On the capacity approach to non-attainability of Hardy's inequality in $\mathbb{R}^N$

Daniele Cassani; Bernhard Ruf; Cristina Tarsi

doi:10.3934/dcdss.2019017

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April 2019, 12(2): 245-250. doi: 10.3934/dcdss.2019017

On the capacity approach to non-attainability of Hardy's inequality in $\mathbb{R}^N$

Daniele Cassani ^1,2,, , Bernhard Ruf ^3, and Cristina Tarsi ^3,

1.	Dip. di Scienza e Alta Tecnologia, Università degli Studi dell'Insubria
2.	RISM-Riemann International School of Mathematics, via G.B. Vico 46, 21100 - Varese, Italy
3.	Dip. di Matematica, Università degli Studi di Milano, via C. Saldini 50, 20133 - Milano, Italy

* Corresponding author: Daniele Cassani

In honor of Vicentiu Rădulescu on the occasion of his 60^th birthday, with friendship and admiration

Received August 2017 Revised December 2017 Published August 2018

In this note we exploit nonlinear capacity estimates in the spirit of Mitidieri-Pohozaev [15] in the context of Lorentz spaces. This from one side yields a simple proof, though non-optimal, of non-attainability of Hardy's inequality in $\mathbb{R}^N$, on the other side gives a partial positive answer to a conjecture raised in [15].

Keywords: Functional inequalities, Hardy inequality, Lorentz spaces, Liouville theorems, non-existence results.

Mathematics Subject Classification: Primary: 35A23; Secondary: 46E30, 35B53.

Citation: Daniele Cassani, Bernhard Ruf, Cristina Tarsi. On the capacity approach to non-attainability of Hardy's inequality in $\mathbb{R}^N$. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 245-250. doi: 10.3934/dcdss.2019017

References:

[1]	A. Alvino, Sulla diseguaglianza di Sobolev in spazi di Lorentz, Boll. Un. Mat. Ital. A, 14 (1977), 148-156.
[2]	C. Bennett and R. Sharpley, Interpolation of Operators, Pure and Applied Mathematics, 129, Boston Academic Press, 1988.
[3]	H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469.
[4]	D. Cassani, B. Ruf and C. Tarsi, Equivalent and attained version of Hardy's inequality in $\mathbb{R}^n$, preprint, (2017), arXiv: 1711.03763.
[5]	D. Cassani, F. Sani and C. Tarsi, Equivalent Moser type inequalities in $\mathbb{R}^2$ and the zero mass case, J. Funct. Anal., 267 (2014), 4236-4263. doi: 10.1016/j.jfa.2014.09.022.
[6]	A. Cianchi and A. Ferone, Hardy inequalities with non-standard remainder terms, Ann. Inst. H. Poincare. An. Non Lineaire, 25 (2008), 889-906. doi: 10.1016/j.anihpc.2007.05.003.
[7]	S. Costea, Sobolev-Lorentz spaces in the Euclidean setting and counterexamples, Nonlinear Anal., 152 (2017), 149-182. doi: 10.1016/j.na.2017.01.001.
[8]	E. B. Davies, A review of Hardy inequalities, Oper. Theory Adv. Appl., 110 (1999), 55-67.
[9]	B. Devyver, M. Fraas and Y. Pinchover, Optimal hardy weight for second-order elliptic operator: An answer to a problem of Agmon, J. Funct. Anal., 266 (2014), 4422-4489. doi: 10.1016/j.jfa.2014.01.017.
[10]	S. Filippas and A. Tertikas, Optimizing improved Hardy inequalities, J. Funct. Anal., 192 (2002), 186-233. doi: 10.1006/jfan.2001.3900.
[11]	N. Ghoussoub and A. Moradifam, Functional Inequalities: New Perspectives and New Applications. Mathematical Surveys and Monographs, 187 Amer. Math. Soc., Providence, RI, 2013. doi: 10.1090/surv/187.
[12]	A. Kufner, L. Maligranda and L.-E. Persson, The prehistory of the Hardy inequality, Amer. Math. Monthly, 113 (2006), 715-732. doi: 10.1080/00029890.2006.11920356.
[13]	G. G. Lorentz, Some new functional spaces, Ann. of Math., 51 (1950), 37-55. doi: 10.2307/1969496.
[14]	V. Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Second ed., Grundlehren der Mathematischen Wissenschaften 342, Springer, 2011. doi: 10.1007/978-3-642-15564-2.
[15]	E. Mitidieri and S. Pohozaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math., 234 (2001), 1-362.
[16]	B. Opic and A. Kufner, Hardy-type Inequalities, Pitman Research Notes in Mathematics Series, 219, 1990.

show all references

References:

[1]	A. Alvino, Sulla diseguaglianza di Sobolev in spazi di Lorentz, Boll. Un. Mat. Ital. A, 14 (1977), 148-156.
[2]	C. Bennett and R. Sharpley, Interpolation of Operators, Pure and Applied Mathematics, 129, Boston Academic Press, 1988.
[3]	H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469.
[4]	D. Cassani, B. Ruf and C. Tarsi, Equivalent and attained version of Hardy's inequality in $\mathbb{R}^n$, preprint, (2017), arXiv: 1711.03763.
[5]	D. Cassani, F. Sani and C. Tarsi, Equivalent Moser type inequalities in $\mathbb{R}^2$ and the zero mass case, J. Funct. Anal., 267 (2014), 4236-4263. doi: 10.1016/j.jfa.2014.09.022.
[6]	A. Cianchi and A. Ferone, Hardy inequalities with non-standard remainder terms, Ann. Inst. H. Poincare. An. Non Lineaire, 25 (2008), 889-906. doi: 10.1016/j.anihpc.2007.05.003.
[7]	S. Costea, Sobolev-Lorentz spaces in the Euclidean setting and counterexamples, Nonlinear Anal., 152 (2017), 149-182. doi: 10.1016/j.na.2017.01.001.
[8]	E. B. Davies, A review of Hardy inequalities, Oper. Theory Adv. Appl., 110 (1999), 55-67.
[9]	B. Devyver, M. Fraas and Y. Pinchover, Optimal hardy weight for second-order elliptic operator: An answer to a problem of Agmon, J. Funct. Anal., 266 (2014), 4422-4489. doi: 10.1016/j.jfa.2014.01.017.
[10]	S. Filippas and A. Tertikas, Optimizing improved Hardy inequalities, J. Funct. Anal., 192 (2002), 186-233. doi: 10.1006/jfan.2001.3900.
[11]	N. Ghoussoub and A. Moradifam, Functional Inequalities: New Perspectives and New Applications. Mathematical Surveys and Monographs, 187 Amer. Math. Soc., Providence, RI, 2013. doi: 10.1090/surv/187.
[12]	A. Kufner, L. Maligranda and L.-E. Persson, The prehistory of the Hardy inequality, Amer. Math. Monthly, 113 (2006), 715-732. doi: 10.1080/00029890.2006.11920356.
[13]	G. G. Lorentz, Some new functional spaces, Ann. of Math., 51 (1950), 37-55. doi: 10.2307/1969496.
[14]	V. Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Second ed., Grundlehren der Mathematischen Wissenschaften 342, Springer, 2011. doi: 10.1007/978-3-642-15564-2.
[15]	E. Mitidieri and S. Pohozaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math., 234 (2001), 1-362.
[16]	B. Opic and A. Kufner, Hardy-type Inequalities, Pitman Research Notes in Mathematics Series, 219, 1990.

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On the capacity approach to non-attainability of Hardy's inequality in $\mathbb{R}^N$

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