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 of Rn$\Omega \subset \mathbb {R}<^>n$ into N subsets Wj$W_j$ of locally finite perimeter in omega, whose main term is n-ary sumation j=1N integral omega boolean AND partial differential Wja(x)dHn-1(x)$\sum _{j=1}<^>N \int _{\Omega \cap \partial W_j} a(x) d\mathcal {H}<^>{n-1}(x)$. Here the positive bounded function a may, for instance, be related to the Landscape function of some Schrodinger 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 boolean AND partial differential Wj$\Omega \cap \partial W_j$.