Year: 2024
Type: article
Abstract: This work is concerned with Hamilton-Jacobi equations of evolution type posed in domains and supplemented with boundary conditions. Hamiltonians are coercive but are neither convex nor quasiconvex. We analyse boundary conditions when understood in the sense of viscosity solutions. This analysis is based on the study of boundary conditions of evolution type. More precisely, we give a new formula for the relaxed boundary conditions derived by J. Guerand (J. Differ. Equations, 2017). This new point of view unveils a connection between the relaxation operator and the classical Godunov flux from the theory of conservation laws. We apply our methods to two classical boundary value problems. It is shown that the relaxed Neumann boundary condition is expressed in terms of Godunov's flux while the relaxed Dirichlet boundary condition reduces to an obstacle problem at the boundary associated with the lower non-increasing envelope of the Hamiltonian.
Source: Pure and Applied Analysis
Authors: Nicolas Forcadel, Cyril Imbert, Régis Monneau
Institutions: Laboratoire de Mathématiques de l'INSA de Rouen, Institut National des Sciences Appliquées Rouen Normandie, École Normale Supérieure - PSL, Département de mathématiques et applications, École nationale des ponts et chaussées, Centre de Recherche en Mathématiques de la Décision, CERMICS, Pôle de recherche et d'enseignement supérieur Université Paris-Est
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Agence Nationale de la Recherche
OpenAlex ID: W4386907257