Regularity of Minimizers of Shape Optimization Problems involving Perimeter - Université Rennes 2 Accéder directement au contenu
Article Dans Une Revue Journal de Mathématiques Pures et Appliquées Année : 2018

Regularity of Minimizers of Shape Optimization Problems involving Perimeter

Résumé

We prove existence and regularity of optimal shapes for the problem$$\min\Big\{P(\Omega)+\mathcal{G}(\Omega):\ \Omega\subset D,\ |\Omega|=m\Big\},$$where $P$ denotes the perimeter, $|\cdot|$ is the volume, and the functional $\mathcal{G}$ is either one of the following:
  • the Dirichlet energy $E_f$, with respect to a (possibly sign-changing) function $f\in L^p$;
  • a spectral functional of the form $F(\lambda_{1},\dots,\lambda_{k})$, where $\lambda_k$ is the $k$th eigenvalue of the Dirichlet Laplacian and $F:\mathbb{R}^k\to\mathbb{R}$ is Lipschitz continuous and increasing in each variable.
The domain $D$ is the whole space $\mathbb{R}^d$ or a bounded domain. We also give general assumptions on the functional $\mathcal{G}$ so that the result remains valid.
Fichier principal
Vignette du fichier
dePhiLamPieVel_revised.pdf (458.43 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-01318549 , version 1 (19-05-2016)
hal-01318549 , version 2 (19-09-2016)

Identifiants

Citer

Guido de Philippis, Jimmy Lamboley, Michel Pierre, Bozhidar Velichkov. Regularity of Minimizers of Shape Optimization Problems involving Perimeter. Journal de Mathématiques Pures et Appliquées, 2018, 109, pp.147-181. ⟨10.1016/j.matpur.2017.05.021⟩. ⟨hal-01318549v2⟩
481 Consultations
410 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More