In this paper, we study the local gradient regularity of non-negative weak solutions to doubly nonlinear parabolic partial differential equations of the type \begin{ align*} \partial_t u^q - \mbox{ div}\, A(x,t,Du)=0 \qquad\mbox{ in $\Omega_T$}, \end{ align*} with $q>0$, $\Omega_T=\Omega\times(0,T)\subset\mathbb{ R}^{ n+1}$ a space-time cylinder, and $A=A(x,t,\xi)$ a vector field satisfying standard $p$-growth conditions. Our main result establishes the local H\"older continuity of the spatial gradient of non-negative weak solutions in the super-critical fast diffusion regime $$0<p-1<q<\frac{ n(p-1)}{ (n-p)_+}.$$ This result is achieved by utilizing a time-insensitive Harnack inequality and Schauder estimates that are developed for equations of parabolic $p$-Laplacian type. Additionally, we establish a local $L^{ \infty}$-bound for the spatial gradient.
Categories: math.AP