Partial differential equations, Geometry and Harmonic analysis

Nicola Garofalo,
Giulio Tralli,

Research areas:
The research group focuses on the study of qualitative and quantitative properties of the solutions to partial differential equations of elliptic and parabolic type. The differential operators under consideration appear frequently in the study of natural phenomena, covering a spectrum that ranges from classical mechanics, to the physics of subatomic particles, robotics, kinetic theory of gases and finance. Among the various research topics, we investigate existence and regularity properties of solutions to linear and nonlinear second order equations, free boundary problems of obstacle type with emphasis on problems of Signorini type in which the obstacle lives on a lower-dimensional manifold, uniqueness results for prescribed curvature problems, analysis of nonlocal operators, unique continuation problems, questions of potential theory for boundary value problems. We are particularly interested in the interplay between the underlying geometry of the operator and the local and global behaviour of the solutions. A recurring aspect is the lack of ellipticity of the operator, which is tied to the lack of coercivity in the corresponding variational problem. A model class consists in the so-called operators of Kolmogorov-Fokker-Planck type. They are non-elliptic in the diffusive part with a drift term having linear growth which induces a non-trivial geometry in the ambient space. For this class we study some harmonic analysis aspects suitably adapted to the geometry of the equations under consideration, and some functional inequalities (of Sobolev and isoperimetric type) with associated energies defined using the fractional powers of the operators.

Keywords: PDEs of elliptic and parabolic type, Geometric analysis in Riemannian and sub-Riemannian manifolds, Harmonic analysis for non-symmetric semigroups