Boundary Estimates for Doubly Nonlinear Parabolic Equations

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