<p>We prove that the key property in models of Nonlinear Elasticity which corresponds to the non-interpenetration of matter, i.e. injectivity a.e., can be achieved in the class of weak limits of homeomorphisms under very minimal assumptions. Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3041_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega \subset \mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> be a domain and let <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3041_Article_IEq2.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">$p>\left\lfloor \frac{n}{2}\right\rfloor $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mfenced close="⌋" open="⌊"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mfenced> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3041_Article_IEq3.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$n\ge 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3041_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$p\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3041_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">$n=2,3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. Assume that <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3041_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">$f_k\in W^{1,p}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>f</mi> <mi>k</mi> </msub> <mo>∈</mo> <msup> <mi>W</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>p</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> is a sequence of homeomorphisms such that <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3041_Article_IEq7.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">$f_k\rightharpoonup f$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>f</mi> <mi>k</mi> </msub> <mo>⇀</mo> <mi>f</mi> </mrow> </math></EquationSource> </InlineEquation> weakly in <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3041_Article_IEq8.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">$W^{1,p}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>W</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>p</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> and assume that <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3041_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">$J_f>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>J</mi> <mi>f</mi> </msub> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> a.e. Then we show that <i>f</i> is injective a.e.</p>

Weak limits of Sobolev homeomorphisms are one to one almost everywhere

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2025年6月6日

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Ondřej Bouchala,
Stanislav Hencl,
Zheng Zhu

摘要

We prove that the key property in models of Nonlinear Elasticity which corresponds to the non-interpenetration of matter, i.e. injectivity a.e., can be achieved in the class of weak limits of homeomorphisms under very minimal assumptions. Let $\Omega \subset \mathbb {R}^n$ $Ω \subset R^{n}$ be a domain and let $p>\left\lfloor \frac{n}{2}\right\rfloor $ $p > ⌊\frac{n}{2}⌋$ for $n\ge 4$ $n \geq 4$ or $p\ge 1$ $p \geq 1$ for $n=2,3$ $n = 2, 3$ . Assume that $f_k\in W^{1,p}$ $f_{k} \in W^{1, p}$ is a sequence of homeomorphisms such that $f_k\rightharpoonup f$ $f_{k} ⇀ f$ weakly in $W^{1,p}$ $W^{1, p}$ and assume that $J_f>0$ $J_{f} > 0$ a.e. Then we show that f is injective a.e.