  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
$\Omega \cap \partial W_j$.
