Mass concentration in rescaled first order integral functionals

Antonin Monteil; Paul Pegon

doi:10.5802/jep.257

Mass concentration in rescaled first order integral functionals
[Concentration de masse dans des fonctionnelles intégrales d’ordre

1

rééchelonnées]

Antonin Monteil ¹ ; Paul Pegon ²

¹ Université Paris-Est Créteil Val-de-Marne, LAMA, 61 avenue du Général de Gaulle, 94010 Créteil, France
² Université Paris-Dauphine, CEREMADE & INRIA Paris, MOKAPLAN, France

Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 431-472.

Résumé
Abstract

Nous considérons des problèmes de minimisation locaux d’ordre $1$ de la forme $min \int_{ℝ^{N}} f (u, \nabla u)$ sous contrainte de masse $\int_{ℝ^{N}} u = m$ . Nous prouvons que la fonction d’énergie minimale $H (m)$ est toujours concave, et que des rééchelonnements appropriés de l’énergie, dépendant d’un petit paramètre $ε$ , $Γ$ -convergent vers la $H$ -masse, définie pour les mesures atomiques $\sum_{i} m_{i} δ_{x_{i}}$ par $\sum_{i} H (m_{i})$ . Nous considérons également des lagrangiens dépendant de $ε$ , et des lagrangiens et $H$ -masses spatialement inhomogènes. Notre résultat est valable sous de faibles hypothèses sur $f$ , et couvre les $α$ -masses en toute dimension $N \geq 2$ pour des exposants $α$ au-dessus d’un seuil critique, et toutes les $H$ -masses concaves en dimension $N = 1$ . Notre résultat donne en particulier la concentration des fluides de Cahn-Hilliard en gouttelettes, et est lié à l’approximation du transport branché par des énergies elliptiques.

We consider first order local minimization problems of the form $min \int_{ℝ^{N}} f (u, \nabla u)$ under a mass constraint $\int_{ℝ^{N}} u = m$ . We prove that the minimal energy function $H (m)$ is always concave, and that relevant rescalings of the energy, depending on a small parameter $ε$ , $Γ$ -converge towards the $H$ -mass, defined for atomic measures $\sum_{i} m_{i} δ_{x_{i}}$ as $\sum_{i} H (m_{i})$ . We also consider Lagrangians depending on $ε$ , as well as space-inhomogeneous Lagrangians and $H$ -masses. Our result holds under mild assumptions on $f$ , and covers in particular $α$ -masses in any dimension $N \geq 2$ for exponents $α$ above a critical threshold, and all concave $H$ -masses in dimension $N = 1$ . Our result yields in particular the concentration of Cahn-Hilliard fluids into droplets, and is related to the approximation of branched transport by elliptic energies.

Reçu le : 2022-03-02
Accepté le : 2024-01-16
Publié le : 2024-02-19

DOI : 10.5802/jep.257

Classification : 28A33, 49J45, 46E35, 49Q20, 76T99, 49Q22, 49J10
Keywords: $\Gamma $-convergence, semicontinuity, integral functionals, convergence of measures, concentration-compactness, Cahn-Hilliard fluids, branched transport
Mot clés : $\Gamma $-convergence, semi-continuité, fonctionnelles intégrales, convergence des mesures, concentration-compacité, fluides de Cahn-Hilliard, transport branché

Affiliations des auteurs :

Antonin Monteil ¹ ; Paul Pegon ²

¹ Université Paris-Est Créteil Val-de-Marne, LAMA, 61 avenue du Général de Gaulle, 94010 Créteil, France
² Université Paris-Dauphine, CEREMADE & INRIA Paris, MOKAPLAN, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Mass concentration in rescaled~first~order~integral functionals},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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Antonin Monteil; Paul Pegon. Mass concentration in rescaled first order integral functionals. Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 431-472. doi : 10.5802/jep.257. https://jep.centre-mersenne.org/articles/10.5802/jep.257/

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