Comptes Rendus
Mathematical analysis/Partial differential equations
Uniqueness in a class of Hamilton–Jacobi equations with constraints
Comptes Rendus. Mathématique, Volume 353 (2015) no. 6, pp. 489-494.

In this note, we discuss a class of time-dependent Hamilton–Jacobi equations depending on a function of time, this function being chosen in order to keep the maximum of the solution of the constant value 0. The main result of the note is that the full problem has a unique classical solution. The motivation is a selection–mutation model that, in the limit of small diffusion, exhibits concentration on the zero-level set of the solution to the Hamilton–Jacobi equation. The uniqueness result that we prove implies strong convergence and error estimates for the selection–mutation model.

Dans cette note, on discute une classe d'équations de Hamilton–Jacobi dépendant du temps et une fonction inconnue du temps choisie pour que le maximum de la solution de l'équation de Hamilton–Jacobi prenne tout le temps la valeur 0. Le résultat principal de cette note est que le problème complet admet une unique solution classique. La motivation est un modèle de sélection–mutation qui, dans la limite d'une diffusivité nulle, présente une concentration sur la ligne de niveau 0 de la solution de l'équation de Hamilton–Jacobi. Le résultat d'unicité que nous démontrons implique une convergence forte, avec estimations d'erreur pour le modèle de sélection–mutation.

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Accepted:
Published online:
DOI: 10.1016/j.crma.2015.03.005
Sepideh Mirrahimi 1; Jean-Michel Roquejoffre 1

1 Institut de mathématiques (UMR CNRS 5219), Université Paul-Sabatier, 118, route de Narbonne, 31062 Toulouse cedex, France
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Sepideh Mirrahimi; Jean-Michel Roquejoffre. Uniqueness in a class of Hamilton–Jacobi equations with constraints. Comptes Rendus. Mathématique, Volume 353 (2015) no. 6, pp. 489-494. doi : 10.1016/j.crma.2015.03.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.03.005/

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