A general notion of uniform ellipticity and the regularity of the stress field for elliptic equations in divergence form

For solutions of ${rm div},(DF(Du))=f$ we show that the quasiconformality of $zmapsto DF(z)$ is the key property leading to the Sobolev regularity of the stress field $DF(Du)$, in relation with the summability of $f$. This class of nonlinearities encodes in a general way the notion of uniform ellipticity and encompasses all known instances where the stress field is known to be Sobolev regular. We provide examples showing the optimality of this assumption and present three applications: the study of the strong locality of the operator ${rm div},(DF(Du))$, a nonlinear Cordes condition for equations in divergence form, and some partial results on the $C^{p}$-conjecture.