Calculus of Variations and Geometric Measure Theory

G. Palatucci - M. Piccinini

Nonlinear fractional equations in the Heisenberg group

created by piccinini on 27 Apr 2023
modified on 01 Mar 2024


Published Paper

Inserted: 27 apr 2023
Last Updated: 1 mar 2024

Journal: Bruno Pini Math. Anal. Semin.
Year: 2024

ArXiv: 2307.16763 PDF


We deal with a wide class of nonlinear nonlocal equations led by integro-differential operators of order $(s,p)$, with summability exponent $p \in (1,\infty)$ and differentiability exponent $s\in (0,1)$, whose prototype is the fractional subLaplacian in the Heisenberg group. We present very recent boundedness and regularity estimates (up to the boundary) for the involved weak solutions, and we introduce the nonlocal counterpart of the Perron Method in the Heisenberg group, by recalling some results on the fractional obstacle problem. Throughout the paper we also list various related open problems

Keywords: Heisenberg group, obstacle problems, Caccioppoli estimates, Nonlocal operators, fractional subLaplacian, De Giorgi-Nash-Moser theory, Perron's method