权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Reserch on the stability of solutions of geometric evolution equation using group equivariance

利用群等方差研究几何演化方程解的稳定性

基本信息

批准号：
17540188
负责人：
NAGASAWA Takeyuki
金额：
$ 2.41万
依托单位：
Saitama University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2007
项目状态：
已结题

项目摘要

In this research we investigate the gradient flow with constraints for functionals defined for family of curves and surfaces as geometric evolution equations. The gradient flow, which decreases the value of functionals via deformation, is one of the method for finding critical points. Various shapes in the nature should be stable in some sense. Functionals are the measure of stability, and therefore the limit of gradient flow should be stable in this sense.Nagasawa and Kohsaka consider the Willmore functional for surfaces with prescribed area and enclosed volume (the Helfrich variational problem), and construct the associate gradient flow (the Helfrich flow), and analyze the structure of center manifold near sphere. On the stationary problem for the problem, solutions bifurcating from sphere with more 2, 4, 6 and 8 are constructed by Nagasawa. We reduce the bifurcation equation by use of the group equivariance but not the equivariant branching lemma of the bifurcation theory. Furthermore Nagasawa considers the Helfrich flow for plane curves. There an approximate problem such that the constraints are realized as a singular limit is proposed. It is shown that the uniform estimates for solutions for approximate problem and their convergence.In many case, equations of gradient flow are parabolic type. Koike investigates the maximum principle and comparison results for fully nonlinear parabolic and elliptic equations. Ohta studies the stability of solutions for evolution equation of hyperbolic type. Sakamoto investigates the CR-structure of manifolds. Kohsaka studies the nonlinear stability of stationary solutions for surface diffusion with boundary conditions. Solutions of geometric variational problem are weak solution of a nonlinear equation. Hence it is important to analyze their regularity. Tachikawa studies the regularity theory of minimal critical points for integral functional with discontinuous coefficients.

在本研究中，我们研究了定义在曲线曲面族上的泛函作为几何演化方程的带约束的梯度流。梯度流是一种通过形变来减小泛函值的方法，它是寻找临界点的一种方法。自然界中的各种形状在某种意义上应该是稳定的。泛函是稳定性的度量，因此梯度流的极限在这个意义上应该是稳定的，Nagasawa和Kohsawa考虑了具有给定面积和封闭体积的曲面的Willmore泛函（Helfrich变分问题），构造了相应的梯度流（Helfrich流），并分析了球面附近中心流形的结构。关于该问题的平稳性问题，Nagasawa构造了从球面分支出多个2，4，6和8的解。我们利用群等变性而不是分歧理论中的等变分支引理来约化分歧方程。此外，Nagasawa认为平面曲线的Helfrich流。提出了一个近似问题，使得约束条件可以用奇异极限来表示.证明了近似问题解的一致估计及其收敛性，在许多情况下，梯度流方程是抛物型的。小池研究了完全非线性抛物型和椭圆型方程的最大值原理和比较结果。Ohta研究了双曲型发展方程解的稳定性。坂本研究了流形的CR-结构。Kohlane研究了具有边界条件的表面扩散定态解的非线性稳定性。几何变分问题的解是非线性方程的弱解。因此，分析其规律性是非常重要的。Tachikawa研究了具有间断系数的积分泛函极小临界点的正则性理论。