## Abstract

This paper deals with a variation of the classical isoperimetric problem in dimension N≥ 2 for a two-phase piecewise constant density whose discontinuity interface is a given hyperplane. We introduce a weighted perimeter functional with three different weights, one for the hyperplane and one for each of the two open half-spaces in which R^{N} gets partitioned. We then consider the problem of characterizing the sets Ω that minimize this weighted perimeter functional under the additional constraint that the volumes of the portions of Ω in the two half-spaces are given. It is shown that the problem admits two kinds of minimizers, which will be called type I and type II, respectively. These minimizers are made of the union of two spherical domes whose angle of incidence satisfies some kind of “Snell’s law”. Finally, we provide a complete classification of the minimizers depending on the various parameters of the problem.

Original language | English |
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Pages (from-to) | 7750-7772 |

Number of pages | 23 |

Journal | Journal of Geometric Analysis |

Volume | 31 |

Issue number | 8 |

DOIs | |

Publication status | Published - 2021 Aug |

## Keywords

- Constrained minimization problem
- Dido’s problem
- Isoperimetric problem
- Weighted manifold
- two-phase

## ASJC Scopus subject areas

- Geometry and Topology