Boundary Estimates for Doubly Nonlinear Parabolic Equations

We consider non-negative, weak solutions to the doubly nonlinear parabolic equation
\[
\partial_t (u^q) − div(|Du|^{p−2}Du)  = 0
\]
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 Ω_T , they satisfy proper Carleson estimates. Assuming further regularity for the boundary of the domain Ω_T , we obtain a power-like decay at the boundary and a boundary Harnack inequality.