Monge solutions of time-dependent Hamilton-Jacobi equations in metric spaces

TL;DR

This paper extends the concept of Monge solutions to time-dependent Hamilton-Jacobi equations in metric spaces, establishing their existence, uniqueness, and relation to viscosity solutions under Lipschitz regularity.

Contribution

It introduces a new notion of Monge solutions for time-dependent equations in metric spaces and proves their fundamental properties.

Findings

Existence and uniqueness of bounded Lipschitz Monge solutions

Equivalence with existing metric viscosity solutions

Reformulation as stationary problem under Lipschitz initial data

Abstract

As a classical notion equivalent to viscosity solutions, Monge solutions are well understood for stationary Hamilton-Jacobi equations in Euclidean spaces and have been recently studied in general metric spaces. In this paper, we introduce a notion of Monge solutions for time-dependent Hamilton-Jacobi equations in metric spaces. The key idea is to reformulate the equation as a stationary problem under the assumption of Lipschitz regularity for the initial data. We establish the uniqueness and existence of bounded Lipschitz Monge solutions to the initial value problem and discuss their equivalence with existing notions of metric viscosity solutions.