On Bernis’ Interpolation Inequalities in Multiple Space Dimensions

  • Günther Grün

    Universität Erlangen-Nünberg, Germany


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

Ωun4u6+Ωun2u2D2u2CΩunΔu2\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 uH2(Ω)u \in H^2 (\Omega) with vanishing normal derivatives on the boundary Ω\partial \Omega­. These inequalities imply that ΩΔun+222\int_{\Omega}|\bigtriangledown \Delta u \frac {n+2}{2} |^2 can be controlled by ΩunΔu2\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.

Cite this article

Günther Grün, On Bernis’ Interpolation Inequalities in Multiple Space Dimensions. Z. Anal. Anwend. 20 (2001), no. 4, pp. 987–998

DOI 10.4171/ZAA/1055