Integral representation of local left--invariant functionals in Carnot groups

TL;DR

This paper establishes a representation theorem for left-invariant functionals in Carnot groups and derives a $ ext{Gamma}$-convergence result for a related class of functionals, advancing the understanding of variational problems in sub-Riemannian geometry.

Contribution

It introduces a new integral representation for local left-invariant functionals in Carnot groups, extending the theoretical framework for variational analysis in these geometric structures.

Findings

Proves a representation theorem for left-invariant functionals in Carnot groups.

Provides a $ ext{Gamma}$-convergence result for a subclass of these functionals.

Enhances the mathematical tools available for variational problems in sub-Riemannian geometry.

Abstract

The aim of this note is to prove a representation theorem for left--invariant functionals in Carnot groups. As a direct consequence, we can also provide a $\Gamma$-convergence result for a smaller class of functionals.