复Monge-Ampere方程
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:
- 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>
- 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.
- $\phi$在$\Omega_\mu$内取得极大从而得到估计.
- 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
- Ladyzhenskaya, O. Ao, and No N. Ural’tseva. Linear and quasilinear equations of elliptic type. (1973): 576. Chap9 section 2. ↩
- Lieberman, G. M. (1996). Second order parabolic differential equations. World scientific. ↩
- Guan, B. The Dirichlet problem for complex Monge-Ampere equations and applications. (2009). ↩
- 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. ↩
