Existence and uniqueness of viscosity solutions to the infinity Laplacian relative to a class of Grushin-type vector fields

Year 2023, Volume: 6 Issue: 2, 77 - 89, 15.06.2023

Zachary Forrest , Thomas Bieske

https://doi.org/10.33205/cma.1245581

Cited By: 4

https://izlik.org/JA28JT59BR

Abstract

In this paper we pose the $\infty$-Laplace Equation as a Dirichlet Problem in a class of Grushin-type spaces whose vector fields are of the form
\begin{equation*}
X_k(p):=\sigma_k(p)\frac{\partial}{\partial x_k}
\end{equation*}
and $\sigma_k$ is not a polynomial for indices $m+1 \leq k \leq n$. Solutions to the $\infty$-Laplacian in the viscosity sense have been shown to exist and be unique in [3], when $\sigma_k$ is a polynomial; we extend these results by exploiting the relationship between Grushin-type and Euclidean second-order jets and utilizing estimates on the viscosity derivatives of sub- and supersolutions in order to produce a comparison principle for semicontinuous functions.

Keywords

Infinite-Laplace equation , viscosity solution , Grushin-type spaces

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