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

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{$\int_{B}V(x)|\nabla u |^{2}dx \geq \int_{B} W(x) u^{2}dx+b\int_{\partial B}u^2 ds$forallu$\in H^1(B)$,} \end{equation*} and \begin{equation*} \label{two} \hbox{$\int_{B}V(x)|\Delta u |^{2}dx \geq \int_{B} W(x)|\nabla u|^{2}dx+b\int_{\partial B}|\nabla u|^2 ds$forallu$\in H^2(B)$.} \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.