Nonlinear analysis and calculus of variations
The modeling of many physical problems leads to the search for solutions of ordinary or partial differential equations. In most cases the problems one deals with are nonlinear, because such are the phenomena that they model. Many techniques have been developed to treat nonlinear phenomena from a mathematical point of view, and they can roughly be classified in topological and variational methods. The combined use of these techniques allows one to deal with a virtually unlimited number of problems. We can list, as examples of lines of research within the group, critical point theory and its applications to elliptic problems and differential geometry, spectral and bifurcation problems, parabolic evolution equations and phase transitions, Schrödinger equations and their numerous applications, problems from calculus of variations and optimization, variational convergence, hyperbolic problems such as the nonlinear wave equation and its generalization. Techniques from Microlocal Analysis are used too, with applications to the mathematical aspects of Quantum Physics and Time-frequency Analysis.
Among subjects of applied nature one can cite fracture mechanics, elastoplasticity, Stefan problems, micromagnetism, Bose-Einstein condensates, rate-independent systems. The research carried out by the group is devoted both to “abstract” methodological aspects and to the applications to real world problems.