Higher integrability of solutions to elliptic equations under additional sign constraints

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.