Liouville Type Theorems for Lichnerowicz Equations and Ginzburg-Landau Equation: Survey
Author(s)
Li Ma
ABSTRACT
In this survey paper, we firstly review some existence aspects of Lichnerowicz equation and Ginzburg-Landau equations. We then discuss the uniform bounds for both equations in Rn. In the last part of this report, we consider the Liouville type theorems for Lichnerowicz equation and Ginzburg-Landau equations in Rn via two approaches from the use of maximum principle and the monotonicity formula
In this survey paper, we firstly review some existence aspects of Lichnerowicz equation and Ginzburg-Landau equations. We then discuss the uniform bounds for both equations in Rn. In the last part of this report, we consider the Liouville type theorems for Lichnerowicz equation and Ginzburg-Landau equations in Rn via two approaches from the use of maximum principle and the monotonicity formula
Cite this paper
nullL. Ma, "Liouville Type Theorems for Lichnerowicz Equations and Ginzburg-Landau Equation: Survey," Advances in Pure Mathematics, Vol. 1 No. 3, 2011, pp. 99-104. doi: 10.4236/apm.2011.13022.
nullL. Ma, "Liouville Type Theorems for Lichnerowicz Equations and Ginzburg-Landau Equation: Survey," Advances in Pure Mathematics, Vol. 1 No. 3, 2011, pp. 99-104. doi: 10.4236/apm.2011.13022.
References
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[1] N. D. Alikakos, “Some Basic Facts on the System ,” Procceedings of the American Mathematical Society, Vol. 139, No. 1, January 2011, pp. 153-162. doi:10.1090/S0002-9939-2010-10453-7 [2] F. Bethuel, P. Gravejat and J.-C. Saut, “Traveling Waves for the Gross-Pitaevskii Equation II,” Communications in Mathemati-cal Physics, Vol. 285, No. 2, 2009, pp. 567-651. doi:10.1007/s00220-008-0614-2 [3] H. Brezis, “Comments on Two Notes by L. Ma and X. Xu,” Comptes Rendus Mathematique, 2011. [4] A. Chaljub-Simon and Y. Cho-quet-Bruhat, “Global Solutions of the Lichnerowicz Equation in General Relativity on an Asymptotically Euclidean Com-plete Manifold,” General Relativity and Gravitation, Vol. 12, No. 2, 1980, pp. 175-185. doi:10.1007/BF00756471 [5] F. Bethuel, H. Brezis and F. Helein, “Ginzburg-Landau Vortices,” Birkhauser, Basel and Boston, 1994. [6] W. X. Chen and C. M. Li, “An Integral System and the Lane-Emden Conjecture,” Discrete Continuous Dynamical Systems, Vol. 24, No. 4, 2009, pp. 1167-1184. doi:10.3934/dcds.2009.24.1167 [7] Y. Choquet-Bruhat, J. Isenberg and D. Pollack, “The Einstein-Scalar Field Constraints on Asymptotically Euclidean Manifolds,” Chinese Annals of Mathematics, Series B, Vol. 27, No. 1, 2006, pp. 31-52. doi:10.1007/s11401-005-0280-z [8] Y. Choquet-Bruhat, J. Isenberg and D. Pollack, “The Constraint Equations for the Einstein-Scalar Field System on Compact Manifolds,” Classi-cal and Quantum Gravity, Vol. 24, No. 4, 2007, pp. 809-828. doi:10.1088/0264-9381/24/4/004 [9] Y. H. Du and L. Ma, “Logistic Equations on by a Squeezing Method Involving Boundary Blow-up Solutions,” Journal of London Mathemati-cal Society, Vol. 64, No. 2, 2001, pp. 107-124. doi:10.1017/S0024610701002289 [10] A. Farina. “Fi-nite-Energy Solutions, Quantization Effects and Liouville-Type Results for a Variant of the Ginzburg -Landau Systems in RK,” Comptes rendus de l'Académie des Sciences, Série 1, Mathé-matique, Vol. 325, No. 5, 1997, pp. 487-491 [11] E. Hebey, F. Pacard and D. Pollack, “A Variational Analysis of Ein-stein-Scalar Field Lichnerowicz Equations on Compact Rie-mannian Manifolds,” Communications in Mathematical Phys-ics, Vol. 278, No. 1, 2008, pp. 117-132. doi:10.1007/s00220-007-0377-1 [12] E. Hebey, “Existence, Stability and Instability for Einstein-Scalar Field Lichnerowicz Equations,” Two hours lectures, Institute for Advanced Study, Princeton, October 2008. [13] J. Davila, “Global Regularity for a Singular Equation and Local Minimizers of a Nondiffer-entiable Functional,” Communications in Contemporary Mathematics, Vol. 6, No. 1, 2004, pp. 165-193. doi:10.1142/S0219199704001240 [14] F. H. Lin, “Static and Moving Vortices in Ginzburg-Landau Theories,” In: T. N. Knoxville, Ed., Nonlinear Partial Differential Equations in Geometry and Physics, Progress in Nonlinear Differential Equations and their Applications, Birkh?user, Basel, Vol. 29, 1997, pp. 71-111. [15] F. H. Lin and J. C. Wei, “Traveling Wave Solutions of the Schrodinger Map Equation,” Communi-cations on Pure and Applied Mathematics, Vol. 63, No. 12, 2010, pp. 1585-1621. doi:10.1002/cpa.20338 [16] L. Ma, “Liouville Type Theorem and Uniform Bound for the Lichnerowicz Equation and the Ginzburg-Landau Equation,” Comptes Rendus Mathematique, Vol. 348, No. 17, 2010, pp. 993-996. doi:10.1016/j.crma.2010.07.031 [17] L. Ma, “Three Remarks on Mean Field Equations,” Pacific Journal of Mathematics, Vol. 242, No. 1, 2009, pp. 167-171. doi:10.2140/pjm.2009.242.167 [18] L. Ma and X. W. Xu, “Uniform Bound and a Non-Existence Result for Lichnerowicz Equation in the Whole N-Space,” Comptes Rendus Mathe-matique, Vol. 347, No. 13-14, 2009, pp. 805-808. doi:10.1016/j.crma.2009.04.017 [19] L. Modica, “Monotonic-ity of the Energy for Entire Solutions of Semilinear Elliptic Equations,” In: F. Colombini, A. Marino and L. Modica, Eds., Partial Differential Equations and the Calculus of Variations, Birkhauser, Boston, Vol. 2, 1989, pp. 843-850. [20] E. Sandier and S. Serfaty., “Vortices in the Magnetic Ginzburg-Landau Model,” Birkhauser, Basel, 1997. [21] P. Souplet, “The Proof of the Lane-Emden Conjecture in Four Space Dimensions,” Advances in Mathematics, Vol. 221, No. 5, 2009, pp. 1409-1427. doi:10.1016/j.aim.2009.02.014 [22] M. del Pino, etc., “Varia-tional Reduction for Ginzburg-Landau Vortices,” Journal of Functional Analysis, Vol. 239, No. 2, 2006, pp. 497-541. doi:10.1016/j.jfa.2006.07.006 [23] P. Polacik, P. Souplet and P. Quittner, “Singularity and Decay Estimates in Superlinear Problems via Liouville-Type Theorems, Part 1: Elliptic Equa-tions and Systems,” Duke Mathematical Journal, Vol. 139, No. 3, 2007, pp. 555-579. doi:10.1215/S0012-7094-07-13935-8 [24] J. Serrin, “Entire Solutions of Nonlinear Poisson Equations,” Proceedings of London Mathematical Society, Vol. s3-24, No. 2, 1972, pp. 343-366. [25] Y. L. Xin, “Geometry of Harmonic Maps. Se-ries: Progress in Nonlinear Differential Equations and their Applications,” Birkhauser Boston, Inc., Boston, Vol. 23, 1996, pp. 241.