In spatial dimensions , we derive estimates of the form
for functions with vanishing normal derivatives on the boundary . These inequalities imply that can be controlled by . 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. 987–998DOI 10.4171/ZAA/1055