Improved Hardy and Rellich inequalities on Riemannian manifolds
HTML articles powered by AMS MathViewer
- by Ismail Kombe and Murad Özaydin PDF
- Trans. Amer. Math. Soc. 361 (2009), 6191-6203 Request permission
Abstract:
In this paper we establish improved Hardy and Rellich type inequalities on a Riemannian manifold $M$. Furthermore, we also obtain sharp constants for improved Hardy and Rellich type inequalities on the hyperbolic space $\mathbb {H}^n$.References
- B. Abdellaoui, E. Colorado, and I. Peral, Some improved Caffarelli-Kohn-Nirenberg inequalities, Calc. Var. Partial Differential Equations 23 (2005), no. 3, 327–345. MR 2142067, DOI 10.1007/s00526-004-0303-8
- Pierre Baras and Jerome A. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc. 284 (1984), no. 1, 121–139. MR 742415, DOI 10.1090/S0002-9947-1984-0742415-3
- G. Barbatis, Best constants for higher-order Rellich inequalities in $L^p(\Omega )$, Math. Z. 255 (2007), no. 4, 877–896. MR 2274540, DOI 10.1007/s00209-006-0056-5
- G. Barbatis, S. Filippas, and A. Tertikas, A unified approach to improved $L^p$ Hardy inequalities with best constants, Trans. Amer. Math. Soc. 356 (2004), no. 6, 2169–2196. MR 2048514, DOI 10.1090/S0002-9947-03-03389-0
- Haim Brezis and Juan Luis Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid 10 (1997), no. 2, 443–469. MR 1605678
- Xavier Cabré and Yvan Martel, Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 11, 973–978 (French, with English and French summaries). MR 1733904, DOI 10.1016/S0764-4442(00)88588-2
- G. Carron, Inégalités de Hardy sur les variétés riemanniennes non-compactes, J. Math. Pures Appl. (9) 76 (1997), no. 10, 883–891 (French, with English and French summaries). MR 1489943, DOI 10.1016/S0021-7824(97)89976-X
- E. B. Davies and A. M. Hinz, Explicit constants for Rellich inequalities in $L_p(\Omega )$, Math. Z. 227 (1998), no. 3, 511–523. MR 1612685, DOI 10.1007/PL00004389
- Manfredo Perdigão do Carmo and Changyu Xia, Complete manifolds with non-negative Ricci curvature and the Caffarelli-Kohn-Nirenberg inequalities, Compos. Math. 140 (2004), no. 3, 818–826. MR 2041783, DOI 10.1112/S0010437X03000745
- Louis Dupaigne, A nonlinear elliptic PDE with the inverse square potential, J. Anal. Math. 86 (2002), 359–398. MR 1894489, DOI 10.1007/BF02786656
- Gerald B. Folland and Alladi Sitaram, The uncertainty principle: a mathematical survey, J. Fourier Anal. Appl. 3 (1997), no. 3, 207–238. MR 1448337, DOI 10.1007/BF02649110
- Eugene B. Fabes, Carlos E. Kenig, and Raul P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 (1982), no. 1, 77–116. MR 643158, DOI 10.1080/03605308208820218
- J. P. García Azorero and I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations 144 (1998), no. 2, 441–476. MR 1616905, DOI 10.1006/jdeq.1997.3375
- Gabriele Grillo, Hardy and Rellich-type inequalities for metrics defined by vector fields, Potential Anal. 18 (2003), no. 3, 187–217. MR 1953228, DOI 10.1023/A:1020963702912
- W. Heisenberg, Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Z. Physik 43 (1927), 172-198.
- I. Kombe, Hardy, Rellich and uncertainty principle inequalities on Carnot groups, preprint.
- Peter Li and Jiaping Wang, Weighted Poincaré inequality and rigidity of complete manifolds, Ann. Sci. École Norm. Sup. (4) 39 (2006), no. 6, 921–982 (English, with English and French summaries). MR 2316978, DOI 10.1016/j.ansens.2006.11.001
- Peter Lindqvist, On the equation $\textrm {div}\,(|\nabla u|^{p-2}\nabla u)+\lambda |u|^{p-2}u=0$, Proc. Amer. Math. Soc. 109 (1990), no. 1, 157–164. MR 1007505, DOI 10.1090/S0002-9939-1990-1007505-7
- Vladimir M. Miklyukov and Matti K. Vuorinen, Hardy’s inequality for $W^{1,p}_0$-functions on Riemannian manifolds, Proc. Amer. Math. Soc. 127 (1999), no. 9, 2745–2754. MR 1600117, DOI 10.1090/S0002-9939-99-04849-2
- V. Minerbe, Weighted Sobolev inequalities and Ricci flat manifolds, preprint.
- I. Peral and J. L. Vázquez, On the stability or instability of the singular solution of the semilinear heat equation with exponential reaction term, Arch. Rational Mech. Anal. 129 (1995), no. 3, 201–224. MR 1328476, DOI 10.1007/BF00383673
- Franz Rellich, Perturbation theory of eigenvalue problems, Gordon and Breach Science Publishers, New York-London-Paris, 1969. Assisted by J. Berkowitz; With a preface by Jacob T. Schwartz. MR 0240668
- Li Min Sun, An uncertainty principle on hyperbolic space, Proc. Amer. Math. Soc. 121 (1994), no. 2, 471–479. MR 1186137, DOI 10.1090/S0002-9939-1994-1186137-8
- Juan Luis Vazquez and Enrike Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal. 173 (2000), no. 1, 103–153. MR 1760280, DOI 10.1006/jfan.1999.3556
Additional Information
- Ismail Kombe
- Affiliation: Department of Mathematics, Dawson-Loeffler Science & Mathematics Bldg., Oklahoma City University, 2501 N. Blackwelder, Oklahoma City, Oklahoma 73106-1493
- MR Author ID: 720054
- Email: ikombe@okcu.edu
- Murad Özaydin
- Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019-0315
- Email: mozaydin@math.ou.edu
- Received by editor(s): March 13, 2007
- Published electronically: July 22, 2009
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 361 (2009), 6191-6203
- MSC (2000): Primary 26D10; Secondary 53C21
- DOI: https://doi.org/10.1090/S0002-9947-09-04642-X
- MathSciNet review: 2538592