Boundary vortices in thin magnetic films
type: Item
instance of: scholarly article
title: Boundary vortices in thin magnetic films
zbMATH Open document ID: 1151.35006
published in: Calculus of Variations and Partial Differential Equations
publication date: 2006-06-14T00:00:00Z
review text: 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)]).
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 82D40
zbMATH DE Number: 5032406
zbMATH Keywords: Ginzburg-Landau type vortices
zbMATH Keywords: thin magnetic film
author: Matthias Kurzke
reviewed by: Gheorghe Moroşanu
MaRDI profile type: Publication
full work available at URL: s00526-005-0331-z
OpenAlex ID: W1971367876
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DOI: 10.1007/S00526-005-0331-Z
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description: scientific article