Elliptic boundary value problems in convex polygons with low regularity boundary data via the unified method

TL;DR

This paper develops a unified integral method to solve elliptic boundary value problems in convex polygons with low regularity boundary data, establishing existence of solutions and characterizing corner singularities.

Contribution

It introduces a novel integral representation and global relation framework that handles distributional boundary data in elliptic problems within convex polygons.

Findings

Existence of solutions with distributional boundary data.

Characterization of corner singularities via the global relation.

Framework applicable to Dirichlet, Neumann, and Robin problems.

Abstract

We use novel integral representations developed by the second author to prove certain rigorous results concerning elliptic boundary value problems in convex polygons. Central to this approach is the so-called global relation, which is a non-local equation in the Fourier space that relates the known boundary data to the unknown boundary values. Assuming that the global relation is satisfied in the weakest possible sense, i.e. in a distributional sense, we prove there exist solutions to Dirichlet, Neumann and Robin boundary value problems with distributional boundary data. We also show that the analysis of the global relation characterises in a straightforward manner the possible existence of both integrable and non-integrable corner-singularities.