Up-to-boundary pointwise gradient estimates for very singular quasilinear elliptic equations with mixed data

TL;DR

This paper develops boundary pointwise gradient estimates for weak solutions to highly singular quasilinear elliptic equations with mixed data, extending understanding of solution regularity up to the boundary in Reifenberg-flat domains.

Contribution

It provides new boundary gradient estimates for a class of very singular quasilinear elliptic equations with mixed data, advancing regularity theory in non-smooth domains.

Findings

Established boundary pointwise gradient estimates for solutions.

Extended regularity results to Reifenberg-flat domains.

Handled equations with mixed divergence form data.

Abstract

This paper establishes pointwise estimates up to boundary for the gradient of weak solutions to a class of very singular quasilinear elliptic equations with mixed data: \begin{cases} -\operatorname{div}(A(x,D u))=g-\operatorname{div} f \quad & \mathrm{in} \quad \Omega \\ u= 0 \quad & \text{on} \ \partial \Omega, \end{cases} where $\Omega \subset \mathbb{R}^n$ is sufficiently flat in the sense of Reifenberg.