Calculus of Variations and Geometric Measure Theory

G. De Philippis - J. Lamboley - M. Pierre - B. Velichkov

Regularity of Minimizers of Shape Optimization Problems involving Perimeter

created by dephilipp on 20 May 2016
modified by velichkov on 21 Apr 2018


Published Paper

Inserted: 20 may 2016
Last Updated: 21 apr 2018

Journal: J. Math. Pures Appl.
Year: 2016

ArXiv: 1605.06294 PDF


We prove existence and regularity of optimal shapes for the problem$$\min\Big\{P(\Omega)+\mathcal{G}(\Omega):\ \Omega\subset D,\
=m\Big\},$$where $P$ denotes the perimeter, $
$ 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.