A nonlocal singular perturbation problem with periodic well potential

Matthias Kurzke

ESAIM: Control, Optimisation and Calculus of Variations (2006)

  • Volume: 12, Issue: 1, page 52-63
  • ISSN: 1292-8119

Abstract

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For a one-dimensional nonlocal nonconvex singular perturbation problem with a noncoercive periodic well potential, we prove a Γ -convergence theorem and show compactness up to translation in all L p and the optimal Orlicz space for sequences of bounded energy. This generalizes work of Alberti, Bouchitté and Seppecher (1994) for the coercive two-well case. The theorem has applications to a certain thin-film limit of the micromagnetic energy.

How to cite

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Kurzke, Matthias. "A nonlocal singular perturbation problem with periodic well potential." ESAIM: Control, Optimisation and Calculus of Variations 12.1 (2006): 52-63. <http://eudml.org/doc/245421>.

@article{Kurzke2006,
abstract = {For a one-dimensional nonlocal nonconvex singular perturbation problem with a noncoercive periodic well potential, we prove a $\Gamma $-convergence theorem and show compactness up to translation in all $L^p$ and the optimal Orlicz space for sequences of bounded energy. This generalizes work of Alberti, Bouchitté and Seppecher (1994) for the coercive two-well case. The theorem has applications to a certain thin-film limit of the micromagnetic energy.},
author = {Kurzke, Matthias},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {gamma-convergence; nonlocal variational problem; micromagnetism; Gamma-convergence},
language = {eng},
number = {1},
pages = {52-63},
publisher = {EDP-Sciences},
title = {A nonlocal singular perturbation problem with periodic well potential},
url = {http://eudml.org/doc/245421},
volume = {12},
year = {2006},
}

TY - JOUR
AU - Kurzke, Matthias
TI - A nonlocal singular perturbation problem with periodic well potential
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2006
PB - EDP-Sciences
VL - 12
IS - 1
SP - 52
EP - 63
AB - For a one-dimensional nonlocal nonconvex singular perturbation problem with a noncoercive periodic well potential, we prove a $\Gamma $-convergence theorem and show compactness up to translation in all $L^p$ and the optimal Orlicz space for sequences of bounded energy. This generalizes work of Alberti, Bouchitté and Seppecher (1994) for the coercive two-well case. The theorem has applications to a certain thin-film limit of the micromagnetic energy.
LA - eng
KW - gamma-convergence; nonlocal variational problem; micromagnetism; Gamma-convergence
UR - http://eudml.org/doc/245421
ER -

References

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  9. [9] S. Müller, Variational models for microstructure and phase transitions, in Calculus of variations and geometric evolution problems (Cetraro, 1996), Springer, Berlin. Lect. Notes Math. 1713 (1999) 85–210. Zbl0968.74050
  10. [10] J.C.C. Nitsche, Vorlesungen über Minimalflächen. Grundlehren der mathematischen Wissenschaften 199 (1975). Zbl0319.53003MR448224
  11. [11] P. Pedregal, Parametrized measures and variational principles, Progre. Nonlinear Differ. Equ. Appl. 30 (1997). Zbl0879.49017MR1452107
  12. [12] C. Pommerenke, Boundary behaviour of conformal maps. Grundlehren der mathematischen Wissenschaften 299 (1992). Zbl0762.30001MR1217706
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