Mixed fully nonlinear local and nonlocal elliptic operators in Heisenberg group

TL;DR

This paper proves the existence, comparison principle, and regularity of viscosity solutions for a mixed local and nonlocal elliptic PDE on the Heisenberg group, involving Pucci's operator and fractional sub-Laplacian.

Contribution

It introduces a framework for analyzing mixed fully nonlinear local and nonlocal elliptic operators on the Heisenberg group, establishing fundamental properties of solutions.

Findings

Established comparison principle for the mixed operator

Proved existence of viscosity solutions

Demonstrated regularity properties of solutions

Abstract

We establish the comparison principle, existence and regularity of viscosity solutions to the following problem concerning the mixed operator: \begin{align} \begin{cases} \alpha\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}u\big)-\beta\big(-\Delta_{\mathbbm{H}^N}\big)^su=f &\text{in } \,{\Omega}, u=g &\text{in } \,\mathbbm{H}^N\setminus\Omega, \end{cases} \end{align} where $\mathcal{M}_{\lambda,\Lambda}^+$ is the extremal Pucci's operator and $(-\Delta_{\mathbbm{H}^N})^s$ denotes the fractional sub-Laplacian on Heisenberg group. Here $\alpha\geq 0$ and $\beta>0$ are constants.