This chapter investigates the precise regularity that is to be expected for non-negative solutions of the PME that have free boundaries. Section 19.1 describes the results of Caffarelli, Vázquez, and Wolanski that show that non-negative solutions with compactly supported initial data have pressures that are Lipschitz continuous functions after a certain time which depends on the data. Section 19.2 presents focusing solutions constructed by Aronson et al., which develop a singularity for speed in finite time, precisely where a hole in the initial support is filled. Section 19.3 proves the transfer of regularity that happens in the PME: Lipschitz continuity with respect to the space variable implies the same type of continuity in time. Section 19.4 reports on more refined regularity results for large times. Section 19.5 discusses the conservation of the initial regularity when the data are smooth; conservation can happen either locally in time or globally in time. Then, the chapter presents the property of asymptotic concavity: it is proved that the pressure of a compactly supported solution becomes a concave function in its support for all large times, hence the free boundary (and all the level sets) are convex hypersurfaces.
Keywords: PME, non-negative solutions, free boundaries, Lipschitz continuity, regularity