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Partial Differential Equations for Vector-Valued Functiions and Utilization of Symbolic Computation Systems

矢量值函数的偏微分方程和符号计算系统的利用

基本信息

批准号：
10640154
负责人：
ITO Hiroya
金额：
$ 0.64万
依托单位：
The University of Electro-Communications
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640154/
关键词：
matrix polynomial Barnett-Lothe's tensors elliptic system boundary-value problem elastic wave equation Rayleigh wave Korn's inequality moving crack problem シンボル Poincare不等式滑り境界条件亀裂問題弾性体方程式表面波圧電体方程式 Stroh形式

项目摘要

Main results in this research are as follows :1. Conditions for homogeneous first order quadratic forms with constant coefficients for smooth vector valued functions with compact supports in a bounded domain or a slab to be uniformly positive were examined. We have obtained, in certain cases, an almost necessary, and sufficient condition in terms of' real simple devisors of the matrix polynomial determined by the quadratic form considered.2. Researchers in applied mechanics often deal with anisotropic elastic materials by means of Barnett Lothe's tensors, which we have derived in a new process from the viewpoint of the theory of ordinary differential equations. This derivation of Barnett-Lothe's tensors is so natural for treating elliptic systems ( I-e., partial differential equations for vector-valued functions) that they play fundamental roles in boundary-value and initial-boundary-value problems for (not necessarily strongly) elliptic systems. They are effective also in studying sub … More sonic Rayleigh waves for the corresponding wave equations.3. Korn's inequality for vector fields satisfying nonpenetrating boundary condition (or its dual) were examined. The usual Korn's inequality holds for vector fields which are tangent (or normal) to the boundary if and only if the domain considered is not rotationally symmetric; this is a known result for the three dimensional case. We have clarified how. this result is extended to the general dimensional case and/or the case where we consider the strain energy with general Lame's constants.4. The 'moving crack problem' for an elastic wave equation was studied. Knowing, in advance, how the crack in the interior of an elastic material expands with time, we have obtained the speed limits of the crack tips for the initial-boundary-value problem for the corresponding elastic wave equation to admit a weak solution; the speed limits are characterized in terms of the speeds of subsonic Rayleigh waves (and the limiting speeds) in various directions. Less

主要研究结果如下：1.研究了有界区域或板上具有紧支集的光滑向量值函数的常系数齐次一阶二次型一致正的条件。在一定的情形下，我们得到了由所考虑的二次型所确定的矩阵多项式的一个真实的简单导数的几乎充要条件.应用力学研究各向异性弹性材料时，常采用巴内特洛兹张量，本文从常微分方程理论出发，用新的方法导出了洛兹张量。Barnett-Lothe张量的这种推导对于处理椭圆系统是非常自然的（即，偏微分方程向量值函数），他们发挥基本作用的边界值和初边值问题（不一定强）椭圆系统。他们也是有效的研究小组 ...更多信息声波瑞利波的相应波动方程.研究了满足非穿透边界条件（或其对偶）的向量场的Korn不等式.通常的科恩不等式对于与边界相切（或垂直）的向量场成立，当且仅当所考虑的域不是旋转对称的;这是三维情况下的已知结果。我们已经澄清了如何。该结果被推广到一般尺寸的情况和/或考虑具有一般拉梅常数的应变能的情况。4.研究了弹性波动方程中的运动裂纹问题。我们事先知道了弹性材料内部的裂纹是如何随时间扩展的，就得到了相应的弹性波方程的初边值问题的裂纹尖端的速度极限，使其具有弱解;该速度极限的特征在于亚音速瑞利波在各个方向上的速度（和极限速度）。少