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doi: 10.3934/cpaa.2022002
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Improved Hardy-Rellich inequalities

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 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.

Citation: Biagio Cassano, Lucrezia Cossetti, Luca Fanelli. Improved Hardy-Rellich inequalities. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2022002
References:
[1]

A. BalinskyA. 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.  Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[10]

L. FanelliV. FelliM. 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.  Google Scholar

[11]

L. FanelliD. KrejčiříkA. 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.  Google Scholar

[12]

V. FelliE. 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.  Google Scholar

[13]

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

[14]

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

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[18]

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

[19]

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

[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.  Google Scholar

[21]

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

[22]

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

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[28]

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

show all references

References:
[1]

A. BalinskyA. 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.  Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[10]

L. FanelliV. FelliM. 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.  Google Scholar

[11]

L. FanelliD. KrejčiříkA. 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.  Google Scholar

[12]

V. FelliE. 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.  Google Scholar

[13]

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

[14]

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

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[18]

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

[19]

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

[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.  Google Scholar

[21]

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

[22]

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

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[28]

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

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