Optimal weighted Hardy-Rellich inequalities on $H^2\cap H_0^1$

Abstract

We give necessary and sufficient conditions on a pair of positive radial functions $V$ and $W$ on a ball $B$ of radius $R$ in $R^n$, $n\ge 1$, so that the following inequalities hold \begin{equation*} \label{two} \hbox{${\int }_{B}V(x)|\nabla u{|}^{2}dx\ge {\int }_{B}W(x)u^{2}dx+b{\int }_{\partial B}{u}^{2}ds$ for all u $\in H^1(B)$,} \end{equation*} and \begin{equation*} \label{two} \hbox{${\int }_{B}V(x)|\Delta u{|}^{2}dx\ge {\int }_{B}W(x)|\nabla u{|}^{2}dx+b{\int }_{\partial B}|\nabla u{|}^{2}ds$ for all u $\in H^2(B)$.} \end{equation*} Then we present various classes of optimal weighted Hardy-Rellich inequalities on $H^2\cap H_0^1$. The proofs are based on decomposition into spherical harmonics. These types inequalities are important in the study of fourth order elliptic equations with Navier boundary condition and systems of second order elliptic equations.

References in corpus (1)

• On the best possible remaining term in the Hardy Inequality

Cited by in corpus (6)

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• Embeddings of weighted Sobolev spaces and a weighted fourth order elliptic equation
• Radial and non radial ground states for a class of dilation invariant fourth order semilinear elliptic equations on ${\mathbb{R}}^n$
• Entire solutions for a class of fourth order semilinear elliptic equations with weights