American Institute of Mathematical Sciences

Advanced
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

[1]

Delia Schiera. Existence and non-existence results for variational higher order elliptic systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5145-5161. doi: 10.3934/dcds.2018227
[2]

Neal Bez, Sanghyuk Lee, Shohei Nakamura, Yoshihiro Sawano. Sharpness of the Brascamp–Lieb inequality in Lorentz spaces. Electronic Research Announcements, 2017, 24: 53-63. doi: 10.3934/era.2017.24.006
[3]

Xiaoxue Ji, Pengcheng Niu, Pengyan Wang. Non-existence results for cooperative semi-linear fractional system via direct method of moving spheres. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1111-1128. doi: 10.3934/cpaa.2020051
[4]

SYLWIA DUDEK, IWONA SKRZYPCZAK. Liouville theorems for elliptic problems in variable exponent spaces. Communications on Pure & Applied Analysis, 2017, 16 (2) : 513-532. doi: 10.3934/cpaa.2017026
[5]

Lorenzo D'Ambrosio, Enzo Mitidieri. Hardy-Littlewood-Sobolev systems and related Liouville theorems. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 653-671. doi: 10.3934/dcdss.2014.7.653
[6]

Yingshu Lü. Symmetry and non-existence of solutions to an integral system. Communications on Pure & Applied Analysis, 2018, 17 (3) : 807-821. doi: 10.3934/cpaa.2018041
[7]

Yutian Lei. Liouville theorems and classification results for a nonlocal Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5351-5377. doi: 10.3934/dcds.2018236
[8]

Genggeng Huang, Congming Li, Ximing Yin. Existence of the maximizing pair for the discrete Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 935-942. doi: 10.3934/dcds.2015.35.935
[9]

Xumin Wang. Singular Hardy-Trudinger-Moser inequality and the existence of extremals on the unit disc. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2717-2733. doi: 10.3934/cpaa.2019121
[10]

Keisuke Matsuya, Tetsuji Tokihiro. Existence and non-existence of global solutions for a discrete semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 209-220. doi: 10.3934/dcds.2011.31.209
[11]

Jitsuro Sugie, Tadayuki Hara. Existence and non-existence of homoclinic trajectories of the Liénard system. Discrete & Continuous Dynamical Systems - A, 1996, 2 (2) : 237-254. doi: 10.3934/dcds.1996.2.237
[12]

Elias M. Guio, Ricardo Sa Earp. Existence and non-existence for a mean curvature equation in hyperbolic space. Communications on Pure & Applied Analysis, 2005, 4 (3) : 549-568. doi: 10.3934/cpaa.2005.4.549
[13]

Alberto Bressan, Truyen Nguyen. Non-existence and non-uniqueness for multidimensional sticky particle systems. Kinetic & Related Models, 2014, 7 (2) : 205-218. doi: 10.3934/krm.2014.7.205
[14]

Jingbo Dou, Huaiyu Zhou. Liouville theorems for fractional Hénon equation and system on $\mathbb{R}^n$. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1915-1927. doi: 10.3934/cpaa.2015.14.1915
[15]

Leszek Gasiński. Existence results for quasilinear hemivariational inequalities at resonance. Conference Publications, 2007, 2007 (Special) : 409-418. doi: 10.3934/proc.2007.2007.409
[16]

Ran Zhuo, Wenxiong Chen, Xuewei Cui, Zixia Yuan. Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1125-1141. doi: 10.3934/dcds.2016.36.1125
[17]

Shu-Yu Hsu. Non-existence and behaviour at infinity of solutions of some elliptic equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 769-786. doi: 10.3934/dcds.2004.10.769
[18]

Ivan Landjev, Assia Rousseva. The non-existence of $(104,22;3,5)$-arcs. Advances in Mathematics of Communications, 2016, 10 (3) : 601-611. doi: 10.3934/amc.2016029
[19]

J. F. Toland. Non-existence of global energy minimisers in Stokes waves problems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3211-3217. doi: 10.3934/dcds.2014.34.3211
[20]

Fuqin Sun, Mingxin Wang. Non-existence of global solutions for nonlinear strongly damped hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 949-958. doi: 10.3934/dcds.2005.12.949

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (78)
  • HTML views (90)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences