Potential characterizations of geodesic balls on hyperbolic spaces: a moving plane approach. (English) Zbl 1510.35190
Summary: We consider the overdetermined problems in terms of the Riesz and Bessel potentials on hyperbolic space \(\mathbb{H}^n\). Taking advantage of the Helgason-Fourier analysis on the hyperbolic space, we apply the moving plane method in integral form to the corresponding integral equations and show that the solution is constant on the boundary of the domain if and only if the domain is a geodesic ball, and therefore, the solution is radially symmetric. Moreover, fractional-order equations involving the Laplace-Beltrami operator on the hyperbolic space are also considered by using their Green’s function estimates. Our operators also include the well-known GJMS operators on the hyperbolic space.
MSC:
| 35N25 | Overdetermined boundary value problems for PDEs and systems of PDEs |
| 43A80 | Analysis on other specific Lie groups |
| 42B37 | Harmonic analysis and PDEs |
| 22E46 | Semisimple Lie groups and their representations |
| 33C90 | Applications of hypergeometric functions |
Keywords:
overdetermined problem; symmetry; moving plane method in hyperbolic spaces; Helgason-Fourier analysis; GJMS and fractional order operatorsReferences:
| [1] | Alexandrov, AD, A characteristic property of spheres, Ann. Mater. Pura Appl., 4, 58, 303-315 (1962) · Zbl 0107.15603 · doi:10.1007/BF02413056 |
| [2] | Almeida, L.; Damascelli, L.; Ge, Y., A few symmetry results for nonlinear elliptic PDE on noncompact manifolds, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19, 3, 313-342 (2002) · Zbl 1029.35096 · doi:10.1016/s0294-1449(01)00091-9 |
| [3] | Almeida, L.; Ge, Y.; Orlandi, G., Some connections between symmetry results for semilinear PDE in real and hyperbolic spaces, J. Math. Anal. Appl., 311, 626-634 (2005) · Zbl 1175.35046 · doi:10.1016/j.jmaa.2005.03.018 |
| [4] | Anker, JP; Ji, L., Heat kernel and Green function estimates on noncompact symmetric spaces, Geom. Funct. Anal., 9, 1035-1091 (1999) · Zbl 0942.43005 · doi:10.1007/s000390050107 |
| [5] | Beckner, W., On Lie groups and hyperbolic symmetry—from Kunze-Stein phenomena to Riesz potentials, Nonlinear Anal., 126, 394-414 (2015) · Zbl 1344.22002 · doi:10.1016/j.na.2015.06.009 |
| [6] | Berestycki, H.; Nirenberg, L., On the method of moving planes and the sliding method, Bol. Soc. Bras. Mater., 22, 1-37 (1991) · Zbl 0784.35025 · doi:10.1007/BF01244896 |
| [7] | Brézis, H.; Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure Appl. Anal., 36, 437-477 (1983) · Zbl 0541.35029 |
| [8] | Caffarelli, L.; Gidas, B.; Spruck, J., Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Commun. Pure Appl. Math., 42, 3, 271-297 (1989) · Zbl 0702.35085 · doi:10.1002/cpa.3160420304 |
| [9] | Chen, W.; Li, C., Methods on Nonlinear Elliptic Equations, AIMS Series on Differential Equations & Dynamical Systems, 299 (2010), New York: American Institute of Mathematical Sciences, New York · Zbl 1214.35023 |
| [10] | Chen, W.; Li, C.; Ou, B., Classification of solutions for an integral equation, Commun. Pure Appl. Math., 59, 3, 330-343 (2006) · Zbl 1093.45001 · doi:10.1002/cpa.20116 |
| [11] | Fefferman, C.; Graham, CR, The Ambient Metric, Annals of Mathematics Studies, Number 178 (2012), Princeton: Princeton University Press, Princeton · Zbl 1243.53004 |
| [12] | Fefferman, C.; Graham, CR, Juhl’s formulae for GJMS operators and Q-curvatures, J. Am. Math. Soc., 26, 4, 1191-1207 (2013) · Zbl 1276.53042 · doi:10.1090/S0894-0347-2013-00765-1 |
| [13] | Fraenkel, LE, Introduction to Maximum Principles and Symmetry in Elliptic Problems, Cambridge Tracts in Mathematics 128 (2000), Cambridge: Cambridge University Press, Cambridge · Zbl 0947.35002 |
| [14] | Fragala, I.; Gazzola, F., Partially overdetermined elliptic boundary value problems, J. Differ. Equ., 245, 1299-1322 (2008) · Zbl 1156.35049 · doi:10.1016/j.jde.2008.06.014 |
| [15] | Gidas, B.; Ni, W.; Nirenberg, L., Symmetry and related properties via the maximum principle, Commun. Math. Phys., 68, 3, 209-243 (1979) · Zbl 0425.35020 · doi:10.1007/BF01221125 |
| [16] | Graham, CR; Jenne, R.; Mason, LJ; Sparling, GAJ, Conformally invariant powers of the Laplacian, I: existence, J. Lond. Math. Soc. (2), 46, 3, 557-565 (1992) · Zbl 0726.53010 · doi:10.1112/jlms/s2-46.3.557 |
| [17] | Han, X.; Lu, G.; Zhu, J., Characterization of balls in terms of Bessel-potential integral equation, J. Differ. Equ., 252, 1589-1602 (2012) · Zbl 1229.31005 · doi:10.1016/j.jde.2011.07.037 |
| [18] | Helgason, S.; Helgason, S., Groups and geometric analysis, Mathematical Surveys and Monographs (1984), Providence: American Mathematical Society, Providence · Zbl 0177.50601 |
| [19] | Helgason, S.; Helgason, S., Geometric analysis on symmetric spaces, Mathematical Surveys and Monographs (2008), Providence: American Mathematical Society, Providence · Zbl 1157.43003 |
| [20] | Juhl, A., Explicit formulas for GJMS-operators and Q-curvatures, Geom. Funct. Anal., 23, 1278-1370 (2013) · Zbl 1293.53049 · doi:10.1007/s00039-013-0232-9 |
| [21] | Kumaresan, S.; Prajapat, JV, Analogue of Gidas-Ni-Nirenberg Result in Hyperbolic Space and Sphere, 107-112 (1998), Trieste: Rendiconti dellIstituto di Matematica dell Universita di Trieste, Trieste · Zbl 0930.35072 |
| [22] | Kumaresan, S.; Prajapat, JV, Serrin’s result for hyperbolic space and sphere, Duke Math. J., 91, 17-28 (1998) · Zbl 0941.35029 · doi:10.1215/S0012-7094-98-09102-5 |
| [23] | Li, H.-Q.: Centered Hardy-Littlewood maximal function on hyperbolic spaces. Preprint at http://arxiv.org/abs/1304.3261v2 |
| [24] | Li, J.; Lu, G.; Yang, Q., Sharp Adam and Hardy-Adams inequalities of any fractional order on hyperbolic spaces of all dimensions, Trans. Am. Math. Soc., 373, 1 (2020) · Zbl 1451.46036 · doi:10.1090/tran/7986 |
| [25] | Li, J.; Lu, G.; Yang, Q., Higher order Brezis-Nirenberg problem on hyperbolic spaces: existence, nonexistence and symmetry of solutions, Adv. Math., 399, 108259 (2022) · Zbl 1496.35211 · doi:10.1016/j.aim.2022.108259 |
| [26] | Li, J., Lu, G., Wang, J.: Symmetry of solutions of higher and fractional order semilinear equations on hyperbolic spaces. Preprint at http://arxiv.org/abs/2210.02278 |
| [27] | Liu, G., Sharp higher-order Sobolev inequalities in the hyperbolic space \({\mathbb{H} }^n\), Calc. Var. Part. Differ. Equ., 47, 3-4, 567-588 (2013) · Zbl 1284.46029 · doi:10.1007/s00526-012-0528-x |
| [28] | Lu, G.; Yang, Q., Paneitz operators on hyperbolic spaces and high order Hardy-Sobolev-Maz’ya inequalities on half spaces, Am. J. Math., 141, 6, 1777-1816 (2019) · Zbl 1431.26013 · doi:10.1353/ajm.2019.0047 |
| [29] | Lu, G.; Yang, Q., Green’s functions of Paneitz and GJMS operators on hyperbolic spaces and sharp Hardy-Sobolev-Maz’ya inequalities on half spaces, Adv. Math., 398, 108156, 42 (2022) · Zbl 1493.46057 |
| [30] | Lu, G.; Zhu, J., Axial symmetry and regularity of solutions to an integral equation in a half-space, Pac. J. Math., 253, 2, 455-473 (2011) · Zbl 1239.45005 · doi:10.2140/pjm.2011.253.455 |
| [31] | Lu, G.; Zhu, J., An overdetermined problem in Riesz-potential and fractional Laplacian, Nonlinear Anal. Theory Methods Appl., 75, 3036-3048 (2012) · Zbl 1236.31004 · doi:10.1016/j.na.2011.11.036 |
| [32] | Matsumoto, H., Closed form formulae for the heat kernels and the Green functions for the Laplacians on the symmetric spaces of rank one, Bull. Sci. Math., 125, 553-581 (2001) · Zbl 0994.60075 · doi:10.1016/S0007-4497(01)01099-5 |
| [33] | Reichel, W., Characterization of balls by Riesz-potentials, Ann. Mater., 188, 235-245 (2009) · Zbl 1180.31008 |
| [34] | Serrin, J., A symmetry problem in potential theory, Arch. Ration. Mech., 43, 304-318 (1971) · Zbl 0222.31007 · doi:10.1007/BF00250468 |
| [35] | Weinberger, H., Remark on the preceding paper of Serrin, Arch. Ration. Mech. Anal., 43, 319-320 (1971) · Zbl 0222.31008 · doi:10.1007/BF00250469 |
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.
