Annalisa Baldi

Hypoellipticity, fundamental solution and Liouville type theorem for matrix-valued differential operators in Carnot groups

Let L be a non-negative self-adjointN N matrix-valued operator of ordera Q on a Carnot group G. HereQ is the homogeneous dimension of G. The aim of this paper is to investigate the relationship between hypoellipticity and maximal hypoellipticity (i.e. sharp L 2 estimates in appropriate Sobolev spaces), L p -maximal hypoellipticity (i.e. sharp L p estimates in appropriate Sobolev spaces for 1 < p <1), and what we call maximal subellipticity of L (which is basically a sharp higher order energy estimate).

Mumford–Shah-type functionals associated with doubling metric measures

In this paper we study Mumford–Shah-type functionals associated with doubling metric measures or strong A ∞ weights in the setting of the perimeter theory in the sense of Ambrosio and Miranda in metric spaces. We prove an existence theorem in a suitably defined class of special BV functions.

A Γ-CONVERGENCE APPROACH TO NON-PERIODIC HOMOGENIZATION OF STRONGLY ANISOTROPIC FUNCTIONALS

In this work we present a homogenization result for a class of degenerate elliptic functionals mimicking strongly anisotropic media. We study the limit as e→0 of the functionals where, for any e>0, αe:ℝn×ℝn→ℝ, αe(x,ξ)≈ p/2-1, Ae∈Mn×n(ℝ) being measurable non-negative matrices such that almost everywhere, p>1. To take into account the anisotropy of the media we consider two families of weight functions reasonably different, λe and Λe, possibly degenerate or singular, such that: The convergence to the homogenized problem is obtained by a classical approach of Γ-convergence.

Maxwell’s Equations in Anisotropic Media and Carnot Groups as Variational Limits

Abstract Let G be a free Carnot group (i.e. a connected simply connected nilpotent stratified free Lie group) of step 2. In this paper, we prove that the variational functional generated by “intrinsic” Maxwell’s equations in G is the Γ-limit of a sequence of classical (i.e. Euclidean) variational functionals associated with strongly anisotropic dielectric permittivity and magnetic permeability in the Euclidean space.

Gagliardo-Nirenberg Inequalities for Horizontal Vector Fields in the Engel Group and in the Seven-Dimensional Quaternionic Heisenberg Group

Recently, Bourgain and Brezis and Lanzani and Stein considered a class of div-curl inequalities in de Rham’s complex. In this note we prove the natural counterpart of these inequalities for horizontal vector fields in the Engel group and in the seven-dimensional quaternionic Heisenberg group.