Search for an approach to general critical points of variational problems materials science

寻找材料科学变分问题一般关键点的方法

• 批准号: 13554003
• 负责人: KIKUCHI Norio
• 金额: $ 6.72万
• 依托单位: Keio University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-13554003/
• 关键词: discrete Morse flow; Morse(variational) flow; Kashiwagi algorism; Rothe's approximation; local estimate; DeGiorgi-Nash-estimate; Campanato estimate; Gehring-Giaquinta-Modica Higher integrability; discrete Morse fow; Morse(variational)flow; 国際研究者

For the construction of Morse (variational) flows, we introduced the "Discrete Morse Flows" method, which has been adopted and named the Minimizing Movements method by De Giorgi. This approach can he used in the construction of Morse flows ; starting from initial data we look for minimizers of a series of variational functionals introduced inductively. By using this method, we construct Morse flows to variational problems of harmonic map type. We have noticed this method can still be used under weaker assumptions on the initial, and boundary data. We are trying to approach the construction problems of Morse flows to harmonic map variational problems between metric manifolds which have no smoothness of the coefficients of the second order differential operators. In the analysis of Discrete Morse Flows, we carry out local estimates of time-discrete partial differential equations of elliptic-parabolic type, and we have achieved De Giorgi-Nash type Holder estimates and Campanato estimates, and the higher integrability of the gradients of Gehring type. Such estimates are proved to hold independently of the approximation scheme, which enables us to construct Morse flows through Discrete Morse Flows. In conclusion, as a result of this research, we have recognized the characteristics of the Discrete Morse Flows method that can be made use of in the problem of constructing Morse flows for which Leray-Schauder theory with Schauder estimates cannot be directly applied. M.Misawa has treated the regularity and construction of p-harmonic map flows and S.Omata has made mathematical and numerical analysis of Morse flows to several kinds of variational problems. M.Kashiwagi has constructed an algorithm of nonlinear optimizations and composed its software available in http ://srg.pi.cuc.ac.jp/〜kasiwagi/numeric/20040221/

为了构建Morse（变化）流，我们引入了“离散的摩尔斯流量”方法，该方法已被De Giorgi采用并命名了最小化运动方法。他可以在摩尔斯流的构造中使用这种方法。从初始数据开始，我们查找了归纳引入的一系列变异功能的最小化器。通过使用此方法，我们将摩尔斯构建到谐波图类型的各种问题。我们已经注意到，在初始数据和边界数据的较弱假设下，此方法仍然可以使用。我们正在尝试将摩尔斯流量的构造问题处理到公制歧管之间的谐波图变异问题，这些公制歧管没有二阶差异操作员的系数平稳性。在分析离散摩尔斯流动的分析中，我们对椭圆形抛物性类型的时差部分微分方程进行了局部估计，并且我们实现了De Giorgi-Nash类型持有人的估计和Campanato估计，以及GEHRING类型梯度的较高整合。提供了这样的估计，可以独立于近似方案，这使我们能够通过离散的摩尔斯流动构建摩尔斯流。总之，由于这项研究的结果，我们已经认识到可以在构建Morse流量的问题中使用离散的摩尔斯流量方法的特征，而该问题无法直接应用schauder估计的Leray-Schauder理论。 M.Misawa处理了p谐波图流的规律性和构建，而S.omata已将摩尔斯流量的数学和数值分析变成了几种变异问题。 M.Kashiwagi已构建了一种非线性优化算法，并在http://srg.pi.cuc.ac.ac.jp/~kasiwagi/numeric/20040221/中构成了其软件。

项目成果

MASASHI MISAWA: "Local ldolder regularity of gradients for evolutional p-Laplacian systems"Annali.Math.pura et.appl.. (to appear). (2002)

MASASHI MISAWA：“演化 p-拉普拉斯系统的局部梯度规律”Annali.Math.pura et.appl..（即将出现）。

Masashi Misawa: "L^q estimates o fgradients for evolutional -Laplacian systems"(Be submitted).

Masashi Misawa：“进化拉普拉斯系统的 L^q 梯度估计”（已提交）。

N.Kikuchi, J.Kacur: "Convergence of Rothe's method in Holder spaces"Applications of Mathematics. 48No.5. 353-365 (2003)

N.Kikuchi、J.Kacur：“Rothe 方法在 Holder 空间中的收敛性”数学应用。

Jun-ichi, Norio Kikuchi: "Morrey-norm estimates of the solutions to the Rothe' s scheme to parabolic partial differential systems with uniformly continuous coefficients"(Be submitted).

Jun-ichi、Norio Kikuchi：“具有均匀连续系数的抛物型偏微分系统的 Rothe 方案解的 Morrey 范数估计”（已提交）。

N.Kikuchi: "Convergence of Rothe's method in Holder spaces"Applications of Mathematics. (to appear).

N.Kikuchi：“Rothe 方法在 Holder 空间中的收敛性”数学应用。