Alberti representations, rectifiability of metric spaces and higher integrability of measures satisfying a PDE

TL;DR

This paper establishes a new criterion for rectifiability of sets in metric spaces using Alberti representations and demonstrates higher integrability of measures satisfying PDE constraints, advancing understanding in geometric measure theory.

Contribution

It introduces a sharp sufficient condition for rectifiability based on Alberti representations and extends higher integrability results for PDE-constrained measures.

Findings

Rectifiability characterized by independent Alberti representations.

Higher integrability of PDE-satisfying measures in Euclidean spaces.

Strengthening of previous rectifiability criteria in metric spaces.

Abstract

We give a sufficient condition for a Borel subset $E\subset X$ of a complete metric space with $\mathcal{H}^n(E)<\infty$ to be $n$-rectifiable. This condition involves a decomposition of $E$ into rectifiable curves known as an Alberti representation. Precisely, we show that if $\mathcal{H}^n|_E$ has $n$ independent Alberti representations, then $E$ is $n$-rectifiable. This is a sharp strengthening of prior results of Bate and Li. It has been known for some time that such a result answers many open questions concerning rectifiability in metric spaces, which we discuss. An important step of our proof is to establish the higher integrability of measures on Euclidean space satisfying a PDE constraint. These results provide a quantitative generalisation of recent work of De Philippis and Rindler and are of independent interest.