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

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

2009-10-07

arxiv.org/abs/0910.1185v1

Mathematics

Analysis of PDEs

Scientific paper

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.

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