Sharp upper bounds for a singular perturbation problem related to micromagnetics

Poliakovsky, Arkady

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 4, pp. 673-701

We construct an upper bound for the following family of functionals ${\left\{E_{\epsilon }\right\}}_{\epsilon >0}$, which arises in the study of micromagnetics:

$$E_{\epsilon }\left(u\right)={\int }_{\Omega }{\epsilon \left|\nabla u\right|}^2+\frac{1}{\epsilon }{\int }_{{ℝ}^2}{\left|H_u\right|}^2.$$

Here $\Omega$ is a bounded domain in ${ℝ}^2$, $u\in H^1(\Omega ,S^1)$ (corresponding to the magnetization) and $H_u$, the demagnetizing field created by $u$, is given by

$$\left\{\begin{array}{cc}\mathrm{div}(\tilde{u}+H_u)=0\hfill & \text{in}{ℝ}^2,\hfill \\ \mathrm{curl}{H}_u=0\hfill & \text{in}{ℝ}^2,\hfill \end{array}\right.$$

where $\tilde{u}$ is the extension of $u$ by $0$ in ${ℝ}^2\setminus \Omega$. Our upper bound coincides with the lower bound obtained by Rivière and Serfaty.

Classification : 49J45, 35B25, 35J20

[1] L. Ambrosio, C. De Lellis and C. Mantegazza, Line energies for gradient vector fields in the plane, Calc. Var. Partial Differential Equations 9 (1999), 327-355. | Zbl | MR

[2] L. Ambrosio, N. Fusco and D. Pallara, “Functions of Bounded Variation and Free Discontinuity Problems”, Oxford Mathematical Monographs, Oxford University Press, New York, 2000. | Zbl | MR

[3] P. Aviles and Y. Giga, A mathematical problem related to the physical theory of liquid crystal configurations, Proc. Centre Math. Anal. Austral. Nat. Univ. 12 (1987), 1-16. | MR

[4] P. Aviles and Y. Giga, On lower semicontinuity of a defect energy obtained by a singular limit of the Ginzburg-Landau type energy for gradient fields, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), 1-17. | Zbl | MR

[5] S. Conti and C. De Lellis, Sharp upper bounds for a variational problem with singular perturbation, Math. Ann. 338 (2007), 119-146. | Zbl | MR

[6] J. Dávila and R. Ignat, Lifting of $BV$ functions with values in $S^1$, C. R. Math. Acad. Sci. Paris 337 (2003) 159-164. | Zbl | MR

[7] C. De Lellis, An example in the gradient theory of phase transitions ESAIM Control Optim. Calc. Var. 7 (2002), 285-289 (electronic). | Zbl | MR | Numdam

[8] A. Desimone, S. Müller, R. V. Kohn and F. Otto, A compactness result in the gradient theory of phase transitions, Proc. Roy. Soc. Edinburgh Sect. A 131 (2001), 833-844. | Zbl | MR

[9] N. M. Ercolani, R. Indik, A. C. Newell and T. Passot, The geometry of the phase diffusion equation, J. Nonlinear Sci. 10 (2000), 223-274. | Zbl | MR

[10] L. C. Evans, “Partial Differential Equations”, Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, 1998. | Zbl | MR

[11] L. C. Evans and R. F. Gariepy, “Measure Theory and Fine Properties of Functions”, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. | Zbl | MR

[12] D. Gilbarg and N. Trudinger, “Elliptic Partial Differential Equations of Elliptic Type”, 2nd ed., Springer-Verlag, Berlin-Heidelberg, 1983. | Zbl | MR

[13] E. Giusti, “Minimal Surfaces and Functions of Bounded Variation”, Monographs in Mathematics, Vol. 80, Birkhäuser Verlag, Basel, 1984. | Zbl | MR

[14] W. Jin and R.V. Kohn, Singular perturbation and the energy of folds, J. Nonlinear Sci. 10 (2000), 355-390. | Zbl | MR

[15] A. Poliakovsky, A method for establishing upper bounds for singular perturbation problems, C. R. Math. Acad. Sci. Paris 341 (2005), 97-102. | Zbl | MR

[16] A. Poliakovsky, Upper bounds for singular perturbation problems involving gradient fields, J. Eur. Math. Soc. 9 (2007), 1-43. | Zbl | MR

[17] A. Poliakovsky, A general technique to prove upper bounds for singular perturbation problems, submitted to Journal d'Analyse. | Zbl

[18] T. Rivière and S. Serfaty, Limiting domain wall energy for a problem related to micromagnetics, Comm. Pure Appl. Math. 54 (2001), 294-338. | Zbl | MR

[19] T. Rivière and S. Serfaty, Compactness, kinetic formulation and entropies for a problem related to mocromagnetics, Comm. Partial Differential Equations 28 (2003), 249-269. | Zbl | MR

[20] A. I. Volpert and S. I. Hudjaev, “Analysis in Classes of Discontinuous Functions and Equations of Mathematical Physics”, Martinus Nijhoff Publishers, Dordrecht, 1985. | Zbl | MR