# [Lecture notes]复Monge-Ampere方程

### Introduction

##### Numann问题的边界梯度估计

$$\begin{cases} \Delta u=f(x),& x\in\Omega,\\ \left.\frac{\pt u}{\pt n}\right|_{\pt\Omega}=a(x), \end{cases}$$

1. 令$w=u(x)+a(x) d$, $d=|x-x_0|$, $x_0\in\pt\Omega$. 若$\pt\Omega\in C^2$, 则由GT书Chap14知$d(x)\in C^2$, 当$x\in\Omega_\mu=\set{x\in\Omega|d(x,\pt\Omega< \mu)}$, $\mu$充分小. 这里我们已将$a(x)$延拓成$C^1(\bar\Omega)$./li>
2. 我们只需对$w$作近边估计, 因为内估计已有. 令
$$\phi(x)=\log|\nabla w|^2+g(d)+h(u),$$
取好的$g,h$使得$\phi$不在$\pt\Omega$达到极大.
3. $\phi$在$\Omega_\mu$内取得极大从而得到估计.
4. 若$\phi$在$\pt\Omega_{\mu_0}达到极大, 用内估计.$

$$g(d)=e^{\alpha_0 d},\quad h(t)=-\frac{1}{2}\log\left(\frac{1}{2}-\frac{t}{4M}\right),\quad M=1+\sup_\Omega |u|.$$

##### Calabi-Yau估计

The notes of 20130910

##### 切像与无边梯度估计

The notes of 20130917

The notes of 20130924

1. Ladyzhenskaya, O. Ao, and No N. Ural’tseva. Linear and quasilinear equations of elliptic type. (1973): 576. Chap9 section 2.
2. Lieberman, G. M. (1996). Second order parabolic differential equations. World scientific.
3. Guan, B. The Dirichlet problem for complex Monge-Ampere equations and applications. (2009).
4. Caffarelli, L., et al. “The dirichlet problem for nonlinear second‐order elliptic equations. II. Complex monge‐ampère, and uniformaly elliptic, equations.” Communications on pure and applied mathematics 38.2 (1985): 209-252.