标题： 谱极小分割的无界区域 显示英文标题

标题： Spectral minimal partitions of unbounded domains

摘要： 我们研究在$d$维空间中构造$k$-谱最小分割的问题，其中要最小化的能量泛函是适当薛定谔算子$-\Delta +V$的谱的下确界的$p$-范数 ($1 \le p \le \infty$)，在分割单元（细胞）的边界上具有狄利克雷条件。本文的主要创新之处在于，这些区域可以是无界的，包括体积无限的区域。 首先，我们通过一个涉及整个区域上薛定谔算子本质谱的下确界$\Sigma$以及所有$k-1$分割的下界能量的阈值，证明了所有$k$分割中最小能量的精确上界。在该阈值严格以下，我们发展了一种类似集中紧性原理的论证，证明最优分割存在，并且每个单元都具有基态（即每个单元上的谱的下确界是一个简单孤立特征值）。其次，对于$p<\infty$，当能量和阈值水平相同时，我们表明可能存在或不存在最小分割。此外，即使存在，它们也可能不具有基态。第三，对于$p=\infty$，最小分割总是存在，甚至在阈值水平上，但这些分割可能具有或不具有基态。此外，在阈值以下，我们可以始终构造一个最小分割，这是一个等分割。在阈值处，我们证明谱最小分割不一定需要是等分割。我们给出了各种域和势的例子，以说明在此设置中出现的新现象。 显示英文摘要

摘要： We study the problem of constructing $k$-spectral minimal partitions of domains in $d$ dimensions, where the energy functional to be minimized is a $p$-norm ($1 \le p \le \infty$) of the infimum of the spectrum of a suitable Schr\"odinger operator $-\Delta +V$, with Dirichlet conditions on the boundary of the partition elements (cells). The main novelty of this paper is that the domains may be unbounded, including of infinite volume. First, we prove a sharp upper bound for the infimal energy among all $k$-partitions by a threshold value which involves the infimum $\Sigma$ of the essential spectrum of the Schr\"odinger operator on the whole domain as well as the infimal energy among all $k-1$-partitions. Strictly below such threshold, we develop a concentration-compactness-type argument showing optimal partitions exist, and each cell admits ground states (i.e., the infimum of the spectrum on each cell is a simple isolated eigenvalue). Second, for $p<\infty$, when the energy and the threshold level coincide, we show there may or may not be minimizing partitions. Moreover, even when these exist, they may not have ground states. Third, for $p=\infty$, minimal partitions always exist, even at the threshold level, but these may or may not admit ground states. Moreover, below the threshold, we can always construct a minimizer, which is an equipartition. At the threshold value we show that spectral minimal partitions may not need to be equipartitions. We give a variety of examples of both domains and potentials to illustrate the new phenomena that occur in this setting.