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March  2022, 21(3): 867-889. doi: 10.3934/cpaa.2022002

Improved Hardy-Rellich inequalities

Biagio Cassano 1,2, , Lucrezia Cossetti 3, and  Luca Fanelli 4,,

1. 

Dipartimento di Matematica, Università degli Studi di Bari "A. Moro", Via Orabona 4, 70125 Bari, Italy
2. 

Dipartimento di Matematica e Fisica, Università degli Studi della Campania "Luigi Vanvitelli", Viale Lincoln 5, 81100 Caserta, Italy
3. 

Fakultät für Mathematik, Institut für Analysis, Karlsruher Institut für Technologie (KIT), Englerstraße 2, 76131 Karlsruhe, Germany
4. 

Ikerbasque, & , Departamento de Matematicas, Universidad del País Vasco/Euskal Herriko Unibertsitatea (UPV/EHU), Aptdo. 644, 48080, Bilbao, Spain

*Corresponding author

Received  June 2021 Revised  November 2021 Published  March 2022 Early access  December 2021

Fund Project: B. C. is member of GNAMPA (INDAM) and he is supported by Fondo Sociale Europeo – Programma Operativo Nazionale Ricerca e Innovazione 2014-2020, progetto PON: progetto AIM1892920-attività 2, linea 2.1. The research of L.C. is supported by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173

We investigate Hardy-Rellich inequalities for perturbed Laplacians. In particular, we show that a non-trivial angular perturbation of the free operator typically improves the inequality, and may also provide an estimate which does not hold in the free case. The main examples are related to the introduction of a magnetic field: this is a manifestation of the diamagnetic phenomenon, which has been observed by Laptev and Weidl in [21] for the Hardy inequality, later by Evans and Lewis in [9] for the Rellich inequality; however, to the best of our knowledge, the so called Hardy-Rellich inequality has not yet been investigated in this regards. After showing the optimal inequality, we prove that the best constant is not attained by any function in the domain of the estimate.

Keywords: Hardy-Rellich inequalities, Biharmonic operators, magnetic fields, sharp inequalities, spherical harmonics decomposition.
Mathematics Subject Classification: Primary: 35A23, 26D10; Secondary: 47A63, 47F05.
Citation: Biagio Cassano, Lucrezia Cossetti, Luca Fanelli. Improved Hardy-Rellich inequalities. Communications on Pure and Applied Analysis, 2022, 21 (3) : 867-889. doi: 10.3934/cpaa.2022002
References:
[1]

A. Balinsky, A. Laptev and A. Sobolev, Generalized Hardy inequality for the magnetic Dirichlet forms, J. Stat. Phys., 116 (2004), 507-521.  doi: 10.1023/B:JOSS.0000037228.35518.ca.

[2]

W. Beckner, Weighted inequalities and Stein-Weiss potentials, Forum Math., 20 (2008), 587-606.  doi: 10.1515/FORUM.2008.030.

[3]

D. M. Bennett, An extension of Rellich's inequality, Proc. Amer. Math. Soc., 106 (1989), 987-993.  doi: 10.2307/2047283.

[4]

B. Cassano and F. Pizzichillo, Self-adjoint extensions for the Dirac operator with Coulomb-type spherically symmetric potentials, Lett. Math. Phys., 108 (2018), 2635-2667.  doi: 10.1007/s11005-018-1093-9.

[5]

C. Cazacu, A new proof of the Hardy-Rellich inequality in any dimension, Proc. Roy. Soc. Edinb. A, 150 (2020), 2894-2904.  doi: 10.1017/prm.2019.50.

[6]

C. Cazacu and D. Krejčiřík, The Hardy inequality and the heat equation with magnetic field in any dimension, Commun. Partial Differ. Equ., 41 (2016), 1056-1088.  doi: 10.1080/03605302.2016.1179317.

[7]

Y. Colin de Verdière and F. Truc, Confining quantum particles with a purely magnetic field, Ann. I Fourier, 60 (2010), 2333-2356. 

[8]

D. G. Costa, On Hardy-Rellich type inequalities in $ \mathbb{R}^N$, Appl. Math. Lett., 22 (2009), 902-905.  doi: 10.1016/j.aml.2008.02.018.

[9]

W. D. Evans and R. T. Lewis, On the Rellich inequality with magnetic potentials, Math. Z., 251 (2005), 267-284.  doi: 10.1007/s00209-005-0798-5.

[10]

L. Fanelli, V. Felli, M. Fontelos and A. Primo, Time decay of scaling invariant electromagnetic Schrödinger equations on the plane, Commun. Math. Phys., 337 (2015), 1515-1533.  doi: 10.1007/s00220-015-2291-2.

[11]

L. Fanelli, D. Krejčiřík, A. Laptev and L. Vega, On the improvement of the Hardy inequality due to singular magnetic fields, Commun. Partial Differ. Equ., 45 (2020), 1202-1212.  doi: 10.1080/03605302.2020.1763399.

[12]

V. Felli, E. Marchini and S. Terracini, On the behavior of solutions to Schrödinger equations with dipole-type potentials near the singularity, Discret. Contin. Dynam. Syst., 21 (2007), 91-119.  doi: 10.3934/dcds.2008.21.91.

[13]

R. L. Frank and M. Loss, Which magnetic fields support a zero mode?, arXiv: 2012.13646.

[14]

F. Gesztesy and L. Littlejohn, Factorizations and Hardy-Rellich-type inequalities, arXiv: 1701.08929.

[15]

N. Ghoussoub and A. Moradifam, Bessel pairs and optimal Hardy and Hardy-Rellich inequalities, Math. Ann., 349 (2011), 1-57.  doi: 10.1007/s00208-010-0510-x.

[16]

N. Hamamoto, Sharp uncertainty principle inequality for solenoidal fields, arXiv: 2104.02351.

[17]

N. Hamamoto and F. Takahashi, Sharp Hardy-Leray and Rellich-Leray inequalities for curl-free vector fields, Math. Ann., 379 (2021), 719-742.  doi: 10.1007/s00208-019-01945-x.

[18]

N. Hamamoto and F. Takahashi, A curl-free improvement of the Rellich-Hardy inequality with weight, arXiv: 2101.01878.

[19]

N. Ioku, M. Ishiwata and T. Ozawa, Sharp remainder of a critical Hardy inequality, Arch. Math., 106 (2016), 65-71.  doi: 10.1007/s00013-015-0841-7.

[20]

I. Kombe and M. Ozaydin, Improved Hardy and Rellich inequalities on Riemannian manifolds, Trans. Amer. Math. Soc., 361 (2009), 6191-6203.  doi: 10.1090/S0002-9947-09-04642-X.

[21]

A. Laptev and T. Weidl, Hardy inequalities for magnetic Dirichlet forms, Oper. Theory Adv. Appl., 108 (1999), 299-305. 

[22]

E. H. Lieb and M. Loss, Analysis, Second Edition, American Mathematical Society, Providence, Rhode Island, 2001.

[23]

V. H. Nguyen, New sharp Hardy and Rellich type inequalities on Cartan-Hadamard manifolds and their improvements, Proc. Roy. Soc. Edinburgh Sect. A., 150 (2020), 2952-2981.  doi: 10.1017/prm.2019.37.

[24]

F. Rellich and J. Berkowitz, Perturbation theory of eigenvalue problems, Gordon and Breach Science Publishers, New York, London, Paris, 1969.

[25]

K. Schmüdgen, Unbounded Self-Adjoint Operators on Hilbert Space, Graduate Texts in Mathematics, Springer, Dordrecht, 2012. doi: 10.1007/978-94-007-4753-1.

[26]

A. Tertikas and N. B. Zographopoulos, Best constants in the Hardy-Rellich inequalities and related improvements, Adv. Math., 206 (2007), 407-459.  doi: 10.1016/j.aim.2006.05.011.

[27]

J. C. Thomas, Some Problems Associated with Sum and Integral Inequalities, Ph.D. thesis, Cardiff University in Wales, 2007.

[28]

D. Yafaev, Sharp constants in the Hardy-Rellich inequalities, J. Funct. Anal., 168 (1999), 121-144.  doi: 10.1006/jfan.1999.3462.

show all references

References:
[1]

A. Balinsky, A. Laptev and A. Sobolev, Generalized Hardy inequality for the magnetic Dirichlet forms, J. Stat. Phys., 116 (2004), 507-521.  doi: 10.1023/B:JOSS.0000037228.35518.ca.

[2]

W. Beckner, Weighted inequalities and Stein-Weiss potentials, Forum Math., 20 (2008), 587-606.  doi: 10.1515/FORUM.2008.030.

[3]

D. M. Bennett, An extension of Rellich's inequality, Proc. Amer. Math. Soc., 106 (1989), 987-993.  doi: 10.2307/2047283.

[4]

B. Cassano and F. Pizzichillo, Self-adjoint extensions for the Dirac operator with Coulomb-type spherically symmetric potentials, Lett. Math. Phys., 108 (2018), 2635-2667.  doi: 10.1007/s11005-018-1093-9.

[5]

C. Cazacu, A new proof of the Hardy-Rellich inequality in any dimension, Proc. Roy. Soc. Edinb. A, 150 (2020), 2894-2904.  doi: 10.1017/prm.2019.50.

[6]

C. Cazacu and D. Krejčiřík, The Hardy inequality and the heat equation with magnetic field in any dimension, Commun. Partial Differ. Equ., 41 (2016), 1056-1088.  doi: 10.1080/03605302.2016.1179317.

[7]

Y. Colin de Verdière and F. Truc, Confining quantum particles with a purely magnetic field, Ann. I Fourier, 60 (2010), 2333-2356. 

[8]

D. G. Costa, On Hardy-Rellich type inequalities in $ \mathbb{R}^N$, Appl. Math. Lett., 22 (2009), 902-905.  doi: 10.1016/j.aml.2008.02.018.

[9]

W. D. Evans and R. T. Lewis, On the Rellich inequality with magnetic potentials, Math. Z., 251 (2005), 267-284.  doi: 10.1007/s00209-005-0798-5.

[10]

L. Fanelli, V. Felli, M. Fontelos and A. Primo, Time decay of scaling invariant electromagnetic Schrödinger equations on the plane, Commun. Math. Phys., 337 (2015), 1515-1533.  doi: 10.1007/s00220-015-2291-2.

[11]

L. Fanelli, D. Krejčiřík, A. Laptev and L. Vega, On the improvement of the Hardy inequality due to singular magnetic fields, Commun. Partial Differ. Equ., 45 (2020), 1202-1212.  doi: 10.1080/03605302.2020.1763399.

[12]

V. Felli, E. Marchini and S. Terracini, On the behavior of solutions to Schrödinger equations with dipole-type potentials near the singularity, Discret. Contin. Dynam. Syst., 21 (2007), 91-119.  doi: 10.3934/dcds.2008.21.91.

[13]

R. L. Frank and M. Loss, Which magnetic fields support a zero mode?, arXiv: 2012.13646.

[14]

F. Gesztesy and L. Littlejohn, Factorizations and Hardy-Rellich-type inequalities, arXiv: 1701.08929.

[15]

N. Ghoussoub and A. Moradifam, Bessel pairs and optimal Hardy and Hardy-Rellich inequalities, Math. Ann., 349 (2011), 1-57.  doi: 10.1007/s00208-010-0510-x.

[16]

N. Hamamoto, Sharp uncertainty principle inequality for solenoidal fields, arXiv: 2104.02351.

[17]

N. Hamamoto and F. Takahashi, Sharp Hardy-Leray and Rellich-Leray inequalities for curl-free vector fields, Math. Ann., 379 (2021), 719-742.  doi: 10.1007/s00208-019-01945-x.

[18]

N. Hamamoto and F. Takahashi, A curl-free improvement of the Rellich-Hardy inequality with weight, arXiv: 2101.01878.

[19]

N. Ioku, M. Ishiwata and T. Ozawa, Sharp remainder of a critical Hardy inequality, Arch. Math., 106 (2016), 65-71.  doi: 10.1007/s00013-015-0841-7.

[20]

I. Kombe and M. Ozaydin, Improved Hardy and Rellich inequalities on Riemannian manifolds, Trans. Amer. Math. Soc., 361 (2009), 6191-6203.  doi: 10.1090/S0002-9947-09-04642-X.

[21]

A. Laptev and T. Weidl, Hardy inequalities for magnetic Dirichlet forms, Oper. Theory Adv. Appl., 108 (1999), 299-305. 

[22]

E. H. Lieb and M. Loss, Analysis, Second Edition, American Mathematical Society, Providence, Rhode Island, 2001.

[23]

V. H. Nguyen, New sharp Hardy and Rellich type inequalities on Cartan-Hadamard manifolds and their improvements, Proc. Roy. Soc. Edinburgh Sect. A., 150 (2020), 2952-2981.  doi: 10.1017/prm.2019.37.

[24]

F. Rellich and J. Berkowitz, Perturbation theory of eigenvalue problems, Gordon and Breach Science Publishers, New York, London, Paris, 1969.

[25]

K. Schmüdgen, Unbounded Self-Adjoint Operators on Hilbert Space, Graduate Texts in Mathematics, Springer, Dordrecht, 2012. doi: 10.1007/978-94-007-4753-1.

[26]

A. Tertikas and N. B. Zographopoulos, Best constants in the Hardy-Rellich inequalities and related improvements, Adv. Math., 206 (2007), 407-459.  doi: 10.1016/j.aim.2006.05.011.

[27]

J. C. Thomas, Some Problems Associated with Sum and Integral Inequalities, Ph.D. thesis, Cardiff University in Wales, 2007.

[28]

D. Yafaev, Sharp constants in the Hardy-Rellich inequalities, J. Funct. Anal., 168 (1999), 121-144.  doi: 10.1006/jfan.1999.3462.

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