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$W^{2,1}$ approximation of planar Sobolev homeomorphisms by smooth diffeomorphisms

Luigi D'Onofrio

Abstract

The approximation of Sobolev homeomorphisms by smooth diffeomorphisms is well understood in first-order spaces $W^{1,p}$, but remains largely open in the second-order space $W^{2,1}$ due to a fundamental tension between curvature control and injectivity. In this paper we isolate and resolve the local analytical component of this problem. We construct explicit local regularisations both across flat interfaces and near multi-cell vertices, and prove convergence in $W^{2,1}$ together with quantitative preservation of the Jacobian. The resulting maps are $C^{1}$ on the whole domain and smooth inside each cell of the partition; in particular they are $C^{2}$ away from the interfaces. These local constructions are combined into a global smoothing theorem: any piecewise quadratic $C^{1}$-compatible planar homeomorphism satisfying a quantitative bi-Lipschitz condition can be approximated in $W^{2,1}$ by maps that are $C^{1}$, injective, and have positive Jacobian. As a consequence, we show that the general $W^{2,1}$ approximation problem reduces to a purely geometric question: the construction of piecewise quadratic approximations with quantitative injectivity and nondegeneracy.

$W^{2,1}$ approximation of planar Sobolev homeomorphisms by smooth diffeomorphisms

Abstract

The approximation of Sobolev homeomorphisms by smooth diffeomorphisms is well understood in first-order spaces , but remains largely open in the second-order space due to a fundamental tension between curvature control and injectivity. In this paper we isolate and resolve the local analytical component of this problem. We construct explicit local regularisations both across flat interfaces and near multi-cell vertices, and prove convergence in together with quantitative preservation of the Jacobian. The resulting maps are on the whole domain and smooth inside each cell of the partition; in particular they are away from the interfaces. These local constructions are combined into a global smoothing theorem: any piecewise quadratic -compatible planar homeomorphism satisfying a quantitative bi-Lipschitz condition can be approximated in by maps that are , injective, and have positive Jacobian. As a consequence, we show that the general approximation problem reduces to a purely geometric question: the construction of piecewise quadratic approximations with quantitative injectivity and nondegeneracy.

Paper Structure

This paper contains 12 sections, 9 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Smoothing across a flat interface
  4. Geometry and mismatch structure
  5. The smoothing construction
  6. Smoothing near a vertex
  7. Second-order structure at the vertex
  8. The vertex smoothing construction
  9. Global smoothing of piecewise quadratic homeomorphisms
  10. Setting and assumptions
  11. Main global result
  12. Approximation theorem in $W^{2,1}$

Key Result

Lemma 2.1

Let $\eta\in C^\infty(\mathbb{R})$ and set $\eta_\varepsilon(t)=\eta(t/\varepsilon)$. Then where $C_m = \lVert \eta^{(m)} \rVert_{L^\infty(\mathbb{R})}$. Similarly, if $\chi\in C^\infty_c(\mathbb{R}^2)$ and $\chi_\varepsilon(x)=\chi(x/\varepsilon)$, then $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (21)

  • Lemma 2.1: Scaled cut-off bounds
  • Lemma 2.2: Sobolev embedding
  • Lemma 2.3: Quantitative separation stability
  • proof
  • Lemma 3.1: Structure of the mismatch
  • proof
  • Proposition 3.2: Flat-interface smoothing
  • proof
  • Lemma 4.1: Vanishing of the mismatch at the origin
  • proof
  • ...and 11 more