Solutions of the Differential Inequality with a~Null Lagrangian: Regularity and Removability of Singularities

TL;DR

This paper establishes new regularity and singularity removal results for solutions of a differential inequality involving a null Lagrangian, enhancing understanding of their stability and smoothness properties.

Contribution

It introduces a self-improving regularity theorem for solutions of a specific differential inequality involving quasiconvex functions and null Lagrangians, with applications to stability and singularity removability.

Findings

Proved a self-improving regularity theorem for solutions.

Improved stability and Hölder regularity results.

Established a theorem on removability of singularities.

Abstract

We prove a theorem on self-improving regularity for derivatives of solutions of the inequality $F(v'(x))\le KG(v'(x))$ constructed by means of a quasiconvex function $F$ and a null Lagrangian $G$. We apply this theorem to improve the stability and H\"older regularity results of \cite{Egor2008} and to establish a theorem on removability of singularities for solutions of this inequality.