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We have decided to discontinue the publication of preprints on our preprint server end of 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

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.

Received:
21.07.98
Published:
21.07.98
MSC Codes:
35A15, 35J65, 49J45, 49K20
Keywords:
volume constraints, free boundary problems

Related publications

inJournal
1999 Repository Open Access
Luigi Ambrosio, Irene Fonseca, Paolo Marcellini and Luc Tartar

On a volume-constrained variational problem

In: Archive for rational mechanics and analysis, 149 (1999) 1, pp. 23-47