On exterior differential systems involving differentials of H\"{o}lder functions

TL;DR

This paper extends Frobenius theorem to certain Pfaff-type systems involving rough signals and H"older continuous functions, providing a framework for solving exterior differential systems with irregular forms.

Contribution

It introduces a novel approach to analyze integral manifolds for differential systems with rough signals, expanding classical geometric analysis to irregular contexts.

Findings

Extended Frobenius theorem for rough differential systems

Framework for integral manifolds with H"older continuous data

Tools for solving exterior differential systems with distributional derivatives

Abstract

We study the validity of an extension of Frobenius theorem on integral manifolds for some classes of Pfaff-type systems of partial differential equations involving multidimensional "rough" signals, i.e. "differentials" of given H\"older continuous functions interpreted in a suitable way, similarly to Young Differential Equations in Rough Paths theory. This can be seen as a tool to study solvability of exterior differential systems involving rough differential forms, i.e. the forms involving weak (distributional) derivatives of highly irregular (e.g. H\"older continuous) functions; the solutions (integral manifolds) being also some very weakly regular geometric structures.