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ISSN 1088-6850(e) ISSN 0002-9947(p)

Improved Hardy and Rellich inequalities on Riemannian manifolds

Author(s): Ismail Kombe; Murad Özaydin
Journal: Trans. Amer. Math. Soc. 361 (2009), 6191-6203.
MSC (2000): Primary 26D10; Secondary 53C21
Posted: July 22, 2009
MathSciNet review: 2538592
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Abstract: In this paper we establish improved Hardy and Rellich type inequalities on a Riemannian manifold . Furthermore, we also obtain sharp constants for improved Hardy and Rellich type inequalities on the hyperbolic space $\mathbb{H}^n$ .

References:

1.: B. Abdellaoui, D. Colorado, I. Peral, Some improved Caffarelli-Kohn-Nirenberg inequalities, Calc. Var. Partial Differential Equations 23 (2005), no. 3, 327-345. MR 2142067 (2006c:26025)
2.: P. Baras and J. A. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc. 284 (1984), 121-139. MR 742415 (85f:35099)
3.: G. Barbatis, Best constants for higher-order Rellich inequalities in $L^p(\Omega)$ , Math. Z. 255 (2007), no. 4, 877-896. MR 2274540
4.: G. Barbatis, S. Filippas and A. Tertikas, A unified approach to improved Hardy inequalities with best constants, Trans. Amer. Math. Soc. 356 (2004), no. 6, 2169-2196. MR 2048514 (2005a:26016)
5.: H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complutense Madrid 10 (1997), 443-469. MR 1605678 (99a:35081)
6.: X. Cabré and Y. Martel, Existence versus explosion instantané pour des équations de la chaleur linéaires avec potentiel singulier, C. R. Acad. Sci. Paris Sér. I. Math. 329 (1999), 973-978. MR 1733904 (2000j:35117)
7.: G. Carron, Inégalités de Hardy sur les variétés riemanniennes non-compactes, J. Math. Pures Appl. (9) 76 (1997), no. 10, 883-891. MR 1489943 (99c:53026)
8.: E. B. Davies and A. M. Hinz, Explicit constants for Rellich inequalities in $L\sb p(\Omega)$ , Math. Z. 227 (1998), no. 3, 511-523. MR 1612685 (99e:58169)
9.: M. P. do Carmo and C. Xia, Complete manifolds with non-negative Ricci curvature and the Caffarelli-Kohn-Nirenberg inequalities, Compos. Math. 140 (2004), no. 3, 818-826. MR 2041783 (2005b:53052)
10.: L. Dupaigne, A nonlinear elliptic PDE with the inverse-square potential, J. Anal. Math. 86 (2002), 359-398. MR 1894489 (2003c:35046)
11.: G. B. Folland and A. Sitaram, The uncertainty principle: A mathematical survey, J. Fourier Anal. Appl. 3 (1997), 207-238. MR 1448337 (98f:42006)
12.: E. Fabes, C. Kenig and R. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 (1982), 77-116. MR 643158 (84i:35070)
13.: J. Garcia Azorero and I. Peral, Hardy inequalities and some critical elliptic and parabolic problems, J. Diff. Equations 144 (1998), 441-476. MR 1616905 (99f:35099)
14.: G. Grillo, Hardy and Rellich-type inequalities for metrics defined by vector fields, Potential Analysis 18 (2003), 187-217. MR 1953228 (2003j:46043)
15.: W. Heisenberg, Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Z. Physik 43 (1927), 172-198.
16.: I. Kombe, Hardy, Rellich and uncertainty principle inequalities on Carnot groups, preprint.
17.: P. Li and J. Wang, Weighted Poincaré inequality and rigidity of complete manifolds, Ann. Sci. École Norm. Sup. (4) 39 (2006), no. 6, 921-982. MR 2316978
18.: P. Lindqvist, On the equation div $(\vert\nabla u\vert^{p-2}\nabla u)+\lambda\vert u\vert^{p-2}u=0$ , Proc. Amer. Math. Soc. 109 (1990), 157-164. MR 1007505 (90h:35088)
19.: V. M. Miklyukov and M. L. Vuorinen, Hardy's inequality for $W\sp {1,p}\sb 0$ -functions on Riemannian manifolds, Proc. Amer. Math. Soc. 127 (1999), no. 9, 2745-2754. MR 1600117 (99m:58196)
20.: V. Minerbe, Weighted Sobolev inequalities and Ricci flat manifolds, preprint.
21.: 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), 201-224. MR 1328476 (96b:35023)
22.: F. Rellich, ``Perturbation theory of eigenvalue problems'', Gordon and Breach, New York, 1969. MR 0240668 (39:2014)
23.: L. Sun, An Uncertainty Principle on Hyperbolic Space, Proc. Amer. Math. Soc. 121 (1994), 471-479. MR 1186137 (94h:43005)
24.: J. L. Vázquez and E. Zuazua, The Hardy constant and the asymptotic behaviour of the heat equation with an inverse-square potential , J. Funct. Anal. 173 (2000), 103-153. MR 1760280 (2001j:35122)

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