A nonlocal singular perturbation problem with periodic well potential

Matthias Kurzke

ESAIM: Control, Optimisation and Calculus of Variations (2005)

  • 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 Lp 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 (2005): 52-63. <http://eudml.org/doc/90790>.

@article{Kurzke2005,
abstract = { 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 Lp 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},
language = {eng},
month = {12},
number = {1},
pages = {52-63},
publisher = {EDP Sciences},
title = {A nonlocal singular perturbation problem with periodic well potential},
url = {http://eudml.org/doc/90790},
volume = {12},
year = {2005},
}

TY - JOUR
AU - Kurzke, Matthias
TI - A nonlocal singular perturbation problem with periodic well potential
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2005/12//
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 Γ-convergence theorem and show compactness up to translation in all Lp 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
UR - http://eudml.org/doc/90790
ER -

References

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  1. G. Alberti, G. Bouchitté and P. Seppecher, Un résultat de perturbations singulières avec la norme H 1 / 2 . C. R. Acad. Sci. Paris Sér. I Math.319 (1994) 333–338.  
  2. G. Alberti, G. Bouchitté and P. Seppecher, Phase transition with the line-tension effect. Arch. Rational Mech. Anal.144 (1998) 1–46.  Zbl0915.76093
  3. A. Garroni and S. Müller, A variational model for dislocations in the line-tension limit. Preprint 76, Max Planck Institute for Mathematics in the Sciences (2004).  Zbl1158.74365
  4. A.M. Garsia and E. Rodemich, Monotonicity of certain functionals under rearrangement. Ann. Inst. Fourier (Grenoble) 24 (1974) VI 67–116.  Zbl0274.26006
  5. R.V. Kohn and V.V. Slastikov, Another thin-film limit of micromagnetics. Arch. Rat. Mech. Anal., to appear.  Zbl1074.78012
  6. M. Kurzke, Analysis of boundary vortices in thin magnetic films. Ph.D. Thesis, Universität Leipzig (2004).  Zbl1151.35006
  7. E.H. Lieb and M. Loss, Analysis, second edition, Graduate Studies in Mathematics14 (2001).  
  8. L. Modica, The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal.98 (1987) 123–142.  Zbl0616.76004
  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.  
  10. J.C.C. Nitsche, Vorlesungen über Minimalflächen. Grundlehren der mathematischen Wissenschaften199 (1975).  
  11. P. Pedregal, Parametrized measures and variational principles, Progre. Nonlinear Differ. Equ. Appl.30 (1997).  
  12. C. Pommerenke, Boundary behaviour of conformal maps. Grundlehren der mathematischen Wissenschaften299 (1992).  Zbl0762.30001
  13. M.E. Taylor, Partial differential equations. III, Appl. Math. Sci.117 (1997).  
  14. J.F. Toland, Stokes waves in Hardy spaces and as distributions. J. Math. Pures Appl.ic> 79 (2000) 901–917.  Zbl0976.35052

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