Nonlocal Harnack inequalities in the Heisenberg group

Abstract

We deal with a wide class of nonlinear integro-differential problems in the Heisenberg-Weyl group $$\mathbb {H}^n$$ 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 $$p=2$$ 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.

The obstacle problem and the Perron Method for nonlinear fractional equations in the Heisenberg group

“…Moreover,it is worth mentioning [16], where the authors proved, via semigroup theory, an asymptotic behaviour of fractional subLaplacians on Carnot groups when the differentiability exponent ր 1. An analogous result has been obtained in [28] via Taylor polynomials for the particular case of the Heisenberg group and in the same paper has been used to prove a stability property of the Harnack inequality when goes to 1.…”

“…However, to our knowledge, there is not much literature about this problems in the non-Euclidean setting of Carnot groups. Then, we trust that the present paper, together with the aforementioned ones [25,28], could be a starting point in investigating nonlocal and nonlinear (or fractional double phase) operators with more complex non-Euclidean underlying geometry and their mean value formulas. Moreover, we hope that Theorem 1.4 could be an incentive in developing an axiomatic Potential Theory for more general nonlinear and nonlocal operators on Carnot group, as the well known results for subelliptic Laplacians presented in the monograph [4].…”

“…This quantity is the analogue in the Heisenberg framework of the one introduced in [12,13]; see also [25,28]. The tail function has been crucial in studying the long-range interactions that naturally arise when dealing with operator as in (1.1); we refer to [27] for a survey in such a framework.…”

“…In particular, (1.6) is needed in the proof of Theorem 1.1 to show that the operator , defined in (1.3), is monotone (e. g. satisfies condition (3.2)). As for conditions (1.4) and (1.5), they are crucial in showing that some regularity properties of weak solutions to (1.7) such as boundedness, Hölder continuity and Harnack inequality hold true; see the recent papers [25,28]. In [28] it is proven a generalization of the nonlocal Harnack inequality obtained by Ferrari and Franchi in [15] for the linear case when = 2 via the celebrated Caffarelli-Silvestre extension (see [7]), which in the more general nonlinear framework (when ≠ 2) is not applicable.…”

“…We also mention that all the theorems proven in the present manuscript do hold when ranges form 1 to −2 1− . Such a bound on the growth exponent arises naturally from the underlying geometrical structure of the Heisenberg group, as already seen in the literature; see e. g. [25,28]. Specifically, it is necessary that ranges over 1 to −2…”