Quasiconvexity and self-improving size estimates

TL;DR

This paper proves a self-improving integrability property for certain quasiconcave functions related to the determinant, extending known bounds from null Lagrangians to a broader class.

Contribution

It establishes that M"uller's $L ext{log}L$ bound applies to quasiconcave, homogeneous functions like the determinant, broadening the scope beyond null Lagrangians.

Findings

M"uller's $L ext{log}L$ bound holds for quasiconcave functions.

The result applies specifically to the determinant function.

Contrast with Hardy space bounds valid only for null Lagrangians.

Abstract

We show that M\"uller's $L\log L$ bound $$F(Du)\geq 0,\,Du\in L^p_{\mathrm{loc}}(\mathbb{R}^n)\implies F(Du)\in L\log L_{\mathrm{loc}}(\mathbb{R}^n)$$ for $F =\det$ and $p=n$ holds for quasiconcave $F$ which are homogeneous of degree $p>1$. This contrasts similar Hardy space bounds which hold only for null Lagrangians.