).