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

Authors : Thomas BİESKE, Zachary FORREST

Pages : 77-89

Doi:10.33205/cma.1245581

View : 132 | Download : 56

Publication Date : 2023-06-15

Article Type : Research Paper

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_kinsert ignore into journalissuearticles values(p);:=\\sigma_kinsert ignore into journalissuearticles values(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