Boundary vortices in thin magnetic films
Consider the functional \[ E_{\varepsilon}(u) = \frac{1}{2}{\int}_\Omega {| \nabla u| }^2 + \frac{1}{2\varepsilon}{\int}_{\partial \Omega}{\sin}^2(u-g)\,d{\mathcal{H}}^1, \] where \(\Omega\) is a simply connected domain in \({\mathbb{R}}^2\), and \(g: \partial \Omega \rightarrow \mathbb{R}\) is a function such that \(e^{ig}: \partial \Omega \rightarrow S^1\) is a map of degree \(\neq 0\). The author studies the behavior of \(E_{\varepsilon}(u)\), as \(\varepsilon\) tends to zero, thus describing the formation of boundary vortices in thin magnetic films. Convergence results are obtained for sequences of minimizers and stationary points of not too high energy. The limit functions are harmonic ones with boundary singularities. For some cases, in particular for minimizers, an asymptotic expansion for the energy is given, showing that the singular part of the energy depends only on the number of singularities, while their interaction energy is described by a renormalized energy occurring as the first nonsingular term of the expansion (like in the case of the Ginzburg-Landau energy; see the book by \textit{F. Bethuel, H. Brézis} and \textit{F. Hélein} [Ginzburg-Landau vortices. Boston, MA: Birkhäuser (1994; Zbl 0802.35142)]).
scientific article