Stable domains for higher order elliptic operators

Jean-François Grosjean; Antoine Lemenant; Rémy  Mougenot

doi:10.5802/crmath.630

Research article - Partial differential equations

Stable domains for higher order elliptic operators

Jean-François Grosjean ¹; Antoine Lemenant ¹ ; Rémy Mougenot ¹

¹ Université de Lorraine, CNRS, IECL, F-54000 Nancy, France

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1189-1203.

Abstract
Résumé

This paper is devoted to prove that any domain satisfying a $(δ_{0}, r_{0})$ -capacitary condition of first order is automatically $(m, p)$ -stable for all $m ⩾ 1$ and $p > 1$ , and for any dimension $N ⩾ 1$ . In particular, this includes regular enough domains such as $𝒞^{1}$ -domains, Lipschitz domains, Reifenberg-flat domains, but is sufficiently weak to also include cusp points. Our result extends some of the results of Hayouni and Pierre valid only for $N = 2, 3$ , and partially extends the results of Bucur and Zolésio for higher order operators, with a different and simpler proof.

Dans cet article nous démontrons que tout domaine satisfaisant une condition de $(δ_{0}, r_{0})$ -capacité de premier ordre, est automatiquement $(m, p)$ stable pour tout $m \geq 1$ et pour tout $p > 1$ . En particulier, ceci inclus tous les domaines suffisamment réguliers tels que les domaines $C^{1}$ , Lipschitz, Reifenberg-plat, mais la condition est suffisamment faible pour inclure des points de type cusp. Notre résultat généralise des résultats antérieurs de Hayouni et Pierre valables seulement en dimension $N = 2, 3$ et étend aussi des résultats antérieurs de Bucur et Zolésio pour des opérateurs d’ordre supérieurs, avec une preuve plus simple et différente.

Received: 2023-12-01
Revised: 2024-02-16
Accepted: 2024-03-10
Published online: 2024-11-05

Zbl

DOI: 10.5802/crmath.630

Classification: 49G05, 35J20, 49Q20
Keywords: Capacity, stable domains, $\gamma _m$-convergence, Mosco-convergence, shape optimisation
Mots-clés : Capacité, domaines stables, $\gamma _m$-convergence, Mosco-convergence, optimisation de forme

Author's affiliations:

Jean-François Grosjean ¹; Antoine Lemenant ¹; Rémy Mougenot ¹

¹ Université de Lorraine, CNRS, IECL, F-54000 Nancy, France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Jean-Fran\c{c}ois Grosjean and Antoine Lemenant and R\'emy  Mougenot},
     title = {Stable domains for higher order elliptic operators},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1189--1203},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.630},
     zbl = {07939452},
     language = {en},
}

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Jean-François Grosjean; Antoine Lemenant; Rémy  Mougenot. Stable domains for higher order elliptic operators. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1189-1203. doi : 10.5802/crmath.630. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.630/

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