<p>Given any <i>f</i>, a locally finitely piecewise affine homeomorphism of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Omega \subset \mathbb {R}^d$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> onto <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Delta \subset \mathbb {R}^d$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Δ</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> (for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$d=3, 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>) such that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$f\in W^{1,p}(\Omega , \mathbb {R}^d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <msup> <mi>W</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>p</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$f^{-1}\in W^{1,q}(\Delta , \mathbb {R}^d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>f</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>∈</mo> <msup> <mi>W</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>q</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mo>,</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$1\le p,q < \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> and any <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\varepsilon >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> we construct a diffeomorphism <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\tilde{f}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>f</mi> <mo stretchy="false">~</mo> </mover> </math></EquationSource> </InlineEquation> such that <Equation ID="Equ23"> <EquationSource Format="TEX">$\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}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mrow> <mo stretchy="false">‖</mo> <mi>f</mi> <mo>-</mo> </mrow> <mover accent="true"> <mi>f</mi> <mo stretchy="false">~</mo> </mover> <msub> <mrow> <mo stretchy="false">‖</mo> </mrow> <mrow> <msup> <mi>W</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>p</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mo stretchy="false">‖</mo> <msup> <mi>f</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>-</mo> <msup> <mover accent="true"> <mi>f</mi> <mo stretchy="false">~</mo> </mover> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">‖</mo> </mrow> <mrow> <msup> <mi>W</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>q</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mo>,</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </msub> <mo><</mo> <mi>ε</mi> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation></p>

Diffeomorphic Approximation of Piecewise Affine Homeomorphisms

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2026年3月31日

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Daniel Campbell,
Luigi D’Onofrio,
Tomáš Vítek

摘要

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