Monge solutions for discontinuous Hamilton-Jacobi equations in Carnot groups

TL;DR

This paper investigates Monge solutions for stationary Hamilton-Jacobi equations with discontinuous Hamiltonians within Carnot groups, establishing foundational results like existence, uniqueness, and stability, and connecting Monge and viscosity solutions.

Contribution

It introduces the study of Monge solutions in the context of discontinuous Hamiltonians on Carnot groups, proving key properties and equivalences with viscosity solutions.

Findings

Established equivalence between Monge and viscosity solutions in continuous setting

Proved existence and uniqueness for the Dirichlet problem

Demonstrated comparison principle and stability results

Abstract

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.