Extremal values of the (fractional) Weinstein functional on the hyperbolic space
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Mayukh Mukherjee
Abstract
We study Weinstein functionals, first defined in [33], mainly on the hyperbolic space
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-1161620
Funding statement: The author was partially supported by NSF grant DMS-1161620.
Acknowledgements
This project was completed when I was a Ph.D. student at UNC Chapel Hill under the guidance of Michael E. Taylor. I wish to thank Jeremy Marzuola for going through a draft copy of this write-up and making several important suggestions, and also for teaching me about the fractional G-N inequality. I wish to thank the anonymous referee for her/his valuable advice.
References
[1] N. Badr, Gagliardo–Nirenberg inequalities on manifolds, J. Math. Anal. Appl. 49 (2009), no. 2, 493–502. 10.1016/j.jmaa.2008.09.013Suche in Google Scholar
[2] A. Balakrishnan, Fractional powers of closed operators and the semigroups generated by them, Pacific J. Math. 10 (1960), 419–437. 10.2140/pjm.1960.10.419Suche in Google Scholar
[3] V. Banica, The nonlinear Schrödinger equation on hyperbolic space, Comm. Partial Differential Equations 32 (2007), 1643–1677. 10.1080/03605300600854332Suche in Google Scholar
[4] F. Baudoin, Online research and lecture notes, https://fabricebaudoin.wordpress.com/2013/10/16/lecture-22-sobolev-inequality-and-volume-growth/. Suche in Google Scholar
[5] C. Brouttelande, The Best-constant problem for a family of Gagliardo–Nirenberg inequalities on a compact Riemannian manifold, Proc. Edinb. Math. Soc. (2) 46 (2003), 117–146. 10.1017/S0013091501000426Suche in Google Scholar
[6] S.-Y. Cheng, P. Li and S.-T. Yau, On the upper estimate of the heat kernel of a complete Riemannian manifold, Amer. J. Math. 103 (1981), no. 5, 1021–1063. 10.2307/2374257Suche in Google Scholar
[7] H. Christianson and J. Marzuola, Existence and stability of solitons for the nonlinear Schrödinger equation on hyperbolic space, Nonlinearity 23 (2010), 89–106. 10.1088/0951-7715/23/1/005Suche in Google Scholar
[8] H. Christianson, J. Marzuola, J. Metcalfe and M. Taylor, Nonlinear bound states on weakly homogeneous spaces, Comm. Partial Differential Equations 39 (2014), no. 1, 34–97. 10.1080/03605302.2013.845044Suche in Google Scholar
[9] H. Christianson, J. Marzuola, J. Metcalfe and M. Taylor, Nonlinear bound states for equations with fractional Laplacian operators on weakly homogeneous spaces, in preparation. Suche in Google Scholar
[10] D. Cordero-Erausquin, B. Nazaret and C. Villani, A mass-transportation approach to sharp Sobolev and Gagliardo–Nirenberg inequalities, Adv. Math. 182 (2004), no. 2, 307–332. 10.1016/S0001-8708(03)00080-XSuche in Google Scholar
[11] E. Davies and B. Simon, Ultracontractive semigroups and some problems in analysis, Aspects of Mathematics and its Applications, North Holland, Amsterdam (1986), 265–280. 10.1016/S0924-6509(09)70260-0Suche in Google Scholar
[12] A. Debiard, B. Gaveau and E. Mazet, Theorems de comparison en geometrie Riemannienne, Publ. Res. Inst. Math. Sci. 12 (1976), 391–425. 10.2977/prims/1195190722Suche in Google Scholar
[13] M. Del Pino and J. Dolbeault, Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions, J. Math. Pures Appl. (9) 81 (2002), 847–875. 10.1016/S0021-7824(02)01266-7Suche in Google Scholar
[14] C. Draghici, Rearrangement inequalities with application to ratios of heat kernels, Potential Anal. 22 (2005), no. 4, 351–374. 10.1007/s11118-004-1328-5Suche in Google Scholar
[15] O. Druet and E. Hebey, The AB program in geometric analysis: Sharp Sobolev inequalities and related problems, Mem. Amer. Math. Soc. 160 (2002), Paper No. 761. 10.1090/memo/0761Suche in Google Scholar
[16]
R. Frank and E. Lenzmann,
Uniqueness of non-linear ground states for fractional Laplacians in
[17] R. Gangolli, Asymptotic behavior of spectra of compact quotients of certain symmetric spaces, Acta Math. 121 (1968), 151–192. 10.1007/BF02391912Suche in Google Scholar
[18] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics Math., Springer, Berlin, 2001. 10.1007/978-3-642-61798-0Suche in Google Scholar
[19] A. Grigor’yan, Heat kernel upper bounds on a complete non-compact manifold, Rev. Mat. Iberoam. 10 (1994), no. 2, 395–452. 10.4171/RMI/157Suche in Google Scholar
[20] M. Haase, The Functional Calculus for Sectorial Operators, Oper. Theory Adv. Appl. 169, Birkhäuser, Basel, 2006. 10.1007/3-7643-7698-8Suche in Google Scholar
[21] B. Hall and M. Stenzel, Sharp bounds on the heat kernel on certain symmetric spaces of non-compact type, Contemp. Math. 317 (2003), 117–135. 10.1090/conm/317/05523Suche in Google Scholar
[22] M. Harris, Numerically computing bound states, Master’s thesis, University of North Carolina, Chapel Hill, 2013, http://www.unc.edu/~marzuola/. Suche in Google Scholar
[23] E. Hebey, Sobolev Spaces on Riemannian Manifolds, Lecture Notes in Math. 1635, Springer, Berlin, 1996. 10.1007/BFb0092907Suche in Google Scholar
[24] M. Ledoux, On improved Sobolev embedding theorems, Math. Res. Lett. 10 (2003), 659–669. 10.4310/MRL.2003.v10.n5.a9Suche in Google Scholar
[25] E. Lieb and M. Loss, Analysis, 2nd ed., Grad. Stud. Math. 14, American Mathematical Society, Providence, 2001. Suche in Google Scholar
[26]
G. Mancini and K. Sandeep,
On a semilinear equation in
[27] M. Mukherjee, Nonlinear travelling waves on non-Euclidean spaces, preprint (2013), http://arxiv.org/abs/1311.5279. Suche in Google Scholar
[28] J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958), 931–954. 10.2307/2372841Suche in Google Scholar
[29] R. Schoen and S-T. Yau, Lectures on Differential Geometry, International Press, Cambridge, 2010. Suche in Google Scholar
[30] R. Strichartz, Analysis of the Laplacian on the complete Riemannian manifold, J. Funct. Anal. 52 (1983), 48–79. 10.1016/0022-1236(83)90090-3Suche in Google Scholar
[31] N. Varopoulos, Hardy–Littlewood theory for semigroups, J. Funct. Anal. 63 (1985), no. 2, 240–260. 10.1016/0022-1236(85)90087-4Suche in Google Scholar
[32] N. Varopoulos, L. Saloff-Coste and T. Coulhon, Analysis and Geometry on Groups, Cambridge University Press, Cambridge, 1992. 10.1017/CBO9780511662485Suche in Google Scholar
[33] M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1983), 567–575. 10.1007/BF01208265Suche in Google Scholar
[34] K. Yosida, Functional Analysis, Springer, Berlin, 1965. 10.1007/978-3-662-25762-3Suche in Google Scholar
© 2017 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Hurewicz fibrations, almost submetries and critical points of smooth maps
- Lower bounds for regular genus and gem-complexity of PL 4-manifolds
- Diffusion semigroup on manifolds with time-dependent metrics
- Blow-up algebras, determinantal ideals, and Dedekind–Mertens-like formulas
- A computational approach to Milnor fiber cohomology
- Metrical universality for groups
- The second and third moment of L(1/2,χ) in the hyperelliptic ensemble
- Slender domains and compact domains
- Topological 2-generation of automorphism groups of countable ultrahomogeneous graphs
- A note on local Hardy spaces
- On the value group of a model of Peano Arithmetic
- Extremal values of the (fractional) Weinstein functional on the hyperbolic space
- Maximal subsemigroups of some semigroups of order-preserving mappings on a countably infinite set
- Unitals in shift planes of odd order
Artikel in diesem Heft
- Frontmatter
- Hurewicz fibrations, almost submetries and critical points of smooth maps
- Lower bounds for regular genus and gem-complexity of PL 4-manifolds
- Diffusion semigroup on manifolds with time-dependent metrics
- Blow-up algebras, determinantal ideals, and Dedekind–Mertens-like formulas
- A computational approach to Milnor fiber cohomology
- Metrical universality for groups
- The second and third moment of L(1/2,χ) in the hyperelliptic ensemble
- Slender domains and compact domains
- Topological 2-generation of automorphism groups of countable ultrahomogeneous graphs
- A note on local Hardy spaces
- On the value group of a model of Peano Arithmetic
- Extremal values of the (fractional) Weinstein functional on the hyperbolic space
- Maximal subsemigroups of some semigroups of order-preserving mappings on a countably infinite set
- Unitals in shift planes of odd order
