On certain variant of strongly nonlinear interpolation inequality in dimension n

Abstract

We obtain the inequality $${\int }_{\Omega }|\nabla u(x){|}^{p}h(u(x))dx\le C(n,p){\int }_{\Omega }{\left(\sqrt{|{\nabla }^{(2)}u(x)||{\mathcal{T}}_{h,C}(u(x))|}\right)}^{p}h(u(x))dx,$$ where $\Omega \subseteq {\mathbf{R}}^n$ and $n\ge 2$, $u:\Omega \to \mathbf{R}$ is in certain subset in second order Sobolev space $W_{loc}^{2,1}(\Omega )$, ${\nabla }^{(2)}u$ is the Hessian matrix of $u$, ${\mathcal{T}}_{h,C}(u)$ is certain transformation of the continuous function $h(\cdot )$. Such inequality is the generalization of similar inequality holding in one dimension, obtained earlier by second author and Peszek.

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