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A new proof of the Hardy–Rellich inequality in any dimension

Part of: Inequalities Partial differential equations

Published online by Cambridge University Press:  19 August 2019

Cristian Cazacu
Cristian Cazacu*
Affiliation:
Faculty of Mathematics and Computer Science & The Research Institute of the University of Bucharest (ICUB), University of Bucharest, 14 Academiei Street, 010014 Bucharest, Romania ([email protected])
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Abstract

The Hardy-Rellich inequality in the whole space with the best constant was firstly proved by Tertikas and Zographopoulos in Adv. Math. (2007) in higher dimensions N ⩾ 5. Then it was extended to lower dimensions N ∈ {3, 4} by Beckner in Forum Math. (2008) and Ghoussoub-Moradifam in Math. Ann. (2011) by applying totally different techniques.

In this note, we refine the method implemented by Tertikas and Zographopoulos, based on spherical harmonics decomposition, to give an easy and compact proof of the optimal Hardy–Rellich inequality in any dimension N ⩾ 3. In addition, we provide minimizing sequences which were not explicitly mentioned in the quoted papers in lower dimensions N ∈ {3, 4}, emphasizing their symmetry breaking. We also show that the best constant is not attained in the proper functional space.

Keywords

Hardy inequalityspherical coordinates

MSC classification

Primary: 35A23: Inequalities involving derivatives and differential and integral operators, inequalities for integrals
Secondary: 26D10: Inequalities involving derivatives and differential and integral operators
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 6 , December 2020 , pp. 2894 - 2904
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Copyright
Copyright © 2019 The Royal Society of Edinburgh

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References

1Agmon, S.. Bounds on exponential decay of eigenfunctions of Schrödinger operators. Schrödinger operators (Como, 1984), Lecture Notes in Math., vol. 1159, pp. 1–38 (Berlin: Springer, 1985).CrossRefGoogle Scholar
2Allegretto, W.. On the equivalence of two types of oscillation for elliptic operators. Pacific J. Math. 55 (1974), 319–328.CrossRefGoogle Scholar
3Beckner, W.. Weighted inequalities and Stein–Weiss potentials. Forum Math. 20 (2008), 587–606.CrossRefGoogle Scholar
4Ghoussoub, N. and Moradifam, A.. Bessel pairs and optimal Hardy and Hardy–Rellich inequalities. Math. Ann. 349 (2011), 1–57.Google Scholar
5Hamamoto, N. and Takahashi, F.. Sharp Hardy–Leray and Rellich–Leray inequalities for curl-free vector fields, arXiv:1808.09614.Google Scholar
6Piepenbrink, J.. Nonoscillatory elliptic equations. J. Differ. Equ. 15 (1974), 541–550.Google Scholar
7Tertikas, A. and Zographopoulos, N. B.. Best constants in the Hardy–Rellich inequalities and related improvements. Adv. Math. 206 (2007), 407–459.CrossRefGoogle Scholar