Monge solutions of time-dependent Hamilton-Jacobi equations in metric spaces
Qing Liu, Made Benny Prasetya Wiranata
TL;DR
The paper advances the theory of Monge solutions by introducing time-dependent Monge solutions for Hamilton–Jacobi equations in general metric spaces, formulating them on the space-time product and enforcing $|D^-v|=f+k$ after a time-shift $v=u+kt$. It establishes well-posedness—existence and uniqueness—for both eikonal-type and general superlinear Hamiltonians under convexity, monotonicity, and coercivity assumptions, using a direct Monge-slope equivalence rather than doubling variables. It proves dynamic programming representations and Lipschitz regularity, and rigorously shows equivalence with curve-based and slope-based metric viscosity solutions in the time-independent cases, thereby unifying three solution notions in this setting. The results extend Monge-solution methods to time-dependent problems on general length spaces, providing robust well-posedness without relying on test-function arguments, and align with Hopf–Lax representations in the appropriate limits. This yields a solid foundation for analysis of first-order HJ dynamics in metric spaces with potential applications to optimal transport, mean field games, and network dynamics where a linear structure may be absent.
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.