  This paper is devoted to the analysis of multiphase shape optimization
problems, which can formally be written as
$\min\Big\{{g}(F_1(\Omega_1),\dots,F_h(\Omega_h))+
m\vert\,\bigcup_{i=1}^h\Omega_i\vert :\ \Omega_i\subset D,\ \Omega_i\cap
\Omega_j =\emptyset\Big\},$ where $D\subset\mathcal{R}^d$ is a given bounded
open set, $\vert\Omega_i\vert$ is the Lebesgue measure of $\Omega_i$ and $m$ is
a positive constant. For a large class of such functionals, we analyse
qualitative properties of the cells $\Omega_i$ and the interaction between
them. Each cell is itself subsolution for a (single phase) shape optimization
problem, from which we deduce properties like finite perimeter, inner density,
separation by open sets, absence of triple junction points, etc.
  As main examples we consider functionals involving the eigenvalues of the
Dirichlet Laplacian of each cell, i.e. $F_i=\lambda_{k_i}$.
