The $\infty(x)$-equation in Riemannian Vector Fields

TL;DR

This paper establishes the existence and uniqueness of viscosity solutions for the $ abla_ ext{infty(x)}$-Laplace equation within Riemannian vector fields, adapting jet methods to Riemannian geometry.

Contribution

It introduces a novel approach using Riemannian jets to prove existence and uniqueness, overcoming limitations of Euclidean methods in Riemannian settings.

Findings

Proved existence and uniqueness of viscosity solutions in Riemannian vector fields.

Developed Riemannian jet techniques tailored to the geometry.

Highlighted differences between Euclidean and Riemannian jet methods.

Abstract

We employ Riemannian jets which are adapted to the Riemannian geometry to obtain the existence-uniqueness of viscosity solutions to the $\infty(x)$-Laplace equation in Riemannian vector fields. Due to the differences between Euclidean jets and Riemannian jets, the Euclidean method of proof is not valid in this environment.