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On the generalized Hardy-Rellich inequalities

Part of: Linear function spaces and their duals Partial differential equations

Published online by Cambridge University Press:  26 January 2019

T.V. Anoop ,
Ujjal Das  and
Abhishek Sarkar
T.V. Anoop
Affiliation:
Department of Mathematics, Indian Institute of Technology Madras, Chennai600036, India (anoop@iitm.ac.in)
Ujjal Das
Affiliation:
The Institute of Mathematical Sciences, Chennai600113, India (ujjaldas@imsc.res.in, ujjal.rupam.das@gmail.com)
Abhishek Sarkar
Affiliation:
NTIS, University of West Bohemia Technická 8, 306 14 Plzeň, Czech Republic (sarkara@ntis.zcu.cz)
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Abstract

In this paper, we look for the weight functions (say g) that admit the following generalized Hardy-Rellich type inequality:

$$\int_\Omega g (x)u^2 dx \les C\int_\Omega \vert \Delta u \vert ^2 dx,\quad \forall u\in {\rm {\cal D}}_0^{2,2} (\Omega ),$$
for some constant C > 0, where Ω is an open set in ℝN with N ⩾ 1. We find various classes of such weight functions, depending on the dimension N and the geometry of Ω. Firstly, we use the Muckenhoupt condition for the one-dimensional weighted Hardy inequalities and a symmetrization inequality to obtain admissible weights in certain Lorentz-Zygmund spaces. Secondly, using the fundamental theorem of integration we obtain the weight functions in certain weighted Lebesgue spaces. As a consequence of our results, we obtain simple proofs for the embeddings of ${\cal D}_0^{2,2} $ into certain Lorentz-Zygmund spaces proved by Hansson and later by Brezis and Wainger.

Keywords

Generalized Hardy-Rellich inequalityMuckenhoupt conditionsymmetrizationLorentz spacesLorentz-Zygmund spacesexterior domains

MSC classification

Primary: 35A23: Inequalities involving derivatives and differential and integral operators, inequalities for integrals
Secondary: 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46E35: Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 2 , April 2020 , pp. 897 - 919
Copyright
Copyright © Royal Society of Edinburgh 2019

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