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A Frobenius integrability theorem for plane fields generated by quasiconformal deformations

Slobodan N. Simić

Abstract

We generalize the classical Frobenius integrability theorem to plane fields of class $C^Q$, a regularity class introduced by Reimann [Rei76] for vector fields in Euclidean spaces. A $C^Q$ vector field is uniquely integrable and its flow is a quasiconformal deformation. We show that an a.e. involutive $C^Q$ plane field (defined in a suitable way) in $\mathbb{R}^n$ is integrable, with integral manifolds of class $C^1$.

A Frobenius integrability theorem for plane fields generated by quasiconformal deformations

Abstract

We generalize the classical Frobenius integrability theorem to plane fields of class , a regularity class introduced by Reimann [Rei76] for vector fields in Euclidean spaces. A vector field is uniquely integrable and its flow is a quasiconformal deformation. We show that an a.e. involutive plane field (defined in a suitable way) in is integrable, with integral manifolds of class .
Paper Structure (3 sections, 4 theorems, 28 equations)

This paper contains 3 sections, 4 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Preliminaries from the DiPerna-Lions-Ambrosio theory
  3. Proof of the main result

Key Result

Theorem 3

A continuous vector field $f : \mathbb{R}^n \to \mathbb{R}^n$$(n \geq 2)$ is of class $C^Q$ if and only if:

Theorems & Definitions (12)

  • Definition 1
  • Definition 2
  • Theorem 3
  • Definition 4
  • Definition 5
  • Definition 6: Regular Lagrangian flows colombo+tione+2021
  • Lemma 7
  • proof
  • Corollary 8
  • Lemma 9
  • ...and 2 more