$C^{1,\alpha}$-regularity for Mixed Local and Nonlocal Degenerate Elliptic Equations in the Heisenberg Group

TL;DR

This paper proves $C^{1,eta}$ regularity for solutions to mixed local and nonlocal degenerate elliptic equations on the Heisenberg group, advancing understanding of regularity in sub-Riemannian geometries.

Contribution

It introduces a new iteration scheme combining horizontal differences and fractional Sobolev inequalities to establish regularity results in the Heisenberg group.

Findings

Established H"older continuity of solutions.

Proved $C^{1,eta}$ regularity for weak solutions.

Developed a Morrey-type iteration scheme.

Abstract

The regularity theory for equations combining both local and nonlocal operators in sub-Riemannian geometries is a huge challenge. In this paper, we investigate the $C^{1,\alpha}$-regularity of weak solutions to mixed local and nonlocal degenerate elliptic equations on the Heisenberg group. We first derive a sophisticated iteration scheme of Morrey-type by leveraging horizontal difference combined with the fractional Sobolev-type inequality on the Heisenberg group. Then, the H\"{o}lder continuity of the weak solutions is established by applying the local boundedness, the iteration scheme of Morrey-type, an iterative method and the Morrey inequality. Finally, we use the H\"{o}lder continuity in conjunction with Theorem 1.2 from Mukherjee and Zhong\cite{MZ21} to prove the $C^{1,\alpha}$-regularity of weak solutions.