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

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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