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

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

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].

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. CassaniF. 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. DevyverM. 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. KufnerL. 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. CassaniF. 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. DevyverM. 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. KufnerL. 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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