# A nonlocal singular perturbation problem with periodic well potential

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

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

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topKurzke, 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

top- [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. Zbl0845.49008
- [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. Zbl1074.78012MR2186425
- [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 Mathematics 14 (2001). Zbl0966.26002MR1817225
- [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. Zbl0968.74050
- [10] J.C.C. Nitsche, Vorlesungen über Minimalflächen. Grundlehren der mathematischen Wissenschaften 199 (1975). Zbl0319.53003MR448224
- [11] P. Pedregal, Parametrized measures and variational principles, Progre. Nonlinear Differ. Equ. Appl. 30 (1997). Zbl0879.49017MR1452107
- [12] C. Pommerenke, Boundary behaviour of conformal maps. Grundlehren der mathematischen Wissenschaften 299 (1992). Zbl0762.30001MR1217706
- [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. 79 (2000) 901–917. Zbl0976.35052

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