### 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) |
---|---|

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

- Mathematics(all)
- Applied Mathematics

## Fingerprint Dive into the research topics of 'Regularity of minimizers of shape optimization problems involving perimeter'. Together they form a unique fingerprint.

## Cite this

De Philippis, G., Lamboley, J., Pierre, M., & Velichkov, B. (2018). Regularity of minimizers of shape optimization problems involving perimeter.

*Journal des Mathematiques Pures et Appliquees*,*109*, 147-181. https://doi.org/10.1016/j.matpur.2017.05.021