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.
@article{ASNSP_2004_5_3_2_367_0, author = {Chen, Chiun-Chuan and Lin, Chang-Shou and Wang, Guofang}, title = {Concentration phenomena of two-vortex solutions in a {Chern-Simons} model}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {367--397}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 3}, number = {2}, year = {2004}, mrnumber = {2075988}, zbl = {1170.35413}, language = {en}, url = {http://archive.numdam.org/item/ASNSP_2004_5_3_2_367_0/} }
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 SP - 367 EP - 397 VL - 3 IS - 2 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2004_5_3_2_367_0/ LA - en ID - ASNSP_2004_5_3_2_367_0 ER -
%0 Journal Article %A Chen, Chiun-Chuan %A Lin, Chang-Shou %A Wang, Guofang %T Concentration phenomena of two-vortex solutions in a Chern-Simons model %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2004 %P 367-397 %V 3 %N 2 %I Scuola Normale Superiore, Pisa %U http://archive.numdam.org/item/ASNSP_2004_5_3_2_367_0/ %G en %F ASNSP_2004_5_3_2_367_0
Chen, Chiun-Chuan; Lin, Chang-Shou; Wang, Guofang. Concentration phenomena of two-vortex solutions in a Chern-Simons model. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 3 (2004) no. 2, pp. 367-397. http://archive.numdam.org/item/ASNSP_2004_5_3_2_367_0/
[1] “Complex analysis”, 2nd edition, McGraw-Hill Book Co., New York, 1966. | MR | Zbl
,[2] Liouville type equations with singular data and their application to periodic multivortices for the electroweak theory, Comm. Math. Phys. 229 (2002), 3-47. | MR | Zbl
- ,[3] Uniform estimates and blow-up behavior for solutions of in two dimensions, Comm. Partial Differential Equation 16 (1991), 1223-1253. | MR | Zbl
- ,[4] Surfaces of mean curvature one in hyperbolic space, Astérisque 154-155 (1987), 321-347. | Numdam | MR | Zbl
,[5] Vortex condensation in the Chern-Simons Higgs model: an existence theorem, Comm. Math. Phys. 168 (1995), 321-336. | MR | Zbl
- ,[6] A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description, Comm. Math. Phys. 143 (1992), 501-525. | MR | Zbl
- - - ,[7] Non-topological multivortex solutions to the self-dual Chern-Simons-Higgs equations, Comm. Math. Phys. 231 (2002), 189-221. | MR | Zbl
- - ,[8] Rotational symmetry of solutions of some nonlinear problems in statistical mechanics and in geometry, Comm. Math. Phys. 160 (1994), 217-238. | MR | Zbl
- ,[9] Prescribing Gaussian curvature on , Acta Math. 159 (1987), 215-259. | MR | Zbl
- ,[10] On the symmetry of blowup solutions to a mean field equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 18 (2001), 271-296. | Numdam | MR | Zbl
- ,[11] Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces, Comm. Pure Appl. Math. 4 (2002), 728-771. | MR | Zbl
- ,[12] Topological Degree for a Mean Field Equation on Riemann Surfaces, Comm. Pure Appl. Math. 56 (2003), 1667-1707. | MR | Zbl
and ,[13] Asymptotic radial symmetry for solutions of in a punctured disc, Pacific J. Math. 163 (1994), 269-276. | MR | Zbl
- ,[14] Multiplicity results of two-vortex Chern-Simons-Higgs model on the two-sphere, Comm. Math. Helv. 74 (1999), 118-142. | MR | Zbl
- - - ,[15] The differential equation of on a compact Riemann surface, Asian J. Math., 1 (1997), 230-248. | MR | Zbl
- - - ,[16] Self duality equations for Ginzburg-Landau and Seiberg-Witten type functional with 6th order potenliatls, Comm. Math. Phys. 217 (2001), 383-407. | MR | Zbl
- - - - ,[17] “Self-dual Chern-Simons Theories”, Lecture Notes in Physics m36, Springer-Verlag, Berlin, 1995. | Zbl
,[18] Multivortex solutions of the Abelian Chern Simons theory, Phys. Rev. Letter 64 (1990), 2230-2233. | MR | Zbl
- - ,[19] Selfdual Chern Simons vortices, Phys. Rev. Lett. 64 (1990), 2234-2237. | MR | Zbl
- ,[20] Harnack type inequality: the method of moving planes, Comm. Math. Phys. 200 (1999), 421-444. | MR | Zbl
,[21] Topological degree for mean field equations on , Duke Math. J. 104 (2000), 501-536. | MR | Zbl
,[22] Uniqueness of solutions to the mean field equations for the spherical Onsager vortex, Arch. Ration. Mech. Anal. 153 (2000), 153-176. | MR | Zbl
,[23] Non-topological -vortex condensates for the self-dual chern-Simons theory, Comm. Pure Appl. Math. 56 (2003), 1752-1780. | MR | Zbl
,[24] Double vortex condensates in the Chern-Simons-Higgs theory, Calc. Var. Partial Differential Equations 9 (1999), 31-94. | MR | Zbl
- ,[25] On a sharp Sobolev-type inequality on two-dimensional compact manifolds, Arch. Ration. Mech. Anal. 145 (1998), 161-195. | MR | Zbl
- ,[26] On a class of elliptic problems in : symmetry and uniqueness results, Proc. Roy. Soc. Edinburgh Sect. A 131 (2001), 967-985. | MR | Zbl
- ,[27] Topological solutions in the self-dual Chern-Simons theory: existence and approximation, Ann. Inst. H. Poincaré Anal. Non Linéaire 12 (1995), 75-97. | Numdam | MR | Zbl
- ,[28] Arbitrary -vortex solutions to the first order Ginzburg-Landau equations, Comm. Math. Phys. 72 (1980), 277-292. | MR | Zbl
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