Discontinuous eikonal equations in metric measure spaces

TL;DR

This paper extends the theory of eikonal equations to metric measure spaces with discontinuous, unbounded inhomogeneous terms, establishing existence, uniqueness, and regularity of solutions using a novel metric approach.

Contribution

It introduces a new metric based on optimal control to handle discontinuous terms, enabling the analysis of eikonal equations in more general metric measure spaces.

Findings

Established existence and uniqueness of solutions for discontinuous eikonal equations.

Proved Holder continuity of solutions under certain regularity conditions.

Developed a new metric framework to transform the problem into a continuous form.

Abstract

In this paper, we study the eikonal equation in metric measure spaces, where the inhomogeneous term is allowed to be discontinuous, unbounded and merely $p$-integrable in the domain with a finite $p$. For continuous eikonal equations, it is known that the notion of Monge solutions is equivalent to the standard definition of viscosity solutions. Generalizing the notion of Monge solutions in our setting, we establish uniqueness and existence results for the associated Dirichlet boundary problem. The key in our approach is to adopt a new metric, based on the optimal control interpretation, which integrates the discontinuous term and converts the eikonal equation to a standard continuous form. We also discuss the Holder continuity of the unique solution with respect to the original metric under regularity assumptions on the space and the inhomogeneous term.