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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.
@article{COCV_2012__18_1_259_0,
author = {Kr\"omer, Stefan},
title = {Dimension reduction for functionals on solenoidal vector fields},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {259--276},
publisher = {EDP-Sciences},
volume = {18},
number = {1},
year = {2012},
doi = {10.1051/cocv/2010051},
mrnumber = {2887935},
zbl = {1251.49018},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2010051/}
}
TY - JOUR AU - Krömer, Stefan TI - Dimension reduction for functionals on solenoidal vector fields JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2012 SP - 259 EP - 276 VL - 18 IS - 1 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv/2010051/ DO - 10.1051/cocv/2010051 LA - en ID - COCV_2012__18_1_259_0 ER -
%0 Journal Article %A Krömer, Stefan %T Dimension reduction for functionals on solenoidal vector fields %J ESAIM: Control, Optimisation and Calculus of Variations %D 2012 %P 259-276 %V 18 %N 1 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv/2010051/ %R 10.1051/cocv/2010051 %G en %F COCV_2012__18_1_259_0
Krömer, Stefan. Dimension reduction for functionals on solenoidal vector fields. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 1, pp. 259-276. doi: 10.1051/cocv/2010051
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