Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-28T13:42:50.508Z Has data issue: false hasContentIssue false
  • English
  • Français

A nonlocal singular perturbation problem with periodic well potential

Published online by Cambridge University Press:  15 December 2005

Matthias Kurzke
Matthias Kurzke*
Affiliation:
Institute for Mathematics and its Applications, University of Minnesota, 400 Lind Hall, 207 Church Street SE, Minneapolis, MN 55455, USA; kurzke@ima.umn.edu
Get access

Abstract

For a one-dimensional nonlocal nonconvex singular perturbation problemwith a noncoercive periodic well potential,we prove a Γ-convergence theorem and show compactnessup to translationin all Lp and the optimal Orlicz space for sequences of boundedenergy. 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 ofthe micromagnetic energy.

Keywords

Gamma-convergencenonlocal variational problemmicromagnetism
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 12 , Issue 1 , January 2006 , pp. 52 - 63
Copyright
© EDP Sciences, SMAI, 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alberti, G., Bouchitté, G. and Seppecher, P., Un résultat de perturbations singulières avec la norme $H\sp {1/2}$ . C. R. Acad. Sci. Paris Sér. I Math. 319 (1994) 333–338.
Alberti, G., Bouchitté, G. and Seppecher, P., Phase transition with the line-tension effect. Arch. Rational Mech. Anal. 144 (1998) 1–46. CrossRef
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).
A.M. Garsia and E. Rodemich, Monotonicity of certain functionals under rearrangement. Ann. Inst. Fourier (Grenoble) 24 (1974) VI 67–116.
R.V. Kohn and V.V. Slastikov, Another thin-film limit of micromagnetics. Arch. Rat. Mech. Anal., to appear.
M. Kurzke, Analysis of boundary vortices in thin magnetic films. Ph.D. Thesis, Universität Leipzig (2004).
E.H. Lieb and M. Loss, Analysis, second edition, Graduate Studies in Mathematics 14 (2001).
Modica, L., The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal. 98 (1987) 123–142. CrossRef
Müller, S., 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. CrossRef
J.C.C. Nitsche, Vorlesungen über Minimalflächen. Grundlehren der mathematischen Wissenschaften 199 (1975).
P. Pedregal, Parametrized measures and variational principles, Progre. Nonlinear Differ. Equ. Appl. 30 (1997).
C. Pommerenke, Boundary behaviour of conformal maps. Grundlehren der mathematischen Wissenschaften 299 (1992).
M.E. Taylor, Partial differential equations. III, Appl. Math. Sci. 117 (1997).
Toland, J.F., Stokes waves in Hardy spaces and as distributions. J. Math. Pures Appl. ic> 79 (2000) 901–917. CrossRef