A remark on staircase laminates in restricted sets

TL;DR

This paper extends the convex integration method using staircase laminates and revisits a proof related to Meyers' regularity, offering insights into optimal regularity results.

Contribution

It introduces a slight extension to the staircase laminate toolbox within convex integration and applies it to a known regularity proof.

Findings

Extended the convex integration toolbox with staircase laminates.

Revisited and provided insights into Meyers' regularity proof.

Enhanced understanding of optimal regularity in elliptic PDEs.

Abstract

We slightly extend the convex integration via staircase laminate toolbox recently developed by Kleiner, M\"uller, Sz\'{e}kelyhidi, and Xie. As an example we revisit the proof by Astala-Faraco-Sz\'{e}kelyhidi on optimal Meyers' regularity theory via this framework.