Analysis of fundamental properties of elastic equations

在这个研究项目中，我们最初的目的是了解弹性方程的解的性质，并澄清这些性质的根。我们看到，尽管弹性波方程是一类双曲型方程，但它们都在标量波方程附近，这是因为弹性算子的正性和对称性。作为这方面的主要结果之一，我们证明了弹性算子可以用与标量值椭圆算子相同的方式表示为一阶算子的乘积形式。对于一般形式的系统不能期望这个表达式。众所周知，弹性方程中存在瑞利波，而标量方程中不存在瑞利波。我们综合地展示了这种存在对散射理论的影响，构造了计入瑞利波的Lax-Phillips类型的一般散射理论，并对该理论中的个别主题进行了研究，得到了散射核的表示。这种表示是解决逆散射问题的一个基本而有用的公式。对于标量方程，还得到了能量衰减的一个特殊结果。也就是说，证明了如果加入非各向同性的耗散项，总能量不一定衰减。这对弹性波的行为提出了一个有趣的建议。

In this research project we aimed initially to know properties of solutions of the elastic equations and to clarify roots of those properties. And, using the clarification, we intended to study individual topics concerned with elastic waves.We have seen that the elastic wave equations are near the scalar-valued wave equation although they are one of kinds of hyperbolic systems, and that this is because of positivity and symmetry of the elastic operators. As one of main results concerning this, we have proved that the elastic operators can be expressed of product form of first order operators in the same way as the scalar-valued elliptic operators. This expression cannot be expected for systems of the general form.It is known that there exists the Rayleigh wave in the elastic equations, which does not occur in the scalar-valued equations. We have shown synthetically how this existence affects scattering theories and have constructed a general scattering theory of the Lax-Phillips type accounting the Rayleigh wave.Furthermore, we have studied individual topics in this theory, and have obtained a representation of the scattering kernel. This representation is a fundamental and useful formula to solve inverse scattering problems.A special result on the energy decay has been got also for the scalar-valued equation. Namely it is proved that the total energy does not decay necessarily if the dissipative term is added of non isotropy. This gives an interesting suggestion on behavior of the elastic waves.