带有辛黑塞矩阵的凸函数
Convex functions with symplectic Hessian
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中文总结 AI 辅助
该研究针对R²ᵐ上满足det(Hess(u))=1且黑塞矩阵属于Sp(2m,ℝ)的实Monge-Ampère方程凸解，证明三阶导数估计，推广Nitsche的经典证明并改进相关常数与估计，还构造了具特殊几何行为的解的例子。
中文摘要 AI 辅助
我们证明了，在R²ᵐ的开集上，对于实Monge-Ampère方程det(Hess(u))=1的凸解，若其黑塞矩阵Hess(u)在每一点都属于Sp(2m,ℝ)（实辛群），则存在三阶导数估计。我们的方法是将Nitsche关于R²上实Monge-Ampère方程Bernstein定理的经典证明，做几何解释并推广到更高维。对于m=1，我们还改进了Nitsche的常数以及Calabi的一些估计，并构造了具有有趣几何行为的解的例子。
英文摘要
We prove a third-order derivative estimate for convex solutions to the real Monge-Ampère equation ${\rm det}\,{\rm Hess}(u) = 1$ on an open set in $\mathbb{R}^{2m}$ under the additional assumption that ${\rm Hess}(u)$ lies in ${\rm Sp}(2m,\mathbb{R})$ at every point. Our method is a geometric interpretation and extension to higher dimensions of Nitsche's classical proof of the Bernstein theorem for the real Monge-Ampère equation on $\mathbb{R}^2$. For $m = 1$ we also improve Nitsche's constant as well as some estimates due to Calabi, and we construct examples of solutions with interesting geometric behavior.
