Non-linear Gagliardo--Nirenberg inequality involving a second-order elliptic operator in non-divergent form
Agnieszka Kałamajska, Dalimil Peša, Tomáš Roskovec
TL;DR
The paper develops a nonlinear Gagliardo–Nirenberg-type framework for second-order elliptic operators in non-divergence form on bounded Lipschitz domains, establishing a weighted energy identity and corresponding inequalities involving a nonlinearity h and its primitive H. Central to the approach are weight transforms T_H, G_H and the second antiderivative ∼H, which yield integral bounds of the form ∫Ω ∥∇u∥_A² h(u) dx ≤ ∫Ω |Pu| |H(u)| dx + Θ, with Θ a boundary term. Through Opial-type inequalities and various structural assumptions (including sign conditions on div A and boundary vanishing of ∼H), the authors derive simplifications that remove the 𝒢_H(u) term and provide chain-rule-type upper bounds for P(∼H(u)). The work connects PDE methods with probability and potential theory, offering a toolkit for a priori estimates of nonlinear PDEs and insights into generators of analytic semigroups, with potential extensions to non-local operators and harmonic extensions. Overall, the results contribute a rigorous nonlinear energy framework for elliptic operators in non-divergence form, linking functional-analytic, probabilistic, and potential-theoretic perspectives to derive new inequalities and boundary representations.
Abstract
We obtain the inequalities of the form $$\int_Ω|\nabla u(x)|^2h(u(x))\,{\rm d} x\leq C\int_Ω \left( \sqrt{ |P u(x)||{\cal T}_{H}(u(x))|}\right)^{2}h(u(x))\,{\rm d} x +Θ,$$ where $Ω\subset \mathbf{R}^n$ is a bounded Lipschitz domain, $u\in W^{2,1}_{\rm loc}(Ω)$ 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 $Θ$ 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.