Boundary Estimates for Doubly Nonlinear Parabolic Equations

We consider non-negative, weak solutions to the doubly nonlinear parabolic equation ₜ uq-div (|Du|^p-2Du) =0 in the super-critical fast diffusion regime 0<p-1<q<N (p-1) (N-p) _+. We show that when solutions vanish continuously at the Lipschitz boundary of a parabolic cylinder T, they satisfy proper Carleson estimates. Assuming further regularity for the boundary of the domain T, we obtain a power-like decay at the boundary and a boundary Harnack inequality.