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arXiv · 2507.06495

On the Gromov--Hausdorff stability of metric viscosity solutions

Shimpei Makida
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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.

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Shimpei Makida. 2025-07-09. On the Gromov--Hausdorff stability of metric viscosity solutions. https://arxiv.org/abs/2507.06495

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