Nonlocal Harnack inequalities in the Heisenberg group

TL;DR

This paper establishes general Harnack inequalities for nonlinear integro-differential problems in the Heisenberg group, with applications to various fields involving nonlocal interactions and non-Euclidean geometries.

Contribution

It introduces new Harnack inequalities for weak solutions of fractional subLaplacian equations in the Heisenberg group, including asymptotic analysis as the fractional parameter approaches 1.

Findings

Proved Harnack inequalities for a broad class of problems in the Heisenberg group.

Analyzed the asymptotic behavior of the fractional subLaplacian as the differentiability parameter approaches 1.

Demonstrated robustness of inequalities in the case p=2.

Abstract

We deal with a wide class of nonlinear integro-differential problems in the Heisenberg-Weyl group $\mathbb{H}^n$, whose prototype is the Dirichlet problem for the $p$-fractional subLaplace equation. These problems arise in many different contexts in quantum mechanics, in ferromagnetic analysis, in phase transition problems, in image segmentations models, and so on, when non-Euclidean geometry frameworks and nonlocal long-range interactions do naturally occur. We prove general Harnack inequalities for the related weak solutions. Also, in the case when the growth exponent is $p=2$, we investigate the asymptotic behavior of the fractional subLaplacian operator, and the robustness of the aforementioned Harnack estimates as the differentiability exponent $s$ goes to $1$.