The EMS Publishing House is now EMS Press and has its new home at ems.press.

Please find all EMS Press journals and articles on the new platform.

Zeitschrift für Analysis und ihre Anwendungen


Full-Text PDF (217 KB) | Metadata | Table of Contents | ZAA summary
Volume 20, Issue 4, 2001, pp. 987–998
DOI: 10.4171/ZAA/1055

Published online: 2001-12-31

On Bernis’ Interpolation Inequalities in Multiple Space Dimensions

Günther Grün[1]

(1) Universität Erlangen-Nünberg, Germany

In spatial dimensions $d < 6$, we derive estimates of the form $$\int_{\Omega} u^{n–4} |\bigtriangledown u|^6 + \int_{\Omega} u^{n–2}|\bigtriangledown u|^2|D^2u|^2 ≤ C \int_{\Omega} u^n|\bigtriangledown \Delta u|^2$$ 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.

Keywords: Interpolation inequalities, fourth order degenerate parabolic equations, thin films

Grün Günther: On Bernis’ Interpolation Inequalities in Multiple Space Dimensions. Z. Anal. Anwend. 20 (2001), 987-998. doi: 10.4171/ZAA/1055