The Double Queen Dido’s Problem

This paper deals with a variation of the classical isoperimetric problem in dimension $$N\ge 2$$ 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.