Gradient estimates for oblique derivative problems via the maximum principle

TL;DR

This paper establishes gradient estimates for solutions to oblique derivative problems in elliptic and parabolic PDEs, extending previous results and introducing new estimates for mean curvature evolution with Neumann data.

Contribution

The paper provides new gradient estimates for a broad class of elliptic and parabolic PDEs, including novel results for mean curvature evolution with nonzero Neumann boundary conditions.

Findings

Gradient estimates for elliptic and parabolic quasilinear PDEs.

Extension of maximum principle techniques to oblique derivative problems.

New estimates for mean curvature evolution with Neumann data.

Abstract

We prove gradient estimates for solutions of the oblique derivative problem for a large class of elliptic and parabolic quasilinear PDEs. In particular, we expand on previous work of the author using a maximum principle argument. In addition, we prove estimates for the parabolic problem of evolution via mean curvature with nonzero Neumann data, which seem to be new.