Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A unified approach to improved $L^p$ Hardy inequalities with best constants

Authors: G. Barbatis, S. Filippas and A. Tertikas
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 2169-2196
MSC (2000): Primary 35J20, 26D10; Secondary 46E35, 35Pxx
Published electronically: December 9, 2003
MathSciNet review: 2048514
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We present a unified approach to improved $L^p$ Hardy inequalities in $\mathbf{R}^N$. We consider Hardy potentials that involve either the distance from a point, or the distance from the boundary, or even the intermediate case where the distance is taken from a surface of codimension $1<k<N$. In our main result, we add to the right hand side of the classical Hardy inequality a weighted $L^p$ norm with optimal weight and best constant. We also prove nonhomogeneous improved Hardy inequalities, where the right hand side involves weighted $L^q$ norms, $q \neq p$.

References [Enhancements On Off] (What's this?)

  • [AS] Ambrosio L. and Soner H.M. Level set approach to mean curvature flow in arbitrary codimension. J. Diff. Geometry 43 (1996) 693-737. MR 97k:58047
  • [BC] Baras P. and Cohen L. Complete blow-up after $T_{{\rm max}}$for the solution of a semilinear heat equation. J. Funct. Anal. 71 (1987) 142-174. MR 88e:35105
  • [BL] Brezis H. and Lieb E.H. Sobolev inequalities with remainder terms. J. Funct. Anal. 62 (1985) 73-86. MR 86i:46033
  • [BM] Brezis H. and Marcus M. Hardy's inequalities revisited. Ann. Scuola Norm. Pisa 25 (1997) 217-237. MR 99m:46075
  • [BMS] Brezis H., Marcus M. and Shafrir I. Extremal functions for Hardy's inequality with weight. J. Funct. Anal. 171 (2000) 177-191. MR 2001g:46068
  • [BV] Brezis H. and Vázquez J.-L. Blow-up solutions of some nonlinear elliptic problems. Rev. Mat. Univ. Comp. Madrid 10 (1997) 443-469. MR 99a:35081
  • [CM] Cabré X. and Martel Y. Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier. C.R. Acad. Sci. Paris Ser. I Math. 329 (1999) 973-978. MR 2000j:35117
  • [D] Davies E.B. A review of Hardy inequalities. Oper. Theory Adv. Appl. 110 (1998) 55-67. MR 2001f:35166
  • [DH] Davies E.B. and Hinz A.M. Explicit constants for Rellich inequalities in $L_p(\Omega)$. Math. Z. 227 (1998) 511-523. MR 99e:58169
  • [DM] Davies E.B. and Mandouvalos N. The hyperbolic geometry and spectrum of irregular domains. Nonlinearity 3 (1990) 913-945. MR 92c:58148
  • [EG] Evans L.C. and Gariepy R.F. Measure theory and fine properties of functions. CRC Press 1992. MR 93f:28001
  • [FT] Filippas S. and Tertikas A. Optimizing improved Hardy inequalities. J. Funct. Anal. 192 (2002) 186-233. MR 2003f:46045
  • [FHT] Fleckinger J., Harrell II M. E. and Thelin F. Boundary behavior and estimates for solutions of equations containing the $p$-Laplacian. Electron. J. Diff. Equations 38 (1999) 1-19. MR 2000k:35083
  • [GGM] Gazzola F., Grunau H.-Ch. and Mitidieri E. Hardy inequalities with optimal constants and remainder terms. Trans. Amer. Math. Soc., this issue.
  • [GP] Garcia J.P. and Peral I. Hardy inequalities and some critical elliptic and parabolic problems. J. Diff. Equations 144 (1998) 441-476. MR 99f:35099
  • [HKM] Heinonen J, Kilpelinen T. and Martio O. Nonlinear potential theory of degenerate elliptic equations. The Clarendon Press, Oxford University Press, New York, 1993. MR 94e:31003
  • [HLP] Hardy G., Pólya G. and Littlewood J.E. Inequalities. 2nd edition, Cambridge University Press 1952. MR 13:727e
  • [L] Lindqvist P. On the equation $\operatorname{div}(\vert\nabla u\vert^{p-2} \nabla u)+\lambda\vert u\vert^{p-2}u=0$. Proc. Amer. Math. Soc. 109 (1990) 157-164. MR 90h:35088
  • [MMP] Marcus M., Mizel V.J. and Pinchover Y. On the best constant for Hardy's inequality in $\mathbf{R}^n$. Trans. Amer. Math. Soc. 350 (1998) 3237-3255. MR 98k:26035
  • [MS] Matskewich T. and Sobolevskii P.E. The best possible constant in generalized Hardy's inequality for convex domain in $\mathbf{R}^n$. Nonlinear Anal., Theory, Methods & Appl., 28 (1997) 1601-1610. MR 98a:26019
  • [M] Maz'ja V.G. Sobolev spaces. Springer 1985. MR 87g:46056
  • [OK] Opic B. and Kufner A. Hardy-type inequalities. Pitman Research Notes in Math., vol. 219, Longman 1990. MR 92b:26028
  • [PV] Peral I. and Vázquez J.L. 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 96b:35023
  • [S] Sneider R. Convex bodies: The Brunn-Minkowski theory. Encyclopedia of Math. and Its Applications 44, Cambridge University Press, 1993. MR 94d:52007
  • [T] Tertikas A. Critical phenomena in linear elliptic problems. J. Funct. Anal. 154 (1998) 42-66. MR 99d:35039
  • [V] Vazquez J.L. Domain of existence and blowup for the exponential reaction-diffusion equation. Indiana Univ. Math. J. 48 (1999) 677-709. MR 2001c:35114
  • [VZ] Vazquez J.L. and Zuazua E., The Hardy inequality and the asymptotic behavior of the heat equation with an inverse-square potential. J. Funct. Anal. 173 (2000) 103-153. MR 2001j:35122

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35J20, 26D10, 46E35, 35Pxx

Retrieve articles in all journals with MSC (2000): 35J20, 26D10, 46E35, 35Pxx

Additional Information

G. Barbatis
Affiliation: Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece

S. Filippas
Affiliation: Department of Applied Mathematics, University of Crete, 71409 Heraklion, Greece

A. Tertikas
Affiliation: Department of Mathematics, University of Crete, 71409 Heraklion, Greece and Institute of Applied and Computational Mathematics, FORTH, 71110 Heraklion, Greece

Keywords: Hardy inequalities, best constants, distance function, weighted norms
Received by editor(s): February 28, 2001
Published electronically: December 9, 2003
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society