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 $\int_{\Omega}|\bigtriangledown \Delta u \frac {n+2}{2} |^2$ can be controlled by $\int –{\Omega} u^n| \bigtriangledown \Delta u|^2$. 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.