A Sharp Rellich Inequality on the Sphere
Abstract
:1. Introduction
2. The Proof of the Main Result
Funding
Conflicts of Interest
References
- Rellich, F. Perturbation Theory of Eigenvalue Problems; Gordon and Breach: New York, NY, USA, 1969. [Google Scholar]
- Brezis, H.; Vázquez, J.L. Blow-up solutions of some nonlinear elliptic problems. Rev. Mat. Univ. Comp. Madrid 1997, 10, 443–469. [Google Scholar]
- Costa, D.G. On Hardy-Rellich type inequalities in . Appl. Math. Lett. 2009, 22, 902–905. [Google Scholar] [CrossRef]
- Davies, E.B.; Hinz, A.M. Explicit constants for Rellich inequalities in Lp(Ω). Math. Z. 1998, 227, 511–523. [Google Scholar] [CrossRef]
- Gazzola, F.; Grunau, H.-C.; Mitidieri, E. Hardy inequalities with optimal constants and remainder terms. Trans. Am. Math. Soc. 2004, 356, 2149–2168. [Google Scholar] [CrossRef]
- Tertikas, A.; Zographopoulos, N. Best constants in the Hardy-Rellich Inequalities and Related Improvements. Adv. Math. 2007, 209, 407–459. [Google Scholar] [CrossRef]
- Kombe, I.; Özaydin, M. Improved Hardy and Rellich inequalities on Riemannian manifolds. Trans. Am. Math. Soc. 2009, 361, 6191–6203. [Google Scholar] [CrossRef] [Green Version]
- Grillo, G. Hardy and Rellich-type inequalities for metrics defined by vector fields. Potential Anal. 2003, 18, 187–217. [Google Scholar] [CrossRef]
- Ghoussoub, N.; Moradifam, A. Bessel pairs and optimal Hardy and Hardy-Rellich inequalities. Math. Ann. 2011, 349, 1–57. [Google Scholar] [CrossRef]
- Kombe, I.; Özaydin, M. Hardy-Poincaré, Rellich and uncertainty principle inequalities on Riemannian manifolds. Trans. Am. Math. Soc. 2013, 365, 5035–5050. [Google Scholar] [CrossRef] [Green Version]
- Du, F.; Mao, J. Hardy and Rellich type inequalities on metric measure spaces. J. Math. Anal. Appl. 2015, 429, 354–365. [Google Scholar] [CrossRef]
- Abolarinwa, A.; Apata, T. Lp-Hardy-Rellich and uncertainty principle inequalities on the sphere. Adv. Oper. Theory 2018, 3, 745–762. [Google Scholar] [CrossRef]
- Dai, F.; Xu, Y. The Hardy-Rellich Inequality and Uncertainty Principle Inequalities on the Sphere. Constr. Approx. 2014, 40, 141–171. [Google Scholar] [CrossRef]
- Xiao, Y. Some Hardy inequalities on the sphere. J. Math. Inequal. 2016, 10, 793–805. [Google Scholar] [CrossRef]
- Bai, Z.; Shen, Y.; Shui, N.; Guo, X. An introduction to Riemann Geometry; Higher Education Press: Beijing, China, 2004; Volume 12. [Google Scholar]
- Yang, Q.; Su, D.; Kong, Y. Hardy inequalities on Riemannian manifolds with negative curvature. Commun. Contemp. Math. 2014, 16, 1350043. [Google Scholar] [CrossRef]
© 2018 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Yin, S. A Sharp Rellich Inequality on the Sphere. Mathematics 2018, 6, 288. https://doi.org/10.3390/math6120288
Yin S. A Sharp Rellich Inequality on the Sphere. Mathematics. 2018; 6(12):288. https://doi.org/10.3390/math6120288
Chicago/Turabian StyleYin, Songting. 2018. "A Sharp Rellich Inequality on the Sphere" Mathematics 6, no. 12: 288. https://doi.org/10.3390/math6120288
APA StyleYin, S. (2018). A Sharp Rellich Inequality on the Sphere. Mathematics, 6(12), 288. https://doi.org/10.3390/math6120288