Analisi Matematica

Nonlinear Hyperbolic Equations in Fluid Dynamics: The research concerns the analysis of free boundary problems for nonlinear equations of hyperbolic type in Fluid Dynamics and Magneto-Hydrodynamics (MHD). We have studied the existence and stability conditions of compressible vortex sheets, of contact discontinuities and current-vortex sheets in MHD, the existence and stability of plasma-vacuum interface problems. Free discontinuity problems: The research concerns the study of free discontinuity problems in spaces of functions of bounded variation. The main applications are in material science, in particular to fracture (and to plasticity. A recent new application is suggested by shape optimization problems with Robin boundary conditions. Models for mechanics of solids: Various models describing phase change phenomena, as well as damage, and contact with adhesion between solids, are analyzed. For the associated evolutionary PDE systems, we address the issues of existence, uniqueness and regularity of solutions, as well as their long-time behaviour. Spectral properties of differential operators: This research deals with spectral and scattering properties of various differential operators arising from quantum physics. The main attention is payed to the analysis of the discrete spectrum, the long time behavior of the associated semi-groups and their connections with certain functional inequalities.