An optimal partition problem for the localization of eigenfunctions

We study the minimizers of a functional on set partitions domain $\Omega \subset R^n$ into $N$ subsets $W_j$ 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 positive bounded function $a$ may for instance be related to Landscape some Schr{\o}dinger operator. prove existence through equivalence with weak formulation, and local Ahlfors regularity uniform rectifiability boundaries W_j$.