For a one-dimensional nonlocal nonconvex singular perturbation problem with a noncoercive periodic well potential, we prove a -convergence theorem and show compactness up to translation in all 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.
Mots-clés : gamma-convergence, nonlocal variational problem, micromagnetism
@article{COCV_2006__12_1_52_0,
author = {Kurzke, Matthias},
title = {A nonlocal singular perturbation problem with periodic well potential},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {52--63},
publisher = {EDP-Sciences},
volume = {12},
number = {1},
year = {2006},
doi = {10.1051/cocv:2005037},
mrnumber = {2192068},
zbl = {1107.49016},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2005037/}
}
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 SP - 52 EP - 63 VL - 12 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2005037/ DO - 10.1051/cocv:2005037 LA - en ID - COCV_2006__12_1_52_0 ER -
%0 Journal Article %A Kurzke, Matthias %T A nonlocal singular perturbation problem with periodic well potential %J ESAIM: Control, Optimisation and Calculus of Variations %D 2006 %P 52-63 %V 12 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2005037/ %R 10.1051/cocv:2005037 %G en %F COCV_2006__12_1_52_0
Kurzke, Matthias. A nonlocal singular perturbation problem with periodic well potential. ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 1, pp. 52-63. doi : 10.1051/cocv:2005037. http://www.numdam.org/articles/10.1051/cocv:2005037/
[1] , and, Un résultat de perturbations singulières avec la norme . C. R. Acad. Sci. Paris Sér. I Math. 319 (1994) 333-338. | Zbl
[2] , and, Phase transition with the line-tension effect. Arch. Rational Mech. Anal. 144 (1998) 1-46. | Zbl
[3] and, A variational model for dislocations in the line-tension limit. Preprint 76, Max Planck Institute for Mathematics in the Sciences (2004).
[4] and, Monotonicity of certain functionals under rearrangement. Ann. Inst. Fourier (Grenoble) 24 (1974) VI 67-116. | Numdam | Zbl
[5] and, Another thin-film limit of micromagnetics. Arch. Rat. Mech. Anal., to appear. | MR | Zbl
[6] , Analysis of boundary vortices in thin magnetic films. Ph.D. Thesis, Universität Leipzig (2004).
[7] and, Analysis, second edition, Graduate Studies in Mathematics 14 (2001). | MR | Zbl
[8] , The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal. 98 (1987) 123-142. | Zbl
[9] , 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. | Zbl
[10] , Vorlesungen über Minimalflächen. Grundlehren der mathematischen Wissenschaften 199 (1975). | MR | Zbl
[11] , Parametrized measures and variational principles, Progre. Nonlinear Differ. Equ. Appl. 30 (1997). | MR | Zbl
[12] , Boundary behaviour of conformal maps. Grundlehren der mathematischen Wissenschaften 299 (1992). | MR | Zbl
[13] , Partial differential equations. III, Appl. Math. Sci. 117 (1997).
[14] , Stokes waves in Hardy spaces and as distributions. J. Math. Pures Appl. 79 (2000) 901-917. | Zbl
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