Homogenization of a moving boundary problem with prescribed normal velocity

The analysis and homogenization of a moving boundary problem for a highly heterogeneous, periodic two-phase medium is considered. In this context, the normal velocity governing the motion of the interface separating the two competing phases is assumed to be prescribed. Parametrizing the boundary motion via a height function, the so-called Direct Mapping Method is employed to construct a coordinate transform characterizing the changes with respect to the initial setup of the geometry. Utilizing this transform, well-posedness of the problem is established. After characterizing the limit behavior (with respect to the heterogeneity parameter $\varepsilon\to0$) of the functions related to the transformation, the homogenized problem of the heterogeneous two-scale problem is deduced.