论文标题
完全非线性部分微分方程的$ l^\ infty $估计
On $L^\infty$ estimates for fully nonlinear partial differential equations
论文作者
论文摘要
对于非kähler歧管上的完全非线性PDE的一般类别获得了尖锐的$ l^\ infty $估计值，并补充了作者早些时候与F. Tong合作为Kähler案共同开发的理论。关键的想法仍然是与辅助蒙格 - 安培方程的比较，但是这次是在具有差异边界条件的球上，因此它总是可以允许独特的解决方案。该方法不仅适用于紧凑的Hermitian流形，而且还适用于Dirichlet问题，以打开对其注入性半径的正下限，$（n-1）$形式方程，甚至用于不可构成的几乎可以几乎复杂或符号歧管。这是任何一般性的非线性方程式中的第一种方法，当碰巧可用于特定方程式时，通常会在其他方法上进行改进。
Sharp $L^\infty$ estimates are obtained for general classes of fully non-linear PDE's on non-Kähler manifolds, complementing the theory developed earlier by the authors in joint work with F. Tong for the Kähler case. The key idea is still a comparison with an auxiliary Monge-Ampère equation, but this time on a ball with Dirichlet boundary conditions, so that it always admits a unique solution. The method applies not just to compact Hermitian manifolds, but also to the Dirichlet problem, to open manifolds with a positive lower bound on their injectivity radii, to $(n-1)$ form equations, and even to non-integrable almost-complex or symplectic manifolds. It is the first method applicable in any generality to large classes of non-linear equations, and it usually improves on other methods when they happen to be available for specific equations.
