Optimal weighted Hardy–Rellich inequalities on H2 ∩ H01.

This paper is a continuation of [N. Ghoussoub and A. Moradifam, ‘On the best possible remaining term in the improved Hardy inequality’, Proc. Natl. Acad. Sci. USA 105 (2008) 13746–13751] and [N. Ghoussoub and A. Moradifam, ‘Bessel pairs and optimal Hardy and Hardy–Rellich inequalities’, Math. Ann. 349 (2011) 1–57], where the authors introduced a general approach for improved Hardy and Hardy–Rellich-type inequalities. In this paper, we present equivalent conditions on a pair of positive radial functions V and W on a ball B in Rn, n≥1, and b∈ℝ so that the following inequalities hold and Then we present various classes of optimal weighted Hardy–Rellich inequalities on H2 ∩ H01. The proofs are based on decomposition into spherical harmonics. This type of inequalities is important in the study of the systems of second-order elliptic equations as well as fourth-order elliptic equations with Navier boundary condition. [ABSTRACT FROM PUBLISHER]