On certain variant of strongly nonlinear multidimensional interpolation inequality

Abstract

We obtain the inequality \[ \int_{\Omega}|\nabla u(x)|^ph(u(x))\,dx \leq C(n,p)\int_{\Omega} \Big( \sqrt{ |\nabla^{(2)} u(x)| |{\mathcal T}_{h,C}(u(x))|}\Big)^{p}h(u(x))\,dx, \] where $\Omega\subset \mathbb R^n$ and $n\ge 2$, $u\colon\Omega\rightarrow \mathbb R$ is in certain subset in second order Sobolev space $W^{2,1}_{\rm loc}(\Omega)$, $\nabla^{(2)} u$ is the Hessian matrix of $u$, ${\mathcal T}_{h,C}(u)$ is a certain transformation of the continuous function $h(\,\cdot\,)$. Such inequality is the generalization of a similar inequality holding in one dimension, obtained earlier by second author and Peszek.