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

Yuhki Hosoya

TL;DR

This work extends Frobenius-type integrability to the case where the guiding field $g$ is not necessarily smooth. By focusing on normal solutions and the differentiability of their level sets, it proves that Jacobi’s integrability condition is necessary and sufficient for the local existence of a $C^1$ normal solution around each point; when $g$ is $C^k$, the corresponding level sets attain $C^{k+1}$ regularity. The results reveal an intrinsic asymmetry: solutions may be limited in differentiability while integral manifolds remain smoother, and they provide a constructive framework (via $E^{x_n}$ and related lemmas) to obtain the normal solutions. An application to optimization shows how quasi-convexity of solutions interacts with KKT-type conditions, offering practical criteria for optimality under differential constraints. Overall, the paper links Jacobi integrability with the geometry of level sets and their role in quasi-convex optimization, advancing both foliation theory and variational analysis.

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.

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

TL;DR

This work extends Frobenius-type integrability to the case where the guiding field is not necessarily smooth. By focusing on normal solutions and the differentiability of their level sets, it proves that Jacobi’s integrability condition is necessary and sufficient for the local existence of a normal solution around each point; when is , the corresponding level sets attain regularity. The results reveal an intrinsic asymmetry: solutions may be limited in differentiability while integral manifolds remain smoother, and they provide a constructive framework (via and related lemmas) to obtain the normal solutions. An application to optimization shows how quasi-convexity of solutions interacts with KKT-type conditions, offering practical criteria for optimality under differential constraints. Overall, the paper links Jacobi integrability with the geometry of level sets and their role in quasi-convex optimization, advancing both foliation theory and variational analysis.

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 . When the system is , we can only ensure the existence of a solution, but the integral manifolds must be . In addition, we see a counterexample in which the system is , but there is no 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.
Paper Structure (17 sections, 13 theorems, 65 equations)

This paper contains 17 sections, 13 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. General Symbols
  4. Lipschitz Functions
  5. Differentiable Manifolds
  6. Ordinary Differential Equations
  7. Convex Analysis
  8. Results
  9. Normal Solutions to the Total Differential Equation
  10. Key Lemmas
  11. Invariance of the Level Sets
  12. Main Result
  13. Application
  14. Examples
  15. The Example in the Introduction
  16. ...and 2 more sections

Key Result

Proposition 1

Suppose that $U\subset \mathbb{R}^n$ is open and $g:U\to \mathbb{R}^n\setminus \{0\}$ is locally Lipschitz. If $V\subset U$ is open and $u:V\to \mathbb{R}, \lambda:V\to \mathbb{R}$ are a pair of a $C^1$ nondegenerate function and a continuous non-zero function that satisfies the relationship (TDE) e

Theorems & Definitions (13)

  • Proposition 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Proposition 2
  • Corollary 1
  • Corollary 2
  • Theorem 1
  • ...and 3 more