Variational approach to various nonlinear problems in geometry and physics

LI Jiayu

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PDF(178 KB)
Front. Math. China ›› 2008, Vol. 3 ›› Issue (2) : 205-220. DOI: 10.1007/s11464-008-0013-1

Variational approach to various nonlinear problems in geometry and physics

  • LI Jiayu
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Abstract

In this survey, we will summarize the existence results of nonlinear partial differential equations which arises from geometry or physics by using variational method. We use the method to study Kazdan-Warner problem, Chern-Simons-Higgs model, Toda systems, and the prescribed Q-curvature problem in 4-dimension.

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LI Jiayu. Variational approach to various nonlinear problems in geometry and physics. Front. Math. China, 2008, 3(2): 205‒220 https://doi.org/10.1007/s11464-008-0013-1

References

1. Chang A S Y Yang P Prescribing Gaussian curvatureon S2Acta Math 1987 159214259. doi:10.1007/BF02392560
2. Chang A S Y Yang P Extremal metrics of zeta functionaldeterminants on 4-manifoldsAnn of Math 1995 142172212. doi:10.2307/2118613
3. Chang K C Liu J Q On Nirenberg's problemInternational J Math 1993 43557. doi:10.1142/S0129167X93000042
4. Chen W X Ding W Y Scalar curvature on S2Trans Amer Math Soc 1987 303365382. doi:10.2307/2000798
5. Ding W Jost J Li J et al.The differential equation Δu = 8π - 8πheu on a compact Riemann surfaceAsian J Math 1997 1230248
6. Ding W Jost J Li J et al.An analysis of the two-vortex case in the Chern-SimonsHiggs modelCal Var and PDE 1998 78797. doi:10.1007/s005260050100
7. Ding W Jost J Li J et al.Existence results for mean field equationsAnn Inst H Poincaré Anal Non Linéaire 1999 16653666. doi:10.1016/S0294‐1449(99)80031‐6
8. Djadli Z Malchiodi A Existence of conformal metricswith constant Q-curvatureAnnof Math (in press)
9. Escobar J F Schoen R M Conformal metrics with prescribedscalar curvatureInvent Math 1986 86243253. doi:10.1007/BF01389071
10. Hong C W Abest constant and the Gaussian curvatureProc Amer Math Soc 1986 97737747. doi:10.2307/2045939
11. Hong J Kim Y Pac P Y Multivortex solutions of the Abelian Chern-Simons theoryPhys Rev Lett 1990 6422302233. doi:10.1103/PhysRevLett.64.2230
12. Jackiw R Weinberg E Self-dual Chern-Simons vorticesPhys Rev Lett 1990 6422342237. doi:10.1103/PhysRevLett.64.2234
13. Jost J Wang G Analytic aspects of the Todasystem. I. A Moser-Trudinger inequalityComm Pure Appl Math 2001 54(11)12891319. doi:10.1002/cpa.10004
14. Kazdan J Warner F Curvature functions for compact2-manifoldsAnn Math 1974 991447. doi:10.2307/1971012
15. Lan X Li J Asymptotic behavior of theChern-Simons Higgs 6-th theoryComm PartialDiff Equat 2007 3214731492. doi:10.1080/03605300701629419
16. Li J Li Y Solutions for Toda systemson Riemann surfacesAnn Scuola Norm SupPisa Cl Sci 2005 5703728
17. Malchiodi A Compactnessof solutions to some geometric fourth-order equations 2004, math/0410140
18. Marcello L Margherita N SU(3) Chern-Simons vortex theoryand Toda systemsJ Diff Equat 2002 184(2)443474. doi:10.1006/jdeq.2001.4148
19. Moser J A sharpform of an inequality of N. Trudinger.IndianaUniv Math J 1971 2010771092. doi:10.1512/iumj.1971.20.20101
20. Schoen R M Conformaldeformation of a Riemannian metric to constant scalar curvatureJ Diff Geom 1984 20479495
21. Struwe M Tarantello G On multivortex solutions inChern-Simons gauge theoryBoll Unione MatItal Sez B Artic Ric Mat 1998 1109121
22. Tarantello G Multiplecondensate solutions for the Chern-Simons-Higgs theoryJ Math Phys 1996 3737693796. doi:10.1063/1.531601
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