The Double Queen Dido’s Problem

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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 RN 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 languageEnglish
Pages (from-to)7750-7772
Number of pages23
JournalJournal of Geometric Analysis
Volume31
Issue number8
DOIs
Publication statusPublished - 2021 Aug

Keywords

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

ASJC Scopus subject areas

  • Geometry and Topology

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