Sharp upper bounds for a singular perturbation problem related to micromagnetics
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 4, p. 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: ${E}_{\epsilon }\left(u\right)={\int }_{\Omega }{\epsilon |\nabla u|}^{2}+\frac{1}{\epsilon }{\int }_{{ℝ}^{2}}{|{H}_{u}|}^{2}.$ Here $\Omega$ is a bounded domain in ${ℝ}^{2}$, $u\in {H}^{1}\left(\Omega ,{S}^{1}\right)$ (corresponding to the magnetization) and ${H}_{u}$, the demagnetizing field created by $u$, is given by $\left\{\begin{array}{cc}\mathrm{div}\phantom{\rule{0.166667em}{0ex}}\left(\stackrel{˜}{u}+{H}_{u}\right)=0\phantom{\rule{1em}{0ex}}\hfill & \text{in}\phantom{\rule{4pt}{0ex}}{ℝ}^{2}\phantom{\rule{0.166667em}{0ex}},\hfill \\ \mathrm{curl}\phantom{\rule{0.166667em}{0ex}}{H}_{u}=0\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\hfill & \phantom{\rule{4pt}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}{ℝ}^{2}\phantom{\rule{0.166667em}{0ex}},\hfill \end{array}\right\$ where $\stackrel{˜}{u}$ 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
@article{ASNSP_2007_5_6_4_673_0,
author = {Poliakovsky, Arkady},
title = {Sharp upper bounds for a singular perturbation problem related to micromagnetics},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 6},
number = {4},
year = {2007},
pages = {673-701},
zbl = {1150.49006},
mrnumber = {2394415},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2007_5_6_4_673_0}
}

Poliakovsky, Arkady. Sharp upper bounds for a singular perturbation problem related to micromagnetics. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 4, pp. 673-701. http://www.numdam.org/item/ASNSP_2007_5_6_4_673_0/

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