Geometric and qualitative analysis in free boundary problems

Several real-world problems across physics, industry, and biology involve partial differential equations that incorporate a priori unknown sets, termed ”free boundaries.” Such kind of problems (Free boundary problem ”FBP”) plays pivotal roles in diverse applications, ranging from heat conduction and finance to tumor growth and fluid dynamics. Mathematical modelling of FBP’s boils down to non-standard partial differential equations, where the domain of a solution u is a priori unknown and is given by {u > 0}. These problems become more involved when a solution is given in unbounded domains (global solutions) or by a vector-valued function, where each component may be interpreted as a density of a species in interactions (competition or collaboration) with others. Three fundamental questions on the solutions of such FBP’s are: Existence, Geometry and Rigidity. Analysis of global solutions and vector-valued free boundary problems have recently gained significant attention and are recognized as active areas of research. However, they are still in a nascent stage, requiring further development. Existence theories for both the scalar and the vectorial case will be studied in specific geometric contexts, such as symmetric or convex regimes. The exploration encompasses highly singular terms in the scalar setting, that arise in the propagation of high activation energy fronts with ignition, limits of boundary layers, and singularly perturbed systems. Furthermore, The rigidity and classification of solutions will be examined in global settings for equations lacking monotonicity. The approach is based on well-established moving-plane and sliding techniques, as well as a recent tool of asymptotic expansion analysis.