Abstract

We construct an upper bound for the following family of functionals $\lbrace E_\varepsilon \rbrace _{\varepsilon >0}$, which arises in the study of micromagnetics: $$ E_\varepsilon =\int _\Omega \varepsilon |\nabla u|^2+\frac{1}{\varepsilon }\int _{\mathbb{R}^2}|H_u|^2. $$ Here $\Omega $ is a bounded domain in $\mathbb{R}^2$, $u\in H^1$ and $H_u$, the demagnetizing field created by $u$, is given by $$ {\left\lbrace \begin{array}{ll} {\rm div}\,=0\quad &\text{in }\mathbb{R}^2\,,\\ {\rm curl}\, H_u=0\quad \quad \quad &\text{ in }\mathbb{R}^2\,, \end{array}\right.} $$ 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