The $\infty(x)$-equation in Grushin-type Spaces

TL;DR

This paper establishes the existence and uniqueness of viscosity solutions for the $ abla_ ext{infty(x)}$-Laplace equation within Grushin-type spaces by utilizing specialized Grushin jets, addressing challenges posed by non-Euclidean geometry.

Contribution

It introduces a novel approach using Grushin jets to prove existence and uniqueness of solutions in non-Euclidean Grushin-type spaces, overcoming limitations of Euclidean methods.

Findings

Existence and uniqueness of viscosity solutions are proven.

Grushin jets are effectively adapted to the geometry of Grushin-type spaces.

Euclidean proof techniques are invalid in this setting, necessitating new methods.

Abstract

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