On a Volume Constrained Variational Problem
Luigi Ambrosio, Irene Fonseca, Paolo Marcellini, and Luc Tartar
Abstract
Existence of minimizers for a volume constrained energy
where
is proved for the case in which 
are extremal points of a compact, convex set
in
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, 
,
and the 
-limit as the sum of the measures of the 2 phases
tends to 
is identified. Minimizers are fully characterized when
N=1, and candidates for solutions are studied for the circle and
the square in the plane.
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