The obstacle problem and the Perron Method for nonlinear fractional equations in the Heisenberg group

TL;DR

This paper investigates the obstacle problem for nonlinear fractional operators in the Heisenberg group, establishing existence, uniqueness, regularity of solutions, and the Perron method for generalized solutions.

Contribution

It introduces new results on existence, uniqueness, and regularity of solutions to the obstacle problem in the Heisenberg group, and extends the Perron method for generalized solutions.

Findings

Proved existence and uniqueness of solutions.

Established regularity properties like boundedness and H"older continuity.

Demonstrated the existence of the Perron solution for general boundary data.

Abstract

We study the obstacle problem related to a wide class of nonlinear integro-differential operators, whose model is the fractional subLaplacian in the Heisenberg group. We prove both the existence and uniqueness of the solution, and that solutions inherit regularity properties of the obstacle such as boundedness, continuity and H\"older continuity up to the boundary. We also prove some independent properties of weak supersolutions to the class of problems we are dealing with. Armed with the aforementioned results, we finally investigate the Perron-Wiener-Brelot generalized solution by proving its existence for very general boundary data.