Nonlocal Harnack inequalities in the Heisenberg group

Abstract

AbstractWe deal with a wide class of nonlinear integro-differential problems in the Heisenberg-Weyl group $$\mathbb {H}^n$$Hn, whose prototype is the Dirichlet problem for thep-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$$p=2$$p=2, we investigate the asymptotic behavior of the fractional subLaplacian operator, and the robustness of the aforementioned Harnack estimates as the differentiability exponentsgoes to 1.