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A remark on Hardy type inequalities with remainder terms
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 |
$\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 $(N-2)^2/4$.
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. |
[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. |
[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, ().
|
[6] |
H. Brezis and J. L. Vazquez, Blow-up solutions of some nonlinear elliptic problems,, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443.
|
[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. |
[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. |
[9] |
S. Filippas, V. G. Maz'ja and A. Tertikas, Sharp Hardy-Sobolev inequalities,, C. R. Math. Acad. Sci. Paris, 339 (2004), 483.
|
[10] |
S. Filippas and A. Tertikas, Optimizing improved Hardy inequalities,, J. Funct. Anal., 192 (2002), 186.
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.
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.
doi: doi:10.1073/pnas.0803703105. |
[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. |
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. |
[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. |
[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, ().
|
[6] |
H. Brezis and J. L. Vazquez, Blow-up solutions of some nonlinear elliptic problems,, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443.
|
[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. |
[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. |
[9] |
S. Filippas, V. G. Maz'ja and A. Tertikas, Sharp Hardy-Sobolev inequalities,, C. R. Math. Acad. Sci. Paris, 339 (2004), 483.
|
[10] |
S. Filippas and A. Tertikas, Optimizing improved Hardy inequalities,, J. Funct. Anal., 192 (2002), 186.
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.
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.
doi: doi:10.1073/pnas.0803703105. |
[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. |
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