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On the energy of type-II superconductors in the mixed phase. (English) Zbl 0964.49006
The paper deals with the asymptotic study of minimizers of the Ginzburg-Landau energy functional with applied magnetic field. The framework corresponds to high values of the Ginzburg-Landau parameter \(\kappa\) and to a prescribed external constant magnetic field \(h_{ex}\), with \(h_{ex}\) varying between the critical fields \(H_{c_1}\) and \(H_{c_3}\). The main result describes the vortex structure of minimizers of the Ginzburg-Landau energy as \(\kappa\rightarrow +\infty\). More precisely, the authors give the leading term in the asymptotic expansion of the minimal energy and show that energy minimizers have vortices whose density tends to be uniform and equal to \(h_{ex}\). The proofs are based on refined elliptic estimates and they combine various upper and lower bounds of the energy on “good” and “bad” subdomains.

MSC:
49J35 Existence of solutions for minimax problems
58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
82D55 Statistical mechanics of superconductors
35Q55 NLS equations (nonlinear Schrödinger equations)
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