Study on Regularity and Singularity of Minimal Surfaces in Higher Dimensions and The Evolution

高维极小曲面的正则性、奇异性及其演化研究

基本信息

批准号：
15540210
负责人：
MISAWA Masashi
金额：
$ 2.05万
依托单位：
Kumamoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540210/
关键词：
minimal surface The evolution of minimal surface constant mean curvature surface Free boundary problem regularity singularity unstable solutions p-harmonic maps m-調和写像相境界

项目摘要

We obtain the following results and prepare the papers to be published in some Journal.(1)Existence and regularity for the evolution of constant mean curvature surfaces in high dimensionIn high dimension where the domain dimension is equal to or greater than 3, the mean curvature of the parametric surfaces is given by the m-Laplace operator of the map which is the parametrization of the surface.We show that, If the initial boundary data is of small image in some sense, there exists a time-global weak solution The solution has the image of the same size as the datum, and its gradients are H"older continuous except some closed set in the domain. The size of the except set for regularity is estimated in the Hausdorff measure of some dimension.To show the existence of a weak solution, we use the variational method called discrete Morse semi-flow, which is the minimization of the family of the functionals, of which the Euler-Lagrange equations are the time-discrete equations of the Rothe ty … More pe.To have the regularity of a weak solution, we use the fundamental regularity theorem for the evolution of p-Laplace operator with lower order term of the critical growth on the gradient, which was obtained by Masashi Misawa in 2002.(2)Regularity and singularity for a singular perturbation problemWe study a singular perturbation problem in a phase transition., and in particular, we study the regularity of the interface which is the level set of the limit function, of the singular perturbation problem. To investigate the regularity and singularity of the interface of the limit function, we make device of the formula for the scaled energy, called monotonicity formula.(3)Free boundary problem for minimal surfaces in high dimensionWe study the free boundary problem for minimal surfaces in high dimension. The existence of a solution is proved by the variational method, in particular, the minimax method combined with some approximation., and the solution is nearly unstable. We also study the relation of the unstable solution with the singularity of the evolution of minimal surfaces in high dimension.. It is shown that there exists a time-global weak solution of the evolution of minimal surfaces with free boundaries in high dimension, and that the solution and its gradient is H"older continuous except finitely many times. Moreover, the singular time is characterized by the existence of a non-constant minimal surface with free boundaries.We will try to study the free boundary problem for p-harmonic maps with values into smooth compact Riemannian manifold, the evolution, of p-harmonic maps, and moreover the wave equations and wave maps into smooth compact Riemannian manifold. Less

我们获得以下结果并准备要在某些日记中发表的论文。（1）在高尺寸的高尺寸中恒定平均曲率表面演变的存在和规律性，其中域尺寸等于或大于3，参数表面的平均曲率是由映射的某些映射符号所示的，如果是supariest of Surfient of Supariest of Surece of Surece的均值。时间全球弱解决方案该解决方案的图像与数据的大小相同，并且其梯度是h“较大的连续性除外，除了域中的某些封闭设置外，域的尺寸是在某个维度的hausdorff测量中估计的规律性。 Euler-Lagrange equations are the time-discrete Equations of the Rothe ty … More pe.To have the regularity of a weak solution, we use the fundamental regularity theorem for the evolution of p-Laplace operator with lower order term of the critical growth on the gradient, which was obtained by Masashi Misawa in 2002.(2)Regularity and singularity for a singular perturbation problemWe study a singular perturbation problem in a相变。，尤其是我们研究界面的规律性，这是极限函数的水平集，即单数扰动问题的级别。为了研究极限函数界面界面的规律性和奇异性，我们制作了缩放能量的公式的设备，称为单调性公式。（3）高维度高维度中最小表面的自由边界问题研究高维度中最小表面的自由边界问题。溶液的存在是通过变分方法证明的，尤其是最小方法与一些近似值相结合，并且该溶液几乎不稳定。我们还研究了不稳定的解决方案与高维度最小表面演变的奇异性。这表明存在一个时间 - 全球较小表面的弱解决方案，最小界面的演变是在高维度中的自由界限的演变，而梯度及其梯度及其梯度及其较旧的层面是“较老的连续时间。边界。我们将尝试研究带有值的P-Harmonic地图的自由边界问题，以平滑的紧凑型riemannian歧管，P-Harmonic图的进化，此外，波动方程和波浪图成为平滑的紧凑型Riemannian歧管