Ginzburg-Landau equation with DeGennes boundary condition
Let \(\Omega\) be a bounded domain in \(\mathbb{R}^2\) and \(\varepsilon>0\), \(\gamma(\varepsilon)\) be parameters. The authors study the following semilinear elliptic boundary value problem \[ \varepsilon^2\Delta u+(1- u^2)u\quad\text{in }\Omega,\quad {\partial u\over\partial\nu}+ \gamma(\varepsilon)u=0\quad\text{on }\partial\Omega\tag{1} \] and its relation to the minimizing problem of the Ginzburg-Landau type energy functional \[ L(\psi,A)= \int_\Omega\Biggl(|(\nabla- iA)\psi|^2+ |\text{rot }A- H|^2+ {1\over2\varepsilon} (|\psi|^2-1)^2\Biggr) dx+\int_{\partial\Omega} |\psi|^2 dS.\tag{2} \] It is proved that if \(\Omega\) is simply-connected, \(H\equiv 0\) and \(\varepsilon\) is small, the global minimizer \((\psi,A)\) of (2) is expressed by the aid of the unique positive solution of (1), that is \((\psi,A)\equiv(u_\varepsilon e^{\kappa i},\nabla\kappa)\). The behavior of \(u_\varepsilon\) is investigated for small \(\varepsilon>0\).
scientific article; zbMATH DE number 933272