This paper is meant as a (short and partial) introduction to the study of the geometry of Carnot groups and, more generally, of Carnot-Carathéodory spaces associated with a family of Lipschitz continuous vector fields. My personal interest in this field goes back to a series of joint papers with E. Lanconelli, where this notion was exploited for the study of pointwise regularity of weak solutions to degenerate elliptic partial differential equations. As stated in the title, here we are mainly concerned...
We prove that if a Poincaré inequality with two different weights holds on every ball, then a Poincaré inequality with the same weight on both sides holds as well.
In this paper we prove a -convergence result for time-depending variational functionals in a space-time Carnot group arising in the study of Maxwell's equations in the group. Indeed, a Carnot groups (a connected simply connected nilpotent stratified Lie group) can be endowed with a complex of ``intrinsic'' differential forms that provide the natural setting for a class of ``intrinsic'' Maxwell's equations. Our main results states precisely that a the vector potentials of a solution of Maxwell's...
In questa Nota enunciamo, per una classe di equazioni ellittiche del secondo ordine «fortemente degeneri» a coefficienti misurabili, un teorema di hölderianità delle soluzioni deboli che estende il ben noto risultato di De Giorgi e Nash. Tale risuJtato discende dalle proprietà geometriche di opportune famiglie di sfere associate agli operatori.
In questa Nota enunciamo, per una classe di equazioni ellittiche del secondo ordine «fortemente degeneri» a coefficienti misurabili, un teorema di hölderianità delle soluzioni deboli che estende il ben noto risultato di De Giorgi e Nash. Tale risuJtato discende dalle proprietà geometriche di opportune famiglie di sfere associate agli operatori.
We prove a higher integrability result - similar to Gehring's lemma - for the metric space associated with a family of Lipschitz continuous vector fields by means of sub-unit curves. Applications are given to show the higher integrability of the gradient of minimizers of some noncoercive variational functionals.
There have been recent attempts to develop the theory of Sobolev spaces on metric spaces that do not admit any differentiable structure. We prove that certain definitions are equivalent. We also define the spaces in the limiting case .
It is known that degenerate parabolic equations exhibit somehow different phenomena when we compare them with their elliptic counterparts. Thus, the problem of existence and properties of the Green function for degenerate parabolic boundary value problems is not completely solved, even after the contributions of [GN] and [GW4], in the sense that the existence problem is still open, even if the a priori estimates proved in [GN] will be crucial in our approach (...).
In this paper we describe some existence and uniqueness theorems for radial ground states of a class of quasilinear elliptic equations. In particular, the mean curvature operator and the degenerate Laplace operator are considered.
Nous présentons une condition suffisante pour qu’un compact dans le groupe de Heisenberg (muni de sa structure de Carnot-Carathéodory) soit contenu dans une courbe rectifiable. Cette condition est aussi nécessaire dans le cas de courbes régulières (en particulier, des géodésiques) et elle est inspirée du lemme géométrique faible du à Peter Jones dans le cas euclidien. Cette note repose sur l’exposé fait par le troisième auteur (au Séminaire X-EDP) et décrit les principaux résultats de l’article...
In questa Nota si considerano, in opportuni spazi con peso, problemi al contorno in un semispazio per operatori del tipo dove è un operatore ellittico e è un parametro complesso.
In questa nota diamo alcuni risultati su di una classe di problemi al contorno per equazioni ellittiche a coefficienti polinomiali in un semispazio. Si stabilisce 1'esistenza di una parametrice destra e di una parametrice sinistra del problema; si stabiliscono inoltre stime a priori del problema e di quello aggiunto.
In this paper we describe some existence and uniqueness theorems for radial ground states of a class of quasilinear elliptic equations. In particular, the mean curvature operator and the degenerate Laplace operator are considered.
A Carnot group G is a connected, simply connected, nilpotent Lie group with stratified Lie algebra. We study intrinsic Lipschitz graphs and intrinsic differentiable graphs within Carnot groups. Both seem to be the natural analogues inside Carnot groups of the corresponding Euclidean notions. Here ‘natural’ is meant to stress that the intrinsic notions depend only on the structure of the algebra of G. We prove that one codimensional intrinsic Lipschitz graphs are sets with locally finite G-perimeter....
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