## Lower bounds for the energy of unit vector fields and applications.(English)Zbl 0908.58004

J. Funct. Anal. 152, No. 2, 379-403 (1998); erratum ibid. 171, No. 1, 233 (2000).
The author gives a lower bound for the Dirichlet energy of a unit vector field defined in a perforated domain of $$\mathbb{R}^2$$ with nonzero degree on the outer boundary in terms of the total diameter of the holes. Using this, the author also gives lower bounds and compactness results for Ginzburg-Laudau functionals.

### MSC:

 5.8e+16 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
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### References:

 [1] Almeida, L.; Bethuel, F., Seminaire X EDP (1995-1996) [3] Beaulieu, A.; Hadiji, R., On a class of Ginzburg-Landau equations with weight, Panamer. Math. J., 5, 1-33 (1995) · Zbl 0843.49003 [4] Bethuel, F.; Brezis, H.; Hélein, F., Asymptotics for the minimization of a Ginzburg-Landau functional, Calc. Var. Partial Differential Equations, 1, 123-148 (1993) · Zbl 0834.35014 [5] Bethuel, F.; Brezis, H.; Hélein, F., Ginzburg-Landau Vortices. Ginzburg-Landau Vortices, Progress in Nonlinear Differential Equations and Their Applications (1994), Birkhauser: Birkhauser Basel · Zbl 0802.35142 [6] Bethuel, F.; Demengel, F., Extensions for Sobolev maps between manifolds, Calc. Var. Partial Differential Equations, 3, 475-491 (1995) · Zbl 0846.46021 [7] Bethuel, F.; Rivière, T., Vortices for a variational problem related to supra- conductivity, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 12, 243-303 (1995) · Zbl 0842.35119 [8] Boutet de Monvel-Berthier, A.; Georgescu, V.; Purice, R., A boundary value problem related to the Ginzburg-Landau model, Comm. Math. Phys., 141, 1-23 (1991) · Zbl 0742.35045 [9] Brezis, H.; Merle, F.; Rivière, T., Quantisation effects for −$$Δu uu^2^2$$, Arch. Rational Mech. Anal., 126, 35-58 (1994) · Zbl 0809.35019 [10] Han, Z. G.; Li, Y. Y., Degenerate elliptic systems and application to Ginzburg-Landau type equations, part I, Calc. Var. Partial Differential Equations, 4, 171-202 (1996) · Zbl 0847.35055 [11] Han, Z. G.; Shafrir, I., Lower bounds for the energy of $$S^1$$, J. Anal. Math., 68, 295-305 (1995) · Zbl 0852.49028 [13] Lin, F. H., Solutions of the Ginzburg-Landau equations and critical points of the normalized energy, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 12, 599-622 (1995) · Zbl 0845.35052 [14] Rivière, T., Lignes de tourbillons dans le modèle de Higgs abélien, C.R. Acad. Sci. Paris. Série I, 395, 73-76 (1995) · Zbl 0840.35109 [15] Schoen, R.; Uhlenbeck, K., Boundary regularity and the Dirichlet problem for harmonic maps, J. Differential Geom., 18, 253-268 (1983) · Zbl 0547.58020 [16] Struwe, M., On the asymptotic behavior of minimizers of the Ginzburg-Landau model in 2 dimensions, J. Differential Integral Equations, 7 (1994) · Zbl 0809.35031
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