Solvability via viscosity solutions for a model of phase transitions driven by configurational forces

This article is concerned with an initial–boundary value problem for an elliptic–parabolic coupled system arising in martensitic phase transition theory of elastically deformable solid materials, e.g., steel. This model was proposed in Alber and Zhu (2007) [4], , and investigated in Alber and Zhu (2006) [3], the existence of weak solutions which are defined in a standard way, however the key technique used in Alber and Zhu (2006) [3], is not applicable to multi-dimensional problem. One of the motivations of this study is to solve this multi-dimensional problem, and another is to investigate the sharp interface limits. Thus we define weak solutions in a way, which is different from Alber and Zhu (2006) [3], by using the notion of viscosity solution. We do prove successfully the existence of weak solutions in this sense for one-dimensional problem, yet the multi-dimensional problem is still open.