An optimal partition problem for the localization of eigenfunctions

TL;DR

This paper investigates an optimal partition problem for localizing eigenfunctions, establishing the existence and regularity of minimizers for a functional related to Schrödinger operators.

Contribution

It proves the existence of minimizers and their boundary regularity for a partitioning functional connected to eigenfunction localization.

Findings

Existence of minimizers for the partition functional.

Boundaries of minimizers are locally Ahlfors regular.

Boundaries are uniformly rectifiable.

Abstract

We study the minimizers of a functional on the set of partitions of a domain $\Omega \subset R^n$ into $N$ subsets $W_j$ of locally finite perimeter in $\Omega$, whose main term is $\sum_{j=1^N} \int_{\Omega \cap \partial W_j} a(x) dH^{n--1}(x)$. Here the positive bounded function $a$ may for instance be related to the Landscape function of some Schr{\"o}dinger operator. We prove the existence of minimizers through the equivalence with a weak formulation, and the local Ahlfors regularity and uniform rectifiability of the boundaries $\Omega \cap \partial W_j$.