A New Proof of Gradient Estimates for Mean Curvature Equations with Oblique Boundary Conditions

TL;DR

This paper presents a novel proof technique using the maximum principle to establish gradient estimates for mean curvature equations with oblique boundary conditions, including capillary and Neumann problems across various dimensions.

Contribution

It introduces a new proof method for gradient estimates in mean curvature equations with oblique boundary conditions, applicable to capillary and Neumann problems in multiple dimensions.

Findings

New proof of gradient estimates using maximum principle

Applicable to capillary problems with zero gravity in any dimension

Valid for Neumann problems in 2 and 3 dimensions

Abstract

In this paper, we will use the maximum principle to give a new proof of the gradient estimates for mean curvature equations with some oblique derivative problems. Specially, we shall give a new proof for the capillary problem with zero gravity in any dimension $n\geq 2$ and Neumann problem in $n=2, 3$ dimensions.