Stability of metric viscosity solutions under Hausdorff convergence

TL;DR

This paper establishes the stability of metric viscosity solutions to Hamilton--Jacobi equations under Hausdorff and uniform convergence of the underlying spaces, with applications to fractal and network approximations.

Contribution

It demonstrates stability of solutions on general metric spaces under specific convergence conditions, extending previous results to more complex and noncompact spaces.

Findings

Stability holds when sets converge in Hausdorff sense and metrics converge uniformly.

Examples include self-similar sets, junctions of shrinking tubes, and lattice lines with Manhattan distance.

Reduced test functions to squared distance functions to prove stability.

Abstract

This study investigated the stability of Hamilton--Jacobi equation on general metric spaces with a perturbation in some whole space. This type of stability appears in the domain perturbation problem. We find that the stability holds when the set converges in the Hausdorff sense and when the metric converges in some uniform sense. Examples of the perturbed space satisfying these assumptions include network approximation of self-similar sets such as the Sierpi\'{n}ski gasket, junction of shrinking tubes, and lattice lines with the Manhattan distance. We also give supplemental results on time-dependent or noncompact case. Stability can be achieved when the class of test function of metric viscosity solutions is reduced to the squared distance functions, whose proof is also given.