Rectifiability of sets of solutions of first order systems of partial differential equations

Abstract

We find sufficient conditions on a set $\mathcal{M}\subset {\mathbf{R}}^n\times \mathcal{L}({\mathbf{R}}^n,{\mathbf{R}}^m)$ ensuring that the set of functions such that $(F(x),DF(x))\in \mathcal{M}$ is rectifiable. We also prove a more general version in which the set to which $DF(x)$ is costrained can depend also on $F(x))$, and show the relation with some classical rigidity statements such as Liouville's theorem on conformal maps