
Journal of Convex Analysis 31 (2024), No. 2, 619670 Copyright Heldermann Verlag 2024 Equivalence between Strict Viscosity Solution and Viscosity Solution in the Wasserstein Space and Regular Extension of the Hamiltonian in L^{2}_{P} Chloé Jimenez Univ Brest, CNRS UMR 6205, Laboratoire de Mathématiques de Bretagne Atlantique, Brest, France chloe.jimenez@univbrest.fr [Abstractpdf] \newcommand{\p}{\mathbb P} This article aims to build bridges between several notions of viscosity solution of first order dynamic HamiltonJacobi equations. The first main result states that, under assumptions, the definitions of GangboNguyenTudorascu and MarigondaQuincampoix are equivalent. Secondly, to make the link with Lions' definition of solution, we build a regular extension of the Hamiltonian in $L^2_\p\times L^2_\p$. This extension allows to give an existence result of viscosity solution in the sense of GangboNguyenTudorascu, as a corollary of the existence result in $L^2_\p\times L^2_\p$. We also give a comparison principle for rearrangement invariant solutions of the extended equation. Finally we illustrate the interest of the extended equation by an example in MultiAgent Control. Keywords: Optimal transport, viscosity solutions, HamiltonJacobi equations, multiagent optimal control. MSC: 49L25. [ Fulltextpdf (324 KB)] for subscribers only. 