Gradient regularity for a class of doubly nonlinear parabolic partial differential equations

TL;DR

This paper proves local gradient regularity and H"older continuity for solutions to a class of doubly nonlinear parabolic PDEs, extending understanding of their regularity properties in the super-critical fast diffusion regime.

Contribution

It establishes the local H"older continuity of the spatial gradient for solutions to doubly nonlinear parabolic PDEs in a new super-critical regime, using novel Harnack inequalities and Schauder estimates.

Findings

Proves local H"older continuity of the gradient.

Establishes local $L^{ abla u}$-bounds for solutions.

Develops new regularity tools for doubly nonlinear equations.

Abstract

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.