Sobolev improvements on sharp Rellich inequalities

There are two Rellich inequalities for the bilaplacian, that is for $\int (\mathrm{\Delta }u{)}^{2}dx$, the one involving $\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }$ and the other involving $\mathrm{\mid }u\mathrm{\mid }$ at the RHS. In this article we consider these inequalities with sharp constants and obtain sharp Sobolev-type improvements. More precisely, in our first result we improve the Rellich inequality with $\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }$ obtained recently by Cazacu in dimensions $n=3,4$ by a sharp Sobolev term thus complementing existing results for the case $n\ge 5$. In the second theorem the sharp constant of the Sobolev improvement for the Rellich inequality with $\mathrm{\mid }u\mathrm{\mid }$ is obtained.