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On a problem related to vortex nucleation of superconductivity. (English) Zbl 1064.35057

The paper considers a system of stationary Ginzburg-Landau equations, which describes the distribution of the complex order parameter and real vector potential of the magnetic field in a type-II superconductor. It is known that, with the increase of external magnetic field, the superconducting (Meissner) state undergoes a transition to a mixed state. However, there is a region of values of the magnetic field in which the superconducting state, although being less stable than the mixed one, is still stable against small perturbations. In this situation, the transition to the mixed state takes place through penetration of vortices into the superconductor, which occurs at the boundary of the sample. It was previously shown, in a formal way (assuming that the penetration depth \(\lambda\) of the magnetic field is vanishing), that the vortices penetrate via points at which the local curvature of the boundary assumes a largest negative value.
The present paper offers a rigorous proof of this assertion: if the configuration remains locally stable everywhere (which means that the magnitude of the magnetic vectorial potential is smaller than a critical value), points of the maximum of the vectorial potential (which are the spots where the penetration of the vortex into the sample is expected) approach points where the border’s curvature takes its minimum values (the minimum is realized in the algebraic sense). The proof is based on detailed analysis of the structure of the solutions in a boundary layer of the thickness \(\sim \lambda\).

MSC:

35J60 Nonlinear elliptic equations
82D55 Statistical mechanics of superconductors
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