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Fares Essebei

Publications and source records attributed to Fares Essebei.

5 recordsLinked to original sources
arXiv2026

A minimal regularity for the area formula in the Engel group

We prove that the upper blow-up theorem in the Engel group holds for $C^1$ submanifolds. Combining this result with the known negligibility of the singular set, we obtain an integral representation of the spherical measure for all surfaces of class $C^{1,\alpha}$ in the Engel group. A new and central aspect of our method is the suitable use of Stokes' theorem to prove the upper blow-up, which relies on the special algebraic structure of left-invariant forms in the Engel group. Some general tools are also introduced to establish area formulas in arbitrary stratified group.

math.MG↗
arXiv2024

Monge solutions for discontinuous Hamilton-Jacobi equations in Carnot groups

In this paper we study Monge solutions to stationary Hamilton-Jacobi equations associated to discontinuous Hamiltonians in the framework of Carnot groups. After showing the equivalence between Monge and viscosity solutions in the continuous setting, we prove existence and uniqueness for the Dirichlet problem, together with a comparison principle and a stability result.

math.AP↗
arXiv2023

Variational problems concerning length distances in metric spaces

Given a locally compact, complete metric space $({\rm X},{\sf D})$ and an open set $Ω\subseteq{\rm X}$, we study the class of length distances $\sf d$ on $Ω$ that are bounded from above and below by fixed multiples of the ambient distance $\sf D$. More precisely, we prove that the uniform convergence on compact sets of distances in this class is equivalent to the $Γ$-convergence of several associated variational problems. Along the way, we fix some oversights appearing in the previous literature.

math.MG↗
arXiv2022

Variational problems concerning sub-Finsler metrics in Carnot groups

This paper is devoted to the study of geodesic distances defined on a subdomain of a given Carnot group, which are bounded both from above and from below by fixed multiples of the Carnot-Carathéodory distance. We show that the uniform convergence (on compact sets) of these distances can be equivalently characterized in terms of $Γ$-convergence of several kinds of variational problems. Moreover, we investigate the relation between the class of intrinsic distances, their metric derivatives and the sub-Finsler convex metrics defined on the horizontal bundle.

math.AP↗