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A boundary value problem related to the Ginzburg-Landau model. (English) Zbl 0742.35045

Summary: We analyze the Ginzburg-Landau equation for a superconductor in the case of a 2-dimensional model: a cylindrical conductor with a magnetic field parallel to the axis. This amounts to find the extrema of the free energy \[ {\mathcal A}_ \kappa=1/2\int_ \Omega[|(\nabla-iA)\Phi|^ 2+| B_ A|^ 2+\kappa/4(|\Phi|^ 2-1)^ 2]dx, \] where \(\Omega\) is a bounded domain with smooth boundary in \(\mathbb{R}^ 2\), \(A=(A_ 1,A_ 2)\) the vector potential, \(B_ A=\partial_ 1A_ 2- \partial_ 2A_ 1\) the magnetic field, \(\Phi\) a complex field. We describe the connected components of the maximal configuration space, i.e. of the set of all \((A,\Phi)\) with components in the Sobolev space \(H^ 1(\Omega)\) and such that \(|\Phi|=1\) on the boundary, modulo the action of the gauge group. In the critical case \(\kappa=1\) we give a complete description of the minimal configurations in each component.

MSC:

35Q40 PDEs in connection with quantum mechanics
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