Gradient estimates for $p\left(\cdot\right)$-harmonic differential forms

TL;DR

This paper develops gradient bounds and regularity results for $p( abla)$-harmonic differential forms with variable exponents, extending classical theory to nonhomogeneous media.

Contribution

It introduces new regularity estimates for variable-exponent $p$-harmonic forms, including higher integrability and H"older continuity of gradients.

Findings

Derived higher integrability estimates of Meyers type.

Proved H"older continuity of the gradient under stronger assumptions.

Extended classical regularity results to variable-exponent settings.

Abstract

In this paper, we establish gradient bounds for $p(\cdot)$-harmonic differential forms subject to a Coulomb-type gauge condition. For variable exponents satisfying the log-H\"older continuity assumption, we derive higher integrability estimates of Meyers type, ensuring improved regularity beyond the natural energy space. Furthermore, under the stronger assumption of H\"older continuity of the exponent function, we prove that the gradient of solutions exhibits H\"older continuity. These results extend classical regularity theory for constant-exponent $p$-harmonic systems to the variable-exponent setting, which is essential for modeling nonhomogeneous and anisotropic media.