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Ciarlet Nečas condition in fractional Sobolev spaces

Stanislav Hencl, Jaromír Mielec, Kaushik Mohanta

Abstract

Let $s\in(\frac{n}{n+1},1)$, $Ω\subset\mathbb{R}^n$ be an open set and let $f\in W^{s,n/s}(Ω,\mathbb{R}^n)$ be mapping with positive distributional Jacobian $\mathcal{J}_f>0$ which models some deformation in fractional Nonlinear Elasticity. We show change of variables formula in this class and as a consequence we show that the analogue of Ciarlet-Nečas condition $\mathcal{J}_f(Ω)=|f(Ω)|$ implies that our mapping is one-to-one a.e.

Ciarlet Nečas condition in fractional Sobolev spaces

Abstract

Let , be an open set and let be mapping with positive distributional Jacobian which models some deformation in fractional Nonlinear Elasticity. We show change of variables formula in this class and as a consequence we show that the analogue of Ciarlet-Nečas condition implies that our mapping is one-to-one a.e.
Paper Structure (15 sections, 92 equations)

This paper contains 15 sections, 92 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hölder spaces
  4. Distributional Determinant
  5. Degree
  6. Change of Variables and Ciarlet Nečas condition
  7. Stability of Ciarlet Nečas condition under weak convergence
  8. Counterexample to Lusin $(N)$ condition in $W^{s,p}$
  9. Construction of Cantor type sets
  10. Construction of a mapping $f$
  11. Hölder continuity of $f$
  12. Example of mapping with $J_f>0$ a.e. which is not sense preserving in $W^{s,p}$
  13. Definition of Cantor sets and mapping
  14. Hölder continuity of the map $f_k$
  15. Global Hölder continuity of the map

Theorems & Definitions (3)

  • proof : Proof of Theorem \ref{['main']}
  • proof
  • proof