中文摘要：

该文介绍了Laplace方程斜边值问题解的梯度估计的两种证明方法：第一种证明重新整理文献[1]中的梯度估计;第二种证明采用不同于文献[1]的辅助函数得到估计.两种方法都充分利用函数在极大值点的性质,得到边界梯度估计和近边梯度估计,结合文献[2]中已有的梯度内估计,从而得到解的全局梯度估计.

英文摘要：

In this paper, the authors study two proofs for the gradient estimates of the Laplace equations with oblique boundary value condition. For the first proof, the gradient estimates of Lieberman[1] are rearranged; for the second proof, barrier function which is different from [1] is used to obtain the gradient estimates. They both use the property of the maximum value point, and get the near boundary gradient estimates and boundary gradient estimates, combining the given inner gradient estimates in [2], and then they obtain the global gradient estimates.