×

zbMATH — the first resource for mathematics

Analytic aspects of the Toda system. I: A Moser-Trudinger inequality. (English) Zbl 1099.35035
The paper studies the open Toda system and establishes an optimal Moser-Trudinger type inequality for this system. The following main result is proved: Let \(\Sigma\) be a closed surface with area 1 and \((a_{ij})_{N\times N}\) be the Cartan matrix for \(SU(N+1)\). Define a functional \(\Phi^N \: (H^1(\Sigma ))^N \to \mathbb R\) by the formula \(\Phi^N(u)=\frac{1}{2}\sum \limits_{i,j=1}^N\int \limits_{\Sigma}a_{ij}(\nabla u_i\nabla u_j+2M_iu_j)-\sum \limits_{i=1}^NM_i\log \int \limits_{\Sigma }\exp \Big (\sum \limits_{j=1}^Na_{ij}u_j\Big )\), where \(M_i>0\) are given. Then the functional has a lower bound if and only if \(M_j\leq 4\pi\) for any \(j=1,2,\cdots ,N\).

MSC:
35J60 Nonlinear elliptic equations
35J45 Systems of elliptic equations, general (MSC2000)
47J30 Variational methods involving nonlinear operators
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bolton, Math Ann 279 pp 599– (1988) · Zbl 0642.53063 · doi:10.1007/BF01458531
[2] Brezis, Comm Partial Differential Equations 16 pp 1223– (1991) · Zbl 0746.35006 · doi:10.1080/03605309108820797
[3] Caffarelli, Comm Math Phys 168 pp 321– (1995) · Zbl 0846.58063 · doi:10.1007/BF02101552
[4] Caglioti, Comm Math Phys 143 pp 501– (1992) · Zbl 0745.76001 · doi:10.1007/BF02099262
[5] Caglioti, Comm Math Phys 174 pp 229– (1995) · Zbl 0840.76002 · doi:10.1007/BF02099602
[6] Chang, Internat J Math 4 pp 35– (1993) · Zbl 0786.58010 · doi:10.1142/S0129167X93000042
[7] Chang, Acta Math 159 pp 215– (1987) · Zbl 0636.53053 · doi:10.1007/BF02392560
[8] Chang, J Differential Geom 27 pp 259– (1988) · Zbl 0649.53022 · doi:10.4310/jdg/1214441783
[9] Chanillo, Geom Funct Anal 5 pp 924– (1995) · Zbl 0858.35035 · doi:10.1007/BF01902215
[10] Chen, Trans Amer Math Soc 303 pp 365– (1987)
[11] Chen, Duke Math J 63 pp 615– (1991) · Zbl 0768.35025 · doi:10.1215/S0012-7094-91-06325-8
[12] Chipot, J Differential Equations 140 pp 59– (1997) · Zbl 0902.35039 · doi:10.1006/jdeq.1997.3316
[13] On the best constant in a Sobolev inequality on compact 2-manifolds and application. Unpublished manuscript, 1984.
[14] Ding, Comm Math Phys 217 pp 383– (2001) · Zbl 0994.58009 · doi:10.1007/s002200100377
[15] Ding, Asian J Math 1 pp 230– (1997) · Zbl 0955.58010 · doi:10.4310/AJM.1997.v1.n2.a3
[16] Ding, Calc Vari Partial Differential Equations 7 pp 87– (1998) · Zbl 0928.58021 · doi:10.1007/s005260050100
[17] Ding, Ann Inst H Poincaré Anal Non Linéaire 16 pp 653– (1999) · Zbl 0937.35055 · doi:10.1016/S0294-1449(99)80031-6
[18] Ding, Comment Math Helv 74 pp 118– (1999) · Zbl 0913.53032 · doi:10.1007/s000140050079
[19] Self-dual Cher-Simons Theories. Lecture Notes in Physics. New series, Monographs, m36. Springer, New York, 1995.
[20] Eells, Adv in Math 49 pp 217– (1983) · Zbl 0528.58007 · doi:10.1016/0001-8708(83)90062-2
[21] Fontana, Comment Math Helv 68 pp 415– (1993) · Zbl 0844.58082 · doi:10.1007/BF02565828
[22] ; eds. Harmonic maps and integrable systems. Aspects of Mathematics, E23. Friedr. Vieweg & Sohn, Braunschweig, 1994. · doi:10.1007/978-3-663-14092-4
[23] Harmonic maps, loops groups, and integrable systems. London Mathematical Society Student Texts, 38. Cambridge University Press, Cambridge, 1997. · doi:10.1017/CBO9781139174848
[24] Hong, Phys Rev Lett 64 pp 2230– (1990) · Zbl 1014.58500 · doi:10.1103/PhysRevLett.64.2230
[25] Horstman, European J Appl Math 12 pp 159– (2001) · Zbl 1017.92006 · doi:10.1017/S0956792501004363
[26] Jackiw, Phys Rev Lett 64 pp 2234– (1990) · Zbl 1050.81595 · doi:10.1103/PhysRevLett.64.2234
[27] ; Analytic aspects of the Toda system. II. An application to the relativistic SU(3) Chern-Simons model. In preparation.
[28] Jost, Internat Math Res Notices
[29] Kao, Phys Rev D(3) 50 pp 6626– (1994) · doi:10.1103/PhysRevB.50.6626
[30] Kiessling, Comm Pure Appl Math 46 pp 27– (1993) · Zbl 0811.76002 · doi:10.1002/cpa.3160460103
[31] Kostant, Adv in Math 34 pp 195– (1979) · Zbl 0433.22008 · doi:10.1016/0001-8708(79)90057-4
[32] Lee, Phys Rev Lett 66 pp 553– (1991) · Zbl 0968.81533 · doi:10.1103/PhysRevLett.66.553
[33] Lee, Phys Lett B 255 pp 381– (1991) · doi:10.1016/0370-2693(91)90782-L
[34] Li, Duke Math J 80 pp 383– (1995) · Zbl 0846.35050 · doi:10.1215/S0012-7094-95-08016-8
[35] Liouville, J Math Pures Appl 18 pp 71– (1853)
[36] Moser, Indiana Univ Math J 20 pp 1077– (1970) · doi:10.1512/iumj.1971.20.20101
[37] Nolasco, Arch Ration Mech Anal 145 pp 161– (1998) · Zbl 0980.46022 · doi:10.1007/s002050050127
[38] Nolasco, Calc Var Partial Differential Equations 9 pp 31– (1999) · Zbl 0951.58030 · doi:10.1007/s005260050132
[39] Nolasco, Comm Math Phys 213 pp 599– (2000) · Zbl 0998.81047 · doi:10.1007/s002200000252
[40] Struwe, Boll Unione Mat Ital Sez B Artic Ric Mat (8) 1 pp 109– (1998)
[41] Tarantello, J Math Phys 37 pp 3769– (1996) · Zbl 0863.58081 · doi:10.1063/1.531601
[42] Trudinger, J Math Mech 17 pp 473– (1967)
[43] Wang, C R Acad Sci Paris Sér I Math 328 pp 895– (1999) · Zbl 0933.37064 · doi:10.1016/S0764-4442(99)80293-6
[44] Wang, Math Nach
[45] Wang, Comm Math Phys 202 pp 501– (1999) · Zbl 0976.58014 · doi:10.1007/s002200050593
[46] Yang, Comm Math Phys 186 pp 199– (1997) · Zbl 0874.58093 · doi:10.1007/BF02885678
[47] Ye, Calc Var Partial Differential Equations
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.