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Viscosity Solutions of Hamilton-Jacobi Equations in Proper \(\mathrm {CAT(0)}\) Spaces

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Abstract

In this article, we develop a novel notion of viscosity solutions for first order Hamilton-Jacobi equations in proper \(\mathrm {CAT(0)}\) spaces. The notion of viscosity is defined by taking test functions that are locally Lipschitz and can be represented as a difference of two semiconvex functions. Under mild assumptions on the Hamiltonian, we recover the main features of viscosity theory for both the stationary and the time-dependent cases in this setting: the comparison principle and Perron’s method. Finally, we show that this notion of viscosity coincides with the classical one in \(\mathbb {R}^N\) and we give several examples of Hamilton-Jacobi equations in more general \(\mathrm {CAT(0)}\) spaces covered by this setting.

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Funding

This research work has received support from the Normandy Region and the European Union through the ERDF research and innovation program under the grant for the “chaire d’excellence COPTI.” The authors also acknowledge support from the Agence Nationale de la Recherche (ANR), project ANR-22-CE40-0010 COSS. The first author acknowledges support from “FMJH - Fondation Mathématique Jacques Hadamard.”

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Authors and Affiliations

  1. Univ Rennes, INSA de Rennes, CNRS, IRMAR - UMR 6625, Rennes, F-35000, France

    Othmane Jerhaoui

  2. Univ Normandie, INSA de Rouen Normandie, LMI (EA 3226 – FR CNRS 3335), Rouen, F-76000, France

    Othmane Jerhaoui & Hasnaa Zidani

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  1. Othmane Jerhaoui
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Correspondence to Othmane Jerhaoui.

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Jerhaoui, O., Zidani, H. Viscosity Solutions of Hamilton-Jacobi Equations in Proper \(\mathrm {CAT(0)}\) Spaces. J Geom Anal 34, 47 (2024). https://doi.org/10.1007/s12220-023-01484-7

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