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 & 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. doi: doi:10.1090/S0002-9939-01-06132-9. Google Scholar
[2]	A. Alvino, R. Volpicelli and B. Volzone, On Hardy inequality with a remainder term,, Ric. Mat., (). Google Scholar
[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. doi: doi:10.1090/S0002-9947-03-03389-0. Google Scholar
[4]	C. Bennet and R. Sharpley, "Interpolation of Operators,", Pure and Appl. Math. Vol. \textbf{129}, 129 (1988). Google Scholar
[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, (). Google Scholar
[6]	H. Brezis and J. L. Vazquez, Blow-up solutions of some nonlinear elliptic problems,, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443. Google Scholar
[7]	X. Cabré and Y. Martel, Weak eigenfunctions for the linearization of extremal elliptic problems,, J. Funct. Anal., 156 (1998), 30. doi: doi:10.1006/jfan.1997.3171. Google Scholar
[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. doi: doi:10.1017/S0308210500001396. Google Scholar
[9]	S. Filippas, V. G. Maz'ja and A. Tertikas, Sharp Hardy-Sobolev inequalities,, C. R. Math. Acad. Sci. Paris, 339 (2004), 483. Google Scholar
[10]	S. Filippas and A. Tertikas, Optimizing improved Hardy inequalities,, J. Funct. Anal., 192 (2002), 186. doi: doi:10.1006/jfan.2001.3900. Google Scholar
[11]	F. Gazzola, H. C. Grunau and E. Mitidieri, Hardy inequalities with optimal constants and remainder terms,, Trans. Amer. Math. Soc., 356 (2004), 2149. doi: doi:10.1090/S0002-9947-03-03395-6. Google Scholar
[12]	N. Ghossoub and A. Moradifam, On the best possible remaining term in the Hardy inequality,, Proc. Natl. Acad. Sci. USA, 105 (2008), 13746. doi: doi:10.1073/pnas.0803703105. Google Scholar
[13]	G. H. Hardy, Notes on some points in the integral calculus,, Messenger Math., 48 (1919), 107. Google Scholar
[14]	G. H. Hardy, J. E. Littlewood and G. Pólya, "Inequalities,", Cambridge University Press, (1934). Google Scholar
[15]	V. G. Maz'ja, "Sobolev Spaces,", Transl. from the Russian by T. O. Shaposhnikova, (1985). Google Scholar
[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. doi: doi:10.1006/jfan.1999.3556. Google Scholar

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References:

[1]	Adimurthi, N. Chaudhuri and M. Ramaswamy, An improved Hardy-Sobolev inequality and its application,, Proc. Amer. Math. Soc., 130 (2002), 489. doi: doi:10.1090/S0002-9939-01-06132-9. Google Scholar
[2]	A. Alvino, R. Volpicelli and B. Volzone, On Hardy inequality with a remainder term,, Ric. Mat., (). Google Scholar
[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. doi: doi:10.1090/S0002-9947-03-03389-0. Google Scholar
[4]	C. Bennet and R. Sharpley, "Interpolation of Operators,", Pure and Appl. Math. Vol. \textbf{129}, 129 (1988). Google Scholar
[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, (). Google Scholar
[6]	H. Brezis and J. L. Vazquez, Blow-up solutions of some nonlinear elliptic problems,, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443. Google Scholar
[7]	X. Cabré and Y. Martel, Weak eigenfunctions for the linearization of extremal elliptic problems,, J. Funct. Anal., 156 (1998), 30. doi: doi:10.1006/jfan.1997.3171. Google Scholar
[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. doi: doi:10.1017/S0308210500001396. Google Scholar
[9]	S. Filippas, V. G. Maz'ja and A. Tertikas, Sharp Hardy-Sobolev inequalities,, C. R. Math. Acad. Sci. Paris, 339 (2004), 483. Google Scholar
[10]	S. Filippas and A. Tertikas, Optimizing improved Hardy inequalities,, J. Funct. Anal., 192 (2002), 186. doi: doi:10.1006/jfan.2001.3900. Google Scholar
[11]	F. Gazzola, H. C. Grunau and E. Mitidieri, Hardy inequalities with optimal constants and remainder terms,, Trans. Amer. Math. Soc., 356 (2004), 2149. doi: doi:10.1090/S0002-9947-03-03395-6. Google Scholar
[12]	N. Ghossoub and A. Moradifam, On the best possible remaining term in the Hardy inequality,, Proc. Natl. Acad. Sci. USA, 105 (2008), 13746. doi: doi:10.1073/pnas.0803703105. Google Scholar
[13]	G. H. Hardy, Notes on some points in the integral calculus,, Messenger Math., 48 (1919), 107. Google Scholar
[14]	G. H. Hardy, J. E. Littlewood and G. Pólya, "Inequalities,", Cambridge University Press, (1934). Google Scholar
[15]	V. G. Maz'ja, "Sobolev Spaces,", Transl. from the Russian by T. O. Shaposhnikova, (1985). Google Scholar
[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. doi: doi:10.1006/jfan.1999.3556. Google Scholar

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