American Institute of Mathematical Sciences

Advanced
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

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

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.  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

[1]

Bernhard Kawohl. Symmetry results for functions yielding best constants in Sobolev-type inequalities. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 683-690. doi: 10.3934/dcds.2000.6.683
[2]

Roberta Bosi, Jean Dolbeault, Maria J. Esteban. Estimates for the optimal constants in multipolar Hardy inequalities for Schrödinger and Dirac operators. Communications on Pure & Applied Analysis, 2008, 7 (3) : 533-562. doi: 10.3934/cpaa.2008.7.533
[3]

Ezequiel R. Barbosa, Marcos Montenegro. On the geometric dependence of Riemannian Sobolev best constants. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1759-1777. doi: 10.3934/cpaa.2009.8.1759
[4]

Barbara Brandolini, Francesco Chiacchio, Cristina Trombetti. Hardy type inequalities and Gaussian measure. Communications on Pure & Applied Analysis, 2007, 6 (2) : 411-428. doi: 10.3934/cpaa.2007.6.411
[5]

Juan Luis Vázquez, Nikolaos B. Zographopoulos. Hardy type inequalities and hidden energies. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5457-5491. doi: 10.3934/dcds.2013.33.5457
[6]

Lorenzo Brasco, Eleonora Cinti. On fractional Hardy inequalities in convex sets. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4019-4040. doi: 10.3934/dcds.2018175
[7]

Duraisamy Balraj, Muthaiah Marudai, Zoran D. Mitrovic, Ozgur Ege, Veeraraghavan Piramanantham. Existence of best proximity points satisfying two constraint inequalities. Electronic Research Archive, 2020, 28 (1) : 549-557. doi: 10.3934/era.2020028
[8]

Pradeep Boggarapu, Luz Roncal, Sundaram Thangavelu. On extension problem, trace Hardy and Hardy's inequalities for some fractional Laplacians. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2575-2605. doi: 10.3934/cpaa.2019116
[9]

Craig Cowan. Optimal Hardy inequalities for general elliptic operators with improvements. Communications on Pure & Applied Analysis, 2010, 9 (1) : 109-140. doi: 10.3934/cpaa.2010.9.109
[10]

Manli Song, Jinggang Tan. Hardy inequalities for the fractional powers of the Grushin operator. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4699-4726. doi: 10.3934/cpaa.2020192
[11]

Wenxiong Chen, Chao Jin, Congming Li, Jisun Lim. Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations. Conference Publications, 2005, 2005 (Special) : 164-172. doi: 10.3934/proc.2005.2005.164
[12]

Stathis Filippas, Luisa Moschini, Achilles Tertikas. Trace Hardy--Sobolev--Maz'ya inequalities for the half fractional Laplacian. Communications on Pure & Applied Analysis, 2015, 14 (2) : 373-382. doi: 10.3934/cpaa.2015.14.373
[13]

Jerome A. Goldstein, Ismail Kombe, Abdullah Yener. A unified approach to weighted Hardy type inequalities on Carnot groups. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2009-2021. doi: 10.3934/dcds.2017085
[14]

Elvise Berchio, Debdip Ganguly. Improved higher order poincaré inequalities on the hyperbolic space via Hardy-type remainder terms. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1871-1892. doi: 10.3934/cpaa.2016020
[15]

Judith Vancostenoble. Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 761-790. doi: 10.3934/dcdss.2011.4.761
[16]

Gianluca Gorni, Gaetano Zampieri. Lagrangian dynamics by nonlocal constants of motion. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2751-2759. doi: 10.3934/dcdss.2020216
[17]

Dirk Pauly. On Maxwell's and Poincaré's constants. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 607-618. doi: 10.3934/dcdss.2015.8.607
[18]

Toshihiro Iwai, Dmitrií A. Sadovskií, Boris I. Zhilinskií. Angular momentum coupling, Dirac oscillators, and quantum band rearrangements in the presence of momentum reversal symmetries. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020021
[19]

Francesco Fassò, Andrea Giacobbe, Nicola Sansonetto. On the number of weakly Noetherian constants of motion of nonholonomic systems. Journal of Geometric Mechanics, 2009, 1 (4) : 389-416. doi: 10.3934/jgm.2009.1.389
[20]

Marek Janasz, Piotr Pokora. On Seshadri constants and point-curve configurations. Electronic Research Archive, 2020, 28 (2) : 795-805. doi: 10.3934/era.2020040

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (24)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences