Improved Hardy-Rellich inequalities

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 intro...

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Bibliographic Details
Published in:Communications on Pure & Applied Analysis
Main Authors: Cassano, Biagio, Cossetti, Lucrezia, Fanelli, Luca
Format: Article in Journal/Newspaper
Language:English
Published: 2022
Subjects:
Hardy-Rellich inequalitie
Biharmonic operator
magnetic field
sharp inequalitie
spherical harmonics decomposition
laptev
Online Access:http://hdl.handle.net/11591/461291
https://doi.org/10.3934/cpaa.2022002
Description
Summary: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.

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