Gradient estimation of solutions for the second boundary value 问题 of the mean curvature equation

Abstract

In partial differential equations, the <italic>a priori</italic>estimate is very important for the existence of solutions to boundary value problems. In this paper, the gradient estimate of the second boundary value 问题 of the mean curvature equation div$(~\frac{Du}{\sqrt{1+|Du|^{2}}})=f(~x,u,Du)$ is given, and the range of $\alpha$ of the gradient estimation of the 解 for the second boundary value 问题 of the mean curvature equation div$(~\frac{Du}{\sqrt{1+|Du|^{2}}})=(~1+|Du|^{2})^{\alpha~}$ is obtained, which is consistent with the range of the gradient estimation obtained by Simon (1976).