一类Laplace方程的Neumann问题的边界梯度估计

Boundary Gradient Estimation of Neumann Problem for a Class of Laplace Equations

在二阶椭圆偏微分方程中,边值问题仍然是十分重要的问题之一,其中Neumann问题是大家极力想解决的问题。本文主要借助梯度内估计、Hopf引理、极大值原理给出一类Laplace方程的Neumann问题的边界梯度估计的一个证明。主要根据所在领域分为三种情况:1)若φ(x)在■Ωμ0∩Ω上达到极大值则归结为梯度内估计;2)若φ(x)在■Ω上达到极大值则可由Hopf引理可得|Du(x0)|有界;3)对固定小的正常数μ0>0,若φ(x)在Ωμ0上达到极大值则有极大值原理证明|Du(x0)|有界。三种情况采用不同方式证明,综合得到最终的结果:supΩμ0|Du|≤max{M1,M2}。

Boundary value problem is still of great importance in second-order elliptic partial differential equations,among which is Neumann problem that presses for a solution.In this paper,a proof of boundary gradient estimation of Neumann problem for a class of Laplace equation is given by means of internal gradient estimation,Hopf lemma and maximum principle.According to the field,there are mainly three cases:1) if φ(x) reaches the maximum value on φΩμ 0 ∩Ω,it will be reduced to the internal gradient estimation;2) if φ(x) reaches the maximum value on φΩ,then |Du(x0)| can be bounded by Hopf lemma;3) for a fixed small normal number μ0>0,maximum principle proves that |Du(x0)| is bounded if φ(x) reaches the maximum value on Ωμ0.The three cases are proved in different ways and the final result supΩμ0|Du|≤max {M1,M2} is obtained by synthesis.