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On the location of defects in stationary solutions of the Ginzburg-Landau equation in \(\mathbb{R}^ 2\). (English) Zbl 0848.35042

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^2\) with smooth boundary \(\partial \Omega\). Consider the Dirichlet problem \[ \Delta u+ {1\over \varepsilon^2} u(1- |u|^2)= 0\quad \text{in } \Omega,\quad u(x)= g(x)\quad \text{on } \partial \Omega.\tag{1} \] Here, \(u\) and \(g\) are complex valued functions with \(|g|= 1\). Let \(u_\varepsilon\) be a solution of (1) which has a zero \(a_\varepsilon\). The authors assume that, for \(\varepsilon\) small enough, \(a_\varepsilon\) is unique and that there exists a sequence \(\varepsilon_n\), tending to zero, such that \(u_{\varepsilon_n}\to u_0\) and \(a_{\varepsilon_n}\to a_0\) as \(n\to \infty\). Under more basic assumptions about the family \(u_\varepsilon\), they derive the formal outer and inner asymptotic expansions for \(u_\varepsilon\). Then, applying the matching conditions, they show that, at the defect \(a_0\), the gradient of the function \(\psi(x)= \arg u_0(x)- \arg(x- a_0)\) must vanish. Possible extensions to the Neumann problem and to problems whose solutions have \(N\) zeros of degree 1, are mentioned.
The vanishing gradient condition for the location of \(a_0\) was also found, when \(\Omega\) is star shaped, by F. Bethuel, H. Brezis and F. Helein [C. R. Acad. Sci., Paris, Ser. I 317, No. 2, 165-171 (1993; Zbl 0783.35014)], by means of variational methods.
Reviewer: D.Huet (Nancy)

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35C20 Asymptotic expansions of solutions to PDEs
82D55 Statistical mechanics of superconductors

Citations:

Zbl 0783.35014
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