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The double queen Dido’s problem. (English) Zbl 1471.49031

Summary: This paper deals with a variation of the classical isoperimetric problem in dimension \(N\geq 2\) 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 \(\mathbb{R}^N\) gets partitioned. We then consider the problem of characterizing the sets \(\Omega\) that minimize this weighted perimeter functional under the additional constraint that the volumes of the portions of \(\Omega\) 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.

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
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