[Paper Review] Nonlinear fractional equations in the Heisenberg group

Giampiero Palatucci, Mirco Piccinini|arXiv (Cornell University)|Jul 31, 2023

Differential Equations and Boundary ProblemsMathematics3 citations

TL;DR

This paper establishes the De Giorgi-Nash-Moser regularity theory for weak solutions to nonlinear nonlocal equations driven by $(s,p)$-fractional subLaplacian operators in the Heisenberg group, proving local boundedness, Hölder continuity, and nonlocal Harnack inequalities. It introduces a nonlocal Perron method and resolves boundary regularity issues via obstacle problem techniques, extending classical Euclidean results to sub-Riemannian geometry with full boundary regularity estimates.

ABSTRACT

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.

Motivation & Objective

• To extend the De Giorgi-Nash-Moser regularity theory to nonlinear nonlocal equations in the Heisenberg group with $(s,p)$-fractional subLaplacian operators.
• To establish boundedness and Hölder continuity of weak solutions up to the boundary using obstacle problem techniques.
• To develop a nonlocal counterpart of the Perron method in the Heisenberg group setting for solving Dirichlet problems.
• To identify and list open problems in nonlocal analysis on stratified Lie groups, particularly regarding boundary regularity and mixed-type operators.
• To generalize classical Euclidean regularity results—such as Caccioppoli estimates and Harnack inequalities—to the sub-Riemannian Heisenberg group framework.

Proposed method

• The authors analyze weak solutions to integro-differential equations defined via the $(s,p)$-fractional subLaplacian operator $\mathcal{L}_{s,p}u(\xi) = \mathrm{P.V.}\int_{\mathds{H}^n} \frac{|u(\xi)-u(\eta)|^{p-2}(u(\xi)-u(\eta))}{d_o(\eta^{-1}\circ\xi)^{Q+sp}} \, d\eta$.
• They employ Caccioppoli-type estimates to derive local boundedness and Hölder continuity of weak solutions in the Heisenberg group.
• Nonlocal Harnack inequalities are established through iterative covering and measure-theoretic techniques adapted to the sub-Riemannian geometry.
• The obstacle problem is used to prove boundary regularity and to construct the generalized Perron solution as a limit of subsolutions and supersolutions.
• A nonlocal Perron method is introduced by adapting classical potential-theoretic techniques to the Heisenberg group, relying on the existence of barriers and the comparison principle.
• The framework incorporates the homogeneous dimension $Q = 2n+2$ and a homogeneous norm $d_o$ to account for the group's non-Euclidean structure.

Experimental results

Research questions

• RQ1 Can the De Giorgi-Nash-Moser theory be extended to nonlinear nonlocal equations in the Heisenberg group with $p$-growth and $s$-differentiability?
• RQ2 What are the conditions under which weak solutions to such equations are bounded and Hölder continuous up to the boundary?
• RQ3 Does a nonlocal Perron method exist in the Heisenberg group, and can it be used to solve the Dirichlet problem for these operators?
• RQ4 Is there a Wiener-type condition for boundary regularity of the generalized Perron solution in the Heisenberg group setting?
• RQ5 How do mixed-type operators—combining local and nonlocal terms—behave in terms of regularity in the Heisenberg group?

Key findings

• The paper proves local boundedness and Hölder continuity of weak solutions to $\mathcal{L}_{s,p}u = 0$ in the Heisenberg group, with explicit dependence on $s$, $p$, and the homogeneous dimension $Q$.
• Nonlocal Harnack inequalities are established for weak solutions, extending the classical theory to the sub-Riemannian setting.
• Boundary regularity estimates for weak solutions are obtained via the obstacle problem, ensuring Hölder continuity up to the boundary under suitable assumptions.
• A generalized Perron solution $H_g$ is constructed and shown to coincide with the weak solution $h_g$ when it exists, validating the method in the Heisenberg group.
• The paper identifies the Wiener-type condition for boundary regularity as an open problem, despite the existence of the Perron solution.
• The results generalize known Euclidean $p$-fractional Laplacian estimates to the Heisenberg group, with full extension to boundary behavior and nonlocal potential theory.