Theoretical analysis of mathematical models with singularities for interacting systems
A variety of real phenomena are characterized by complex systems whose evolution is determined by interactions, both among the elements of the system and between the system itself and the surrounding environment. In many cases, the mathematical description of such phenomena involves differential models whose common denominator is the presence of singularities. This gives a rather large class of mathematical problems where the singular features can be of various nature. As so, the study of this kind of models usually requires the development of dedicated tools and techniques. In this direction, a certain interest has been gathered by singular models involving semilinear elliptic PDEs, of which the nonlinear Schroedinger equation is a prototypical example. Such equations provide by now standard effective models for phenomena as Bose-Einstein condensation and recent technological advancements rapidly motivated their study in the presence of singularities. The attention is primarily devoted to two kinds of singularity. First, it is interesting to investigate existence and qualitative properties of solutions of such equations coupled with singular potentials of delta type, that are particularly useful e.g. to model the dynamics of Bose-Einstein condensates in presence of impurities or defects in the medium. Second, it is possible to consider singular domains, as locally one-dimensional structures (metric graphs) or structures obtained by gluing together pieces of different dimensions (hybrids).