Optimal weighted Hardy-Rellich inequalities on $H^2\cap H^{1}_{0}$

TL;DR

This paper establishes necessary and sufficient conditions for weighted Hardy-Rellich inequalities involving radial functions on a ball, providing optimal inequalities crucial for analyzing fourth order elliptic equations with Navier boundary conditions.

Contribution

It introduces new criteria for weighted Hardy-Rellich inequalities on $H^2 igcap H^1_0$, including classes of optimal inequalities, using spherical harmonics decomposition.

Findings

Derived necessary and sufficient conditions for inequalities.

Presented classes of optimal weighted Hardy-Rellich inequalities.

Applied results to elliptic equations with Navier boundary conditions.

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 $R^{n}$, $n \geq 1$, so that the following inequalities hold \begin{equation*} \label{two} \hbox{$\int_{B}V(x)|\nabla u |^{2}dx \geq \int_{B} W(x) u^{2}dx+b\int_{\partial B}u^2 ds$ for all u $\in H^1(B)$,} \end{equation*} and \begin{equation*} \label{two} \hbox{$\int_{B}V(x)|\Delta u |^{2}dx \geq \int_{B} W(x)|\nabla u|^{2}dx+b\int_{\partial B}|\nabla u|^2 ds$ for all u $\in H^2(B)$.} \end{equation*} Then we present various classes of optimal weighted Hardy-Rellich inequalities on $H^{2}\cap H^{1}_{0}$. 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.