L_(p)-Minkowski问题周期解的存在性

Existence of periodic solution of L_(p)-Minkowski problem

L_(p)-Minkowski问题是凸几何分析L_(p)-Brunn-Minkowski理论的核心,其实质是分析给定测度是否为凸体的L_(p)表面积测度问题.这个问题可以简化为二阶微分方程的周期解的存在性,L_(p)-Minkowski问题中周期解的存在性问题如下u″+u=h(t)/u^(ρ),其中h>0是连续的周期函数,常数ρ=1-p.利用了二阶微分方程周期解存在的充分条件,通过建立的一个方程的周期解的存在性判据,再利用Sobolev inequality证明了这个二阶微分方程周期解的存在性,得到在一定条件下周期解存在,这个方法在一定程度上扩大p的取值范围.最后给出一个例子,验证文中所得到的主要结果的可行性.

The L_(p)-Minkowski problem was the core of convex geometric analysis,specifically the L_(p)-Brunn-Minkowski theory.It aimed to analyze whether a given measure is the surface area measure of a convex body.This problem could be simplified as the existence of periodic solutions for second-order differential equations.The existence of periodic solutions in the L_(p)-Minkowski problem is stated as follows:u″+u=h(t)/u^(ρ),where h is a continuous periodic function and the constantρ=1-p.By utilizing the sufficient condition for the existence of periodic solutions in second-order differential equations,the existence of a periodic solution for a derived equation was established.This was further proved using Sobolev inequality,demonstrating the existence of periodic solutions for the second-order differential equation under certain conditions.This method expanded the range of values for p to a certain extent.Finally,an example was provided to verify the feasibility of the main results obtained in this paper.