On Berni's interpolation inequalities in multiple space dimensions (Q1598247)

There are derived estimates of the form \[ \int_\Omega u^{n-4}|\nabla u|^6+\int_\Omega u^{n-2}|\nabla u|^2 |D^2u|^2\leq C\int_\Omega u^n|\nabla \Delta u|^2 \] for functions \(u\in H^2(\Omega),\) \(\Omega\subset {\mathbb R}^d,\) \(d<6,\) with vanishing normal derivatives on the smooth boundary \(\partial\Omega.\) This inequality implies that \(\int_\Omega |\nabla\Delta u^{(n+2)/2}|^2\) can be controlled by \(\int_\Omega u^n|\nabla\Delta u|^2.\) The observation allows to prove certain qualitative results, as finite speed of propagation or occurrence of a waiting time phenomenon, for solutions to fourth-order degenerate parabolic equations like the thin film equation.