On certain variant of strongly nonlinear multidimensional interpolation inequality

Abstract

We obtain the inequality \begin{multline*} \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, \end{multline*} 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 $W^{2,1}_{\rom{loc}}(\Omega)$, $\nabla^{(2)} u$ 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.