# A compactness result in thin-film micromagnetics and the optimality of the Néel wall

Journal of the European Mathematical Society (2008)

- Volume: 010, Issue: 4, page 909-956
- ISSN: 1435-9855

## Access Full Article

top## Abstract

top## How to cite

topIgnat, Radu, and Otto, Felix. "A compactness result in thin-film micromagnetics and the optimality of the Néel wall." Journal of the European Mathematical Society 010.4 (2008): 909-956. <http://eudml.org/doc/277514>.

@article{Ignat2008,

abstract = {In this paper, we study a model for the magnetization in thin ferromagnetic films. It comes as a variational problem for $S^1$-valued maps $m^\{\prime \}$ (the magnetization) of two variables $x^\{\prime \}$:
$E_\varepsilon (m^\{\prime \})=\varepsilon \int |\nabla ^\{\prime \}\cdot m^\{\prime \}|^2dx^\{\prime \}+\frac\{1\}\{2\}\int \left||\nabla ^\{\prime \}|^\{-1/2\}\nabla ^\{\prime \}\cdot m^\{\prime \}\right|^2dx^\{\prime \}$. We are interested in the behavior of minimizers as $\varepsilon \rightarrow 0$. They are expected to be $S^1$-valued maps $m^\{\prime \}$ of vanishing distributional divergence $\nabla ^\{\prime \}\cdot m^\{\prime \}=0$, so that appropriate boundary conditions
enforce line discontinuities. For finite $\varepsilon >0$, these line discontinuities are approximated by smooth transition layers, the so-called Néel walls. Néel walls have a line energy density of the order $1/|\ln \varepsilon |$. One of the main results is to show that the boundedness of $\lbrace |\ln \varepsilon |E_\varepsilon (m^\{\prime \}_\varepsilon )\rbrace $ implies the compactness of $\lbrace m^\{\prime \}_\varepsilon \rbrace _\{\varepsilon \downarrow 0\}$, so that indeed limits $m^\{\prime \}$ will be $S^1$-valued and weakly divergence-free. Moreover, we show the optimality of the $1$-d Néel wall under $2$-d perturbations as $\varepsilon \downarrow 0$.},

author = {Ignat, Radu, Otto, Felix},

journal = {Journal of the European Mathematical Society},

keywords = {micromagnetics; Néel wall; compactness; principle of characteristics; micromagnetics; Néel wall; compactness; principle of characteristics},

language = {eng},

number = {4},

pages = {909-956},

publisher = {European Mathematical Society Publishing House},

title = {A compactness result in thin-film micromagnetics and the optimality of the Néel wall},

url = {http://eudml.org/doc/277514},

volume = {010},

year = {2008},

}

TY - JOUR

AU - Ignat, Radu

AU - Otto, Felix

TI - A compactness result in thin-film micromagnetics and the optimality of the Néel wall

JO - Journal of the European Mathematical Society

PY - 2008

PB - European Mathematical Society Publishing House

VL - 010

IS - 4

SP - 909

EP - 956

AB - In this paper, we study a model for the magnetization in thin ferromagnetic films. It comes as a variational problem for $S^1$-valued maps $m^{\prime }$ (the magnetization) of two variables $x^{\prime }$:
$E_\varepsilon (m^{\prime })=\varepsilon \int |\nabla ^{\prime }\cdot m^{\prime }|^2dx^{\prime }+\frac{1}{2}\int \left||\nabla ^{\prime }|^{-1/2}\nabla ^{\prime }\cdot m^{\prime }\right|^2dx^{\prime }$. We are interested in the behavior of minimizers as $\varepsilon \rightarrow 0$. They are expected to be $S^1$-valued maps $m^{\prime }$ of vanishing distributional divergence $\nabla ^{\prime }\cdot m^{\prime }=0$, so that appropriate boundary conditions
enforce line discontinuities. For finite $\varepsilon >0$, these line discontinuities are approximated by smooth transition layers, the so-called Néel walls. Néel walls have a line energy density of the order $1/|\ln \varepsilon |$. One of the main results is to show that the boundedness of $\lbrace |\ln \varepsilon |E_\varepsilon (m^{\prime }_\varepsilon )\rbrace $ implies the compactness of $\lbrace m^{\prime }_\varepsilon \rbrace _{\varepsilon \downarrow 0}$, so that indeed limits $m^{\prime }$ will be $S^1$-valued and weakly divergence-free. Moreover, we show the optimality of the $1$-d Néel wall under $2$-d perturbations as $\varepsilon \downarrow 0$.

LA - eng

KW - micromagnetics; Néel wall; compactness; principle of characteristics; micromagnetics; Néel wall; compactness; principle of characteristics

UR - http://eudml.org/doc/277514

ER -

## Citations in EuDML Documents

top- Radu Ignat, A survey of some new results in ferromagnetic thin films
- Andrew Lorent, A simple proof of the characterization of functions of low Aviles Giga energy on a ball via regularity
- Andrew Lorent, A simple proof of the characterization of functions of low Aviles Giga energy on a ball regularity

## NotesEmbed ?

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