## Abstract

We prove the existence and regularity of optimal shapes for the problem min{P(Ω)+G(Ω):Ω⊂D,|Ω|=m}, where P denotes the perimeter, |⋅| is the volume, and the functional G is either one of the following: • the Dirichlet energy E_{f}, with respect to a (possibly sign-changing) function f∈L^{p};• a spectral functional of the form F(λ_{1},…,λ_{k}), where λ_{k} is the kth eigenvalue of the Dirichlet Laplacian and F:R^{k}→R is locally Lipschitz continuous and increasing in each variable.The domain D is the whole space R^{d} or a bounded domain. We also give general assumptions on the functional G so that the result remains valid.

Original language | English (US) |
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Pages (from-to) | 147-181 |

Number of pages | 35 |

Journal | Journal des Mathematiques Pures et Appliquees |

Volume | 109 |

DOIs | |

State | Published - Jan 2018 |

## Keywords

- Dirichlet energy
- Regularity
- Shape optimization
- Spectral functional

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics