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MiS Preprint
31/1998
On a volume constrained variational problem
Luigi Ambrosio, Irene Fonseca, Paolo Marcellini and Luc Tartar
Abstract
Existence of minimizers for a volume constrained energy $E(u):= \int_a W(\nabla u)dx$ 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 $\Gamma$-limit as the sum of the measures of the 2 phases tends to $L^N(\Omega)$ is identified. Minimizers are fully characterized when N=1, and candidates for solutions are studied for the circle and the square in the plane.
