ESAIM: Control, Optimisation and Calculus of Variations (ESAIM: COCV)

ESAIM: COCV, Vol. 13, N°4, pp. 809-828
DOI: 10.1051/cocv:2007041

$\Gamma$ -convergence of functionals on divergence-free fields

Nadia Ansini¹ and Adriana Garroni²

¹ Section de Mathématiques, EPFL, 1015 Lausanne, Switzerland; nadia.ansini@epfl.ch
² Dip. di Matematica, Univ. di Roma `La Sapienza', P.le A. Moro 2, 00185 Rome, Italy; garroni@mat.uniroma1.it

(Received January 18, 2006. Published online September 5, 2007.)

Abstract
We study the stability of a sequence of integral functionals on divergence-free matrix valued fields following the direct methods of $\Gamma$ -convergence. We prove that the $\Gamma$ -limit is an integral functional on divergence-free matrix valued fields. Moreover, we show that the $\Gamma$ -limit is also stable under volume constraint and various type of boundary conditions.

Mathematics Subject Classification. 35E99, 35J99, 49J45

Key words: ${\cal A}$ -quasiconvexity, divergence-free fields, $\Gamma$ -convergence, homogenization

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