A non-stationary multi-scale oscillating free boundary for the laplace and heat equations

The paper is concerned with the homogenization problem for the Poisson equation in a domain, a part of whose boundary varies in time and is rapidly oscillating in space at each time. Such a problem arises in growing semiconductor film, taking into account the fact that the surface of the film is changing in time by chemical vapor deposition, or by etching. We prove existence and uniqueness of the solution and establish its homogenizated approximation. We obtain an L∞ estimate on the error between the free boundary and the homogenized boundary.