Boundary Estimates for Doubly Nonlinear Parabolic Equations

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