H 1/2 maps with values into the circle : minimal connections, lifting, and the Ginzburg-Landau equation
Publications Mathématiques de l'IHÉS, Tome 99 (2004), pp. 1-115. 
@article{PMIHES_2004__99__1_0,
     author = {Bourgain, Jean and Brezis, Haim and Mironescu, Petru},
     title = {$H^{1/2}$ maps with values into the circle : minimal connections, lifting, and the {Ginzburg-Landau} equation},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {1--115},
     publisher = {Springer},
     volume = {99},
     year = {2004},
     doi = {10.1007/s10240-004-0019-5},
     zbl = {1051.49030},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1007/s10240-004-0019-5/}
}
TY  - JOUR
AU  - Bourgain, Jean
AU  - Brezis, Haim
AU  - Mironescu, Petru
TI  - $H^{1/2}$ maps with values into the circle : minimal connections, lifting, and the Ginzburg-Landau equation
JO  - Publications Mathématiques de l'IHÉS
PY  - 2004
SP  - 1
EP  - 115
VL  - 99
PB  - Springer
UR  - http://archive.numdam.org/articles/10.1007/s10240-004-0019-5/
DO  - 10.1007/s10240-004-0019-5
LA  - en
ID  - PMIHES_2004__99__1_0
ER  - 
%0 Journal Article
%A Bourgain, Jean
%A Brezis, Haim
%A Mironescu, Petru
%T $H^{1/2}$ maps with values into the circle : minimal connections, lifting, and the Ginzburg-Landau equation
%J Publications Mathématiques de l'IHÉS
%D 2004
%P 1-115
%V 99
%I Springer
%U http://archive.numdam.org/articles/10.1007/s10240-004-0019-5/
%R 10.1007/s10240-004-0019-5
%G en
%F PMIHES_2004__99__1_0
Bourgain, Jean; Brezis, Haim; Mironescu, Petru. $H^{1/2}$ maps with values into the circle : minimal connections, lifting, and the Ginzburg-Landau equation. Publications Mathématiques de l'IHÉS, Tome 99 (2004), pp. 1-115. doi : 10.1007/s10240-004-0019-5. http://archive.numdam.org/articles/10.1007/s10240-004-0019-5/

1. R. A. Adams, Sobolev spaces, Acad. Press, 1975. | MR | Zbl

2. F. Almgren, W. Browder, and E. H. Lieb, Co-area, liquid crystals and minimal surfaces, in: Partial differential equations (Tianjin, 1986), Lect. Notes Math. 1306, Springer, 1988. | MR | Zbl

3. F. Bethuel, A characterization of maps in H1(B3,S2) which can be approximated by smooth maps, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 7 (1990), 269-286. | Numdam | MR | Zbl

4. F. Bethuel, Approximations in trace spaces defined between manifolds, Nonlinear Anal. Theory Methods Appl., 24 (1995), 121-130. | MR | Zbl

5. F. Bethuel, J. Bourgain, H. Brezis, and G. Orlandi, W1,p estimate for solutions to the Ginzburg-Landau equation with boundary data in H1/2, C. R. Acad. Sci., Paris, Sér. I, Math., 333 (2001), 1069-1076. | MR | Zbl

6. F. Bethuel, H. Brezis, and J.-M. Coron, Relaxed energies for harmonic maps, in: H. Berestycki, J.-M. Coron, and I. Ekeland (eds.), Variational Problems, pp. 37-52, Birkhäuser, 1990. | MR | Zbl

7. F. Bethuel, H. Brezis, and F. Hélein, Asymptotics for the minimization of a Ginzburg-Landau functional, Calc. Var. Partial Differ. Equ., 1 (1993), 123-148. | MR | Zbl

8. F. Bethuel, H. Brezis, and G. Orlandi, Small energy solutions to the Ginzburg-Landau equation, C. R. Acad. Sci., Paris, Sér. I, 331 (2000), 763-770. | MR | Zbl

9. F. Bethuel, H. Brezis, and G. Orlandi, Asymptotics for the Ginzburg-Landau equation in arbitrary dimensions, J. Funct. Anal., 186 (2001), 432-520. | MR | Zbl

10. F. Bethuel and X. Zheng, Density of smooth functions between two manifolds in Sobolev spaces, J. Funct. Anal., 80 (1988), 60-75. | MR | Zbl

11. J. Bourgain and H. Brezis, On the equation div Y=f and application to control of phases, J. Am. Math. Soc., 16 (2003), 393-426. | MR | Zbl

12. J. Bourgain, H. Brezis, and P. Mironescu, Lifting in Sobolev spaces, J. Anal. Math., 80 (2000), 37-86. | MR | Zbl

13. J. Bourgain, H. Brezis, and P. Mironescu, On the structure of the Sobolev space H1/2 with values into the circle, C. R. Acad. Sci., Paris, Sér. I, 310 (2000), 119-124. | MR | Zbl

14. J. Bourgain, H. Brezis, and P. Mironescu, Another look at Sobolev spaces, in: J. L. Menaldi, E. Rofman, and A. Sulem (eds.), Optimal Control and Partial Differential Equations, pp. 439-455, IOS Press, 2001. | Zbl

15. J. Bourgain, H. Brezis, and P. Mironescu, Limiting embedding theorems for Ws,p when s↗1 and applications, J. Anal. Math., 87 (2002), 77-101. | MR | Zbl

16. J. Bourgain, H. Brezis, and P. Mironescu, Lifting, degree and distibutional Jacobian revisited, to appear in Commun. Pure Appl. Math. | MR | Zbl

17. A. Boutet De Monvel, V. Georgescu, and R. Purice, A boundary value problem related to the Ginzburg-Landau model, Commun. Math. Phys., 142 (1991), 1-23. | MR | Zbl

18. H. Brezis, Liquid crystals and energy estimates for S2-valued maps, in: J. Ericksen and D. Kinderlehrer (eds.), Theory and Applications of Liquid Crystals, pp. 31-52, Springer, 1987. | MR

19. H. Brezis, J.-M. Coron, and E. Lieb, Harmonic maps with defects, Commun. Math. Phys., 107 (1986), 649-705. | MR | Zbl

20. H. Brezis, Y. Y. Li, P. Mironescu, and L. Nirenberg, Degree and Sobolev spaces, Topol. Methods Nonlinear Anal., 13 (1999), 181-190. | MR | Zbl

21. H. Brezis and P. Mironescu, Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces, J. Evolution Equ., 1 (2001), 387-404. | MR | Zbl

22. H. Brezis and L. Nirenberg, Degree Theory and BMO, Part I: Compact manifolds without boundaries, Sel. Math., 1 (1995), 197-263. | MR | Zbl

23. A. Cohen, W. Dahmen, I. Daubechies, and R. Devore, Harmonic analysis of the space BV, Rev. Mat. Iberoam., 19 (2003), 235-263. | MR | Zbl

24. F. Demengel, Une caractérisation des fonctions de W1,1(Bn ,S1) qui peuvent être approchées par des fonctions régulières, C. R. Acad. Sci., Paris, Sér. I, 310 (1990), 553-557. | MR | Zbl

25. M. Escobedo, Some remarks on the density of regular mappings in Sobolev classes of SM-valued functions, Rev. Mat. Univ. Complut. Madrid, 1 (1988), 127-144. | MR | Zbl

26. H. Federer, Geometric measure theory, Springer, 1969. | MR | Zbl

27. M. Giaquinta, G. Modica, and J. Soucek, Cartesian Currents in the Calculus of Variations, vol. II, Springer, 1998. | MR | Zbl

28. F. B. Hang and F. H. Lin, A remark on the Jacobians, Comm. Contemp. Math., 2 (2000), 35-46. | MR | Zbl

29. R. Hardt, D. Kinderlehrer, and F. H. Lin, Stable defects of minimizers of constrained variational principles, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 5 (1988), 297-322. | Numdam | MR | Zbl

30. T. Iwaniec, C. Scott, and B. Stroffolini, Nonlinear Hodge theory on manifolds with boundary, Ann. Mat. Pura Appl., 157 (1999), 37-115. | MR | Zbl

31. R. L. Jerrard and H. M. Soner, Rectifiability of the distributional Jacobian for a class of functions, C. R. Acad. Sci., Paris, Sér. I, 329 (1999), 683-688. | MR | Zbl

32. R. L. Jerrard and H. M. Soner, Functions of bounded higher variation, Indiana Univ. Math. J., 51 (2002), 645-677. | MR | Zbl

33. R. L. Jerrard and H. M. Soner, The Jacobian and the Ginzburg-Landau energy, Calc. Var. Partial Differ. Equ., 14 (2002), 151-191. | MR | Zbl

34. 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 2 (2002), 87-91. | MR | Zbl

35. V. Maz'Ya and T. Shaposhnikova, On the Bourgain, Brezis and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., 195 (2002), 230-238. | Zbl

36. A. Ponce, On the distributions of the form ∑ i (δ p i -δ n i ), J. Funct. Anal., 210 (2004), 391-435; part of the results were announced in a note by the same author: On the distributions of the form ∑ i (δ p i -δ n i ), C. R. Acad. Sci., Paris Sér. I, Math., 336 (2003), 571-576. | MR | Zbl

37. T. Rivière, Line vortices in the U(1)-Higgs model, Control Optim. Calc. Var., 1 (1996), 77-167. | Numdam | MR | Zbl

38. T. Rivière, Dense subsets of H1/2(S2;S1), Ann. Global Anal. Geom., 18 (2000), 517-528. | MR | Zbl

39. E. Sandier, Lower bounds for the energy of unit vector fields and applications, J. Funct. Anal., 152 (1998), 379-403. | MR | Zbl

40. E. Sandier, Ginzburg-Landau minimizers from R n+1 to R n and minimal connections, Indiana Univ. Math. J., 50 (2001), 1807-1844. | MR | Zbl

41. R. Schoen and K. Uhlenbeck, Boundary regularity and the Dirichlet problem for harmonic maps, J. Differ. Geom., 18 (1983), 253-268. | MR | Zbl

42. L. Simon, Lectures on geometric measure theory, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. | MR | Zbl

43. D. Smets, On some infinite sums of integer valued Dirac's masses, C. R. Acad. Sci., Paris, Sér. I, 334 (2002), 371-374.

44. V. A. Solonnikov, Inequalities for functions of the classes W → p (𝐑 𝐧 ), J. Soviet Math., 3 (1975), 549-564. | Zbl

45. H. Triebel, Interpolation theory. Function spaces. Differential operators, Johann Ambrosius Barth, Heidelberg, Leipzig, 1995. | MR | Zbl

46. G. Alberti, S. Baldo, and G. Orlandi, Variational convergence for functionals of Ginzburg-Landau type, to appear. | MR

47. F. Bethuel, G. Orlandi, and D. Smets, On an open problem for Jacobians raised by Bourgain, Brezis and Mironescu, C. R. Acad Sci., Paris, Sér. I, 337 (2003), 381-385. | MR | Zbl

48. F. Bethuel, G. Orlandi, and D. Smets, Approximation with vorticity bounds for the Ginzburg-Landau functional, to appear in Comm. Contemp. Math. | MR | Zbl

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

50. H. Federer and W. H. Fleming, Normal and integral currents, Ann. Math., 72 (1960), 458-520. | MR | Zbl

51. A. Ponce, An estimate in the spirit of Poincaré's inequality, J. Eur. Math. Soc., 6 (2004), 1-15. | Zbl

52. J. Van Schaftingen, On an inequality of Bourgain, Brezis and Mironescu, C. R. Acad Sci., Paris, Sér. I, 338 (2004), 23-26. | MR

Cité par Sources :