An Equivalence Result on the Order of Differentiability in Frobenius' Theorem

TL;DR

This paper investigates the differentiability properties of solutions and integral manifolds in total differential equations, revealing an asymmetry in smoothness levels and establishing conditions for quasi-convex solutions.

Contribution

It provides a new equivalence result relating the order of differentiability of solutions and integral manifolds in Frobenius' theorem without smoothness assumptions.

Findings

Integral manifolds are $C^1$ when the system is Lipschitz.

Solutions are $C^k$ if the system is $C^k$, but integral manifolds are $C^{k+1}.

Counterexample shows a $C^1$ system may lack $C^2$ solutions.

Abstract

This paper examines the simplest case of total differential equations that appears in the theory of foliation structures, without imposing the smoothness assumptions. This leads to a peculiar asymmetry in the differentiability of solutions. To resolve this asymmetry, this paper focuses on the differentiability of the integral manifold. When the system is locally Lipschitz, a solution is ensured to be only locally Lipschitz, but the integral manifolds must be $C^1$. When the system is $C^k$, we can only ensure the existence of a $C^k$ solution, but the integral manifolds must be $C^{k+1}$. In addition, we see a counterexample in which the system is $C^1$, but there is no $C^2$ solution. Moreover, we characterize a minimizer of an optimization problem whose objective function is a quasi-convex solution to a total differential equation. In this connection, we examine two necessary and sufficient conditions for the system in which any solution is quasi-convex.