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

TL;DR

This paper establishes regularity results, including boundedness and H"older continuity, for mixed local and nonlocal degenerate elliptic equations in the Heisenberg group, extending classical theories to a sub-Riemannian setting.

Contribution

It develops new regularity theory for mixed local and nonlocal degenerate elliptic equations in the Heisenberg group, including Harnack inequalities and tail estimates.

Findings

Proved local boundedness of weak subsolutions.

Established H"older continuity of solutions.

Derived Harnack inequalities involving tail terms.

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 H\"{o}lder 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.