A 2-D free boundary problem with onset of a phase and singular elliptic boundary value problems

Numerous models of industrial processes, such as diffusion in glassy polymers or solidification phenomena, lead to general one phase free boundary value problemswith phase onset. The classicalwell-posedness of a fast diffusion approximation to the concerned free boundary value problems is proved. The analysis is performed via a singular change of variables leading to a singular system in a fixed domain. An existence and regularity theory for classical solutions is developed for the relevant underlying class of singular elliptic boundary value problems and is then used to prove the well-posedness for the models considered in which these are coupled to Hamilton-Jacobi or to parabolic evolution equations.