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Lower bounds for generalized Ginzburg–Landau functionals. (English) Zbl 0928.35045

The author studies properties of Ginzburg-Landau functionals expressed by the integral of \[ | \nabla u| ^n/n+(1-| u| ^2)^2/(4\varepsilon^2), \] where \(u\in W^{1,n}(U; R^n)\), \(U\subset R^n\). Such information can be useful in the analysis of PDE’s associated with those functionals. The lower bounds relating the energy to the Brower degree of \(u\) are established, and it is proved that the energy concentrates on a small number of small sets. As a consequence, some compactness theorems are derived. Also, the results are applied to the functional which is related to the magnetic potential.

MSC:

35J50 Variational methods for elliptic systems
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35Q99 Partial differential equations of mathematical physics and other areas of application
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