The twin blow-up method for Hamilton–Jacobi equations in higher dimension

¹ INSA Rouen Normandie, Normandie Univ, LMI UR 3226, 76000 Rouen, France
² Département de mathématiques et applications, ENS-PSL & CNRS 45 rue d’Ulm, 75005 Paris, France
³ CEREMADE - CEntre de REcherches en MAthématiques de la DEcision, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France
⁴ CERMICS - Centre d’Enseignement et de Recherche en Mathématiques et Calcul, Scientifique, 6 et 8 avenue Blaise Pascal, Cité Descartes - Champs sur Mame, 77455 Marne la Vallée Cedex 2, France

^* Corresponding author: Cyril.Imbert@ens.fr

Received: 11 January 2024
Accepted: 9 December 2024

Abstract

In this paper, we show how to extend the twin blow-up method recently developped by the authors (Comptes Rendus. Math., 2024), in order to obtain a new comparison principle for an evolution coercive Hamilton–Jacobi equation posed in a domain of an Euclidian space of any dimension and supplemented with a boundary condition. The method allows dealing with the case where tangential variables and the variable corresponding to the normal gradient of the solution are strongly coupled at the boundary. We elaborate on a method introduced by Lions and Souganidis (Atti Accad. Naz. Lincei, 2017). Their argument relies on a single blow-up procedure after rescaling the semi-solutions to be compared while two simultaneous blow-ups are performed in this work, one for each variable of the classical doubling variable technique. A one-sided Lipschitz estimate satisfied by a combination of the two blow-up limits plays a key role.