Prescribing the \(Q\)-curvature on the sphere with conical singularities. (English) Zbl 1350.53052
Summary: In this paper we investigate the problem of prescribing the \(Q\)-curvature, on the sphere of any dimension with prescribed conical singularities. We also give the asymptotic behaviour of the solutions that we find and we prove their uniqueness in the negative curvature case. We focus mainly on the odd dimensional case, more specifically the three dimensional sphere.
MSC:
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 35Q82 | PDEs in connection with statistical mechanics |
References:
| [1] | K. Akutagawa, The Yamabe problem on stratified spaces,, Geometric and Functional Analusis, 24, 1039 (2014) · Zbl 1310.53034 · doi:10.1007/s00039-014-0298-z |
| [2] | A. Bahri, The scalar curvature problem on the standard three-dimensional sphere,, J. Funct. Anal., 95, 106 (1991) · Zbl 0722.53032 · doi:10.1016/0022-1236(91)90026-2 |
| [3] | D. Bartolucci, Supercritical conformal metrics on surfaces with conical singularities,, Int. Math. Res. Not., 24, 5625 (2011) · Zbl 1254.30066 · doi:10.1093/imrn/rnq285 |
| [4] | W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality,, Ann. Math., 138, 213 (1993) · Zbl 0826.58042 · doi:10.2307/2946638 |
| [5] | P. Billingsley, <em>Convergence of Probability Measures</em>,, J. Wiley and Sons (1968) · Zbl 0172.21201 |
| [6] | T. Branson, Group representations arising from Lorentz conformal geometry,, J. Funct. Anal., 74, 199 (1987) · Zbl 0643.58036 · doi:10.1016/0022-1236(87)90025-5 |
| [7] | S. Y. A. Chang, On a fourth-order partial differential equation in conformal geometry harmonic analysis and partial differential equations,, Chicago Lectures in Math. (1999) |
| [8] | S.-Y. A. Chang, A note on a class of higher order conformally covariant equations,, Discrete Contin. Dynam. Systems, 7, 275 (2001) · Zbl 1014.35025 · doi:10.3934/dcds.2001.7.275 |
| [9] | S. Y. A. Chang, Prescribing Gaussian curvature on \(S^2\),, Acta Math., 159, 215 (1987) · Zbl 0636.53053 · doi:10.1007/BF02392560 |
| [10] | S. Y. A. Chang, The Q-curvature equation in conformal geometry, Géométrie différentielle,, physique mathématique, 322, 23 (2008) · Zbl 1182.53032 |
| [11] | S. Chanillo, Surfaces with prescribed Gauss curvature,, Duke Math. J., 105, 309 (2000) · Zbl 1023.53005 · doi:10.1215/S0012-7094-00-10525-X |
| [12] | A. Carlotto, Weighted barycentric sets and singular Liouville equations on compact surfaces,, J. Funct. Anal., 262, 409 (2012) · Zbl 1232.53035 · doi:10.1016/j.jfa.2011.09.012 |
| [13] | A. Carlotto, A class of existence results for the singular Liouville equation,, C. R. Math. Acad. Sci. Paris, 349, 161 (2011) · Zbl 1216.35067 · doi:10.1016/j.crma.2010.12.016 |
| [14] | W. Chen, Classification of solutions of some nonlinear elliptic equations,, Duke Math. J., 63, 615 (1991) · Zbl 0768.35025 · doi:10.1215/S0012-7094-91-06325-8 |
| [15] | W. Chen, Qualitative Properties of solutions to some non-linear elliptic equations in \(\mathbbR^2\),, Duke Math. J., 71, 427 (1993) · Zbl 0923.35055 · doi:10.1215/S0012-7094-93-07117-7 |
| [16] | Z. Djadli, Existence of conformal metrics with constant Q-curvature,, Ann. of Math., 168, 813 (2008) · Zbl 1186.53050 · doi:10.4007/annals.2008.168.813 |
| [17] | J. Dolbeault, The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions,, Ann. Sc. Norm. Super. Pisa Cl. Sci., VII, 313 (2008) · Zbl 1179.26055 |
| [18] | R. S. Ellis, <em>Entropy, Large Deviations, and Statistical Mechanics</em>,, Springer-Verlag (1985) · Zbl 0566.60097 · doi:10.1007/978-1-4613-8533-2 |
| [19] | J. Glimm, Quantum Physics,, \(2^{nd}\) ed. (1987) |
| [20] | C. Graham, Conformally invariant powers of the Laplacian, I: Existence,, J. London Math. Soc., 46, 557 (1992) · Zbl 0726.53010 · doi:10.1112/jlms/s2-46.3.557 |
| [21] | T. Jin, Existence and asymptotics for solutions of a non-local \(Q\)-curvature equation in dimension three,, Calc. Var. Partial Differential Equations, 52, 469 (2015) · Zbl 1325.35300 · doi:10.1007/s00526-014-0718-9 |
| [22] | J. L. Kazdan, Curvature functions for compact 2-manifolds,, Ann. Math., 99, 14 (1974) · Zbl 0273.53034 · doi:10.2307/1971012 |
| [23] | M. K.-H. Kiessling, Statistical mechanics of classical particles with logarithmic interactions,, Commun. Pure Appl. Math., 46, 27 (1993) · Zbl 0811.76002 · doi:10.1002/cpa.3160460103 |
| [24] | M. K.-H. Kiessling, Statistical mechanics approach to some problems in conformal geometry,, Physica A, 279, 353 (2000) · doi:10.1016/S0378-4371(99)00515-4 |
| [25] | M. K.-H. Kiessling, Typicality analysis for the Newtonian N-body problem on \(S^2\) in the \(N\to \infty\) limit,, J. Stat. Mech. Theory Exp., 01 (2011) · Zbl 1456.82211 |
| [26] | A. Malchiodi, Conformal metrics with constant Q-curvature,, SIGMA Symmetry Integrability Geom. Methods Appl., 3 (2007) · Zbl 1135.35304 · doi:10.3842/SIGMA.2007.120 |
| [27] | A. Malchiodi, Variational methods for singular Liouville equations,, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 21, 349 (2010) · Zbl 1206.35038 · doi:10.4171/RLM/577 |
| [28] | A. Malchiodi, New improved Moser-Trudinger inequalities and singular Liouville equations on compact surfaces,, Geometric and Functional Analysis, 21, 1196 (2011) · Zbl 1235.35094 · doi:10.1007/s00039-011-0134-7 |
| [29] | L. Martinazzi, Classification of solutions to the higher order Liouville’s equation on \(\mathbbR^{2m}\),, Math. Z., 263, 307 (2009) · Zbl 1172.53026 · doi:10.1007/s00209-008-0419-1 |
| [30] | J. Messer, Statistical mechanics of the isothermal Lane-Emden equation,, J. Stat. Phys., 29, 561 (1982) · doi:10.1007/BF01342187 |
| [31] | C. B. Ndiaye, Constant T-curvature conformal metrics on 4-manifolds with boundary,, Pacific J. Math., 240, 151 (2009) · Zbl 1173.35009 · doi:10.2140/pjm.2009.240.151 |
| [32] | L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator,, Comm. Pure Appl. Math., 60, 67 (2007) · Zbl 1141.49035 · doi:10.1002/cpa.20153 |
| [33] | M. Troyanov, Prescribing curvature on compact surfaces with conical singularities,, Trans. Am. Math. Soc., 324, 793 (1991) · Zbl 0724.53023 · doi:10.1090/S0002-9947-1991-1005085-9 |
| [34] | Y. Wang, <em>Curvature and Statistics</em>,, Ph.D. Dissertation (2013) |
| [35] | J. Wei, On conformal deformations of metrics on \(S^n\),, J. Funct. Anal., 157, 292 (1998) · Zbl 0924.58120 · doi:10.1006/jfan.1998.3271 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
