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Dimension reduction for functionals on solenoidal vector fields

Published online by Cambridge University Press:  02 December 2010

Stefan Krömer
Stefan Krömer*
Affiliation:
Universität zu Köln, Köln, Germany. skroemer@math.uni-koeln.de
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Abstract

We study integral functionals constrained to divergence-free vector fields in Lp on a thin domain, under standard p-growth and coercivity assumptions, 1 < p < ∞. We prove that as the thickness of the domain goes to zero, the Gamma-limit with respect to weak convergence in Lp is always given by the associated functional with convexified energy density wherever it is finite. Remarkably, this happens despite the fact that relaxation of nonconvex functionals subject to the limiting constraint can give rise to a nonlocal functional as illustrated in an example.

Keywords

Divergence-free fieldsGamma-convergencedimension reduction

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Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 1 , January 2012 , pp. 259 - 276
Copyright
© EDP Sciences, SMAI, 2010

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