On a Volume‐Constrained Variational Problem

Abstract

. Existence of minimizers for a volume-constrained energy $E(u) := \int_{\Omega} W(\nabla u)\, dx$ where ${\cal L}^N(\{u = z_i\}) = \alpha_i, i = 1, \ldots, P,$ is proved for the case in which $z_i$ are extremal points of a compact, convex set in $\Bbb R^d$ 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 Γ‐limit as the sum of the measures of the 2 phases tends to $\L(\Omega)$ is identified. Minimizers are fully characterized when $N=1$, and candidates for solutions are studied for the circle and the square in the plane.