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Measure preserving maps with bounded total variation

Stefano Bianchini, Luca Talamini

Abstract

Consider a piecewise affine Lipschitz map $φ: Ω\to \mathbb R$, where $Ω\subset \mathbb R^d$ is an open set, and assume that $x \mapsto x + t \nabla φ(x)$ is injective for almost every $t > 0$. In (J.-G. Liu, R.~L. Pego, \emph{Rigidly breaking potential flows and a countable Alexandrov theorem for polytopes}, Pure Appl. Anal., \textbf{7}(4), 2025) the authors conjecture that every such $φ$ must be locally convex. We prove the result assuming additionally $\nabla φ\in BV_{loc}(Ω)$, for a more general class of measure preserving maps.

Measure preserving maps with bounded total variation

Abstract

Consider a piecewise affine Lipschitz map , where is an open set, and assume that is injective for almost every . In (J.-G. Liu, R.~L. Pego, \emph{Rigidly breaking potential flows and a countable Alexandrov theorem for polytopes}, Pure Appl. Anal., \textbf{7}(4), 2025) the authors conjecture that every such must be locally convex. We prove the result assuming additionally , for a more general class of measure preserving maps.
Paper Structure (4 sections, 8 theorems, 89 equations)

This paper contains 4 sections, 8 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Measure preserving maps and main result
  3. Subharmonicity and expanding maps
  4. Optimal transport approach

Key Result

Theorem 1.2

In the above setting, if $\nabla \phi \in BV_{loc}(\Omega)$, $\phi$ must be locally convex.

Theorems & Definitions (17)

  • Conjecture 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Proposition 2.4
  • proof : Proof of Theorem \ref{['thm:main']}
  • Definition 3.1
  • Remark 3.2
  • Proposition 3.3
  • ...and 7 more