An optimal partition problem for the localization of eigenfunctions

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 $\Omega \cap \partial W_j$.