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Extremal values of the (fractional) Weinstein functional on the hyperbolic space

  • Mayukh Mukherjee EMAIL logo
Published/Copyright: September 16, 2016

Abstract

We study Weinstein functionals, first defined in [33], mainly on the hyperbolic space n. We are primarily interested in the existence of Weinstein functional maximizers or, in other words, existence of extremal functions for the best constant of the Gagliardo–Nirenberg inequality. The main result is that the supremum of the Weinstein functional on n is the same as that on n and the related fact that the said supremum is not attained on n, when functions are chosen from the Sobolev space H1(n). This proves a conjecture made in [8] (see also [3]). We also prove an analogous version of the conjecture for the Weinstein functional defined with the fractional Laplacian.

MSC 2010: 35J61; 35H20

Communicated by Christopher D. Sogge


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.2008.09.013Search 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.1960.10.419Search in Google Scholar

[3] V. Banica, The nonlinear Schrödinger equation on hyperbolic space, Comm. Partial Differential Equations 32 (2007), 1643–1677. 10.1080/03605300600854332Search 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/. Search 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/S0013091501000426Search 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. 10.2307/2374257Search 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. 10.1088/0951-7715/23/1/005Search 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. 10.1080/03605302.2013.845044Search 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. Search 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-XSearch 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)70260-0Search 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/1195190722Search 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)01266-7Search 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/s11118-004-1328-5Search 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/0761Search in Google Scholar

[16] R. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in , Acta Math. 210 (2013), no. 2, 261–318. 10.1007/s11511-013-0095-9Search in Google Scholar

[17] R. Gangolli, Asymptotic behavior of spectra of compact quotients of certain symmetric spaces, Acta Math. 121 (1968), 151–192. 10.1007/BF02391912Search in Google Scholar

[18] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics Math., Springer, Berlin, 2001. 10.1007/978-3-642-61798-0Search 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/157Search in Google Scholar

[20] M. Haase, The Functional Calculus for Sectorial Operators, Oper. Theory Adv. Appl. 169, Birkhäuser, Basel, 2006. 10.1007/3-7643-7698-8Search 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/317/05523Search 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/. Search in Google Scholar

[23] E. Hebey, Sobolev Spaces on Riemannian Manifolds, Lecture Notes in Math. 1635, Springer, Berlin, 1996. 10.1007/BFb0092907Search in Google Scholar

[24] M. Ledoux, On improved Sobolev embedding theorems, Math. Res. Lett. 10 (2003), 659–669. 10.4310/MRL.2003.v10.n5.a9Search in Google Scholar

[25] E. Lieb and M. Loss, Analysis, 2nd ed., Grad. Stud. Math. 14, American Mathematical Society, Providence, 2001. Search in Google Scholar

[26] G. Mancini and K. Sandeep, On a semilinear equation in n, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 7 (2008), 635–671. 10.2422/2036-2145.2008.4.03Search in Google Scholar

[27] M. Mukherjee, Nonlinear travelling waves on non-Euclidean spaces, preprint (2013), http://arxiv.org/abs/1311.5279. Search in Google Scholar

[28] J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958), 931–954. 10.2307/2372841Search in Google Scholar

[29] R. Schoen and S-T. Yau, Lectures on Differential Geometry, International Press, Cambridge, 2010. Search 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)90090-3Search 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)90087-4Search in Google Scholar

[32] N. Varopoulos, L. Saloff-Coste and T. Coulhon, Analysis and Geometry on Groups, Cambridge University Press, Cambridge, 1992. 10.1017/CBO9780511662485Search in Google Scholar

[33] M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1983), 567–575. 10.1007/BF01208265Search in Google Scholar

[34] K. Yosida, Functional Analysis, Springer, Berlin, 1965. 10.1007/978-3-662-25762-3Search in Google Scholar

Received: 2015-2-17
Revised: 2016-8-14
Published Online: 2016-9-16
Published in Print: 2017-7-1

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