Improved Hardy-Rellich inequalities

Biagio Cassano; Lucrezia Cossetti; Luca Fanelli

doi:10.3934/cpaa.2022002

Article Contents

2022, Volume 21, Issue 3: 867-889. 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
^*Corresponding author

Received: May 31, 2021

Revised: October 31, 2021

Early access: December 2021

Published: March 2022

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

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Abstract

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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Mathematics Subject Classification: Primary: 35A23, 26D10; Secondary: 47A63, 47F05.

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References

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