The research topics proposed below involve the research group composed also by my collegues F. Punzo and M. Muratori, both at the Department of Mathematics of the Politecnico di Milano.
International networking and possible co-supervision with foreign scholar (e.g. J.L. Vazquez and M. Bonforte at Universidad Autonoma de Madrid) are also possible.
Both topics below have aspects that could be suitable for M. D. Theses as well.
1) Nonlinear elliptic and parabolic equations on the Euclidean space and on manifolds.
We consider nonlinear elliptic equations whose model is the Emden-Fowler equation and nonlinear evolution equations, of diffusive type, whose models are the porous medium equation or the fast diffusion equation. Among the topics we propose to investigate we mention existence, or nonexistence, of solutions, uniqueness issues, asymptotic behavior and qualitative properties of solutions, often in relation with the validity of suitable functional inequalities. These topics are being actively investigated in the Euclidean framework but a number of challenging open problems exist. Extension to non-Euclidean settings, like the manifold one, on which we recently proved the first existing results, can be dealt with and are so far almost completely open.
Detailed description and references
2) Nonlocal, nonlinear elliptic and parabolic equations.
Nonlocal diffusion operators are being widely studied since the recent, pioneering works of Caffarelli, Silvestre, Vázquez. They involve a number of possible generalization of the porous medium equation in which either the generator or the pressure are fractional powers of the Laplacian. Weighted versions of these equations, corresponding to spatially inhomogeneous media, have also been studied. Crucial issues are still open, we mention in particular uniqueness for bounded distributional solutions. Extensions to more general integro-differential operators are also planned.