Calculus of Variations and Free Boundary Problems V

University of Pisa, 28 March 2023
Aula Magna of the Dipartiment of Mathematics


Elisa Affili (Università di Bologna)

Francesca Bianchi (Università di Parma)

Gabriele Cora (Università di Torino)

Giovanni Siclari (Università di Milano-Bicocca)

Stefano Vita (Università di Torino)


9:30 - 10:20   Elisa Affili  

Divergent fractional operators on the Heisenberg group

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.

10:30 - 11:20   Francesca Bianchi  

A sharp Hardy inequality in fractional Sobolev spaces

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).

12:00 - 12:50   Gabriele Cora  

Almost-CMC surfaces of annular type

Abstract. By Alexandrov theorem, the only connected, closed, C2 hypersurfaces of constant (non zero) mean curvature (CMC) embedded in R3 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 R3, 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.

14:30 - 15:20   Giovanni Siclari  

Unique continuation for the spectral fractional Laplacian from the boundary

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.

15:30 - 16:20   Stefano Vita  

Higher order boundary Harnack principle via degenerate equations

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.


ERC VaReg (European Union - Horizon 2020)

Organizers: Roberto Ognibene, Giorgio Tortone and Bozhidar Velichkov