# Optimal Hardy–Rellich inequalities, maximum principle and related eigenvalue problem

Adimurthi, * and Grossi, Massimo and Santra, Sanjiban (2006) Optimal Hardy–Rellich inequalities, maximum principle and related eigenvalue problem. In: Journal of Functional Analysis, 240 (1). pp. 36-83.

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Official URL: http://dx.doi.org/10.1016/j.jfa.2006.07.011

## Abstract

In this paper we deal with three types of problems concerning the Hardy–Rellich's embedding for a bi-Laplacian operator. First we obtain the Hardy–Rellich inequalities in the critical dimension n=4. Then we derive a maximum principle for fourth order operators with singular terms. Then we study the existence, non-existence, simplicity and asymptotic behavior of the first eigenvalue of the Hardy–Rellich operator ${ \Delta}^2$ = $\frac {n^2(n-4)^2}{16}$ $\frac {q(x)}{ |x|^4}$ under various assumptions on the perturbation q.

Item Type: Journal Article Journal of Functional Analysis Elsevier Science Copyright of this article belongs to Elsevier Science. Biharmonic equation;Hardy–Rellich's inequality;Maximum principle;Perturbed eigenvalue problem;Boggio's principle; Dirichlet and Navier boundary conditions; Division of Physical & Mathematical Sciences > Mathematics 28 May 2008 01 Mar 2012 08:51 http://eprints.iisc.ac.in/id/eprint/14072