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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 60th 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 & 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.   Google Scholar

[2]

C. Bennett and R. Sharpley, Interpolation of Operators, Pure and Applied Mathematics, 129, Boston Academic Press, 1988.  Google Scholar

[3]

H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

E. B. Davies, A review of Hardy inequalities, Oper. Theory Adv. Appl., 110 (1999), 55-67.   Google Scholar

[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.  Google Scholar

[10]

S. Filippas and A. Tertikas, Optimizing improved Hardy inequalities, J. Funct. Anal., 192 (2002), 186-233.  doi: 10.1006/jfan.2001.3900.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

G. G. Lorentz, Some new functional spaces, Ann. of Math., 51 (1950), 37-55.  doi: 10.2307/1969496.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[16]

B. Opic and A. Kufner, Hardy-type Inequalities, Pitman Research Notes in Mathematics Series, 219, 1990.  Google Scholar

show all references

References:
[1]

A. Alvino, Sulla diseguaglianza di Sobolev in spazi di Lorentz, Boll. Un. Mat. Ital. A, 14 (1977), 148-156.   Google Scholar

[2]

C. Bennett and R. Sharpley, Interpolation of Operators, Pure and Applied Mathematics, 129, Boston Academic Press, 1988.  Google Scholar

[3]

H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

E. B. Davies, A review of Hardy inequalities, Oper. Theory Adv. Appl., 110 (1999), 55-67.   Google Scholar

[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.  Google Scholar

[10]

S. Filippas and A. Tertikas, Optimizing improved Hardy inequalities, J. Funct. Anal., 192 (2002), 186-233.  doi: 10.1006/jfan.2001.3900.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

G. G. Lorentz, Some new functional spaces, Ann. of Math., 51 (1950), 37-55.  doi: 10.2307/1969496.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[16]

B. Opic and A. Kufner, Hardy-type Inequalities, Pitman Research Notes in Mathematics Series, 219, 1990.  Google Scholar

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