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.

我们获得以下结果并尝试在某些期刊中发布论文。（1）M-Harmonic Maps和M-Harmonic Map Flowwe的自由边界问题显示，M-Harmonic地图的时间解决方案存在于M-Harmonic Map流量的时代解决方案中。在目标歧管的闭合子曼群上具有自由边界的紧凑型歧管，在弱意义上满足M谐波映射方程，并且在较大的时间空间区域中较老，直到其在空间域的边界上。下面的最大存在时间是通过初始基准的M能量估计的，在单数时间（最大存在时间）的奇异行为可​​以通过非恒定的M-Harmonic图来表征目标歧管。目标歧管。（2）M-Harmonic Map Flowit的有限奇异性预计，在单数时，奇异集由有限的多点组成。在论文中，我们制作了某种公式的设备，并试图证明猜想。但是，我们面临着严重的证据差距。我们获得了一个公式，该公式说以固有方式到达M谐波laplace操作员的单调性，并且具有其自身利益。（3）对非差异形式的线性性抛物线系统的先验估计，我们表明先验某些Sobolev空间中的估计值适用于M谐波映射流的线性化抛物线系统以及系统的强溶液的存在。存在结果与Leray-Schauder固定点定理结合使用的反射方法，以显示具有自由边界的M谐波映射流的局部解决方案。

项目成果

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

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.

Cauchy-free boundary problem of the evolution of the m-harmonic maps (preprint).

m 调和映射演化的柯西自由边界问题（预印本）。

• 作者: Chen;Yunmei;Rao;M.;Tonegawa;Y.;Wunderli;Masashi Misawa;Masashi Misawa
• 通讯作者: Masashi Misawa

Sharp-interface limit of the Allen-Cahn action functional in one space dimension

一维艾伦-卡恩作用泛函的锐界面极限

• 发表时间: 2006
• 期刊: Calc. Var. Partial Differential Equations 25, no. 4
• 作者: Kohn Robert V.;Reznikoff Maria G.;Yoshihiro Tonegawa
• 通讯作者: Yoshihiro Tonegawa

A variational problem for affine connections

仿射连接的变分问题

• 发表时间: 2006
• 期刊: Arch. Math. (Basel) 86, no. 5
• 作者: Kohn Robert V.;Reznikoff Maria G.;Yoshihiro Tonegawa;Tohru Nakajima;Osamu Kobayashi
• 通讯作者: Osamu Kobayashi