Partial Differential Equations for VectorValued Functiions and Utilization of Symbolic Computation Systems
Project/Area Number 
10640154

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Basic analysis

Research Institution  The University of ElectroCommunications 
Principal Investigator 
ITO Hiroya The University of ElectroCommunications, Faculty of ElectroCommunications, Associate Professor, 電気通信学部, 助教授 (30211056)

CoInvestigator(Kenkyūbuntansha) 
YOSHIDA Minoru The University of ElectroCommunications, Faculty of ElectroCommunications, Associate Professor, 電気通信学部, 助教授 (00182791)
NAITO Toshiki The University of ElectroCommunications, Faculty of ElectroCommunications, Professor, 電気通信学部, 教授 (60004446)
TAYOSHI Takao The University of ElectroCommunications, Faculty of ElectroCommunications, Professor, 電気通信学部, 教授 (60017382)
MISAWA Masashi The University of ElectroCommunications, Faculty of ElectroCommunications, Lecturer, 電気通信学部, 講師 (40242672)

Project Period (FY) 
1998 – 1999

Project Status 
Completed (Fiscal Year 2001)

Budget Amount *help 
¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 1999: ¥1,000,000 (Direct Cost: ¥1,000,000)

Keywords  matrix polynomial / BarnettLothe's tensors / elliptic system / boundaryvalue 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 
Research Abstract 
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 BarnettLothe's tensors is so natural for treating elliptic systems ( Ie., partial differential equations for vectorvalued functions) that they play fundamental roles in boundaryvalue and initialboundaryvalue problems for (not necessarily strongly) elliptic systems. They are effective also in studying sub
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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 initialboundaryvalue 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

Report
(3 results)
Research Products
(11 results)