Improved estimates for the Ginzburg-Landau equation : the elliptic case
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Volume 4 (2005) no. 2, p. 319-355
We derive estimates for various quantities which are of interest in the analysis of the Ginzburg-Landau equation, and which we bound in terms of the GL-energy E ε and the parameter ε. These estimates are local in nature, and in particular independent of any boundary condition. Most of them improve and extend earlier results on the subject.
Classification:  35J60,  35B40,  35Q40,  31B35,  46E35
@article{ASNSP_2005_5_4_2_319_0,
     author = {Bethuel, Fabrice and Orlandi, Giandomenico and Smets, Didier},
     title = {Improved estimates for the Ginzburg-Landau equation : the elliptic case},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 4},
     number = {2},
     year = {2005},
     pages = {319-355},
     zbl = {1121.35052},
     mrnumber = {2163559},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2005_5_4_2_319_0}
}
Bethuel, Fabrice; Orlandi, Giandomenico; Smets, Didier. Improved estimates for the Ginzburg-Landau equation : the elliptic case. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Volume 4 (2005) no. 2, pp. 319-355. http://www.numdam.org/item/ASNSP_2005_5_4_2_319_0/

[1] D. Adams, A Trace inequality for generalized potentials, Studia Math. 48 (1973), 99-105. | MR 336316 | Zbl 0237.46037

[2] D. Adams, Weighted nonlinear potential theory, Trans. Amer. Math. Soc. 297 (1986), 73-94. | MR 849468 | Zbl 0656.31012

[3] G. Alberti, S. Baldo and G. Orlandi, Variational convergence for functionals of Ginzburg-Landau type, Indiana Univ. Math. J., to appear | MR 2177107 | Zbl 1160.35013

[4] L. Ambrosio and M. Soner, A measure theoretic approach to higher codimension mean curvature flow, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25 (1997), 27-49. | Numdam | MR 1655508 | Zbl 1043.35136

[5] F. Bethuel, J. Bourgain, H. Brezis and G. Orlandi, W 1,p estimates for solutions to the Ginzburg-Landau equations with boundary data in H 1/2 , C.R. Acad. Sci. Paris Sér. I Math. 333 (2001), 1-8. | MR 1881236 | Zbl 1080.35020

[6] F. Bethuel, H. Brezis and F. Hélein, Asymptotics for the minimization of a Ginzburg-Landau functional, Calc. Var. Partial Differential Equations 1 (1993), 123-148. | MR 1261720 | Zbl 0834.35014

[7] F. Bethuel, H. Brezis and F. Hélein, “Ginzburg-Landau vortices”, Birkhäuser, Boston, 1994. | MR 1269538 | Zbl 0802.35142

[8] F. Bethuel, H. Brezis and G. Orlandi, Asymptotics for the Ginzburg-Landau equation in arbitrary dimensions, J. Funct. Anal. 186 (2001), 432-520. Erratum 188 (2002), 548-549. | MR 1864830 | Zbl 1077.35047

[9] F. Bethuel and G. Orlandi, Uniform estimates for the parabolic Ginzburg-Landau equation, ESAIM Control Optim. Calc. Var. 9 (2002), 219-238. | Numdam | MR 1932951 | Zbl 1078.35013

[10] F. Bethuel, G. Orlandi and D. Smets, Vortex rings for the Gross-Pitaevskii equation, J. Eur. Math. Soc. 6 (2004), 17-94. | MR 2041006 | Zbl 1091.35085

[11] F. Bethuel, G. Orlandi and D. Smets, Convergence of the parabolic Ginzburg-Landau equation to motion by mean curvature, Ann. of Math., to appear | MR 1988309 | Zbl 1103.35038

[12] F. Bethuel, G. Orlandi and D. Smets, Motion of concentration sets in Ginzburg-Landau equations, Ann. Sci. Fac. Toulouse 13 (2004), 3-43. | Numdam | MR 2060028 | Zbl 1063.35075

[13] F. Bethuel, G. Orlandi and D. Smets, Approximations with vorticity bounds for the Ginzburg-Landau functional, Commun. Contemp. Math. 6 (2004), 803-832. | MR 2100765 | Zbl 1129.35329

[14] F. Bethuel, G. Orlandi and D. Smets, Collisions and phase-vortex interactions in dissipative Ginzburg-Landau dynamics, Duke Math. J., to appear. | MR 2184569 | Zbl 1087.35008

[15] F. Bethuel and T. Rivière, A minimization problem related to superconductivity, Ann. Inst. H. Poincaré Anal. Non Linéaire 12 (1995), 243-303. | Numdam | MR 1340265 | Zbl 0842.35119

[16] J. Bourgain and H. Brezis, New estimates for the Laplacian, the div-curl and related Hodge systems, C.R. Acad. Sci. Paris Sér. I Math. 338 (2004), 539-543. | MR 2057026 | Zbl 1101.35013

[17] J. Bourgain, H. Brezis and P. Mironescu, On the structure of the Sobolev space H 1 with values in the circle, C.R. Acad. Sci. Paris Série I 331 (2000), 119-124. | MR 1781527 | Zbl 0970.35069

[18] J. Bourgain, H. Brezis and P. Mironescu, H 1/2 maps with values into the circle: minimal connections, lifting and the Ginzburg-Landau equation, Inst. Hantes Études Sci. Publ. Math. 99 (2004), 1-115. | Numdam | MR 2075883 | Zbl 1051.49030

[19] K. Brakke, “The motion of a surface by its mean curvature”, Princeton University Press, Princeton, 1978. | MR 485012 | Zbl 0386.53047

[20] H. Brezis and P. Mironescu, Sur une conjecture de E. De Giorgi relative à l'énergie de Ginzburg-Landau, C. R. Acad. Sci. Paris Série I Math. 319 (1994), 167-170. | MR 1288397 | Zbl 0805.49004

[21] Y. Chen and M. Struwe, Existence and partial regularity results for the heat flow for harmonic maps, Math. Z. 201 (1989), 83-103. | MR 990191 | Zbl 0652.58024

[22] D. Chiron, Boundary problems for the Ginzburg-Landau equation, Comm. Contemp. Math., to appear. | MR 2175092 | Zbl 1124.35081

[23] R. Coifman, P. L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces J. Math. Pures Appl. 72 (1993), 247-286. | MR 1225511 | Zbl 0864.42009

[24] M. Comte and P. Mironescu, The behavior of a Ginzburg-Landau minimizer near its zeroes, Calc. Var. Partial Differential Equations 4 (1996), 323-340. | MR 1393268 | Zbl 0869.35036

[25] R. Hervé, and M. Hervé, Etude qualitative des solutions réelles d'une équation différentielle liée à l'équation de Ginzburg-Landau, Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994), 427-440. | Numdam | MR 1287240 | Zbl 0836.34090

[26] L. Hedberg and T. Wolff, Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenoble) 23 (1983), 161-187. | Numdam | MR 727526 | Zbl 0508.31008

[27] T. Ilmanen, Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature, J. Differential Geom. 38 (1993), 417-461. | MR 1237490 | Zbl 0784.53035

[28] R. L. Jerrard and H. M. Soner, The Jacobian and the Ginzburg-Landau energy, Calc. Var. Partial Differential Equations 14 (2002), 151-191. | MR 1890398 | Zbl 1034.35025

[29] F. H. Lin and T. Rivière, Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents, J. Eur. Math. Soc. 1 (1999), 237-311. Erratum, J. Eur. Math. Soc. 2 (2000), 87-91. | MR 1750451 | Zbl 0939.35056

[30] F. H. Lin and T. Rivière, A quantization property for static Ginzburg-Landau vortices, Comm. Pure Appl. Math. 54 (2001), 206-228. | MR 1794353 | Zbl 1033.58013

[31] B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192 (1974), 261-274. | MR 340523 | Zbl 0289.26010

[32] D. Preiss, Geometry of measures in n : distribution, rectifiability, and densities, Ann. of Math. 125 (1987), 537-643. | MR 890162 | Zbl 0627.28008

[33] T. Rivière, Line vortices in the U(1) Higgs model, ESAIM Control Optim. Calc. Var. 1 (1996), 77-167. | Numdam | MR 1394302 | Zbl 0874.53019

[34] R. Schoen and S. T. Yau, “Lectures on harmonic maps”, International Press, Cambridge, MA, 1997. | MR 1474501 | Zbl 0886.53004

[35] L. Simon, Lectures on Geometric Measure Theory, Proc. of the centre for Math. Anal., Austr. Nat. Univ., 1983. | MR 756417 | Zbl 0546.49019

[36] M. Struwe, On the asymptotic behavior of minimizers of the Ginzburg-Landau model in 2 dimensions, Differential Integral Equations 7 (1994), 1613-1624. | MR 1269674 | Zbl 0809.35031

[37] J. Van Schaftingen, A simple proof of an inequality of Bourgain, Brezis and Mironescu, C.R. Acad. Sci. Paris, Sér. 1 Math. 338 (2004), 23-26. | MR 2038078 | Zbl 1188.26015

[38] W. P. Ziemer, “Weakly differentiable functions. Sobolev spaces and functions of bounded variation”, Graduate Texts in Math. 120, Springer Verlag, New York, 1989. | MR 1014685 | Zbl 0692.46022