<p>We establish the regularity of solutions to the strain tensor equation on a region S where the Gaussian curvature changes sign. Specifically, we demonstrate that solutions possess enhanced smoothness under this geometric condition. Furthermore, we derive the density property: smooth infinitesimal isometries are dense in the space of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2075_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="88" /> </InlineMediaObject> <EquationSource Format="TEX">$W^{2,2}(S,I\!\!R^3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>W</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>I</mi> <mspace width="-0.166667em" /> <mspace width="-0.166667em" /> <msup> <mi>R</mi> <mn>3</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>-regular infinitesimal isometries. This result is proved via a careful approximation argument in Sobolev spaces. Finally, we establish the matching property: sufficiently smooth infinitesimal isometries can be matched with higher-order infinitesimal isometries. These findings serve as critical tools for constructing recovery sequences (via the <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2075_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-lim sup inequality) in the analysis of dimensionally-reduced shell theories within the framework of elasticity.</p>

Strain Tensors and Matching Property on Surfaces with the Gaussian Curvature Having Mixed Signs

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发表日期：

2025年7月7日

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Liang-Biao Chen,
Peng-Fei Yao

摘要

We establish the regularity of solutions to the strain tensor equation on a region S where the Gaussian curvature changes sign. Specifically, we demonstrate that solutions possess enhanced smoothness under this geometric condition. Furthermore, we derive the density property: smooth infinitesimal isometries are dense in the space of $W^{2,2}(S,I\!\!R^3)$ $W^{2, 2} (S, I R^{3})$ -regular infinitesimal isometries. This result is proved via a careful approximation argument in Sobolev spaces. Finally, we establish the matching property: sufficiently smooth infinitesimal isometries can be matched with higher-order infinitesimal isometries. These findings serve as critical tools for constructing recovery sequences (via the $\Gamma $ $Γ$ -lim sup inequality) in the analysis of dimensionally-reduced shell theories within the framework of elasticity.