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

Agnieszka Kałamajska, Dalimil Peša, Tomáš Roskovec

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 $Ω…