Analysis of a non-linear phenomenon mainly on the issue of shape-memory alloy

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[2004] In dynamics of shape memory alloys the relationship between the strain and the stress can not be described by a usual function and is described by the hysteresis operator. In this research project we describe the relationship by the stop operator, which is one type of hysteresis operators. Throughout the previous works we needed some largeness for the viscosity in order to establish the well-posedness of our mathematical model. In this year we can remove the assumption for the viscosity.[2005] As a reviewing process for our mathematical model for shape memory alloys, we have noticed that there were no mathematical results concerned with change of shape of elastic materials. Hence, we have proposed a mathematical for elastic materials, which is given by a free boundary problem. Here, we proved the local existence of a solution in time for the free boundary problem. In the poof the standard approximation method was used.[2006] The result of this year is to obtain the uniqueness result for our free boundary problem. In the whole research project we can not prove the global existence of a solution. We continue to study the dynamics for the elastic material.