Nonlocal Harnack inequalities in the Heisenberg group
Giampiero Palatucci, Mirco Piccinini
Abstract
Abstract We deal with a wide class of nonlinear integro-differential problems in the Heisenberg-Weyl group $$\mathbb {H}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msup></mml:math> , 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 $$p=2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math> , 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.