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Zeitschrift für Analysis und ihre Anwendungen

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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