Available in electronic format
Available in print format		 Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Hardy inequalities with optimal constants and remainder terms

Author(s): Filippo Gazzola; Hans-Christoph Grunau; Enzo Mitidieri
Journal: Trans. Amer. Math. Soc. 356 (2004), 2149-2168.
MSC (2000): Primary 46E35; Secondary 35B50, 35J40
Posted: December 9, 2003
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: We show that the classical Hardy inequalities with optimal constants in the Sobolev spaces $W_0^{1,p}$ and in higher-order Sobolev spaces on a bounded domain $\Omega\subset\mathbb{R} ^n$ can be refined by adding remainder terms which involve $L^p$ norms. In the higher-order case further $L^p$ norms with lower-order singular weights arise. The case $1<p<2$ being more involved requires a different technique and is developed only in the space $W_0^{1,p}$.

References:

[ACR]
Adimurthi, N. Chaudhuri, M. Ramaswamy, An improved Hardy-Sobolev inequality and its application, Proc. Amer. Math. Soc. 130 (2002), 489-505. MR 2002j:35232

[AE]
Adimurthi, M.J. Esteban, An improved Hardy-Sobolev inequality in $W^{1,p}$ and its application to Schrödinger operator, Nonlinear Differ. Eq. Appl. NoDEA, to appear.

[AL]
F. Almgren, E. Lieb, Symmetric decreasing rearrangement is sometimes continuous, J. Amer. Math. Soc. 2 (1989), 683-773. MR 90f:49038

[AGGM]
G. Arioli, F. Gazzola, H.-Ch. Grunau, E. Mitidieri, A semilinear fourth order elliptic problem with exponential nonlinearity, preprint.

[BFT]
G. Barbatis, S. Filippas, A. Tertikas, A unified approach to improved $L^p$ Hardy inequalities with best constants, Trans. Amer. Math. Soc., this issue.

[Bo]
T. Boggio, Sulle funzioni di Green d'ordine $m$, Rend. Circ. Mat. Palermo 20 (1905), 97-135.

[BL]
H. Brezis, E. Lieb, Sobolev inequalities with remainder terms, J. Funct. Anal. 62 (1985), 73-86. MR 86i:46033

[BM]
H. Brezis, M. Marcus, Hardy's inequalities revisited, Ann. Sc. Norm. Sup. Pisa 25 (1997), 217-237. MR 99m:46075

[BMS]
H. Brezis, M. Marcus, I. Shafrir, Extremal functions for Hardy's inequality with weight, J. Funct. Anal. 171 (2000), 177-191. MR 2001g:46068

[BV]
H. Brezis, J. L. Vazquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complutense Madrid 10 (1997), 443-469. MR 99a:35081

[DA]
L. D'Ambrosio, Some Hardy inequalities on the Heisenberg group, Differential Equations, in press.

[DH]
E. B. Davies, A. M. Hinz, Explicit constants for Rellich inequalities in $L_p (\Omega)$, Math. Z. 227 (1998), 511-523. MR 99e:58169

[E]
H. Egnell, Existence and nonexistence results for $m$-Laplace equations involving critical Sobolev exponents, Arch. Ration. Mech. Anal. 104 (1988), 57-77. MR 90e:35069

[FT]
S. Filippas, A. Tertikas, Optimizing improved Hardy inequalities, J. Funct. Anal. 192 (2002), 186-233. MR 2003f:46045

[GG]
F. Gazzola, H.-Ch. Grunau, Critical dimensions and higher order Sobolev inequalities with remainder terms, Nonlin. Diff. Eq. Appl. NoDEA 8 (2001), 35-44. MR 2002d:35024

[H]
G. H. Hardy, Notes on some points in the integral calculus, Messenger Math. 48 (1919), 107-112.

[HLP]
G. H. Hardy, J. E. Littlewood, G. P'olya, Inequalities, Cambridge: University Press, 1934.

[L]
P. Lindqvist, On the equation div $(\vert \nabla u\vert \sp{p-2}\nabla u)+ \lambda \vert u\vert \sp{p-2}u=0$, Proc. Amer. Math. Soc. 109 (1990), 157-164. MR 90h:35088

[MS]
M. Marcus, I. Shafrir, An eigenvalue problem related to Hardy's $L^p$ inequality, Ann. Scuola Norm. Sup. Pisa Ser. IV 29 (2000), 581-604. MR 2002c:49082

[Ma]
V. G. Maz'ya, Sobolev spaces, Transl. from the Russian by T. O. Shaposhnikova, Berlin etc.: Springer-Verlag, 1985. MR 87g:46056

[Mi]
E. Mitidieri, A simple approach to Hardy's inequalities, Math. Notes 67 (2000), 479-486, translation from Mat. Zametki 67 (2000), 563-572. MR 2001f:26022

[Mo]
J. J. Moreau, Décomposition orthogonale d'un espace hilbertien selon deux cônes mutuellement polaires, C. R. Acad. Sci. Paris 255 (1962), 238-240. MR 25:3346

[PS]
P. Pucci, J. Serrin, Critical exponents and critical dimensions for polyharmonic operators, J. Math. Pures Appl. 69 (1990), 55-83. MR 91i:35065

[R]
F. Rellich, Halbbeschränkte Differentialoperatoren höherer Ordnung, in: J. C. H. Gerretsen et al. (eds.), Proceedings of the International Congress of Mathematicians Amsterdam 1954, Vol. III, pp. 243-250, Groningen: Nordhoff, 1956. MR 19:550c

[Ta]
G. Talenti, Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa Ser. IV 3 (1976), 697-718. MR 58:29170

[To]
P. Tolksdorf, Regularity for a more general class of quasilinear equations, J. Differ. Equations 51 (1984), 126-150. MR 85g:35047

[VZ]
J. L. Vazquez, 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 2001j:35122

[V]
R. van der Vorst, Best constant for the embedding of the space $H^2\cap H^1_0 (\Omega)$ into $L^{2N/(N-4)} (\Omega)$, Diff. Int. Eq. 6 (1993), 259-276. MR 94b:46053

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 46E35, 35B50, 35J40

Retrieve articles in all Journals with MSC (2000): 46E35, 35B50, 35J40

Additional Information:

Filippo Gazzola
Affiliation: Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci, 32, I-20133 Milano, Italy
Email: gazzola@mate.polimi.it

Hans-Christoph Grunau
Affiliation: Fakultät für Mathematik, Otto-von-Guericke-Universität, Postfach 4120, D-39016 Magdeburg, Germany
Email: Hans-Christoph.Grunau@mathematik.uni-magdeburg.de

Enzo Mitidieri
Affiliation: Dipartimento di Scienze Matematiche, Via A. Valerio 12/1, Università degli Studi di Trieste, I-34100 Trieste, Italy
Email: mitidier@univ.trieste.it

DOI: 10.1090/S0002-9947-03-03395-6
PII: S 0002-9947(03)03395-6
Received by editor(s): June 20, 2000
Received by editor(s) in revised form: May 8, 2003
Posted: December 9, 2003
Copyright of article: Copyright 2003, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google