Global regularity estimates for $p(x)$-Laplacian variational inequalities with singular or degenerate matrix-valued weights

TL;DR

This paper establishes global gradient bounds for solutions to $p(x)$-Laplacian variational inequalities with singular or degenerate weights, advancing regularity theory under optimal conditions.

Contribution

It develops new weighted regularity estimates for solutions involving degenerate or singular matrix weights, with minimal dependence on structural constants.

Findings

Derived weighted Calderón-Zygmund-type estimates

Established general weighted Orlicz-type estimates

Achieved near-optimal scaling parameters in level-set estimates

Abstract

We establish the global gradient bounds for weak solutions to the elliptic variational inequality with two-sided obstructions, associated with a $p(x)$-Laplacian type operator involving degenerate or singular matrix weights. Under the optimal regularity assumptions on the matrix-valued weight, suitable geometric flatness of the domain, and the prescribed data, we aim to investigate the effects of the problem structure on the level of integrability properties of solutions. To this end, we develop regularity in two regards: weighted Calder\'on-Zygmund-type and general weighted Orlicz-type estimates. A notable feature of our results is that, through a constructive level-set approach, the estimates can be derived with minimal dependence of the scaling parameter on the structural constants. The regularity results are then sharp in the sense that they enable the construction of a level-set estimate with nearly optimal scaling parameters, within admissible parameter sets.