We deal with a wide class of nonlinear integro-differential problems in the Heisenberg-Weyl group~$\mathds{H}^n$, 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 $1$.
Nonlocal Harnack inequalities in the Heisenberg group / Palatucci, Giampiero; Piccinini, Mirco. - In: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 1432-0835. - 61:(2022). [10.1007/s00526-022-02301-9]
Nonlocal Harnack inequalities in the Heisenberg group
Palatucci, Giampiero
;Piccinini, Mirco
2022-01-01
Abstract
We deal with a wide class of nonlinear integro-differential problems in the Heisenberg-Weyl group~$\mathds{H}^n$, 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 $1$.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.