The reach and limits of slope eikonal equations in compact spaces

TL;DR

This paper investigates the conditions under which slope eikonal equations have solutions in compact metric spaces, providing a metric characterization and exploring the scope and limitations of the concept.

Contribution

It offers a purely metric characterization of compact spaces where slope eikonal equations always admit solutions, including examples and counterexamples.

Findings

Characterized spaces where solutions always exist

Identified limitations of slope eikonal equations in certain spaces

Provided examples illustrating the reach of the concept

Abstract

It is a well known fact that the eikonal equation is well posed in complete length spaces. Among the studied notions of solutions in the literature, there is one that can be defined in any metric space using the local (descent) slope and considering pointwise solutions: functionals such that their slope coincides with the prescribed data at every point of the domain. In this work we explore the question ``Can we characterize the class of compact metric spaces in which every slope eikonal equation (under standard assumptions) always admits a pointwise solution?''. We provide a purely metric characterization of these spaces, as well as some interesting examples and counterexamples that illustrate the reach and limitations of the concept.