Variational solutions of the Dirichlet problem, Lebesgue's cusp and non-local properties

TL;DR

This paper explores variational solutions to the Dirichlet problem, demonstrating their properties, relation to Perron solutions, and analyzing non-local behaviors and irregularities like Lebesgue's cusp.

Contribution

It establishes the equivalence of variational and Perron solutions, characterizes solutions with finite energy, and investigates non-local properties and irregular boundary phenomena.

Findings

Variational solutions coincide with Perron solutions.

Finite energy solutions can be characterized by boundary data.

Non-continuity at singular points is a generic, non-local property.

Abstract

A recent result from [AtES24] allows one to define variational solutions of the Dirichlet problem for general continuous boundary data. We establish basic properties of this notion of solution and show that it coincides with the Perron solution. Variational solutions can elegantly be characterised in terms of the given boundary function when the variational solution has finite energy. However, it is impossible to decide in terms of the regularity of the given boundary function when a classical solution exists. We demonstrate this by analysing Lebesgue's cusp, and more precisely Lebesgue's domain which is associated with the potential of a thin rod with mass density going to zero at one end. We also show that the non-continuity of the Perron solution at a singular point is a generic and non-local property.