Stable domains for higher order elliptic operators

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

Comptes Rendus. Mathématique, Tome 362 (2024) no. G10, pp. 1189-1203

This paper is devoted to prove that any domain satisfying a $({\delta }_0,r_0)$-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 $({\delta }_0,r_0)$-capacité de premier ordre, est automatiquement $(m,p)$ stable pour tout $m\ge 1$ 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.

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

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