Limiting domain wall energy for a problem related to micromagnetics. (English) Zbl 1031.35142

Rivière, Tristan; Serfaty, Sylvia

Commun. Pure Appl. Math. 54, No. 3, 294-338 (2001).

This paper concerns the asymptotic limit of a family of functionals related to the theory of micromagnetics in two dimensions. A compactness result for families of uniformly bounded energy is proved. The authors prove that the minimal wall energy is twice the perimeter.

Reviewer: Laura-Iulia Aniţa (Iaşi)

MSC:

35Q60	PDEs in connection with optics and electromagnetic theory
82D40	Statistical mechanics of magnetic materials
35A15	Variational methods applied to PDEs
35J35	Variational methods for higher-order elliptic equations

Keywords:

asymptotic limit; wall energy; eikonal equation; micromagnetics

References:

[1]	Ambrosio, Calc Var Partial Differential Equations 9 pp 327– (1999) · Zbl 0960.49013
[2]	; A mathematical problem related to the physical theory of liquid crystal configurations. Miniconference on geometry and partial differential equations, 2 (Canberra, 1986), 1-16. Proc Centre Math Anal Austral Nat Univ, 12. Australian National University, Canberra, 1987.
[3]	Aviles, Proc Roy Soc Edinburgh Sect A 129 pp 1– (1999) · Zbl 0923.49008
[4]	; ; Ginzburg-Landau vortices. Progress in Nonlinear Differential Equations and Their Applications, 13. Birkhäuser, Boston, 1994.
[5]	Bethuel, J Funct Anal 80 pp 60– (1988) · Zbl 0657.46027
[6]	Inversion de systèmes linéaires pour la simulation des matériaux ferromagnétiques. Singularités d’une configuration d’aimantation. Doctoral dissertation, Université Joseph Fourier, Grenoble, 1996.
[7]	; Differential forms in algebraic topology. Graduate Texts in Mathematics, 82. Springer, New York-Berlin, 1982. · Zbl 0496.55001
[8]	Carbou, C R Acad Sci Paris Sér I Math 314 pp 359– (1992)
[9]	Carbou, Calc Var Partial Differential Equations 5 pp 409– (1997) · Zbl 0889.58022
[10]	Choi, Pacific J Math 181 pp 57– (1997) · Zbl 0885.53004
[11]	; ; ; Magnetic microstructures, a paradigm of multiscale problems. Preprint, 1999.
[12]	DeSimone, Proc Roy Soc Edinburgh, Sec A
[13]	Hardt, Comm Partial Differential Equations 25 pp 1235– (2000) · Zbl 0958.35136
[14]	; Magnetic domains: the analysis of magnetic microstructures. Springer, Berlin-New York, 1998.
[15]	Jabin, C R Acad Sci Paris Sér I Math 331 pp 441– (2000) · Zbl 0965.35159
[16]	; Compactness in Ginzburg-Landau energy by kinetic averaging. To appear. · Zbl 1124.35312
[17]	James, Contin Mech Thermodyn 2 pp 215– (1990)
[18]	Jin, J Nonlinear Sci 10 pp 355– (2000) · Zbl 0973.49009
[19]	Murat, Ann Scuola Norm Sup Pisa Cl Sci (4) 5 pp 489– (1978)
[20]	Murat, J Math Pures Appl (9) 60 pp 309– (1981)
[21]	Compensated compactness and applications to partial differential equations. Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, 136-212. Res Notes in Math, 39. Pitman, Boston-London, 1979.
[22]	van den Berg, J Appl Phys 60 pp 1104– (1986)