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

We consider non-negative weak super-solutions u : Omega(T) -> R->= 0 to the doubly nonlinear equation partial derivative(t) (vertical bar u vertical bar(q-1)u) - divA( x, t, u, Du) = 0 in Omega(T) = Omega x (0, T], where Omega is an bounded open set in R-N for N >= 2, T > 0 and q is a non-negative parameter. Furthermore, the vector field Asatisfies 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 uis located in the function space C-loc(0) [0, T]; L-loc(q+1) loc (Omega) boolean AND n L-loc(p) (0, T; W-loc(1,p) (Omega)). (c) 2024 The Author. Published by Elsevier Inc.