Free Boundary Problems (S. Salsa, N. Soave, G. Verzini)

Free boundary problems (FBP) appear in many areas of mathematics and science in general. Typical steady state examples are constrained energy minimization like obstacle problems, optimal insulation, minimal capacity potential at prescribed volume, FBP arising from variational or quasivariational inequalities, optimal partition problems. Model problems for evolution or moving FBP are the Stefan problem and its variations (e.g. melting of ice or phase transitions in general), seepage through porous media, oxygen diffusion with sorption, flame propagation, flows in jets or cavitations.

An important source of problems involving FBP can be found as limits of models where two or more densities sharing the same support are subject to a strong competition which forces the formation of a segregated pattern: this kind of segregation phenomena appear in stationary as well as in evolutionary context, e.g. in models with strong competition in population dynamics or phase separation in Bose-Einstein condensates in quantum mechanics. The strong competition models are not only interesting by themselves, by they are also useful as they can be used as relaxed, regularized versions of corresponding optimal partition problems. 

In a FBP a solution of a governing boundary value problem (BPV) has to be constructed in a domain whose boundary is not completely known. Thus, the interphase, or free boundary, is part of the unknown. Typical results of interest are: existence and qualitative properties of solutions exhibiting particular patterns, regularity of the solutions of the BVP, regularity of the free boundary, asymptotic estimates as either the competition parameter or the number of densities goes to infinity (hexagonal conjecture for optimal partitions in many subsets). This gives rise to a regularity theory which has been developed during the last three decades. Part of it is due to members of our research group and still leaves many important open questions.

Politecnico di Milano - Dipartimento di Matematica