Smooth approximation of mappings with rank of the derivative at most $1$

TL;DR

This paper proves that locally Lipschitz mappings with rank at most 1 can be uniformly approximated by smooth mappings with the same rank constraint in simply connected domains, using metric space analysis techniques.

Contribution

It establishes a positive approximation result for rank-1 mappings, filling a gap in the conjecture for the case m=1, using novel metric space methods.

Findings

Approximation of rank-1 mappings by smooth functions is possible in simply connected domains.

Counterexamples exist for higher ranks, but not for rank 1.

Techniques involve metric trees and analysis on metric spaces.

Abstract

It was conjectured that if $f\in C^1(\mathbb{R}^n,\mathbb{R}^n)$ satisfies $\operatorname{rank} Df\leq m<n$ everywhere in $\mathbb{R}^n$, then $f$ can be uniformly approximated by $C^\infty$-mappings $g$ satisfying $\operatorname{rank} Dg\leq m$ everywhere. While in general, there are counterexamples to this conjecture, we prove that the answer is in the positive when $m=1$. More precisely, if $m=1$, our result yields an almost-uniform approximation of locally Lipschitz mappings $f:\Omega\to\mathbb{R}^n$, satisfying $\operatorname{rank} Df\leq 1$ a.e., by $C^\infty$-mappings $g$ with $\operatorname{rank} Dg\leq 1$, provided $\Omega\subset\mathbb{R}^n$ is simply connected. The construction of the approximation employs techniques of analysis on metric spaces, including the theory of metric trees ($\mathbb{R}$-trees).