Fine singularity analysis of solutions to the Laplace equation: Berg's effect

TL;DR

This paper investigates Berg's effect, a rare phenomenon where harmonic functions with specific boundary conditions exhibit monotonicity, revealing its fragility and dependence on domain geometry.

Contribution

The study provides a detailed analysis of Berg's effect on special domains, highlighting its rarity and sensitivity to boundary conditions.

Findings

Berg's effect is a rare phenomenon.

It is fragile and easily lost under domain perturbations.

Monotonicity depends on boundary geometry and conditions.

Abstract

We study Berg's effect on special domains. This effect is understood as monotonicity of a harmonic function (with respect to the distance from the center of a flat part of the boundary) restricted to the boundary. The harmonic function must satisfy piecewise constant Neumann boundary conditions. We show that Berg's effect is a rare and fragile phenomenon.