Coercive Hamilton–Jacobi equations in domains: the twin blow-ups method
In this note, we consider an evolution coercive Hamilton–Jacobi equation posed in a domain and supplemented with a boundary condition. We are interested in proving a comparison principle in the case where the time and the (normal) gradient variables are strongly coupled at the boundary. We elaborate...
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Académie des sciences
2024-10-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.591/ |
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| author | Forcadel, Nicolas Imbert, Cyril Monneau, Régis |
| author_facet | Forcadel, Nicolas Imbert, Cyril Monneau, Régis |
| author_sort | Forcadel, Nicolas |
| collection | DOAJ |
| description | In this note, we consider an evolution coercive Hamilton–Jacobi equation posed in a domain and supplemented with a boundary condition. We are interested in proving a comparison principle in the case where the time and the (normal) gradient variables are strongly coupled at the boundary. We elaborate on a method introduced by P.-L. Lions and P. Souganidis (Atti Accad. Naz. Lincei, 2017) to extend their comparison principle to more general boundary conditions and to Hamiltonians that are not globally Lipschitz continuous in the time variable. Their argument relies on a single blow-up procedure after rescaling the semi-solutions to be compared. In this work, two blow-ups are performed simultaneously, one for each variable of the doubling variable method. We show a key one-sided Lipschitz estimate satisfied by a combination of the two blow-up limits. Both blow-up limits are a priori allowed to be infinite separately. For expository reasons, the result is presented here in the framework of space dimension one and the general case is treated in a companion paper. |
| format | Article |
| id | doaj-art-c2269220b0a844f6a05c800b44ce1060 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-10-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-c2269220b0a844f6a05c800b44ce10602025-02-07T11:22:49ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-10-01362G882983910.5802/crmath.59110.5802/crmath.591Coercive Hamilton–Jacobi equations in domains: the twin blow-ups methodForcadel, Nicolas0Imbert, Cyril1Monneau, Régis2LMI - Laboratoire de Mathématiques de l’INSA de Rouen Normandie, 685 Avenue de l’Université 76800 Saint-Etienne-du-Rouvray - FranceDépartement de mathématiques et applications, ENS-PSL & CNRS, 45 rue d’Ulm, 75005 Paris, France.CEREMADE - CEntre de REcherches en MAthématiques de la DEcision, Place du Maréchal de Lattre de Tassigny 75775 - Paris Cedex 16 - France; CERMICS - Centre d’Enseignement et de Recherche en Mathématiques et Calcul Scientifique, 6 et 8 avenue Blaise Pascal Cité Descartes - Champs sur Marne 77455 Marne la Vallée Cedex 2 - FranceIn this note, we consider an evolution coercive Hamilton–Jacobi equation posed in a domain and supplemented with a boundary condition. We are interested in proving a comparison principle in the case where the time and the (normal) gradient variables are strongly coupled at the boundary. We elaborate on a method introduced by P.-L. Lions and P. Souganidis (Atti Accad. Naz. Lincei, 2017) to extend their comparison principle to more general boundary conditions and to Hamiltonians that are not globally Lipschitz continuous in the time variable. Their argument relies on a single blow-up procedure after rescaling the semi-solutions to be compared. In this work, two blow-ups are performed simultaneously, one for each variable of the doubling variable method. We show a key one-sided Lipschitz estimate satisfied by a combination of the two blow-up limits. Both blow-up limits are a priori allowed to be infinite separately. For expository reasons, the result is presented here in the framework of space dimension one and the general case is treated in a companion paper.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.591/ |
| spellingShingle | Forcadel, Nicolas Imbert, Cyril Monneau, Régis Coercive Hamilton–Jacobi equations in domains: the twin blow-ups method Comptes Rendus. Mathématique |
| title | Coercive Hamilton–Jacobi equations in domains: the twin blow-ups method |
| title_full | Coercive Hamilton–Jacobi equations in domains: the twin blow-ups method |
| title_fullStr | Coercive Hamilton–Jacobi equations in domains: the twin blow-ups method |
| title_full_unstemmed | Coercive Hamilton–Jacobi equations in domains: the twin blow-ups method |
| title_short | Coercive Hamilton–Jacobi equations in domains: the twin blow-ups method |
| title_sort | coercive hamilton jacobi equations in domains the twin blow ups method |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.591/ |
| work_keys_str_mv | AT forcadelnicolas coercivehamiltonjacobiequationsindomainsthetwinblowupsmethod AT imbertcyril coercivehamiltonjacobiequationsindomainsthetwinblowupsmethod AT monneauregis coercivehamiltonjacobiequationsindomainsthetwinblowupsmethod |