﻿ 平均曲率型方程的全局梯度估计

# 平均曲率型方程的全局梯度估计Global Gradient Estimate of the Mean Curvature Type Equation

Abstract: In this paper, by selecting the appropriate auxiliary function, three cases are considered and the extremum principle is used to prove the global gradient estimation of the mean curvature type equation.

1. 引言

${a}_{ij}{u}_{ij}=:\left({\delta }_{ij}-\frac{{u}_{i}{u}_{j}}{1+{|\nabla u|}^{2}}\right){u}_{ij}=H\left(x\right)\sqrt{1+{|\nabla u|}^{2}}$ (1)

$|\nabla u\left(0\right)|\le \mathrm{exp}\left({C}_{1}+{C}_{2}\frac{{M}^{2}}{{r}^{2}}\right)$

${a}_{ij}{u}_{ij}=H\left(x\right)\sqrt{1+{|\nabla u|}^{2}}+\alpha u$

${a}_{ij}{u}_{ij}=H\left(x\right)\sqrt{1+{|\nabla u|}^{2}}+\alpha u$

$|H\left(x\right)|\le {C}_{0}$$|\nabla H\left(x\right)|\le {C}_{0}$，则有

$|\nabla u\left(0\right)|\le \mathrm{exp}\left({C}_{1}+{C}_{2}|\alpha |+{C}_{3}\frac{{M}^{2}}{{r}^{2}}+{C}_{3}\frac{{M}^{2}}{{r}^{2}}\right)$

$\Delta u-\frac{{u}_{i}{u}_{j}}{1+{|\nabla u|}^{2}}{u}_{ij}=H\left(x\right)\sqrt{1+{|\nabla u|}^{2}}\text{in}\Omega$

$\underset{\Omega }{\mathrm{sup}}|\nabla u|\le \left(\underset{\partial \Omega }{\mathrm{sup}}|\nabla u|+\underset{\Omega }{\mathrm{sup}}|H|+2\right)\mathrm{exp}\left\{\left(\underset{\Omega }{\mathrm{sup}}|\nabla H|+{c}_{0}\right)\underset{\Omega }{\text{osc}}u\right\}$

${a}_{ij}{u}_{ij}=H\left(x\right)\sqrt{1+{|\nabla u|}^{2}}+\alpha u.$

2. 主要结果

${a}_{ij}{u}_{ij}=H\left(x\right)\sqrt{1+{|\nabla u|}^{2}}+\alpha u$

$\Delta u-\frac{{u}_{i}{u}_{j}}{1+{|\nabla u|}^{2}}{u}_{ij}=H\left(x\right)\sqrt{1+{|\nabla u|}^{2}}+\alpha \text{u}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}\Omega$

$\underset{\Omega }{\mathrm{sup}}|\nabla u|\le \left(\underset{\partial \Omega }{\mathrm{sup}}|\nabla u|+\underset{\Omega }{\mathrm{sup}}|H|\right)\mathrm{exp}\left\{\left(\underset{\Omega }{\mathrm{sup}}|\nabla H|+{c}_{0}\right){|u|}_{{C}^{0}}\right\}$

${a}_{ij}={\delta }_{ij}-\frac{{p}_{i}{p}_{j}}{1+{|p|}^{2}}$

$f\left(x,u,p\right)=H\left(x\right)\sqrt{1+{|p|}^{2}}+\alpha u$

${a}_{ij}\left(Du\right){u}_{ij}=f\left(x,u,Du\right)\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{inΩ}.$ (2)

$\phi \left(x\right)={|Du|}^{2}{e}^{\beta \left({M}_{0}+u\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{inΩ}$

$\phi \left(x\right)$${x}_{0}\in \stackrel{¯}{\Omega }$ 处达到极大值，则 $\forall x\in \text{Ω}$$\phi \left(x\right)\le \phi \left({x}_{0}\right)$

${|Du\left(x\right)|}^{2}{\text{e}}^{\beta \left({M}_{0}+u\left(x\right)\right)}\le {|Du\left({x}_{0}\right)|}^{2}{\text{e}}^{\beta \left({M}_{0}+u\left({x}_{0}\right)\right)}$

${|Du\left(x\right)|}^{2}\le {|Du\left({x}_{0}\right)|}^{2}{\text{e}}^{\beta \left(u\left({x}_{0}\right)-u\left( x \right)}\right)$

$|Du\left(x\right)|\le |Du\left({x}_{0}\right)|{\text{e}}^{\frac{1}{2}\beta \left(u\left({x}_{0}\right)-u\left(x\right)\right)}\le |Du\left({x}_{0}\right)|{\text{e}}^{\beta |u{|}_{{C}^{0}}}$

$|Du\left(x\right)|\le |Du\left({x}_{0}\right)|\mathrm{exp}\left\{\beta {|u|}_{{C}^{0}}\right\}\text{.}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{inΩ}$ (3)

$\phi \left(x\right)$${x}_{0}\in \partial \text{Ω}$ 处达到极大值，则

$|Du\left({x}_{0}\right)|\le \underset{\partial \Omega }{\mathrm{sup}}|Du|\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{inΩ}$ (4)

$\phi \left(x\right)$${x}_{0}\in \text{Ω}$ 处达到极大值，由极值原理

${\left(\mathrm{log}\phi \right)}_{i}\left({x}_{0}\right)=0$${\left(\mathrm{log}\phi \right)}_{ij}\left({x}_{0}\right)\le 0$

${\left(\mathrm{log}\phi \right)}_{i}=\frac{2{u}_{k}{u}_{ki}}{{|Du|}^{2}}+\beta {u}_{i}$

${\left(\mathrm{log}\phi \right)}_{ij}=\frac{2{u}_{k}{u}_{kij}}{{|Du|}^{2}}+\frac{2{u}_{ki}{u}_{kj}}{{|Du|}^{2}}-\frac{4{u}_{k}{u}_{l}{u}_{ki}{u}_{lj}}{{|Du|}^{4}}+\beta {u}_{ij}$

${a}_{ij}{\left(\mathrm{log}\phi \right)}_{ij}=\frac{2{u}_{k}}{{|Du|}^{2}}{a}_{ij}{u}_{kij}+\frac{2}{{|Du|}^{2}}{a}_{ij}{u}_{ki}{u}_{kj}-\frac{4{u}_{k}{u}_{l}}{{|Du|}^{4}}{a}_{ij}{u}_{ki}{u}_{lj}+\beta {a}_{ij}{u}_{ij}$

${a}_{ij}{u}_{kij}+{a}_{ij,{p}_{l}}{u}_{kl}{u}_{ij}={\partial }_{k}f$

${a}_{ij}{\left(log\phi \right)}_{ij}=-\frac{2{u}_{k}}{{|Du|}^{2}}{a}_{ij,{p}_{l}}{u}_{kl}{u}_{ij}+\frac{2}{{|Du|}^{2}}{a}_{ij}{u}_{ki}{u}_{kj}-\frac{4{u}_{k}{u}_{l}}{{|Du|}^{4}}{a}_{ij}{u}_{ki}{u}_{lj}+\frac{2{u}_{k}}{{|Du|}^{2}}{\partial }_{k}f+\beta f.$

$|Du\left({x}_{0}\right)|=\left({u}_{1}\left({x}_{0}\right),0,\cdots ,0\right)$

${a}_{ij}{\left(\mathrm{log}\phi \right)}_{ij}=-\frac{2}{{u}_{1}}{a}_{ij,{p}_{l}}{u}_{1l}{u}_{ij}+\frac{2}{{u}_{1}^{2}}{a}_{ij}{u}_{ki}{u}_{kj}-\frac{4}{{u}_{1}^{2}}{a}_{ij}{u}_{1i}{u}_{1j}+\frac{2}{{u}_{1}}{\partial }_{1}f+\beta f.$

${\left(\mathrm{log}\phi \right)}_{i}=0$，有

${u}_{1i}=-\frac{1}{2}\beta {u}_{1}{u}_{i}$

${u}_{11}=-\frac{1}{2}\beta {u}_{1}^{2}$(5)

${u}_{1i}=0,\text{}i\ge 2,$

$\left({u}_{ij}\left({x}_{0}\right)\right)$ 是对角矩阵。因此有

$\begin{array}{c}{a}_{ij}{\left(\mathrm{log}\phi \right)}_{ij}=-\frac{2}{{u}_{1}}{a}_{ii,{p}_{1}}{u}_{11}{u}_{ii}+\frac{2}{{u}_{1}^{2}}{a}_{ii}{u}_{ii}^{2}-\frac{4}{{u}_{1}^{2}}{a}_{11}{u}_{11}^{2}+\frac{2}{{u}_{1}}{\partial }_{1}f+\beta f\\ =-\frac{2}{{u}_{1}}{a}_{ii,{p}_{1}}{u}_{11}{u}_{ii}-\frac{2}{{u}_{1}^{2}}{a}_{11}{u}_{11}^{2}+{\sum }_{i=2}^{n}\frac{2}{{u}_{1}^{2}}{a}_{ii}{u}_{ii}^{2}+\frac{2}{{u}_{1}}{\partial }_{1}f+\beta f.\end{array}$

${a}_{11}=\frac{1}{1+{u}_{1}^{2}}$${a}_{11,{p}_{1}}=-\frac{2{u}_{1}}{{\left(1+{u}_{1}^{2}\right)}^{2}}$

${a}_{ii}=1\text{}\left(i\ge 2\right)$${a}_{ii,{p}_{1}}=0\text{}\left(i\ge 2\right)$

$\begin{array}{c}{a}_{ij}{\left(\mathrm{log}\phi \right)}_{ij}\ge \frac{4}{{\left(1+{u}_{1}^{2}\right)}^{2}}{u}_{11}^{2}-\frac{2}{{u}_{1}^{2}\left(1+{u}_{1}^{2}\right)}{u}_{11}^{2}+\frac{2}{{u}_{1}}{\partial }_{1}f+\beta f\\ =\frac{2\left({u}_{1}^{2}-1\right)}{{u}_{1}^{2}{\left(1+{u}_{1}^{2}\right)}^{2}}{u}_{11}^{2}+\frac{2}{{u}_{1}}{\partial }_{1}f+\beta f.\end{array}$

${\partial }_{1}f={H}_{1}\sqrt{1+{u}_{1}^{2}}+\frac{H{u}_{1}{u}_{11}}{\sqrt{1+{u}_{1}^{2}}}+\alpha {u}_{1}$

$\begin{array}{c}{a}_{ij}{\left(\mathrm{log}\phi \right)}_{ij}\ge \frac{2\left({u}_{1}^{2}-1\right)}{{u}_{1}^{2}{\left(1+{u}_{1}^{2}\right)}^{2}}{u}_{11}^{2}+\frac{2{H}_{1}}{{u}_{1}}\sqrt{1+{u}_{1}^{2}}+\frac{2H{u}_{11}}{\sqrt{1+{u}_{1}^{2}}}+2\alpha +\beta H\sqrt{1+{u}_{1}^{2}}+\beta \alpha u\\ \ge \frac{2\left({u}_{1}^{2}-1\right)}{{u}_{1}^{2}{\left(1+{u}_{1}^{2}\right)}^{2}}{u}_{11}^{2}+\frac{2{H}_{1}}{{u}_{1}}\sqrt{1+{u}_{1}^{2}}+\frac{2H{u}_{11}}{\sqrt{1+{u}_{1}^{2}}}+\beta H\sqrt{1+{u}_{1}^{2}}.\end{array}$

${a}_{ij}{\left(\mathrm{log}\phi \right)}_{ij}\ge \frac{{\beta }^{2}{u}_{1}^{2}\left({u}_{1}^{2}-1\right)}{2{\left(1+{u}_{1}^{2}\right)}^{2}}+\frac{2{H}_{1}}{{u}_{1}}\sqrt{1+{u}_{1}^{2}}+\frac{-\beta H{u}_{1}^{2}}{\sqrt{1+{u}_{1}^{2}}}+\frac{\beta H+\beta H{u}_{1}^{2}}{\sqrt{1+{u}_{1}^{2}}}$

$\frac{{\beta }^{2}{u}_{1}^{2}\left({u}_{1}^{2}-1\right)}{2{\left(1+{u}_{1}^{2}\right)}^{2}}+\frac{2{H}_{1}}{{u}_{1}}\sqrt{1+{u}_{1}^{2}}+\frac{\beta H}{\sqrt{1+{u}_{1}^{2}}}\le 0$

$\sqrt{1+{u}_{1}^{2}}\left(\frac{{\beta }^{2}{u}_{1}^{2}\left({u}_{1}^{2}-1\right)}{2{\left(1+{u}_{1}^{2}\right)}^{2}}+\frac{2{H}_{1}}{{u}_{1}}\sqrt{1+{u}_{1}^{2}}\right)\le -\beta H$

${u}_{1}^{2}\le 3$，则

$\frac{{u}_{1}^{2}\left({u}_{1}^{2}-1\right)}{2{\left(1+{u}_{1}^{2}\right)}^{2}}=\frac{1}{2}\left(1-\frac{1}{1+{u}_{1}^{2}}\right)\left(1-\frac{2}{1+{u}_{1}^{2}}\right)\le \frac{3}{16}$

$\frac{1+{u}_{1}^{2}}{{u}_{1}^{2}}=1+\frac{1}{{u}_{1}^{2}}\ge \frac{4}{3}$

$\frac{{\beta }^{2}{u}_{1}^{2}\left({u}_{1}^{2}-1\right)}{2{\left(1+{u}_{1}^{2}\right)}^{2}}\le \frac{3}{16}{\beta }^{2}$

$|{u}_{1}|\left(\frac{3}{16}{\beta }^{2}+\frac{4}{\sqrt{3}}|\nabla H|\right)\le \beta |H|$

$\beta ={c}_{0}+\underset{\Omega }{\mathrm{sup}}|\nabla H|$，则

$\begin{array}{c}\frac{3}{16}{\beta }^{2}+\frac{4}{\sqrt{3}}|\nabla H|=\frac{3}{16}{\left({c}_{0}+\underset{\Omega }{\mathrm{sup}}|\nabla H|\right)}^{2}+\frac{4}{\sqrt{3}}|\nabla H|\\ \ge \frac{3}{16}{\left({c}_{0}+\underset{\Omega }{\mathrm{sup}}|\nabla H|\right)}^{2},\end{array}$

$\beta \ge \frac{16}{3}$ 时，就有

$\frac{3}{16}{\beta }^{2}+\frac{4}{\sqrt{3}}|\nabla H|\ge \beta \ge 1$

$|\nabla u\left({x}_{0}\right)|\le \underset{\Omega }{\mathrm{sup}}|H|$

${u}_{1}^{2}\ge 3$，则

$\frac{{u}_{1}^{2}\left({u}_{1}^{2}-1\right)}{2{\left(1+{u}_{1}^{2}\right)}^{2}}=\frac{1}{2}\left(1-\frac{1}{1+{u}_{1}^{2}}\right)\left(1-\frac{2}{1+{u}_{1}^{2}}\right)\ge \frac{3}{16}$

$\begin{array}{c}\frac{3}{16}{\beta }^{2}-\frac{4}{\sqrt{3}}|\nabla H|=\frac{3}{16}{\left({c}_{0}+\underset{\Omega }{\mathrm{sup}}|\nabla H|\right)}^{2}-\frac{4}{\sqrt{3}}|\nabla H|\\ =\frac{3}{16}\left({c}_{0}{}^{2}+{\underset{\Omega }{\mathrm{sup}}}^{2}|\nabla H|+2{c}_{0}\underset{\Omega }{\mathrm{sup}}|\nabla H|\right)-\frac{4}{\sqrt{3}}|\nabla H|\\ =\frac{3}{16}\left({c}_{0}{}^{2}+{\underset{\Omega }{\mathrm{sup}}}^{2}|\nabla H|\right)+\frac{3}{8}{c}_{0}\underset{\Omega }{\mathrm{sup}}|\nabla H|-\frac{4}{\sqrt{3}}|\nabla H|,\end{array}$

$\frac{3}{16}{\beta }^{2}-\frac{4}{\sqrt{3}}|\nabla H|\ge \beta \ge 1$

$|{u}_{1}\left({x}_{0}\right)|\le \underset{\Omega }{\mathrm{sup}}|H|$

$|\nabla u\left({x}_{0}\right)|\le \underset{\Omega }{\mathrm{sup}}|H|$

$|\nabla u\left({x}_{0}\right)|\le \underset{\Omega }{\mathrm{sup}}|H|$ (6)

$|\nabla u\left({x}_{0}\right)|\le \underset{\partial \Omega }{\mathrm{sup}}|\nabla u|+\underset{\Omega }{\mathrm{sup}}|H|$ (7)

[1] Bombieri, E., De Giorgi, E. and Miranda, M. (1969) Una maggiorazone a priori relativa alleipersuperfici minimali non parabolic. Archive for Rational Mechanics and Analysis, 32, 255-267.
https://doi.org/10.1007/BF00281503

[2] Ladyzhenskaya, O.A. and Ural’Tseva, N.N. (1970) Local Estimates for Solution of Non-Uniformly Elliptic and Parabolic Equations. Communications on Pure and Applied Mathematics, 23, 677-703.
https://doi.org/10.1002/cpa.3160230409

[3] Gilbarg, D. and Trudinger, N.S. (2001) Elliptic Partial Differential Equation of Second Order. Springer-Verlag, Berlin.
https://doi.org/10.1007/978-3-642-61798-0

[4] Wang, X.J. (1998) Interior Gradient Estimates for Mean Curvature. Mathematische Zeitschrift, 1, 73­81.
https://doi.org/10.1007/PL00004604

[5] 王聪涵. 平均曲率型方程的内部梯度估计和Liouville型结果[D]: [硕士学位论文]. 新乡: 河南师范大学, 2019.

[6] 奥列尼克. 偏微分方程讲义(第三版) [M]. 北京: 高等教育出版社, 2008: 124-135.

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