# Local minimizers and gamma-convergence for nonlocal perimeters in Carnot groups

@article{Carbotti2020LocalMA,
title={Local minimizers and gamma-convergence for nonlocal perimeters in Carnot groups},
author={Alessandro Carbotti and Sebastiano Don and D. Pallara and A. Pinamonti},
journal={arXiv: Analysis of PDEs},
year={2020}
}
We prove the local minimality of halfspaces in Carnot groups for a class of nonlocal functionals usually addressed as nonlocal perimeters. Moreover, in a class of Carnot groups in which the De Giorgi's rectifiability Theorem holds, we provide a lower bound for the $\Gamma$-liminf of the rescaled energy in terms of the horizontal perimeter.
3 Citations
Gamma-convergence of Gaussian fractional perimeter
• Mathematics
• 2021
We prove the Γ-convergence of the renormalised Gaussian fractional s-perimeter to the Gaussian perimeter as s → 1−. Our definition of fractional perimeter comes from that of the fractional powers ofExpand
A quantitative dimension free isoperimetric inequality for the fractional Gaussian perimeter
• Mathematics
• 2020
The Gaussian isoperimetric inequality states that among all sets with prescribed Gaussian measure, the halfspace is the one with least Gaussian perimeter. This result has been proved independently byExpand
A quantitative dimension free isoperimetric inequality for the Gaussian fractional perimeter
• Mathematics
• 2020
We prove a quantitative isoperimetric inequality for the Gaussian fractional perimeter using extension techniques. Though the exponent of the Fraenkel asymmetry is not sharp, the constant appearingExpand

#### References

SHOWING 1-10 OF 70 REFERENCES
On the structure of finite perimeter sets in step 2 Carnot groups
• Mathematics
• 2003
In this article we study codimension 1 rectifiable sets in Carnot groups and we extend classical De Giorgi ’s rectifiability and divergence theorems to the setting of step 2 groups. Related problemsExpand
Harnack inequality for fractional sub-Laplacians in Carnot groups
• Mathematics
• 2015
In this paper we prove an invariant Harnack inequality on Carnot–Carathéodory balls for fractional powers of sub-Laplacians in Carnot groups. The proof relies on an “abstract” formulation of aExpand
Regularity of sets with constant intrinsic normal in a class of Carnot groups
In this Note, we define a class of stratified Lie groups of arbitrary step (that are called groups of type $\star$'' throughout the paper), and we prove that, in these groups, sets with constantExpand
Rectifiability of Sets of Finite Perimeter in Carnot Groups: Existence of a Tangent Hyperplane
• Mathematics
• 2008
We consider sets of locally finite perimeter in Carnot groups. We show that if E is a set of locally finite perimeter in a Carnot group G then, for almost every x∈G with respect to the perimeterExpand
A rectifiability result for finite-perimeter sets in Carnot groups
• Mathematics
• 2019
In the setting of Carnot groups, we are concerned with the rectifiability problem for subsets that have finite sub-Riemannian perimeter. We introduce a new notion of rectifiability that is possibly,Expand
Nonlocal minimal surfaces
• Mathematics
• 2009
The de Giorgi theory for minimal surfaces consists in studying sets whose indicator function is a (local) minimum of the BV norm. In this paper we replace the BV norm by the $H^\sigma$ norm, withExpand
Rectifiability and perimeter in the Heisenberg group
• Mathematics
• 2001
Abstract. In this paper, we fully extend to the Heisenberg group endowed with its intrinsic Carnot-Carathéodory metric and perimeter the classical De Giorgi's rectifiability divergence theorems.
On the asymptotic behaviour of nonlocal perimeters
• Mathematics
• 2018
We study a class of integral functionals known as nonlocal perimeters, which, intuitively, express a weighted interaction between a set and its complement. The weight is provided by a positive kernelExpand
Surface measures in Carnot-Carathéodory spaces
• Mathematics
• 2001
Abstract. In the framework of Carnot-Carathéodory spaces we study Minkowski content and perimeter, we prove some coarea formulas, and finally we prove some variational approximations of the perimeter.
Fine properties of functions with bounded variation in Carnot-Carath\'eodory spaces
• Mathematics
• 2018
We study properties of functions with bounded variation in Carnot-Ca\-ra\-theo\-do\-ry spaces. We prove their almost everywhere approximate differentiability and we examine their approximateExpand