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.
In the meantime, nonlocal diffusion problems have been the subject of considerable research too and, as FB problems, they possess significant applications in Finance, Industry as well as in Probability and Statistics. The interest in nonlocal problems comes from a simple empiric observation: a large class of physical phenomena are characterized by long-range interactions between particles involved in the diffusion process, which cannot be investigated with models with local diffusion (that is, based on Brownian motion). Models with nonlocal diffusion (for example, the fractional heat equation) seem to be the right tools to fruitfully describe them, since they take into account that the moving particles can jump from a value to another in a time step: this generates probability densities exhibiting fat tails, in deep contrast with standard diffusion.
The MSCA 892017 (LNLFB-Problems) has been mainly developed around this two main areas, with two corresponding parts and four main objectives:
- The first part is devoted to elliptic problems and focuses on two objectives: the first one is to prove
some quantitative regularity estimates and classification results for solutions to a class of one-phase semilinear problems related to combustion theory, while the second is to investigate the singular sets of fractional harmonic maps and relate such singularities to nonlocal minimal sets.
- The second part is devoted to one-phase FB problems of parabolic type with other two objectives: the first one, is to construct solutions of some parabolic non-local one-phase problems while, the second, is to investigate the optimal regularity of such solutions and their FBs.
