Fine singularity analysis of solutions to Laplace equation

TL;DR

This paper conducts a detailed singularity analysis of Laplace equation solutions in polygonal domains with specific boundary conditions, aiming to rigorously prove Berg's effect.

Contribution

It provides a rigorous proof of Berg's effect through fine singularity analysis of Laplace solutions in special polygonal domains with Neumann boundary conditions.

Findings

Confirmed the presence of singularities at polygon corners.

Established conditions under which Berg's effect holds.

Enhanced understanding of boundary behavior in Laplace solutions.

Abstract

We present here a fine singularity analysis of solutions to the Laplace equation in special polygonal domains in the plane. We assume piecewise constant Neumann on one component of the boundary. Our motivation is to find the rigorous proof the so-called Berg's effect.