# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Almeida, L.; Bethuel, F., Seminaire X EDP, (1995-1996) [2] N. André, I. Shafrir, Asymptotic behaviour of minimizers for the Ginzburg- Landau functional with weight, Arch. Rational Mech. Anal. [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, Progress in nonlinear differential equations and their applications, (1994), Birkhauser Basel [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 −δuuu22, Arch. rational mech. anal., 126, 35-58, (1994) [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 ofS1, J. anal. math., 68, 295-305, (1995) [12] R. Jerrard [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.