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Diffeomorphic approximation of piecewise affine homeomorphisms

Daniel Campbell, Luigi D'Onofrio, Tomáš Vítek

TL;DR

The paper proves that in dimensions $d=3,4$ every locally finite piecewise affine homeomorphism $f$ with Sobolev regularity can be approximated by a $C^{\infty}$-diffeomorphism $\tilde f$ so that both $\tilde f$ and its inverse are arbitrarily close to $f$ and $f^{-1}$ in the respective $W^{1,p}$ and $W^{1,q}$ norms. The authors construct this diffeomorphism via a detailed 3D smoothing scheme that modifies along faces, edges, and cells, ensuring positive Jacobian and injectivity, and then extend the finite-case construction to locally finite complexes. A key part of the approach uses rearrangement-invariant function spaces to formulate general, size-controlled approximation results beyond $L^{\infty}$-type norms. The work also discusses the dimensional limitations, tying the possibility of higher-dimensional generalization to the connectivity of the diffeomorphism group of spheres and Smale/Hatcher isotopy results. Overall, the paper provides a rigorous bridge between piecewise affine and smooth diffeomorphisms in 3D and 4D within Sobolev-type frameworks, with implications for geometric analysis and PDE applications.

Abstract

Given any $f$ a locally finitely piecewise affine homeomorphism of $Ω\subset \mathbb{R}^d$ onto $Δ\subset \mathbb{R}^d$ (for $d=3, 4$) such that $f\in W^{1,p}(Ω, \mathbb{R}^d)$ and $f^{-1}\in W^{1,q}(Δ, \mathbb{R}^d)$, $1\leq p ,q < \infty$ and any $ε>0$ we construct a diffeomorphism $\tilde{f}$ such that $$\|f-\tilde{f}\|_{W^{1,p}(Ω,\mathbb{R}^d)} + \|f^{-1}-\tilde{f}^{-1}\|_{W^{1,q}(Δ,\mathbb{R}^d)} < ε.$$

Diffeomorphic approximation of piecewise affine homeomorphisms

TL;DR

The paper proves that in dimensions every locally finite piecewise affine homeomorphism with Sobolev regularity can be approximated by a -diffeomorphism so that both and its inverse are arbitrarily close to and in the respective and norms. The authors construct this diffeomorphism via a detailed 3D smoothing scheme that modifies along faces, edges, and cells, ensuring positive Jacobian and injectivity, and then extend the finite-case construction to locally finite complexes. A key part of the approach uses rearrangement-invariant function spaces to formulate general, size-controlled approximation results beyond -type norms. The work also discusses the dimensional limitations, tying the possibility of higher-dimensional generalization to the connectivity of the diffeomorphism group of spheres and Smale/Hatcher isotopy results. Overall, the paper provides a rigorous bridge between piecewise affine and smooth diffeomorphisms in 3D and 4D within Sobolev-type frameworks, with implications for geometric analysis and PDE applications.

Abstract

Given any a locally finitely piecewise affine homeomorphism of onto (for ) such that and , and any we construct a diffeomorphism such that

Paper Structure

This paper contains 7 sections, 12 theorems, 73 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Rearrangement invariant function spaces
  4. The construction in 3D
  5. Proof of Theorem \ref{['Core']}
  6. Proof of Theorem \ref{['main']}
  7. Remark on the proof for higher dimensions

Key Result

Theorem A

Let $d=3$ or $d=4$. Let $\Omega\subset\mathbb R^d$ be a domain and let $p,q\in[1,\infty)$. Let $f:\Omega\to\mathbb R^d$ be a locally finite piecewise affine homeomorphism. Then, for every $\varepsilon>0$, there exists a $\mathcal{C}^{\infty}$-diffeomorphism $\tilde{f}$ satisfying and

Figures (2)

  • Figure 1: The red areas are the sets where $G_i$. The blue disk represents $B_{\mathbb R^2}(0,r)\times\mathbb R$. If $w_i$ is small compared to $r$ then then $(G_i\cap G_j) \setminus (B_{\mathbb R^2}(0,r)\times\mathbb R) = \emptyset$ and in fact have positive distance from each other. This fact remains true as long as the ratio $\frac{w_i}{r}$ remains small enough (i.e., is not affected by linear scaling). It holds that $g_i = g_{i+1} = f$ when $\theta\in (\theta_i,\theta_{i+1})$ and $(t\cos\theta_i, t\sin\theta_i,z)$ is outside $G_i$ (and outside $B_{\mathbb R^2}(0,r)\times\mathbb R$).
  • Figure 2: In the annulus $P_1$ we shift in the $e_3$ direction to so that hyperplanes disks inside $P_1$ are mapped onto hyperplanes. In $P_2$ we squeeze so that the image of a circle is a circle. In $P_3$ we 'untwist' to achieve a rotation in the disks inside $P_3$ we extend as the rotation.

Theorems & Definitions (21)

  • Theorem A
  • Theorem B
  • Definition 2.1
  • Theorem 2.2
  • Corollary 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • ...and 11 more