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arXiv · 2310.06637

A new approach to weighted Hardy-Rellich inequalities: improvements, symmetrization principle and symmetry breaking

Anh Xuan Do·Nguyen Lam·Guozhen Lu
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Abstract

We investigate necessary and sufficient conditions on the weights for the Hardy-Rellich inequalities to hold, and propose a new way to use the notion of Bessel pair to establish the optimal Hardy-Rellich type inequalities. Our results sharpened earlier Hardy-Rellich and Rellich type inequalities in the literature. We also study several results about the symmetry and symmetry breaking properties of the Rellich type and Hardy-Rellich type inequalities, and then partially answered an open question raised by Ghoussoub and Moradifam. Namely, we will present conditions on the weights such that the Rellich type and Hardy-Rellich type inequalities hold for all functions if and only if the same inequalities hold for all radial functions.

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Anh Xuan Do, Nguyen Lam, Guozhen Lu. 2023-10-10. A new approach to weighted Hardy-Rellich inequalities: improvements, symmetrization principle and symmetry breaking. https://arxiv.org/abs/2310.06637

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