﻿ 一类散度型椭圆方程的霍普夫引理

# 一类散度型椭圆方程的霍普夫引理Hopf’s Lemma for a Class of Elliptic Equations of Divergence Type

Abstract: The Maximum Principle is one of the basic properties of elliptic partial differential equations. Linear elliptic partial differential equations have strong maximum principle, whose proof depends on Hopf’s lemma. This paper obtains Hopf’s lemma for a class of divergence elliptic equations.

1. 预备知识

$\Delta u=\frac{{\partial }^{2}u}{\partial {x}_{1}^{2}}+\frac{{\partial }^{2}u}{\partial {x}_{2}^{2}}+\cdots +\frac{{\partial }^{2}u}{\partial {x}_{n}^{2}}$

$\Delta$ 称为拉普拉斯算子。当 $f=0$ 时，位势方程也称为拉普拉斯方程，其解称为调和函数 [1]。调和函数的基本性质之一是强极值原理，它断言非常值的调和函数最值在闭区域上边界达到，而这个性质的证明需要用到霍普夫引理。拉普拉斯方程的霍普夫引理如下：

$\frac{\partial u}{\partial n}\left({x}_{0}\right)\ge C\left(u\left({x}_{0}\right)-u\left( 0 \right)\right)$

$Lu\equiv {a}_{ij}\left(x\right){D}_{ij}u+{b}_{i}\left(x\right){D}_{i}u+c\left(x\right)u$

2. 主要结论及证明

(1) 在B内， $Lu\le 0$

(2) 存在 ${x}_{0}\in \partial B$ 使 $u\left({x}_{0}\right)\ge 0$ 且对任意的 $x\in B$，有 $u\left(x\right)

$\frac{\partial u}{\partial \upsilon }>0$

$Lu=-\underset{i,j=1}{\overset{n}{\sum }}{\partial }_{i}\left({a}_{ij}\left(x\right){\partial }_{j}u\right)+\underset{i=1}{\overset{n}{\sum }}{b}_{i}\left(x\right)\frac{\partial u}{\partial {x}_{i}}+c\left(x\right)u=f\left(x\right),x\in \Omega$

$\underset{i,j=1}{\overset{N}{\sum }}{a}_{ij}\left(x\right){\xi }_{i}{\xi }_{j}>0$ (1.3)

$-\underset{i,j=1}{\overset{N}{\sum }}{\partial }_{i}\left({a}_{ij}\left(x\right){\partial }_{j}u\right)+\underset{i=1}{\overset{N}{\sum }}{b}_{i}\left(x\right){\partial }_{i}u+c\left(x\right)u\le 0$ (1.4)

$\Omega$ 内，对于 $1\le i\le N$，有 ${b}_{i}\in {L}^{\infty }\left(\Omega \right)$$c\in {L}^{\infty }\left(\Omega \right)$$c\left(x\right)\ge 0$

$u\left(x\right)>u\left( x 0 \right)$

$\frac{\partial u}{\partial \nu }\left({x}_{0}\right)>0$ (1.5)

$\nu$ 为外法向， $\nu \in {R}^{N}$ 以便 $〈\nu ,n〉<0$，n为指向 ${x}_{0}$ 的单位外法向量。

$Lu=-\underset{i,j=1}{\overset{N}{\sum }}{\partial }_{i}\left({a}_{ij}\left(x\right){\partial }_{j}u\right)$

${u}_{0}=u\left({x}_{0}\right)$，则

$L\left(u-{u}_{0}\right)+\underset{i=1}{\overset{N}{\sum }}{b}_{i}{\partial }_{i}\left(u-{u}_{0}\right)+c\left(u-{u}_{0}\right)\le -c{u}_{0}\le 0$

${\Omega }_{\rho }=\rho {e}_{N}-{D}_{\rho }=\left\{x:\frac{\rho }{2}<|x+\rho {e}_{N}|<\rho \right\}$

$L\upsilon +\underset{i=1}{\overset{N}{\sum }}{b}_{i}{\partial }_{i}\upsilon +c\upsilon =0,\text{\hspace{0.17em}}x\in {\Omega }_{\rho }$

$\upsilon =1,\text{\hspace{0.17em}}x\in \partial {\Omega }_{\rho }^{-}$ (2.1)

$\upsilon =0,\text{\hspace{0.17em}}x\in \partial {\Omega }_{\rho }^{+}$

$\partial {\Omega }_{\rho }^{-}=\left\{x:|x+\rho {e}_{N}|=\frac{\rho }{2}\right\}$$\partial {\Omega }_{\rho }^{+}=\left\{x:|x+\rho {e}_{N}|=\rho \right\}$.

$u-{u}_{0}-\epsilon \upsilon \le 0$

$\frac{\partial u}{\partial \nu }\left(0\right)\ge \epsilon \frac{\partial \upsilon }{\partial \nu }\left(0\right)$ (2.2)

$\frac{\partial \upsilon }{\partial \nu }\left(0\right)\to -\infty ,\text{\hspace{0.17em}}\rho \to {0}_{+}$ (2.3)

(b) 当 $u\left({x}_{0}\right)=0$ 时，不需要限制c的符号或对所有 $x\in \Omega$，有 $c\left(x\right)=0$，那么 $u\left({x}_{0}\right)$ 的符号可以是任意的。

NOTES

*通讯作者。

[1] 保继光, 朱汝金. 偏微分方程[M]. 北京: 北京师范大学出版社, 2011: 120.

[2] Han, Q. and Lin, F.H. (2011) El-liptic Partial Differential Equations. American Mathematical Society, 21-27.

[3] Finn, R. and Gilbarg, D. (1957) Asymptotic Behavior and Uniqueness of Plane Subsonic Flows. Communications on Pure and Applied Mathematics, 10, 23-63.
https://doi.org/10.1002/cpa.3160100102

[4] de Lis, S. and José, C. (2015) Hopf Maximum Principle Revisited. Electronic Journal of Differential Equations, 115, 1-9.

[5] Gilbarg, D. and Trudinger, N.S. (1983) Elliptic Partial Differential Equations of Second Order. Springer Verlag.

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