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

Ignat, 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 -