Non-linear Gagliardo--Nirenberg inequality involving a second-order elliptic operator in non-divergent form

Abstract

We obtain the inequalities of the form $${\int }_{\Omega }|\nabla u(x){|}^{2}h(u(x))\mathrm{d}x\le C{\int }_{\Omega }{\left(\sqrt{|Pu(x)||{\mathcal{T}}_H(u(x))|}\right)}^{2}h(u(x))\mathrm{d}x+\Theta ,$$ where $\Omega \subset {\mathbf{R}}^n$ is a bounded Lipschitz domain, $u\in W_{\mathrm{loc}}^{2,1}(\Omega )$ is non-negative, $P$ is a uniformly elliptic operator in non-divergent form, ${\mathcal{T}}_H(\cdot )$ is certain transformation of the monotone $C^1$ function $H(\cdot )$, which is the primitive of the weight $h(\cdot )$, and $\Theta$ is the boundary term which depends on boundary values of $u$ and $\nabla u$, which hold under some additional assumptions. Our results are linked to some results from probability and potential theories, e.g.~to some variants of the Douglas formulae.

References in corpus (1)

• The Douglas formula in $L^p$