| 研究実績の概要 |
We achieved the following in the theory of free boundary problems using the viscosity solution techniques for homogenization and stability pursued in this project:
With Y. Giga (U. of Tokyo), we succeeded in developing a new notion of viscosity solutions for the crystalline mean curvature flow, a model of the growth of crystals, and proved their uniqueness, existence and stability in an arbitrary dimension. This is the first notion of general solutions in three dimensions and it gives an answer to a long standing open problem. It opens up the possibilities for the further study and the development of numerical methods, some of which we are currently working on.
With I. Kim and B. Woodhouse (UCLA), we studied the incompressible limit of the solutions of the porous medium equation with a drift. We established that the limit is a solution of a Hele-Shaw problem with a sharp interface and we were able to identify its free boundary velocity law. This results explains the relationship of these two models of tumor growth and crowd motion. The viscosity solution arguments proved to be crucial in this work.
With a doctoral student at Kanazawa U., G.T.T. Vu, we were able to apply the
homogenization technique to understand the behavior of the Stefan problem, describing for example the melting/freezing transition of water and ice, for large times when the ice has nonuniform properties. A more general result when the heat diffusion is anisotropic is in preparation. This builds on the understanding of combining comparison principle arguments with the variational structure of the problem.
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