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Nonlocal Hamilton–Jacobi Equations on a network with Kirchhoff type conditions
[Équations de Hamilton–Jacobi non-locales sur des réseaux avec conditions de type Kirchhoff aux nœuds]
Guy Barles1 ; Olivier Ley2 ; Erwin Topp3
1Université de Tours, Université d’Orléans, CNRS, Institut Denis Poisson, UMR CNRS 7013, 37200 Tours, France
2Univ Rennes, INSA Rennes, CNRS, IRMAR, UMR CNRS 6625, 35000 Rennes, France
3Universidade Federal do Rio de Janeiro, Instituto de Matemáticas, Rio de Janeiro, RJ, 21941-909, Brazil
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 35 (2026) no. 3, pp. 839-902

In this article, we consider nonlocal Hamilton–Jacobi Equations on networks with Kirchhoff type conditions for the interior vertices and Dirichlet boundary conditions for the boundary ones: we provide general existence and comparison results in the case when the nonlocal operator is an integro-differential operator of order strictly less than $1$. The main originality of these results is to allow these nonlocal terms to have contributions on several different edges of the network. The existence of Lipschitz continuous viscosity solutions is proved in two ways: either by using the vanishing viscosity method or by the usual Perron’s method. The comparison proof relies on arguments introduced by Lions and Souganidis. We also introduce a notion of flux-limited solution, nonlocal analog to the one introduced by Imbert and Monneau, and prove that the solutions of the Kirchhoff problem are flux-limited solutions for a suitable flux-limiter. After treating in details the case when we only have one interior vertex, we extend our approach to treat general networks.

Dans cet article, nous considérons des équations de Hamilton–Jacobi non-locales sur des réseaux avec conditions de type Kirchhoff aux nœuds intérieurs et des conditions de bord de Dirichlet aux extrémités du réseau : nous prouvons des résultats d’existence et de comparaison généraux dans le cas où l’opérateur non-local est un opérateur intégro-différentiel d’ordre strictement plus petit que 1. L’originalité principale de ces résultats est de permettre des contributions des termes non-locaux dans toutes les arêtes du réseau. L’existence d’une solution de viscosité lipschitzienne est établie de deux manières différentes : en utilisant une méthode de viscosité évanescente ou par la méthode de Perron. La preuve du principe de comparaison repose sur des arguments introduits par Lions et Souganidis. Nous introduisons aussi une notion de solution avec limiteur de flux qui est l’analogue non-local ce celle introduite par Imbert et Monneau. Nous prouvons que les deux notions de solutions sont équivalentes. L’étude est faite en détail dans le cas d’un réseau de type jonction mais nous expliquons comment étendre nos résultats au cas des réseaux généraux.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/afst.1865
Classification : 35K55, 35R09, 47G20, 35B40, 33B20
Keywords: nonlocal equations, Hamilton–Jacobi Equations, networks, Kirchhoff conditions, existence, comparison, regularity, junction viscosity solutions, flux-limited solutions

Guy Barles  1   ; Olivier Ley  2   ; Erwin Topp  3

1 Université de Tours, Université d’Orléans, CNRS, Institut Denis Poisson, UMR CNRS 7013, 37200 Tours, France
2 Univ Rennes, INSA Rennes, CNRS, IRMAR, UMR CNRS 6625, 35000 Rennes, France
3 Universidade Federal do Rio de Janeiro, Instituto de Matemáticas, Rio de Janeiro, RJ, 21941-909, Brazil
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
Guy Barles; Olivier Ley; Erwin Topp. Nonlocal Hamilton–Jacobi Equations on a network with Kirchhoff type conditions. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 35 (2026) no. 3, pp. 839-902. doi: 10.5802/afst.1865
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     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
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