Dimension reduction for functionals on solenoidal vector fields

Stefan Krömer

ESAIM: Control, Optimisation and Calculus of Variations (2012)

  • Volume: 18, Issue: 1, page 259-276
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Krömer, Stefan. "Dimension reduction for functionals on solenoidal vector fields." ESAIM: Control, Optimisation and Calculus of Variations 18.1 (2012): 259-276. <http://eudml.org/doc/221928>.

@article{Krömer2012,
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. },
author = {Krömer, Stefan},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Divergence-free fields; Gamma-convergence; dimension reduction; divergence-free fields; -convergence},
language = {eng},
month = {2},
number = {1},
pages = {259-276},
publisher = {EDP Sciences},
title = {Dimension reduction for functionals on solenoidal vector fields},
url = {http://eudml.org/doc/221928},
volume = {18},
year = {2012},
}

TY - JOUR
AU - Krömer, Stefan
TI - Dimension reduction for functionals on solenoidal vector fields
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2012/2//
PB - EDP Sciences
VL - 18
IS - 1
SP - 259
EP - 276
AB - 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.
LA - eng
KW - Divergence-free fields; Gamma-convergence; dimension reduction; divergence-free fields; -convergence
UR - http://eudml.org/doc/221928
ER -

References

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