A remark on Hardy type inequalities with remainder terms

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August 2011, 4(4): 801-807. doi: 10.3934/dcdss.2011.4.801

A remark on Hardy type inequalities with remainder terms

Angelo Alvino ^1, , Roberta Volpicelli ^1, and Bruno Volzone ^2,

1.	Dipartimento di Matematica e Applicazioni "Renato Caccioppoli", Università degli Studi di Napoli "Federico II", Complesso Monte S. Angelo, Via Cintia, 80126 Naples, Italy, Italy
2.	Dipartimento per le Tecnologie, Università degli Studi di Napoli, Italy

Received October 2009 Revised February 2010 Published November 2010

Abstract
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In this paper we focus our attention to some Hardy type inequalities with a remainder term. In particular we find the best value of the constant $h$ for the inequalities

$\int_{\Omega}|\nabla u|^2 dx \geq c \int_{\Omega}\frac{u^2}{|x|^2} dx+ h\int_{\Omega}\frac{u^2}{|x|}dx, \forall u\in H_0^1( \Omega) $

$ \int_{\Omega}|\nabla u|^2dx\geq c\int_{\Omega} \frac{u^2}{|x|^2}dx+ h(\int_{\Omega}|\nabla u| dx)^2, \forall u\in H_0^1 (\Omega)$

where $c\geq 0$ is smaller than the optimal Hardy constant $(N-2)^2/4$.

Keywords: best constants, Hardy inequalities, rearrangements..

Mathematics Subject Classification: Primary: 35J20, 26D10; Secondary: 46E3.

Citation: Angelo Alvino, Roberta Volpicelli, Bruno Volzone. A remark on Hardy type inequalities with remainder terms. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 801-807. doi: 10.3934/dcdss.2011.4.801

References:

[1]	Adimurthi, N. Chaudhuri and M. Ramaswamy, An improved Hardy-Sobolev inequality and its application, Proc. Amer. Math. Soc., 130 (2002), 489-505. doi: doi:10.1090/S0002-9939-01-06132-9.
[2]	A. Alvino, R. Volpicelli and B. Volzone, On Hardy inequality with a remainder term,, Ric. Mat., ().
[3]	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), 2169-2196. doi: doi:10.1090/S0002-9947-03-03389-0.
[4]	C. Bennet and R. Sharpley, "Interpolation of Operators," Pure and Appl. Math. Vol. 129, Academic Press, 1988.
[5]	E. Berchio, F. Gazzola and D. Pierotti, Gelfand type elliptic problem under Steklov boundary conditions,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, ().
[6]	H. Brezis and J. L. Vazquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469.
[7]	X. Cabré and Y. Martel, Weak eigenfunctions for the linearization of extremal elliptic problems, J. Funct. Anal., 156 (1998), 30-56. doi: doi:10.1006/jfan.1997.3171.
[8]	N. Chaudhuri and M. Ramaswamy, Existence of positive solutions of some semilinear elliptic equations with singular coefficients, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 1275-1295. doi: doi:10.1017/S0308210500001396.
[9]	S. Filippas, V. G. Maz'ja and A. Tertikas, Sharp Hardy-Sobolev inequalities, C. R. Math. Acad. Sci. Paris, 339 (2004), 483-486.
[10]	S. Filippas and A. Tertikas, Optimizing improved Hardy inequalities, J. Funct. Anal., 192 (2002), 186-233. doi: doi:10.1006/jfan.2001.3900.
[11]	F. Gazzola, H. C. Grunau and E. Mitidieri, Hardy inequalities with optimal constants and remainder terms, Trans. Amer. Math. Soc., 356 (2004), 2149-2168. doi: doi:10.1090/S0002-9947-03-03395-6.
[12]	N. Ghossoub and A. Moradifam, On the best possible remaining term in the Hardy inequality, Proc. Natl. Acad. Sci. USA, 105 (2008), 13746-13751. doi: doi:10.1073/pnas.0803703105.
[13]	G. H. Hardy, Notes on some points in the integral calculus, Messenger Math., 48 (1919), 107-112.
[14]	G. H. Hardy, J. E. Littlewood and G. Pólya, "Inequalities," Cambridge University Press, 1934.
[15]	V. G. Maz'ja, "Sobolev Spaces," Transl. from the Russian by T. O. Shaposhnikova, Springer-Verlag, 1985.
[16]	J. L. Vazquez, E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal., 173 (2000), 103-153. doi: doi:10.1006/jfan.1999.3556.

show all references

References:

[1]	Adimurthi, N. Chaudhuri and M. Ramaswamy, An improved Hardy-Sobolev inequality and its application, Proc. Amer. Math. Soc., 130 (2002), 489-505. doi: doi:10.1090/S0002-9939-01-06132-9.
[2]	A. Alvino, R. Volpicelli and B. Volzone, On Hardy inequality with a remainder term,, Ric. Mat., ().
[3]	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), 2169-2196. doi: doi:10.1090/S0002-9947-03-03389-0.
[4]	C. Bennet and R. Sharpley, "Interpolation of Operators," Pure and Appl. Math. Vol. 129, Academic Press, 1988.
[5]	E. Berchio, F. Gazzola and D. Pierotti, Gelfand type elliptic problem under Steklov boundary conditions,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, ().
[6]	H. Brezis and J. L. Vazquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469.
[7]	X. Cabré and Y. Martel, Weak eigenfunctions for the linearization of extremal elliptic problems, J. Funct. Anal., 156 (1998), 30-56. doi: doi:10.1006/jfan.1997.3171.
[8]	N. Chaudhuri and M. Ramaswamy, Existence of positive solutions of some semilinear elliptic equations with singular coefficients, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 1275-1295. doi: doi:10.1017/S0308210500001396.
[9]	S. Filippas, V. G. Maz'ja and A. Tertikas, Sharp Hardy-Sobolev inequalities, C. R. Math. Acad. Sci. Paris, 339 (2004), 483-486.
[10]	S. Filippas and A. Tertikas, Optimizing improved Hardy inequalities, J. Funct. Anal., 192 (2002), 186-233. doi: doi:10.1006/jfan.2001.3900.
[11]	F. Gazzola, H. C. Grunau and E. Mitidieri, Hardy inequalities with optimal constants and remainder terms, Trans. Amer. Math. Soc., 356 (2004), 2149-2168. doi: doi:10.1090/S0002-9947-03-03395-6.
[12]	N. Ghossoub and A. Moradifam, On the best possible remaining term in the Hardy inequality, Proc. Natl. Acad. Sci. USA, 105 (2008), 13746-13751. doi: doi:10.1073/pnas.0803703105.
[13]	G. H. Hardy, Notes on some points in the integral calculus, Messenger Math., 48 (1919), 107-112.
[14]	G. H. Hardy, J. E. Littlewood and G. Pólya, "Inequalities," Cambridge University Press, 1934.
[15]	V. G. Maz'ja, "Sobolev Spaces," Transl. from the Russian by T. O. Shaposhnikova, Springer-Verlag, 1985.
[16]	J. L. Vazquez, E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal., 173 (2000), 103-153. doi: doi:10.1006/jfan.1999.3556.

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