抛物型Monge-Ampere方程的整体解的存在性与解的局部估计

Partial differential equation is one of the core fields in fundamental research. The parabolic Monge-Ampere equation has important applications in stochastic theory and differential geometry. Many key issues are equivalent to finding some solutions to parabolic Monge-Ampere equations. This project focuses on the existence of entire ancient solutions to parabolic Monge-Ampere equation, which is closely related to the property of the Gauss-Kronecker curvature around Type II singularity. We will also explore the optimal estimates of solutions to parabolic Monge-Ampere equation. Besides, we will study the optimal Lp estimates of solutions to linearized parabolic Monge-Ampere equation.

偏微分方程是基础研究的核心领域之一。抛物型Monge-Ampere方程在随机理论，微分几何等领域有着重要的应用。许多关键问题都等价于求解相应的抛物型Monge-Ampere方程。本项目将研究抛物型Monge-Ampere方程的古代解的整体存在性，它与Gauss-Kronecker曲率流的第II型奇点密切相关。本项目还将讨论抛物型Monge-Ampere方程的解的最优先验估计。此外，我们将研究线性化抛物型Monge-Ampere方程的解的Lp估计.

抛物型Monge-Ampere方程在随机理论，微分几何等领域有着重要的应用。本项目研究了抛物型Monge-Ampere方程以及线性化抛物型Monge-Ampere方程解的局部估计。我们得到了具有VMO型数据的抛物型Monge-Ampere方程解的局部估计，以及线性化抛物型Monge-Ampere方程解的Lp估计。此外，我们还得到了完备Riemannian流形上的一类椭圆方程正解的梯度估计。