Andrea Pinamonti

Magnetic BV-functions and the Bourgain–Brezis–Mironescu formula

Abstract We prove a general magnetic Bourgain–Brezis–Mironescu formula which extends the one obtained in [37] in the Hilbert case setting. In particular, after developing a rather complete theory of magnetic bounded variation functions, we prove the validity of the formula in this class.

Symmetry results for stable and monotone solutions to fibered systems of PDEs

We study the symmetry properties for solutions of elliptic systems of the type where x ∈ ℝm with 1 ≤ m < N, X = (x, y) ∈ ℝm × ℝN-m, and F1,…,Fn are the derivatives with respect to ξ1,…,ξn of some F = F(x,ξ1,…,ξn) such that for any i = 1,…,n and any fixed (x,ξ1,…,ξi-1,ξi+1,…,ξn) ∈ ℝm × ℝn-1 the map ξi → F(x,ξ1,…,ξi,…,ξn) belongs to C2(ℝ). We obtain a Poincare-type formula for the solutions of the system and we use it to prove a symmetry result both for stable and for monotone solutions.

Nonexistence Results for Semilinear Equations in Carnot Groups

Abstract In this paper, following [3], we provide some nonexistence results for semilinear equations in the the class of Carnot groups of type ★.This class, see [20], contains, in particular, all groups of step 2; like the Heisenberg group, and also Carnot groups of arbitrarly large step. Moreover, we prove some nonexistence results for semilinear equations in the Engel group, which is the simplest Carnot group that is not of type ★.

BV minimizers of the area functional in the Heisenberg group under the bounded slope condition

We consider the area functional for t-graphs in the sub-Riemannian Heisenberg group and study minimizers of the associated Dirichlet problem. We prove that, under a bounded slope condition on the boundary datum, there exists a unique minimizer and that this minimizer is Lipschitz continuous. We also provide an example showing that, in the first Heisenberg group, Lipschitz regularity is sharp even under the bounded slope condition.

A measure zero universal differentiability set in the Heisenberg group

We show that the Heisenberg group

Some characterizations of magnetic Sobolev spaces

\mathbb {H}^n

A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian

Hn contains a measure zero set N such that every Lipschitz function