Lipschitz extensions on generalized Grushin spaces

Abstract

In this paper, we look to extend the concept of viscosity solutions to Grushin-type spaces, which are constructed using R but lack a group structure. The first part of this article is dedicated to background material and the establishment of Grushin maximum principles. This allows us to prove comparison principles, including one for viscosity infinite harmonic functions. After doing so, the final section is used to prove that C1 sub absolute minimizers are viscosity infinite harmonic. (For the definitions of C1 sub and C 2 sub functions, see Definition 1.) This result is inspired by the work of Capogna and the author in [5], in which absolute minimizers in Carnot groups are shown to be viscosity infinite harmonic and C1 sub minimizers in free vector fields are also shown to be viscosity infinite harmonic. The main nicety of the proof here is that the Rothschild–Stein lifting theorem [18] is not needed, for the Taylor polynomial is directly computable. We begin by constructing the Grushin-type spaces. We consider R with coordinates (x1, x2 , . . . , xn) and the vector fields Xi = ρi(x1, x2 , . . . , xi−1) ∂