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

TL;DR

This paper establishes conditions under which solution sets of first-order PDE systems are rectifiable, extending classical rigidity results like Liouville's theorem to more general constrained derivative sets.

Contribution

It provides new sufficient conditions for rectifiability of solution sets of PDEs and generalizes existing rigidity theorems to broader contexts.

Findings

Identifies conditions ensuring rectifiability of solution sets.

Extends classical rigidity results to more general derivative constraints.

Shows the relation with Liouville's theorem on conformal maps.

Abstract

We find sufficient conditions on a set $\mathscr{M}\subset\mathbf{R}^n\times\mathscr{L}(\mathbf{R}^n,\mathbf{R}^m)$ ensuring that the set of functions such that $(F(x),DF(x))\in\mathscr{M}$ 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