Nonlinear Elliptic PDEs (G. Arioli, D. Monticelli,  B. Noris, N. SoaveG. Verzini)

Besides classical potential theory, modern approaches to elliptic equations require a strong background in functional analysis, along with a good knowledge of Sobolev spaces and classical inequalities (e.g. Poincaré, Hölder, Hardy, Rellich). Leaning on these tools, we study existence, uniqueness and qualitative properties of solutions to nonlinear elliptic problems in bounded domains under suitable boundary conditions, as well as in the whole space. In the case of semilinear equations driven by the Laplace operator, a standard way to prove existence of solutions is critical point theory, whereas qualitative properties are obtained by means of the maximum principle (in its different formulations), symmetrization and shooting techniques. Moving from the prototype case of scalar semilinear equations for the Laplace operator, and motivated by several applications arising in physics, biology, and engineering, our research deals with:

Non-local equations, where the standard diffusion is replaced by integro-differential operators like the fractional Laplacian.

Systems of elliptic equations with various type of interaction (cooperative, competitive, local, non-local).

Equations driven by different second-order operators, quasi-linear or fully non-linear.

Nonlinear elliptic equations on Riemannian manifolds Higher-order equations.

Degenerate Equations (S. Biagi, M. Bramanti)

In the class of degenerate equations of elliptic-parabolic type a particularly important one is that of Hörmander's operators, which turn out to be hypoelliptic. Equations of this type arise, for instance, as Kolmogorov-Fokker-Planck equations corresponding to systems of stochastic ODEs, which can model many physical systems, or from the theory of several complex variables. In the study of these equations great progresses have been made in the last decades, exploiting methods from several areas of mathematics: classical techniques of PDEs as well as singular integrals, differential geometry, Lie groups and Lie algebras. In the last ten years new classes of more general operators have been introduced and studied, modeled on Hörmander's vector fields, but also containing nonsmooth ingredients, and therefore no longer hypoelliptic. For these operators a number of interesting results have been already proved, often requiring new tools or new insights, and several challenging problems remain open.

Politecnico di Milano - Dipartimento di Matematica