论文标题
双皇后迪多的问题
The double queen Dido's problem
论文作者
论文摘要
本文介绍了对两相分段恒定密度的经典等级问题的变体，其不连续性接口是给定的超平面。我们引入了一个具有三种不同权重的加权周长功能，一个用于超平面，一个用于两个开放的半空间中的每个，其中$ \ mathbb {r}^n $被分区。然后，我们考虑表征集合$ω$的问题，该集合在给出了两个半空间中$ω$的量的其他约束条件下，将此加权周边功能最小化。结果表明，该问题承认两种最小化器，分别称为I型和II型。这些最小化器是由两个球形圆顶的结合组成的，它们的发射角满足了某种\ texquotedblleft snell的律法\ textquotedblright。最后，我们根据问题的各种参数提供了最小化器的完整分类。
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 $Ω$ 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 \textquotedblleft Snell's law\textquotedblright. Finally, we provide a complete classification of the minimizers depending on the various parameters of the problem.
