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A relaxation result for micromagnetics in SBV

Authors:

Pedro M Santos

Institution:

(1) Instituto Superior Tecnico, Av. Rovisco Pais, 1049-001 Lisbon, Portugal

Abstract:

An integral representation for the functional

$\begin{aligned} {\mathcal F}(m,M)\,:=&\,{\rm inf}\left\{ \liminf_{k \to +\infty}\int_\Omega f(x,m_k(x),\nabla m_k(x))\,dx + \int_{\Omega\cap S(m_k)} |m_k](x)|d{\mathcal H}^{N-1} :\right. \\ &\left.m_k \in SBV(\Omega;{\mathbb R}^N),\quad |m_k(x)|=1\,\,{{\rm a.e.\, in}\,\, \Omega},\right.\hspace*{19pt} \left. \vphantom{\int_{\Omega\cap S(m_k)} |m_k](x)|d{\mathcal H}^{N-1} :}\right.\\&\left. m_k \to m\quad \text{in} \quad L^1(\Omega;{\mathbb R}^N),\quad \nabla m_k \rightharpoonup M\text{ in }L^2(\Omega; {\mathbb R}^N)\right\}\end{aligned}$

is obtained. This problem is motivated by equilibria issues in micromagnetics.

Keywords:

Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) 35D99 35E99 49J45

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