Viscosity Solutions in Martinet Spaces

TL;DR

This paper investigates viscosity solutions within Martinet spaces, establishing their key properties, proving uniqueness for certain elliptic PDEs, and addressing challenges posed by the space's unique geometric structure.

Contribution

It introduces the analysis of viscosity solutions in Martinet spaces, a setting lacking common geometric structures, and proves their uniqueness for specific classes of PDEs.

Findings

Established properties of viscosity solutions in Martinet spaces

Proved uniqueness of solutions for strictly monotone elliptic PDEs

Proved uniqueness for the infinite Laplace equation in Martinet spaces

Abstract

In this paper, we establish the properties of viscosity solutions in Martinet spaces, which lack both the algebraic group law of Carnot groups and the triangular vector fields of Grushin-type spaces. We then prove the uniqueness of viscosity solutions to strictly monotone elliptic PDEs and to the infinite Laplace equation.