Regularity for Mixed Local and Nonlocal Degenerate Elliptic Equations in the Heisenberg Group

TL;DR

This work develops a De Giorgi–Nash–Moser-type regularity theory for mixed local and nonlocal degenerate elliptic equations in the Heisenberg group $H^n$, treated with subelliptic geometry and $1<p<\infty$. By deriving a Caccioppoli-type inequality, logarithmic estimates, and Tail-term controls, the authors prove local boundedness and Hölder continuity of weak solutions. They further establish Harnack and weak Harnack inequalities via expansion of positivity and tail estimates adapted to the nonlocal, sub-Riemannian framework. The results advance regularity theory in Carnot groups by incorporating nonlocal interactions and demonstrate the pivotal role of Tail terms in connecting nonlocal effects to local oscillation behavior.

Abstract

In this paper, we investigate the regularity for mixed local and nonlocal degenerate elliptic equations in the Heisenberg group. Inspired by the De Giorgi-Nash-Moser theory, the local boundedness of weak subsolutions and the Hölder continuity of weak solutions to mixed local and nonlocal degenerate elliptic equations are established by deriving the Caccioppoli type inequality for weak subsolutions and the logarithmic estimates for weak supersolutions. Furthermore, the Harnack inequality for weak solutions and the weak Harnack inequality for weak supersolutions are proved by using the estimates involving a Tail term and expansion of positivity.

Theorems & Definitions (26)

• Theorem 1.1: Local boundedness of weak subsolutions
• Theorem 1.2: Hölder continuity
• Theorem 1.3: Harnack inequality
• Theorem 1.4: Weak Harnack inequality
• Lemma 2.1: Poincaré-type inequality, (1.1) in DLS07
• Lemma 2.2: Sobolev inequality, CDG93
• Lemma 2.3: MPPP23
• Definition 2.4
• Lemma 2.5: GK22
• Lemma 2.6: D93