An optimal partition problem for the localization of eigenfunctions

We study the minimizers of a functional on the set of partitions of a domain $\mathrm{\Omega }\subset R^n$ into $N$ subsets $W_j$ of locally finite perimeter in $\mathrm{\Omega }$, whose main term is $\sum_{j=1^N} \int_{\Omega \cap \partial W_j} a(x) dH^{n–1}(x)$\mathrm{.}Herethepositiveboundedfunction$a$ 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 $\mathrm{\Omega }\cap \mathrm{\partial }{W}_j$.