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

在本研究中，我们研究了以几何演化方程定义的曲线和曲面族泛函约束的梯度流。梯度流是一种寻找临界点的方法，它通过变形降低泛函的值。自然界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，因此在这个意义上，梯度流的极限应该是稳定的。Nagasawa和Kohsaka考虑了具有规定面积和封闭体积的曲面的Willmore泛函（Helfrich变分问题），构造了关联梯度流（Helfrich流），并分析了中心流形近球的结构。在该问题的平稳问题上，Nagasawa构造了多个2、4、6、8球的分岔解。我们利用分岔理论中的群等变分支引理而不是等变分支引理来简化分岔方程。此外，Nagasawa还考虑了平面曲线的helrich流。提出了一个近似问题，使约束以奇异极限的形式实现。证明了近似问题解的一致估计及其收敛性。在许多情况下，梯度流动方程是抛物线型的。小池研究了完全非线性抛物型和椭圆型方程的极大值原理和比较结果。Ohta研究了双曲型演化方程解的稳定性。Sakamoto研究了流形的cr结构。Kohsaka研究了具有边界条件的表面扩散平稳解的非线性稳定性。几何变分问题的解是非线性方程的弱解。因此，分析它们的规律性是很重要的。Tachikawa研究了系数不连续的积分泛函极小临界点的正则性理论。

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.