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

TL;DR

This paper establishes a new class of non-linear Gagliardo--Nirenberg inequalities involving second-order elliptic operators in non-divergent form, connecting PDE analysis with probability and potential theory.

Contribution

It introduces a novel inequality framework for second-order elliptic operators in non-divergent form, extending classical Gagliardo--Nirenberg inequalities with boundary terms and transformations.

Findings

Derived inequalities involving elliptic operators and transformations.

Linked PDE inequalities to probability and potential theory results.

Provided conditions under which boundary terms are controlled.

Abstract

We obtain the inequalities of the form $$\int_{\Omega}|\nabla u(x)|^2h(u(x))\,{\rm d} x\leq C\int_{\Omega} \left( \sqrt{ |P u(x)||{\cal T}_{H}(u(x))|}\right)^{2}h(u(x))\,{\rm d} x +\Theta,$$ where $\Omega\subset \mathbf{R}^n$ is a bounded Lipschitz domain, $u\in W^{2,1}_{\rm loc}(\Omega)$ is non-negative, $P$ is a uniformly elliptic operator in non-divergent form, ${\cal T}_{H}(\cdot )$ 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.