Calculus of Variations and Geometric Measure Theory

G. Palatucci - M. Piccinini

Nonlocal Harnack inequalities in the Heisenberg group

created by piccinini on 21 Jul 2022
modified by palatucci on 05 Jan 2024


Published Paper

Inserted: 21 jul 2022
Last Updated: 5 jan 2024

Journal: Calc. Var. Partial Differential Equations
Volume: 61
Number: Art. 185
Year: 2022

ArXiv: 2207.04051v2 PDF


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 $1$.

Keywords: Hölder continuity, Heisenberg group, Harnack inequalities, fractional Sobolev spaces,, fractional sublaplacian