We will discuss homogenization of free boundary problems in the periodic media, where the free boundary is oscillatory due to the inhomogeneities in the media. One example is the contact line dynamics of liquid droplets on patterned surface. It turns out that, as the oscillation size goes to zero, there exists a unique and stable limiting free boundary problem to which the solutions converge.
We will present the main ideas in the proof, difficulties involved, and remaining questions.