BEHAVIOR OF THE SOLUTIONS OF AN ELLIPTIC BOUNDARY VALUE PROBLEM IN A POLYGONAL OR POLYHEDRAL DOMAIN

Abstract

The aim of this talk is to describe the main results about the regularity (as well as the singularities) of the solutions of an elliptic boundary value problem in domains with a non smooth boundary. Up to now, results concerning general elliptic boundary value problems, are available only if the domain is two-dimensional, mainly with a polygonal boundary. On the contrary few results are known in higher dimension when the domain presents singularities more complicated than conical points. For instance for a domain with edges but with no vertices, only boundary value problems for elliptic operator of the second order have been investigated. I will give a brief survey of recent results on this subject with precise bibliographical references together with some new contributions to the study of a second order elliptic boundary value problem in a three dimensional polyhedron. Except for new results the material is presented without proofs. The topics mentioned above are closely related to the study of mixed boundary value problems for an elliptic equation and of problems with interface conditions (conjugacy problems). For these problems, I will only mention briefly the results which are similar to those concerning our main subject, with bibliographical references.