Summary
The Ginzburg-Landau equations are studied using the techniques of nonlinear functional analysis. In the absence of a magnetic field, for a generic geometry, the existence of a nontrivial branch of solutions is shown, the direct method of the calculus of variations and the Liusternik-Schrmirelmann category theory are used to study the free-energy functional. In the presence of magnetic field, for a cylindric geometry, the existence of a nontrivial branch of solutions is shown and the direct method of calculus of variations is used to study the free-energy functional.
Riassunto
Le tecniche dell’analisi non lineare sono applicate allo studio dell’equazioni di Ginzburg-Landau. In assenza di campo magnetico, per una geometria generica si dimostra l’esistenza di un ramo di soluzioni non banali. Il funzionale dell’energia libera è studiato per mezzo del metodo diretto del calcolo delle variazioni e della teoria della categoria di Liusternik-Schrmirelmann. In presenza di campo magnetico e in geometria cilindrica si dimostra l’esistenza di un ramo di soluzioni non banali. Infine il metodo diretto del calcolo delle variazioni è utilizzato nello studio del funzionale dell'energia libera.
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s is the continuous applicationψ→∣ψ∣2 ψ
−Δ L 2(Ω)→L 2(Ω) is the closed self-adjoint operator such that\(\left( { - \Delta u, \upsilon } \right)_{\mathcal{L}^2 \left( \Omega \right)} = \int\limits_{\partial \Omega } {\frac{{\hbar ^2 }}{{2m}}} \nabla _u \nabla _u d^3 x + \gamma \int\limits_{\partial \Omega } {uvd\sigma } \)
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Part of the work has been done with the support of the Italian National Research Council (GNAFA).
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Micheletti, A., Pace, S. & Zirilli, F. Some rigorous results about Ginzburg-Landau equations. Nuovo Cim B 29, 87–99 (1975). https://doi.org/10.1007/BF02732231
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DOI: https://doi.org/10.1007/BF02732231