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Ginzburg-Landau vortices for thin ferromagnetic films. (English) Zbl 1057.35070
The magnetization of a bounded ferromagnetic body \(\Omega \subset\mathbb{R}^3\) is described by a vector field \(m:\Omega\to\mathbb{R}^3\) satisfying \(| m|=1\) and the energy is given by the functional \[ {\mathcal E}(m)= \frac{\varepsilon^2}{2} \int_\Omega|\nabla m |^2\,dx+\frac 12\int_{\mathbb{R}^3}|\nabla u|^2 \,dx, \] where \(u\) is determined by \(\Delta u=m\) in \(\mathbb{R}^3\) \((m\) is extended by 0 outside of \(\Omega)\). The author studies minimizers of \({\mathcal E}\) in the shape of very thin films \(\Omega=\Omega'\times(0,\varepsilon^2)\) where \(\Omega'\subset \mathbb{R}^2\) is a bounded, open, simply connected domain. A sequence \(\varepsilon_k \to 0\) is determined such that the maps \[ m_k(\chi')=\frac{1} {\varepsilon^2_k} \int_0^{\varepsilon_k^2} m_k(\chi',s)\,ds \quad (\chi'\in\Omega') \] converge weakly in \(H^1_{\text{loc}}\cap V^{1,p}\) and the limit satisfies certain second-order partial differential equations. The article was inspired by analogous investigations with the \(L^2\)-penalization, cf., e.g., F. Bethuel, H. Brézis and F. Hélein [Ginzburg-Landau vortices (Progress in Nonlinear Differential Equations and their Applications 13, Birkhäuser Boston, MA) (1994; Zbl 0802.35142)]. In particular the free boundary value problem for the \(L^2\)-modified functional \[ {\mathcal J}(f)=\frac 12\int\left(|\nabla f|^2+ \frac{1}{2\varepsilon^2} \bigl(| f|^2-1)^2\right)\,dx+ \frac{1} {2\varepsilon^\alpha} \int_{\partial\Omega} (f\cdot\nu)^2\,d\sigma \] also is thoroughly discussed.

MSC:
35Q60 PDEs in connection with optics and electromagnetic theory
82D40 Statistical mechanical studies of magnetic materials
35B25 Singular perturbations in context of PDEs
49J20 Existence theories for optimal control problems involving partial differential equations
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