一类常微分方程的伯恩斯坦定理Ⅱ

Bernstein's Theorem for a Class of Ordinary Differential Equations Ⅱ

本文基于对伪欧氏空间中拉格朗日平均曲率流自相似膨胀解的伯恩斯坦定理研究,不失一般性,即考虑一类二阶常微分方程u″=F(u-1/2tu′),u=u(t)在一定条件下解的形式,若u′(0)=0,且本文对函数F做出限制条件—函数F(u′,u,t)解析,则可得到方程的解必然是二次多项式。同时本文对一类常微分方程的解的经典伯恩斯坦定理首次利用更为简洁直观的方法加以证明,进而完善伪欧氏空间拉格朗日平均曲率流自相似解刚性定理研究。

In this paper,the Bernstein theorem for the self-expansion solution of the Lagrangian mean curvature flow in the pseudo-European space is studied.Without loss of the generality,for a class of second order ordinary differential equations such as u″=F u-1[]2 tu′,u=u(t),and under certain conditions,their solutions are investigated.If u′(0)=0,and the function F is an analytic function,it is shown that the solutions of the equations are quadratic polynomials.At the same time,the classical Bernstein theorem for the solution of a class of equations is proved by using a more concise and intuitionistic method for the first time,and then the study for the self-similarity solution of the pseudo-European space Lagrangian mean curvature is developed.