Beyond Peano's theorem: a variational look at discontinuous ODE systems

TL;DR

This paper introduces a new variational framework for defining solutions to ODE systems that relax the classical continuity requirement, expanding the scope of solvable systems and providing new insights into Sobolev fields.

Contribution

It presents a novel variational approach to ODEs that allows for discontinuous systems, extending classical solution concepts beyond traditional continuity constraints.

Findings

New variational framework for discontinuous ODEs

Illustrative examples involving Sobolev fields

Open questions for further research

Abstract

We propose a framework to define solutions of ODE systems under a novel condition that goes well beyond the usual continuity condition required in the classical theory of ODEs (Peano's or Picard's theorems). We illustrate our results with some simple but enlightening examples, including some facts about Sobolev fields, and mention some relevant questions to proceed with this analysis further.