The asymptotic $p$-Poisson equation as $p \to \infty$ in Carnot-Carath\'eodory spaces

TL;DR

This paper investigates the limiting behavior of solutions to the subelliptic p-Poisson equation in Carnot-Carathéodory spaces as p approaches infinity, revealing that these limits satisfy a hybrid PDE involving the infinity-Laplacian and Eikonal equation.

Contribution

It extends previous results by defining a suitable notion of differentiability and proving that solutions converge to a viscosity solution of a hybrid PDE in Carnot-Carathéodory spaces.

Findings

Limits of solutions solve a hybrid PDE involving infinity-Laplacian and Eikonal equation.

Extended the notion of differentiability in Carnot-Carathéodory spaces.

Proved convergence of solutions as p approaches infinity.

Abstract

In this paper we study the asymptotic behavior of solutions to the subelliptic $p$-Poisson equation as $p\to +\infty$ in Carnot Carath\'eodory spaces. In particular, introducing a suitable notion of differentiability, we extend the celebrated result of Bhattacharya, DiBenedetto and Manfredi [Rend. Sem. Mat. Univ. Politec. Torino, 1989, Special Issue, 15-68] and we prove that limits of such solutions solve in the sense of viscosity a hybrid first and second order PDE involving the $\infty-$Laplacian and the Eikonal equation.