Equivalence of solutions of eikonal equation in metric spaces

TL;DR

This paper establishes the equivalence of various solution concepts for the eikonal and Hamilton-Jacobi equations in complete, rectifiably connected metric spaces, extending classical results to more general settings.

Contribution

It proves the equivalence of multiple solution notions in metric spaces, including curve-based, slope-based, and Monge solutions, without relying on boundary data.

Findings

Equivalence of solution notions in metric spaces.

Reduction to length spaces via intrinsic metric.

Discussion of solution regularity and semi-concavity.

Abstract

In this paper we prove the equivalence between some known notions of solutions to the eikonal equation and more general analogs of the Hamilton-Jacobi equations in complete and rectifiably connected metric spaces. The notions considered are that of curve-based viscosity solutions, slope-based viscosity solutions, and Monge solutions. By using the induced intrinsic (path) metric, we reduce the metric space to a length space and show the equivalence of these solutions to the associated Dirichlet boundary problem. Without utilizing the boundary data, we also localize our argument and directly prove the equivalence for the definitions of solutions. Regularity of solutions related to the Euclidean semi-concavity is discussed as well.