On Bernis’ Interpolation Inequalities in Multiple Space Dimensions

Abstract

In spatial dimensions $d<6$, we derive estimates of the form

for functions $u\in H^2(\Omega )$ with vanishing normal derivatives on the boundary $\partial \Omega$. These inequalities imply that ${\int }_{\Omega }|\nabla \Delta u\frac{n+2}{2}{|}^2$ can be controlled by ${\int }_{\Omega }{u}^n|\nabla \Delta u{|}^2$. This observation will be a key ingredient for the proof of certain qualitative results – e.g. finite speed of propagation or occurrence of a waiting time phenomenon – for solutions to fourth order degenerate parabolic equations like the thin film equation. Our result generalizes – in a slightly modified way – estimates in one space dimension which were obtained by F. Bernis.