MiS Preprint Repository

Existence of minimizers for a volume constrained energy
$E(u):={\int }_{a}W(\mathrm{\nabla }u)dx$
where $L^N(\{u=z_i\})=a_i,i=1,...,P,$ is proved in the case where $z_i$ are extremal points of a compact, convex set in $R^d$ and under suitable assumptions on a class of quasiconvex energy densities W. Optimality properties are studied in the scalar-valued problem where d=1, P=2, $W(\xi )=|\xi {|}^2$, and the $\mathrm{\Gamma }$-limit as the sum of the measures of the 2 phases tends to $L^N(\mathrm{\Omega })$ is identified. Minimizers are fully characterized when N=1, and candidates for solutions are studied for the circle and the square in the plane.