On the Lebesgue measure of the boundary of the evoluted set

TL;DR

This paper investigates conditions under which the boundary of a set evolved through a fixed flow has Lebesgue measure zero, providing theoretical results and counterexamples to demonstrate near-sharpness of these conditions.

Contribution

It establishes new criteria ensuring the evolved set's boundary has measure zero and presents counterexamples illustrating the limits of these criteria.

Findings

Conditions for negligible boundary of evoluted sets

Counterexamples close to the theoretical limits

Theoretical framework for measure-zero boundaries

Abstract

The evoluted set is the set of configurations reached from an initial set via a fixed flow for all times in a fixed interval. We find conditions on the initial set and on the flow ensuring that the evoluted set has negligible boundary (i.e. its Lebesgue measure is zero). We also provide several counterexample showing that the hypotheses of our theorem are close to sharp.