On a volume-constrained variational problem

Existence of minimizers for a volume-constrained energy E(u) := integral(Omega) W(del u)dx where L-N({mu = Z(i)}) = alpha(i), i = 1,..., P, is proved for the case in which 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.