Higher integrability of solutions to elliptic equations under additional sign constraints

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 $({\mathrm{\partial }}_{i}u)$ 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 Müller’s result on the higher integrability of the determinant for maps $u\in W^{1,n}$ such that $\mathrm{d}\mathrm{e}\mathrm{t}(\mathrm{\nabla }u)\ge 0$ (or ${\mathrm{d}\mathrm{e}\mathrm{t}}_{-}(\mathrm{\nabla }u)\in L\log L$). Second, we consider (very weak) solutions to the $p$-Laplace equation that satisfy sign constraints for their partial derivatives, i.e. that $({\mathrm{\partial }}_{i}u{)}_{-}$ is of higher integrability than $({\mathrm{\partial }}_{i}u{)}_{+}$. 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.