伪欧式空间中的拉格朗日平均曲率流研究

Research on the Lagrangian Mean Curvature Flow in Pseudo-Euclidean

摘要: 该文讨论在伪欧氏空间中，带有以下初始条件的拉格朗日平均曲率流方程$\{\begin{array}{l}\frac{\text{d}Y\left(x,t\right)}{\text{d}t}=H\\ Y\left(x,0\right)=Y_0\left(x\right)\end{array}$ 。其中，该方程等价于特殊拉格朗日抛物方程$\{\begin{array}{l}\frac{\partial u}{\partial t}=F_{\tau }\left(D^{2}u\right),t>0,x\in {ℝ}^n\\ u=u_0\left(x\right),t=0,x\in {ℝ}^n\end{array}$ 。通过构造函数，将证明若$0<\tau <\frac{\pi }{4}$ 或$\frac{\pi }{4}<\tau <\frac{\pi }{2}$ ，该抛物方程存在唯一光滑解$u\left(x,t\right)$ ，且存在更高阶导数的衰减估计。另一方面，应用Arzelà-Ascoli定理来获得$u\left(x,t\right)$ 收敛到拉格朗日平均曲率流方程的自膨胀解。

Abstract: In this paper, we consider the Lagrangian mean curvature flow equation in pseudo-Euclidean space with the initial value: $\{\begin{array}{l}\frac{\text{d}Y\left(x,t\right)}{\text{d}t}=H\\ Y\left(x,0\right)=Y_0\left(x\right)\end{array}$ . This equation is equivalent to the special Lagrangian parabolic equation $\{\begin{array}{l}\frac{\partial u}{\partial t}=F_{\tau }\left(D^{2}u\right),t>0,x\in {ℝ}^n\\ u=u_0\left(x\right),t=0,x\in {ℝ}^n\end{array}$ . By constructing a suitable function, it is proven that if $0<\tau <\frac{\pi }{4}$ or $\frac{\pi }{4}<\tau <\frac{\pi }{2}$ , the parabolic equation has a unique smooth solution $u\left(x,t\right)$ and decay estimates for higher-order derivatives exist. On the other hand, the Arzelà-Ascoli theorem is applied to obtain the convergence of $u\left(x,t\right)$ to the self-expanding solution of the Lagrangian mean curvature flow equation.