Free boundary problems
Free Boundary (FB) problems arise in many applications such us Biology, Physics, Finance, Fluid Dynamics: they are usually described by some Partial Differential Equations (PDEs) that exhibit in addition some unknown interfaces. A classical example is the Stefan problem, which describes the melting of an ice block into water: the main idea is to model the space-time variation of the temperature through a parabolic problem, distinguishing between the region where it is zero (ice) and where it is positive (water). The most challenging problem is to study the evolution and regularity of both the temperature distribution and its FB, that is the region which separates the ice from the water. The peculiarity of the "double unknown" makes this class of problems significantly complex: the advancements in this field have been possible thanks to some innovative techniques which combine tools from different areas like PDEs, Calculus of Variations and Geometric Measure Theory.