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 10 Oct 2022

[BibTeX]

Published Paper

Inserted: 21 jul 2022
Last Updated: 10 oct 2022

Journal: Calc. Var. Partial Differential Equations
Year: 2022
Doi: https://doi.org/10.1007/s00526-022-02301-9

ArXiv: 2207.04051v2 PDF

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 $1$.

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


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