Concentration phenomena of two-vortex solutions in a Chern-Simons model

Chiun-Chuan Chen[1]; Chang-Shou Lin[2]; Guofang Wang[3]

  • [1] Department of Mathematics Taiwan University Taipei, Taiwan
  • [2] Department of Mathematics Chung-Cheng University Minghsiung, Chia-Yi 621 Taiwan
  • [3] Max-Planck-Institute for Mathematics in the Sciences Inselstr. 22-26 04103 Leipzig, Germany

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2004)

  • Volume: 3, Issue: 2, page 367-397
  • ISSN: 0391-173X

Abstract

top
By considering an abelian Chern-Simons model, we are led to study the existence of solutions of the Liouville equation with singularities on a flat torus. A non-existence and degree counting for solutions are obtained. The former result has an application in the Chern-Simons model.

How to cite

top

Chen, Chiun-Chuan, Lin, Chang-Shou, and Wang, Guofang. "Concentration phenomena of two-vortex solutions in a Chern-Simons model." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 3.2 (2004): 367-397. <http://eudml.org/doc/84534>.

@article{Chen2004,
abstract = {By considering an abelian Chern-Simons model, we are led to study the existence of solutions of the Liouville equation with singularities on a flat torus. A non-existence and degree counting for solutions are obtained. The former result has an application in the Chern-Simons model.},
affiliation = {Department of Mathematics Taiwan University Taipei, Taiwan; Department of Mathematics Chung-Cheng University Minghsiung, Chia-Yi 621 Taiwan; Max-Planck-Institute for Mathematics in the Sciences Inselstr. 22-26 04103 Leipzig, Germany},
author = {Chen, Chiun-Chuan, Lin, Chang-Shou, Wang, Guofang},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {2},
pages = {367-397},
publisher = {Scuola Normale Superiore, Pisa},
title = {Concentration phenomena of two-vortex solutions in a Chern-Simons model},
url = {http://eudml.org/doc/84534},
volume = {3},
year = {2004},
}

TY - JOUR
AU - Chen, Chiun-Chuan
AU - Lin, Chang-Shou
AU - Wang, Guofang
TI - Concentration phenomena of two-vortex solutions in a Chern-Simons model
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2004
PB - Scuola Normale Superiore, Pisa
VL - 3
IS - 2
SP - 367
EP - 397
AB - By considering an abelian Chern-Simons model, we are led to study the existence of solutions of the Liouville equation with singularities on a flat torus. A non-existence and degree counting for solutions are obtained. The former result has an application in the Chern-Simons model.
LA - eng
UR - http://eudml.org/doc/84534
ER -

References

top
  1. [1] L. Ahlfors, “Complex analysis”, 2nd edition, McGraw-Hill Book Co., New York, 1966. Zbl0395.30001MR510197
  2. [2] D. Bartolucci – G. Tarantello, Liouville type equations with singular data and their application to periodic multivortices for the electroweak theory, Comm. Math. Phys. 229 (2002), 3-47. Zbl1009.58011MR1917672
  3. [3] H. Brezis – F. Merle, Uniform estimates and blow-up behavior for solutions of - Δ u = V ( x ) e u in two dimensions, Comm. Partial Differential Equation 16 (1991), 1223-1253. Zbl0746.35006MR1132783
  4. [4] R. L. Bryant, Surfaces of mean curvature one in hyperbolic space, Astérisque 154-155 (1987), 321-347. Zbl0635.53047MR955072
  5. [5] L. Caffarelli – Y. Yang, Vortex condensation in the Chern-Simons Higgs model: an existence theorem, Comm. Math. Phys. 168 (1995), 321-336. Zbl0846.58063MR1324400
  6. [6] E. Caglioti – P. L. Lions – C. Marchioro – M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description, Comm. Math. Phys. 143 (1992), 501-525. Zbl0745.76001MR1145596
  7. [7] H. Chan – C. C. Fu – C. S. Lin, Non-topological multivortex solutions to the self-dual Chern-Simons-Higgs equations, Comm. Math. Phys. 231 (2002), 189-221. Zbl1018.58008MR1946331
  8. [8] S. Chanillo – M. Kiessling, Rotational symmetry of solutions of some nonlinear problems in statistical mechanics and in geometry, Comm. Math. Phys. 160 (1994), 217-238. Zbl0821.35044MR1262195
  9. [9] S. Y. Chang – P. Yang, Prescribing Gaussian curvature on S 2 , Acta Math. 159 (1987), 215-259. Zbl0636.53053MR908146
  10. [10] C. C. Chen – C. S. Lin, On the symmetry of blowup solutions to a mean field equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 18 (2001), 271-296. Zbl0995.35004MR1831657
  11. [11] C. C. Chen – C. S. Lin, Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces, Comm. Pure Appl. Math. 4 (2002), 728-771. Zbl1040.53046MR1885666
  12. [12] C. C. Chen and C. S. Lin, Topological Degree for a Mean Field Equation on Riemann Surfaces, Comm. Pure Appl. Math. 56 (2003), 1667-1707. Zbl1032.58010MR2001443
  13. [13] K. S. Chou - Tom Y. H. Wan, Asymptotic radial symmetry for solutions of Δ u + e u = 0 in a punctured disc, Pacific J. Math. 163 (1994), 269-276. Zbl0794.35049MR1262297
  14. [14] W. Ding – J. Jost – J. Li – G. Wang, Multiplicity results of two-vortex Chern-Simons-Higgs model on the two-sphere, Comm. Math. Helv. 74 (1999), 118-142. Zbl0913.53032MR1677094
  15. [15] W. Ding – J. Jost – J. Li – G. Wang, The differential equation of Δ u = 8 π - 8 π h e u on a compact Riemann surface, Asian J. Math., 1 (1997), 230-248. Zbl0955.58010MR1491984
  16. [16] W. Ding – J. Jost – J. Li – X. Peng – G. Wang, Self duality equations for Ginzburg-Landau and Seiberg-Witten type functional with 6th order potenliatls, Comm. Math. Phys. 217 (2001), 383-407. Zbl0994.58009MR1821229
  17. [17] G. Dunne, “Self-dual Chern-Simons Theories”, Lecture Notes in Physics m36, Springer-Verlag, Berlin, 1995. Zbl0834.58001
  18. [18] J. Hong – Y. Kim – P. Y. Pac, Multivortex solutions of the Abelian Chern Simons theory, Phys. Rev. Letter 64 (1990), 2230-2233. Zbl1014.58500MR1050529
  19. [19] R. Jackiw – E. J. Weinberg, Selfdual Chern Simons vortices, Phys. Rev. Lett. 64 (1990), 2234-2237. Zbl1050.81595MR1050530
  20. [20] Y. Y. Li, Harnack type inequality: the method of moving planes, Comm. Math. Phys. 200 (1999), 421-444. Zbl0928.35057MR1673972
  21. [21] C. S. Lin, Topological degree for mean field equations on S 2 , Duke Math. J. 104 (2000), 501-536. Zbl0964.35038MR1781481
  22. [22] C. S. Lin, Uniqueness of solutions to the mean field equations for the spherical Onsager vortex, Arch. Ration. Mech. Anal. 153 (2000), 153–176. Zbl0968.35045MR1770683
  23. [23] M. Nolasco, Non-topological N -vortex condensates for the self-dual chern-Simons theory, Comm. Pure Appl. Math. 56 (2003), 1752-1780. Zbl1032.58005MR2001445
  24. [24] M. Nolasco – G. Tarantello, Double vortex condensates in the Chern-Simons-Higgs theory, Calc. Var. Partial Differential Equations 9 (1999), 31-94. Zbl0951.58030MR1710938
  25. [25] M. Nolasco – G. Tarantello, On a sharp Sobolev-type inequality on two-dimensional compact manifolds, Arch. Ration. Mech. Anal. 145 (1998), 161-195. Zbl0980.46022MR1664542
  26. [26] J. Prajapat – G. Tarantello, On a class of elliptic problems in 2 : symmetry and uniqueness results, Proc. Roy. Soc. Edinburgh Sect. A 131 (2001), 967-985. Zbl1009.35018MR1855007
  27. [27] Spruck – Y. Yang, Topological solutions in the self-dual Chern-Simons theory: existence and approximation, Ann. Inst. H. Poincaré Anal. Non Linéaire 12 (1995), 75-97. Zbl0836.35007MR1320569
  28. [28] Taubes, Arbitrary N -vortex solutions to the first order Ginzburg-Landau equations, Comm. Math. Phys. 72 (1980), 277-292. Zbl0451.35101MR573986

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.