[Équations de Hamilton–Jacobi à non-linéarités monotones sur des cônes convexes]
We study the Cauchy problem of a Hamilton–Jacobi equation with the spatial variable in a closed convex cone. A monotonicity assumption on the nonlinearity allows us to prescribe no condition on the boundary of the cone. We show the well-posedness of the equation in the viscosity sense and prove several properties of the solution: monotonicity, Lipschitzness, and representations by variational formulas.
Nous étudions le problème de Cauchy pour une équation de Hamilton–Jacobi où la variable spatiale appartient à un cône convexe fermé. Une hypothèse de monotonicité sur la non-linéarité nous permet de nous affranchir de prescrire une condition au bord du cône. Nous établissons le caractère bien-posé de l’équation au sens des solutions de viscosité et démontrons quelques propriétés : monotonicité, bornes Lipschitz et représentation des solutions par des formules variationnelles.
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Keywords: Hamilton–Jacobi equation, monotonicity, convex cone
Hong-Bin Chen 1 ; Jiaming Xia 2
CC-BY 4.0
@article{AFST_2025_6_34_1_1_0,
author = {Hong-Bin Chen and Jiaming Xia},
title = {Hamilton{\textendash}Jacobi equations with monotone nonlinearities on convex cones},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {1--45},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 34},
number = {1},
year = {2025},
doi = {10.5802/afst.1804},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1804/}
}
TY - JOUR AU - Hong-Bin Chen AU - Jiaming Xia TI - Hamilton–Jacobi equations with monotone nonlinearities on convex cones JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2025 SP - 1 EP - 45 VL - 34 IS - 1 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1804/ DO - 10.5802/afst.1804 LA - en ID - AFST_2025_6_34_1_1_0 ER -
%0 Journal Article %A Hong-Bin Chen %A Jiaming Xia %T Hamilton–Jacobi equations with monotone nonlinearities on convex cones %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2025 %P 1-45 %V 34 %N 1 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1804/ %R 10.5802/afst.1804 %G en %F AFST_2025_6_34_1_1_0
Hong-Bin Chen; Jiaming Xia. Hamilton–Jacobi equations with monotone nonlinearities on convex cones. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 34 (2025) no. 1, pp. 1-45. doi : 10.5802/afst.1804. https://afst.centre-mersenne.org/articles/10.5802/afst.1804/
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