On the Gromov--Hausdorff stability of metric viscosity solutions

TL;DR

This paper proves the stability of metric viscosity solutions to Hamilton-Jacobi equations when the underlying spaces converge in the Gromov--Hausdorff sense, using a novel approach that avoids embedding into a common space.

Contribution

It introduces a new method for establishing stability of solutions directly under Gromov--Hausdorff convergence without embeddings, and provides a PDE-based proof for Kantorovich duality stability.

Findings

Stability of metric viscosity solutions under Gromov--Hausdorff convergence.

A new proof technique avoiding space embeddings.

Stability of dual Kantorovich problems under measured-Gromov--Hausdorff convergence.

Abstract

We establish the stability of metric viscosity solutions to first-order Hamilton--Jacobi equations under Gromov--Hausdorff convergence. Our proof combines a characterization of metric viscosity solutions via quadratic distance functions with a doubling variable method adapted to epsilon-isometries, which allows us to pass to the Gromov--Hausdorff limit without embedding the spaces into a common ambient space. As a byproduct, we give a PDE-based proof of the stability of the dual Kantorovich problems under measured-Gromov--Hausdorff convergence.