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

Abstract

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.

Asymptotic Analysis of Low Energy Extremals with Γ-Convergence in Variable Exponent Lebesgue Spaces

“…Amaziane et al [21] worked with homogenization and minimization problems for a class of variational functionals, using Γ-convergence in the framework of Sobolev spaces with continuous variable exponents. Carbotti et al [22] proved the local minimality of half spaces in Carnot groups and provided a lower bound of Γ-lim inf of the recalled energy in the form of horizontal parameters. Kolditz and Mang [23] studied the numerical solutions of quasi-static phase-field fracture problems based upon the Γconvergence of parameters.…”

Asymptotics of non-local perimeters

“…where 0 < α < 1 and E c is the complement of E. By |x| we denote the Euclidean norm of x ∈ R d . This object is strongly related to the fractional Sobolev norm and it has been intensively studied over last years, see [2], [6], [7], [8], [18], [23], [24], [28], [39], and [26], [10], [11]. We also refer to [19] and [20] for the case of fractional norms related to Feller generators.…”

A new integral decoupling property of sub-Riemannian heat kernels and some notable consequences

“…To provide the reader with a perspective on our results we note that if, as we have done above, one looks at Theorem B as a corollary of Theorem A, then the spherical symmetry of the approximate identities ρ ε (|x − y|), and therefore of the Euclidean heat kernel in (1.8), seems to play a crucial role in the dimensionless characterisations (1.9) and (1.10). With this comment in mind, we mention there has been considerable effort in recent years in extending Theorem A to various non-Euclidean settings, see [3,37,15,19,34,11,29,12,2,31] for a list, far from being exhaustive, of some of the interesting papers in the subject. In these works the approach is similar to that in the Euclidean setting, and this is reflected in the fact that the relevant approximate identities ρ ε either depend on a distance d(x, y), or are asymptotically close in small scales to the well-understood symmetric scenario of R n .…”