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$(INV)$ condition and regularity of the inverse

Anna Doležalová, Stanislav Hencl, Jani Onninen

TL;DR

The paper addresses the regularity of inverses for Sobolev mappings of finite distortion under the $(INV)$ condition. It constructs a generalized inverse $h$ that remains Sobolev with finite distortion and inherits the $(INV)$ property, even when the Jacobian may vanish on sets of positive measure, and extends the result to higher dimensions under $p>n-1$. A higher-dimensional analogue connects the $n$-harmonic energy of the inverse to the inner distortion of the forward map, enriching the variational perspective in Geometric Function Theory and nonlinear elasticity. The work articulates a robust framework for inverse maps in the $(INV)$ class, clarifying structural properties such as topological images, cavities, and a.e. differentiability, with implications for existence of (approximate) homeomorphic minimizers in elasticity models and Teichmüller-type energy problems in higher dimensions.

Abstract

Let $f \colon Ω\to Ω' $ be a Sobolev mapping of finite distortion between planar domains $Ω$ and $Ω'$, satisfying the $(INV)$ condition and coinciding with a homeomorphism near $\partialΩ$. We show that $f$ admits a generalized inverse mapping $h \colon Ω' \to Ω$, which is also a Sobolev mapping of finite distortion and satisfies the $(INV)$ condition. We also establish a higher-dimensional analogue of this result: if a mapping $f \colon Ω\to Ω' $ of finite distortion is in the Sobolev class $W^{1,p}(Ω, \mathbb{R}^n)$ with $p > n-1$ and satisfies the $(INV)$ condition, then $f$ has an inverse in $W^{1,1}(Ω', \mathbb{R}^n)$ that is also of finite distortion. Furthermore, we characterize Sobolev mappings satisfying $(INV)$ whose generalized inverses have finite $n$-harmonic energy.

$(INV)$ condition and regularity of the inverse

TL;DR

The paper addresses the regularity of inverses for Sobolev mappings of finite distortion under the condition. It constructs a generalized inverse that remains Sobolev with finite distortion and inherits the property, even when the Jacobian may vanish on sets of positive measure, and extends the result to higher dimensions under . A higher-dimensional analogue connects the -harmonic energy of the inverse to the inner distortion of the forward map, enriching the variational perspective in Geometric Function Theory and nonlinear elasticity. The work articulates a robust framework for inverse maps in the class, clarifying structural properties such as topological images, cavities, and a.e. differentiability, with implications for existence of (approximate) homeomorphic minimizers in elasticity models and Teichmüller-type energy problems in higher dimensions.

Abstract

Let be a Sobolev mapping of finite distortion between planar domains and , satisfying the condition and coinciding with a homeomorphism near . We show that admits a generalized inverse mapping , which is also a Sobolev mapping of finite distortion and satisfies the condition. We also establish a higher-dimensional analogue of this result: if a mapping of finite distortion is in the Sobolev class with and satisfies the condition, then has an inverse in that is also of finite distortion. Furthermore, we characterize Sobolev mappings satisfying whose generalized inverses have finite -harmonic energy.

Paper Structure

This paper contains 14 sections, 10 theorems, 71 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $(INV)$ condition
  4. Density points
  5. Change of variables
  6. ACL condition
  7. Embedding into Hölder spaces
  8. Hausdorff measure and $p$-capacity
  9. The mappings satisfy the strong $(INV+)$ condition
  10. Variations of the $(INV)$ condition
  11. Counterexamples
  12. Definition of $h$ a.e. in $f_0(\Omega)$
  13. Definition of $h$ in cavities
  14. Definition of $h$ at other points

Key Result

Lemma 2.5

Let $u$ be a function belonging to the Sobolev class $W_{\operatorname{loc}}^{1,p} (\Omega)$ where $\Omega \subseteq \mathbb R^n$ and $p>n$. Then for almost every $x,y \in B_s=B(x,s) \subseteq \Omega$. If $n=1$, then the estimate eq:soboembed also holds for $p=n=1$. Furthermore, for the continuous representative the estimate eq:soboembed holds for every $x,y \in B_s$.

Figures (2)

  • Figure 1: Construction of bad mapping in $(INV)$ which is not a mapping of finite distortion.
  • Figure 2: Finding of big part of $f(G)$ in $B(y,r)$ for the validity of \ref{['goal']} for $n=2$.

Theorems & Definitions (26)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Lemma 3.4
  • proof
  • ...and 16 more