Diffeomorphic Approximation of Piecewise Affine Homeomorphisms
Abstract
Abstract Given any f , a locally finitely piecewise affine homeomorphism of $$\Omega \subset \mathbb {R}^d$$ Ω ⊂ R d onto $$\Delta \subset \mathbb {R}^d$$ Δ ⊂ R d (for $$d=3, 4$$ d = 3 , 4 ) such that $$f\in W^{1,p}(\Omega , \mathbb {R}^d)$$ f ∈ W 1 , p ( Ω , R d ) and $$f^{-1}\in W^{1,q}(\Delta , \mathbb {R}^d)$$ f - 1 ∈ W 1 , q ( Δ , R d ) , $$1\le p,q < \infty $$ 1 ≤ p , q < ∞ and any $$\varepsilon >0$$ ε > 0 we construct a diffeomorphism $$\tilde{f}$$ f ~ such that $$\begin{aligned} \Vert f-\tilde{f}\Vert _{W^{1,p}(\Omega ,\mathbb {R}^d)} + \Vert f^{-1}-\tilde{f}^{-1}\Vert _{W^{1,q}(\Delta ,\mathbb {R}^d)} < \varepsilon . \end{aligned}$$ ‖ f - f ~ ‖ W 1 , p ( Ω ,
Fields of Study & Subjects
Analytic and geometric function theory
Mathematical Dynamics and Fractals
Holomorphic and Operator Theory
Publication details
- Journal / source
- Journal of Geometric Analysis
- Publisher
- Springer Science and Business Media LLC
- Volume
- 36
- Issue
- 5
- Publication date
- 31 March 2026
- Language
- EN