AI 中文总结
本文通过建立无一致椭圆性的Pogorelov型内部$C^2$估计，证明了实扭曲蒙日-安培方程整分裂凸解的刘维尔定理，推广了Streets和Warren的刚性定理。
AI 中文摘要
我们在不假设一致椭圆性的情况下，建立了实扭曲蒙日-安培方程的Pogorelov型内部$C^2$估计。该方程是完全非线性且椭圆的，但既不是凹的，也不是Hessian特征值的函数，因此标准的内部二阶导数估计技术不能直接适用。基于Streets和Warren的偏Legendre变换公式，我们推导了相关的正定矩阵$W(D^2u)$的微分恒等式，并利用一些线性代数来控制极大值原理论证中出现的三阶项。作为应用，我们证明了具有二次增长的整分裂凸解的刘维尔定理，通过用适当的二次增长假设替换他们的一致椭圆性假设，推广了Streets和Warren的刚性定理。
英文摘要
We establish a Pogorelov type interior $C^2$ estimate for the real twisted Monge-Ampere equation without assuming uniform ellipticity. This equation is fully nonlinear and elliptic, but is neither concave nor a function of the eigenvalues of the Hessian, so the standard techniques for interior second-derivative estimates do not directly apply. Building on the partial Legendre transform formulation of Streets and Warren, we derive a differential identity for the associated positive-definite matrix $W(D^2u)$ and use some linear algebra to control the third-order terms arising in the maximum principle argument. As an application, we prove a Liouville theorem for entire split convex solutions with quadratic growth, extending the rigidity theorem of Streets and Warren by replacing their uniform ellipticity hypothesis with suitable quadratic growth assumptions.
Comments28 pages. Comments are welcome!
