Aula Magna of the Dipartiment of Mathematics

*Abstract.* For the fractional Laplacian, a polynomial decay at infinity is required for the definition to make sense.
In the recent articles of Dipierro, Dzhugan, Savin, and Valdinoci, the definition of some fractional operators (among
which the fractional Laplacian) has been extended to diverging functions with a
cutoff procedure that unloads the diverging part on a class of polynomials of suitable
degree.

We present the notion of divergent fractional operators on the first Heisenberg
group, which acts as a prototype for a general Carnot group. Then, we provide
applications to polynomials. In fact, we show that the divergent fractional operators
applied to polynomials of a suitable degree in the sense of the Heisenberg group are
equal to zero. Moreover, we discuss rigidity results of Liouville type in this general
framework.

This is joint work with Fausto Ferrari.

*Abstract.* We consider the fractional Hardy inequality and we look for the
sharp constant on open convex sets of the Euclidean space.
The construction of supersolutions for the associated equation is the key point
to compute the explicit sharp constant for the class of convex sets, under the
restriction sp≥ 1. Moreover, we overcome this limitation in the case p=2.

Some of the results presented are obtained in collaboration with Lorenzo
Brasco (University of Ferrara), Firoj Sk (Okinawa Institute of Science and
Technology) and Anna Chiara Zagati (University of Parma).

*Abstract.* By Alexandrov theorem, the only connected, closed, C^{2} hypersurfaces
of constant (non zero) mean curvature (CMC) embedded in R^{3} are the spheres.
It is therefore natural to ask whether, having prescribed a mean curvature which is not constant instead,
it is possible to construct embedded surfaces of any fixed genus.

In this talk we present some recent results concerning annular type surfaces,
which could provide a useful tool to answer this question.
In particular, we show existence and nonexistence results for annular type
parametric surfaces with prescribed, almost constant mean curvature,
characterized as normal graphs of compact portions of Delaunay surfaces
(unduloids or nodoids) in R^{3}, and whose boundary consists of two
coaxial circles of the same radius. This kind of surface also appears in gas
dynamics, in capillarity phenomena, and in biology.

This is a joint work with A. Iacopetti and P. Caldiroli.

*Abstract.* We study unique continuation properties and the asymptotic behaviour of the spectral fractional
Laplacian on a bounded domain from a point lying on the boundary of the domain. Our
methods are based on a Almgren-type monotonicity formula combined with a blow-up argument.
Since the operator has a global nature we will use a suitable extension results in the spirit
of Caffarelli-Silvestre extension. Furthermore we will show a Sobolev regularity result
needed to obtain a Pohozaev type identity which is the key ingredient to develop a monotonicity formula.

*Abstract.* Aim of this talk is to present regularity results for solutions
to elliptic equations whose coefficients vanish or explode on some fixed manifold.
Then, some applications to boundary Harnack principles will be discussed.