Sobolev improvements on sharp Rellich inequalities

Abstract

There are two Rellich inequalities for the bilaplacian, that is, for $\int (\Delta u{)}^{2}dx$, the one involving $|\nabla u|$ and the other involving $|u|$ 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 $|\nabla u|$ obtained by Beckner 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 $|u|$ is obtained.