A nonlocal singular perturbation problem with periodic well potential
ESAIM: Control, Optimisation and Calculus of Variations, Volume 12 (2006) no. 1, p. 52-63

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.

DOI : https://doi.org/10.1051/cocv:2005037
Classification:  49J45
Keywords: 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},
     publisher = {EDP-Sciences},
     volume = {12},
     number = {1},
     year = {2006},
     pages = {52-63},
     doi = {10.1051/cocv:2005037},
     zbl = {1107.49016},
     mrnumber = {2192068},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2006__12_1_52_0}
}
Kurzke, Matthias. A nonlocal singular perturbation problem with periodic well potential. ESAIM: Control, Optimisation and Calculus of Variations, Volume 12 (2006) no. 1, pp. 52-63. doi : 10.1051/cocv:2005037. http://www.numdam.org/item/COCV_2006__12_1_52_0/

[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. | Zbl 0845.49008

[2] G. Alberti, G. Bouchitté and P. Seppecher, Phase transition with the line-tension effect. Arch. Rational Mech. Anal. 144 (1998) 1-46. | Zbl 0915.76093

[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). | Zbl pre05051266

[4] A.M. Garsia and E. Rodemich, Monotonicity of certain functionals under rearrangement. Ann. Inst. Fourier (Grenoble) 24 (1974) VI 67-116. | Numdam | Zbl 0274.26006

[5] R.V. Kohn and V.V. Slastikov, Another thin-film limit of micromagnetics. Arch. Rat. Mech. Anal., to appear. | MR 2186425 | Zbl 1074.78012

[6] M. Kurzke, Analysis of boundary vortices in thin magnetic films. Ph.D. Thesis, Universität Leipzig (2004).

[7] E.H. Lieb and M. Loss, Analysis, second edition, Graduate Studies in Mathematics 14 (2001). | MR 1817225 | Zbl 0966.26002

[8] L. Modica, The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal. 98 (1987) 123-142. | Zbl 0616.76004

[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. | Zbl 0968.74050

[10] J.C.C. Nitsche, Vorlesungen über Minimalflächen. Grundlehren der mathematischen Wissenschaften 199 (1975). | MR 448224 | Zbl 0319.53003

[11] P. Pedregal, Parametrized measures and variational principles, Progre. Nonlinear Differ. Equ. Appl. 30 (1997). | MR 1452107 | Zbl 0879.49017

[12] C. Pommerenke, Boundary behaviour of conformal maps. Grundlehren der mathematischen Wissenschaften 299 (1992). | MR 1217706 | Zbl 0762.30001

[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. 79 (2000) 901-917. | Zbl 0976.35052