Higher integrability of solutions to elliptic equations under additional sign constraints

TL;DR

This paper demonstrates enhanced regularity of solutions to elliptic equations under sign constraints, including higher integrability of determinants and derivatives, using advanced Lipschitz truncation techniques.

Contribution

It extends higher integrability results by incorporating sign constraints and develops a novel asymmetric Lipschitz truncation method.

Findings

Higher integrability of determinants for maps with non-negative Jacobian determinants.

Sign constraints on derivatives lead to improved regularity of solutions.

Introduction of asymmetric Lipschitz truncation technique.

Abstract

Solutions to elliptic equations often exhibit higher regularity properties such as \emph{higher integrability}. That is, for instance, a solution $u$ to a system that a priori only satisfies $ u \in W^{1,r}$ is more regular and even in the Sobolev space $W^{1,s}$ for some $s>r$. Under additional constraints of the sign of specific terms such as $(\partial_i u)$ this improvement of regularity can be sharpened further. In this work, we consider two examples of such higher integrability results: First, we show a version of M\"uller's result on the higher integrability of the determinant for maps $u \in W^{1,n} $ such that $\mathrm{det}(\nabla u) \geq 0$ (or $ \mathrm{det}_-(\nabla u) \in L \log L$). Second, we consider (very weak) solutions to the $p$-Laplace equation that satisfy sign constraints for their partial derivatives, i.e. that $(\partial_i u)_- $ is of higher integrability than $(\partial_i u)_+$. To prove our results, we use the method of Lipschitz truncation; for the second example we further develop a variation of this technique, the \emph{asymmetric} Lipschitz truncation.