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{\bf Dimension Reduction for Functionals on Solenoidal Vector Fields}\\

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Stefan Kr\"omer\\
Mathematisches Institut\\
Universitaet zu Koeln\\
D-50923 Koeln\\
Germany\\
skroemer@math.uni-koeln.de
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\noindent {\bf Abstract}: We study integral functionals constrained to
divergence-free vector fields in $L^p$ on a thin domain, under standard
$p$-growth and coercivity assumptions, $1<p<\infty$. We prove that as the
thickness of the domain goes to zero, the Gamma-limit with respect to weak
convergence in $L^p$ 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.

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