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

• 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

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

我们得到了以下结果，并准备在一些期刊上发表论文。（1）高维常平均曲率曲面演化的存在性和正则性在高维区域维数等于或大于3的情况下，参数曲面的平均曲率由曲面的参数化映射的m-Laplace算子给出，我们证明了，如果初始边界数据在某种意义下是小图像，存在一个时间整体弱解，该解具有与基准面相同大小的象，其梯度除区域内的某些闭集外是H“older连续的。本文在Hausdorff测度下估计了正则性例外集的大小.为了证明弱解的存在性，我们使用了称为离散莫尔斯半流的变分方法，它是泛函族的极小化，其中Euler-Lagrange方程是Rothe型的时间离散方程 ...更多信息 利用Masashi Misawa在2002年得到的关于p-Laplace算子在梯度上的临界增长的低阶项的发展的基本正则性定理，得到了弱解的正则性.（2）奇异摄动问题的正则性和奇异性研究了相变中的奇异摄动问题，特别地，我们研究了奇异摄动问题的极限函数水平集界面的正则性。为了研究极限函数界面的正则性和奇异性，我们利用了标度能量的单调性公式。（3）高维极小曲面的自由边界问题我们研究了高维极小曲面的自由边界问题。用变分方法，特别是极小极大方法结合一些近似证明了解的存在性，并且溶液几乎不稳定。我们还研究了高维极小曲面演化的奇异性与不稳定解的关系。证明了高维自由边界极小曲面演化的时间整体弱解的存在性，且该解及其梯度除n次外是H“older连续的.我们将尝试研究光滑紧致黎曼流形上p-调和映射的自由边界问题，p-调和映射的演化，以及波动方程和波动映射到光滑紧致黎曼流形上的问题。少

项目成果

On Stable Critical Points for a Singular Perturbation Problem

• DOI: 10.4310/cag.2005.v13.n2.a7
• 发表时间: 2003
• 期刊: Communications in Analysis and Geometry
• 影响因子: 0.7
• 作者: Y. Tonegawa

Integrality of varifolds in the singular limit of reaction-diffusion equations

• DOI: 10.32917/hmj/1150997978
• 发表时间: 2003-11
• 期刊: Hiroshima Mathematical Journal
• 影响因子: 0.2
• 作者: Y. Tonegawa

Singular Fano compactifications of C^3(I)

C^3(I) 的奇异 Fano 紧化

• 发表时间: 2004
• 期刊: Math.Z. 248
• 作者: Masashi Misawa;Masashi Misawa;Yoshihiro Tonegawa;Masashi Misawa;Masashi Misawa;三沢 正史;利根川 吉広;Masashi Misawa;Masashi Misawa;Mikio Furushima
• 通讯作者: Mikio Furushima

利根川 吉広: "Domain dependent monotonicity formula for a singular perturbation problem"Indiana Univ.Math.J.. 52, no.1. 69-83 (2003)

利根川义博：“奇异扰动问题的域相关单调性公式”Indiana Univ.Math.J.. 52，no.1（2003）。

L^q-estimates of gradients for evolutional p-Laplacian system

演化 p-拉普拉斯系统的 L^q 梯度估计

• 发表时间: 2005
• 期刊: J.Differential Equations (In press)
• 作者: Masashi Misawa;Masashi Misawa
• 通讯作者: Masashi Misawa