Study on Regularity and Singularity of a weak solution to the m-harmonic maps and the evolution

m调和映射弱解的正则性和奇异性及其演化研究

• 批准号: 17540199
• 负责人: MISAWA Masashi
• 金额: $ 2.24万
• 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: 2005
• 资助国家: 日本
• 起止时间: 2005 至 2006
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-17540199/
• 关键词: m-harmonic map; m-harmonic map flow; regularity; singularity; energy concentration; free boundary; 特異摂動問題; 自由境界問題

We obtain the following results and try to publish the papers in some Journal.(1) Free boundary problem for m-harmonic maps and m-harmonic map flowWe show the existence of the local in time solution of the m-harmonic map flow into smooth compact manifold with free boundary on a closed submanifold of the target manifold, which satisfies the m-harmonic map flow equation in the weak sense and is H"older continuous with its gradient in time-space region up to the boundary of the space domain. The maximal existence time of the solution is estimated below by the m-energy of the initial datum. Also, the singular behavior of the solution at the singular time (maximal existence time) can be characterized by a non-constant m-harmonic maps into the target manifold. The m-harmonic map is defined on m-dimensional sphere, or m-dimensional ball with free boundary, and they are called m-harmonic sphere, or m-harmonic disk, respectively. These solutions are exactly minimal submanifolds in the target manifold.(2) Finite singularity of the m-harmonic map flowIt is expected that the singular set at the singular time is consist of finitely many points. In the paper, we make device of some formula and try to prove the conjecture. However, we are faced with a serious gap of the proof., which is now studied by us to be overcome. We obtain the formula which says the monotonicity of the scaled energy in the intrinsic way to the m-harmonic Laplace operator and is of its own interest.(3) A priori estimates for the linearized parabolic system of non-divergence formWe show the a priori estimates in some Sobolev space hold for the linearized parabolic system of the m-harmonic map flow and the existence of a strong solution of the system. The existence result is combined with the Leray-Schauder fixed point theorem aid the reflection method to show the local in time solution of the m-harmonic map flow with free boundary.

我们获得了以下结果，并尝试在一些期刊上发表论文。(1) m-调和映射流和m-调和映射流的自由边界问题我们证明了m-调和映射流在目标流形的一个封闭子流形上的自由边界光滑紧致流形的局部时间解的存在性，它满足弱意义上的m-调和映射流方程，并且在时空区域上具有梯度直至空间域边界的H′老连续。解的最大存在时间由初始基准的m能量估计如下。此外，解在奇异时间（最大存在时间）的奇异行为可以用一个非常数的m调和映射来表征。m调和映射定义在m维球面上，或有自由边界的m维球上，分别称为m调和球面，或m调和盘。这些解是目标流形中的最小子流形。(2) m调和映射流的有限奇异性，期望奇异时刻的奇异集由有限多个点组成。本文对一些公式进行了验证，并对其进行了证明。然而，我们面临着一个严重的证据缺口。，这是我们现在研究要克服的问题。我们得到了m调和拉普拉斯算子固有方式下标度能量单调性的公式，它本身是有意义的。(3)非发散形式线性化抛物型系统的先验估计我们给出了m调和映射流线性化抛物型系统在Sobolev空间hold下的先验估计以及该系统强解的存在性。将存在性结果与Leray-Schauder不动点定理结合，利用反射法给出了具有自由边界的m-谐映射流的局部时间解。

项目成果

Singular points of harmonic maps from 4-dimensional domains into 3-spheres.

调和奇点从 4 维域映射到 3 维球体。

• 发表时间: 2006
• 期刊: Duke Mathematical Journal 132・no.3
• 作者: Toru Nakajima;Toru Nakajima
• 通讯作者: Toru Nakajima

Partial regularity for a selective smoothing functional for image restoration in BV space

BV 空间中图像恢复的选择性平滑函数的部分正则性

• 发表时间: 2005
• 期刊: SIAM Journal on Mathematical Analysis 37, no.4
• 作者: Chen;Yunmei;Rao;M.;Tonegawa;Y.;Wunderli;T.
• 通讯作者: T.

A diffused interface whose chemical potential lies in Sobolev spaces

化学势位于索博列夫空间的扩散界面

• 发表时间: 2005
• 期刊: Ann. Scuola Norm. Sup. Pisa Cl. Sci. 5(4)
• 作者: Tonegawa;Yoshihiro
• 通讯作者: Yoshihiro

Lq estimates of gradients for evolutional p-Laplacian systems

• DOI: 10.1016/j.jde.2004.11.003
• 发表时间: 2005-12
• 期刊: Journal of Differential Equations
• 影响因子: 2.4
• 作者: M. Misawa

The evolution of minimal surfaces with free boundaries in higher dimensions,

在更高维度中具有自由边界的最小表面的演变，

• 发表时间: 2005
• 期刊: "The 5th East Asia PDE Conference" T.Suzuki eds, GAKUTO Internatinal Ser.Math.Sci.Appl. 22
• 作者: M.Abe;M.Furushima;T.Shima;M Otani;M.Misawa;M.Misawa
• 通讯作者: M.Misawa