[Lecture notes]复Monge-Ampere方程

复Monge-Ampere方程 Lecturer:Ma XiNan

Introduction

Numann问题的边界梯度估计

考虑方程

$$
\begin{cases}
\Delta u=f(x),& x\in\Omega,\\
\left.\frac{\pt u}{\pt n}\right|_{\pt\Omega}=a(x),
\end{cases}
$$
我们相用极大值原理来做切向估计. 回忆, 内估计在HanQing-Lin FangHua的书上有(Chap2, Prop3.1, Prop3.2).

关键步骤:

  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$\pt\Omega_{\mu_0}达到极大, 用内估计.$

最后我们取的$g,h$为

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

可以参考Ladyzhenskaya1以及Lieberman2.

Calabi-Yau估计

参考:Fu, J-X., and S-T. Yau. “The theory of superstring with flux on non-Kähler manifolds and the complex Monge-Ampere equation.” Journal of Differential Geometry 78.3 (2008): 369-428.

The notes of 20130910

切像与无边梯度估计

主要参考文章Bo Guan3以及Caffarelli, L.; Kohn, J. J.; Nirenberg, L.; Spruck, J.合作的文章4.

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.

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