The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

Sharp upper bounds for a singular perturbation problem related to micromagnetics

Arkady Poliakovsky

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2007)

  • Volume: 6, Issue: 4, page 673-701
  • ISSN: 0391-173X

Abstract

top
We construct an upper bound for the following family of functionals { E ε } ε > 0 , which arises in the study of micromagnetics: E ε ( u ) = ∫ Ω ε | ∇ u | 2 + 1 ε ∫ ℝ 2 | H u | 2 . Here Ω is a bounded domain in ℝ 2 , u ∈ H 1 ( Ω , S 1 ) (corresponding to the magnetization) and H u , the demagnetizing field created by u , is given by div ( u ˜ + H u ) = 0 in ℝ 2 , curl H u = 0 in ℝ 2 , where u ˜ is the extension of u by 0 in ℝ 2 ∖ Ω . Our upper bound coincides with the lower bound obtained by Rivière and Serfaty.

How to cite

top

Poliakovsky, Arkady. "Sharp upper bounds for a singular perturbation problem related to micromagnetics." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 6.4 (2007): 673-701. <http://eudml.org/doc/272277>.

@article{Poliakovsky2007,
abstract = {We construct an upper bound for the following family of functionals $\lbrace E_\varepsilon \rbrace _\{\varepsilon &gt;0\}$, which arises in the study of micromagnetics:\[ E\_\varepsilon (u)=\int \_\Omega \varepsilon |\nabla u|^2+\frac\{1\}\{\varepsilon \}\int \_\{\mathbb \{R\}^2\}|H\_u|^2. \]Here $\Omega $ is a bounded domain in $\mathbb \{R\}^2$, $u\in H^1(\Omega ,S^1)$ (corresponding to the magnetization) and $H_u$, the demagnetizing field created by $u$, is given by\[ \{\left\lbrace \begin\{array\}\{ll\} \{\rm div\}\,(\tilde\{u\}+H\_u)=0\quad &\text\{in \}\mathbb \{R\}^2\,,\\ \{\rm curl\}\, H\_u=0\quad \quad \quad &\text\{ in \}\mathbb \{R\}^2\,, \end\{array\}\right.\} \]where $\tilde\{u\}$ is the extension of $u$ by $0$ in $\mathbb \{R\}^2\setminus \Omega $. Our upper bound coincides with the lower bound obtained by Rivière and Serfaty.},
author = {Poliakovsky, Arkady},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {micromagnetics; upper bounds},
language = {eng},
number = {4},
pages = {673-701},
publisher = {Scuola Normale Superiore, Pisa},
title = {Sharp upper bounds for a singular perturbation problem related to micromagnetics},
url = {http://eudml.org/doc/272277},
volume = {6},
year = {2007},
}

TY - JOUR
AU - Poliakovsky, Arkady
TI - Sharp upper bounds for a singular perturbation problem related to micromagnetics
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2007
PB - Scuola Normale Superiore, Pisa
VL - 6
IS - 4
SP - 673
EP - 701
AB - We construct an upper bound for the following family of functionals $\lbrace E_\varepsilon \rbrace _{\varepsilon &gt;0}$, which arises in the study of micromagnetics:\[ E_\varepsilon (u)=\int _\Omega \varepsilon |\nabla u|^2+\frac{1}{\varepsilon }\int _{\mathbb {R}^2}|H_u|^2. \]Here $\Omega $ is a bounded domain in $\mathbb {R}^2$, $u\in H^1(\Omega ,S^1)$ (corresponding to the magnetization) and $H_u$, the demagnetizing field created by $u$, is given by\[ {\left\lbrace \begin{array}{ll} {\rm div}\,(\tilde{u}+H_u)=0\quad &\text{in }\mathbb {R}^2\,,\\ {\rm curl}\, H_u=0\quad \quad \quad &\text{ in }\mathbb {R}^2\,, \end{array}\right.} \]where $\tilde{u}$ is the extension of $u$ by $0$ in $\mathbb {R}^2\setminus \Omega $. Our upper bound coincides with the lower bound obtained by Rivière and Serfaty.
LA - eng
KW - micromagnetics; upper bounds
UR - http://eudml.org/doc/272277
ER -

References

top
  1. [1] L. Ambrosio, C. De Lellis and C. Mantegazza, Line energies for gradient vector fields in the plane, Calc. Var. Partial Differential Equations9 (1999), 327–355. Zbl0960.49013MR1731470
  2. [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. Zbl0957.49001MR1857292
  3. [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. MR924423
  4. [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. A129 (1999), 1–17. Zbl0923.49008MR1669225
  5. [5] S. Conti and C. De Lellis, Sharp upper bounds for a variational problem with singular perturbation, Math. Ann.338 (2007), 119–146. Zbl1186.49004MR2295507
  6. [6] J. Dávila and R. Ignat, Lifting of B V functions with values in S 1 , C. R. Math. Acad. Sci. Paris337 (2003) 159–164. Zbl1046.46026MR2001127
  7. [7] C. De Lellis, An example in the gradient theory of phase transitions ESAIM Control Optim. Calc. Var. 7 (2002), 285–289 (electronic). Zbl1037.49010MR1925030
  8. [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. A131 (2001), 833–844. Zbl0986.49009MR1854999
  9. [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. Zbl0981.76087MR1743400
  10. [10] L. C. Evans, “Partial Differential Equations”, Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, 1998. Zbl0902.35002MR2597943
  11. [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. Zbl0804.28001MR1158660
  12. [12] D. Gilbarg and N. Trudinger, “Elliptic Partial Differential Equations of Elliptic Type”, 2nd ed., Springer-Verlag, Berlin-Heidelberg, 1983. Zbl0198.14101MR737190
  13. [13] E. Giusti, “Minimal Surfaces and Functions of Bounded Variation”, Monographs in Mathematics, Vol. 80, Birkhäuser Verlag, Basel, 1984. Zbl0545.49018MR775682
  14. [14] W. Jin and R.V. Kohn, Singular perturbation and the energy of folds, J. Nonlinear Sci.10 (2000), 355–390. Zbl0973.49009MR1752602
  15. [15] A. Poliakovsky, A method for establishing upper bounds for singular perturbation problems, C. R. Math. Acad. Sci. Paris341 (2005), 97–102. Zbl1068.49009MR2153964
  16. [16] A. Poliakovsky, Upper bounds for singular perturbation problems involving gradient fields, J. Eur. Math. Soc.9 (2007), 1–43. Zbl1241.49011MR2283101
  17. [17] A. Poliakovsky, A general technique to prove upper bounds for singular perturbation problems, submitted to Journal d’Analyse. Zbl1153.49018
  18. [18] T. Rivière and S. Serfaty, Limiting domain wall energy for a problem related to micromagnetics, Comm. Pure Appl. Math.54 (2001), 294–338. Zbl1031.35142MR1809740
  19. [19] T. Rivière and S. Serfaty, Compactness, kinetic formulation and entropies for a problem related to mocromagnetics, Comm. Partial Differential Equations28 (2003), 249–269. Zbl1094.35125MR1974456
  20. [20] A. I. Volpert and S. I. Hudjaev, “Analysis in Classes of Discontinuous Functions and Equations of Mathematical Physics”, Martinus Nijhoff Publishers, Dordrecht, 1985. Zbl0564.46025MR785938

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.