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

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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

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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

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