On Lebesgue points of entropy solutions to the eikonal equation

TL;DR

This paper investigates the fine properties of entropy solutions to the eikonal equation in two dimensions, showing that the set of points where the solution's gradient behavior is irregular has codimension at least one.

Contribution

It establishes that entropy solutions to the eikonal equation share key fine properties with BV functions, despite lacking bounded variation.

Findings

Set of non-Lebesgue points has codimension at least one

Entropy solutions share fine properties with BV functions

Solutions can lack bounded variation but still have regularity properties

Abstract

We consider entropy solutions to the eikonal equation $|\nabla u|=1$ in two space dimensions. These solutions are motivated by a class of variational problems and fail in general to have bounded variation. Nevertheless they share with BV functions, several of their fine properties: we show in particular that the set of non-Lebesgue points has co-dimension at least one.