Weak Differentiability to Nonuniform Nonlinear Degenerate Elliptic Systems under $p,q$-growth Condition on the Heisenberg Group

TL;DR

This paper investigates the weak differentiability properties of solutions to certain nonlinear degenerate elliptic systems on the Heisenberg Group, employing fractional difference quotient techniques to establish regularity results in both horizontal and vertical directions.

Contribution

It introduces a novel approach using fractional difference quotients to prove weak differentiability of solutions under $p,q$-growth conditions on the Heisenberg Group, including second order derivatives.

Findings

Weak differentiability in the vertical direction (${L^p}$ with $1<p<4$)

Second order weak differentiability in horizontal directions (${L^2}$)

Weak differentiability of the gradient in the vertical direction (${L^2}$)

Abstract

The paper concerns the weak differentiability of weak solutions to two kinds of nonuniform nonlinear degenerate elliptic systems under the $p,q$-growth condition on the Heisenberg Group. We use the iteration to fractional difference quotients on the Heisenberg Group to get the weak differentiability of weak solution $u$ in the vertical direction (i.e., ${L^p}$($1<p<4$) integrability of $Tu$) and then the second order weak differentiability of weak solution in the horizontal directions (i.e., ${L^2}$ integrability of $\nabla_H^2u$) and weak differentiability of gradient of weak solution in the vertical direction (i.e., ${L^2}$ integrability of $T{\nabla_H}u$)).