Regularity and corner singularities for an oblique derivative problem in a plane polygon

Abstract

This paper analyzes an oblique derivative boundary value problem for the Laplace equation in a bounded planar polygon. Although the regularity theory for elliptic problems in non–smooth do-mains is classical, the available results are often established within very general frameworks. We present a direct, self-contained treatment in the standard Sobolev setting. Under a uniform obliqueness assumption on the boundary vector field, we establish the existence of weak solutions and prove local regularity away from the vertices. Our approach relies on a priori estimates and the invariance of the operator index under homotopy and compact perturbations.