Higher Weak Differentiability to Mixed Local and Nonlocal Degenerate Elliptic Equations in the Heisenberg Group

TL;DR

This paper studies the higher weak differentiability of solutions to mixed local and nonlocal degenerate elliptic equations in the Heisenberg group, using iterative and truncation methods to analyze their gradients.

Contribution

It introduces a novel approach combining fractional difference quotients and truncation techniques to establish higher weak differentiability in the Heisenberg group context.

Findings

Established weak differentiability in the vertical direction.

Extended differentiability to horizontal and vertical gradients.

Proved higher weak differentiability of the solutions' gradients.

Abstract

In this paper, we investigate the higher weak differentiability of solutions to a class of mixed local and nonlocal degenerate elliptic equations in the Heisenberg group $\mathbb{H}^n$. Owing to the non-commutative property and two-step nilpotent Lie algebra structure of $\mathbb{H}^n$, we first employ an iterative scheme involving fractional difference quotients to establish the weak differentiability of solutions in the vertical direction. This is subsequently extended to the horizontal and vertical gradients. Then, by coupling a truncation argument with the difference quotient method, we prove the higher weak differentiability of the gradients of solutions.