## 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