Open AccessImproved Hardy-Rellich inequalities
Author(s) -
Biagio Cassano,
Lucrezia Cossetti,
Luca Fanelli
Publication year - 2022
Publication title -
communications on pure and applied analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.077
H-Index - 42
eISSN - 1553-5258
pISSN - 1534-0392
DOI - 10.3934/cpaa.2022002
Subject(s) - inequality , mathematics , perturbation (astronomy) , pure mathematics , mathematical analysis , physics , quantum mechanics
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.