Weak Harnack inequality for doubly non-linear equations of slow diffusion type

Abstract

We consider non-negative weak super-solutions u:ΩT→R≥0 to the doubly non-linear equation∂t(|u|q−1u)−divA(x,t,u,Du)=0inΩT=Ω×(0,T], where Ω is an bounded open set in RN for N≥2, T>0 and q is a non-negative parameter. Furthermore, the vector field A satisfies standard p-growth assumptions for some p>1. The main novelty of this paper is that we establish the weak Harnack inequality in the entire slow diffusion regime p−q−1>0. Additionally, we only require that the weak super-solution u is located in the function spaceCloc0([0,T];Llocq+1(Ω))∩Llocp(0,T;Wloc1,p(Ω)).