Asymptotic behavior for minimizers of an anisotropic Ginzburg-Landau functional (Q5933744)

The authors study an anisotropic form of the Ginzburg-Landau functional NEWLINE\[NEWLINEE_\varepsilon (u,\Omega)= 1/2\int_\Omega \biggl\{a_{ij} u_{xi} u_{xj}+ \bigl[|u|^2- \beta^2(x)\bigr]^2/ (2\varepsilon^2) \biggr\},NEWLINE\]NEWLINE arising from their theoretical approach to superconductivity. It is closely related to several problems such as the phase transition problem posed in the fundamental paper (*). Here \(\varepsilon>0\) is a ``small'' positive parameter, while \(\Omega\) is a smooth, open and bounded domain in \(\mathbb{R}^2\). The function \(\beta\) represents ``thermal noise''. The map \(g(x): \partial\Omega\to S^1\) is smooth but multi-valued, with topological degree \(d>0\). Let \(W\) denote the set of functions in \(H^1(\mathbb{R}^2, \Omega)\) such that \(u|_{\partial \Omega}= \beta g\), and let \(u_\varepsilon\to W\) be a minimizer of \(E_\varepsilon (u,\Omega)\). The authors prove that there exist finitely many points \(\alpha_i\) such that the minimizers \(u_\varepsilon\) of the functional \(E_\varepsilon (u,\Omega)\) converge to a harmonic map away from points \(\alpha_i\) as \(\varepsilon\to 0\). [(*)\textit{V. L. Ginzburg} and \textit{L. D. Landau}, On the theory of superconductivity, J.E.T.P. 20, 1064 (1950)].