Boundary higher integrability for very weak solutions of quasilinear parabolic equations

TL;DR

This paper establishes boundary higher integrability for the gradients of very weak solutions to quasilinear parabolic equations, extending known results to less regular settings and including systems and higher order equations.

Contribution

It proves boundary higher integrability for very weak solutions of quasilinear parabolic equations without smoothness assumptions on the structure, applicable to systems and higher order cases.

Findings

Gradients satisfy a reverse Hölder inequality near the boundary

Constructed Lipschitz test functions preserving boundary values

Results are new even for linear equations on smooth domains

Abstract

We prove boundary higher integrability for the (spatial) gradient of \emph{very weak} solutions of quasilinear parabolic equations of the form $$u_t - \text{div}\,\mathcal{A}(x,t, \nabla u)=0 \quad \text{on} \ \Omega \times \mathbb{R},$$ where the non-linear structure $\text{div}\,\mathcal{A}(x, t,\nabla u)$ is modelled after the $p$-Laplace operator. To this end, we prove that the gradients satisfy a reverse H\"older inequality near the boundary. In order to do this, we construct a suitable test function which is Lipschitz continuous and preserves the boundary values. \emph{These results are new even for linear parabolic equations on domains with smooth boundary and make no assumptions on the smoothness of $\mathcal{A}(x,t,\nabla u)$}. These results are also applicable for systems as well as higher order parabolic equations.