My main research topics are listed below. I am available for possible degree, master and Ph. D. theses. A good background in Functional Analysis and Elliptic and Parabolic PDEs is welcome, but not really necessary at least for some of the topics. Students can contact me by email for further details and/or for arranging a meeting.
1) LINEAR AND NONLINEAR DIFFUSIONS. The role of equations of parabolic type is ubiquitous when modelling physical and biological systems. Indeed, the study of suitable classes of linear and nonlinear diffusion processes is still showing an outburst of new results and challenging problems. In the nonlinear setting, our research has been mainly focused on the porous media and the fast diffusion equations. The asymptotic behavior of singularly nonlinear differential equations is strongly dependent on the class of initial data under consideration. Detailed results have been given for initial data close to special explicit fundamental solutions, by means of entropy methods and establishing related functional inequalities, often with a clear geometrical meaning. Fine properties of solutions have been studied as well. In particular, propagation of positivity and local smoothing of singular equations have been tackled via techniques of functional-analytic flavor. Extensions to fractional-type operators are being presently studied.
Having in mind the celebrated Li-Yau methods concerning Harnack inequalities and Hölder continuity properties of the heat equation on manifolds, we are concentrating on the role of the curvature in the behavior of solutions to nonlinear parabolic equations on manifolds, where appreciable similarities and sharp differences with respect to the flat case occur at the same time.
2) FUNCTIONAL INEQUALITIES. As mentioned above, certain functional inequalities may have a crucial importance in the study of properties of linear and nonlinear evolution equations. The inequalities we have in mind are e.g. the ones which go under the names of Sobolev, Nash, Gagliardo-Nirenberg, Hardy, Poincaré and so on. The quest for optimal constants in such inequalities is particularly challenging and further difficulties are present when one work in geometrically nontrivial settings (Riemannian manifolds). We aim at discussing some of these topics e.g. in the nonstandard but important setting of the hyperbolic space (a manifold with constant, negative sectional curvature and trivial topology), starting from some results we already obtained for Hardy-Poincaré inequalities.
3) DECAY ESTIMATES FOR SEMIGROUPS AND GROUPS OF OPERATORS. Solutions to heat-type (parabolic) equation, or more challengingly to Schroedinger-type equations, exhibit short-time regularization properties (smoothing effects) and a long-time behaviour that depends in an extremely delicate way e.g. from the details of the generator, from the kind of equation, from the geometric properties of the ambient space. It is for example a classical issue, so far not fully understood, to connect the long-behaviour of solutions to the heat equation posed on a Riemannian manifold and curvature properties of the latter. Dispersive estimates for Schroedinger type equations are even more subtle, since L2 norms are often preserved in time and hence no decay occurs in such norm. We aim at discussing sharp heat kernel bounds on noncompact manifolds and the relationships among functional inequalities, spectral properties of the generator and asymptotics of the heat kernel on manifolds with nontrivial topologies, and at studying Schroedinger-type equations with scaling critical potentials, a particularly delicate case.