In this paper, we investigate the global higher regularity properties of weak solutions for a linear elliptic system coupled with a nonlinear Maxwell-type system defined on Lipschitz domains. The regularity result is established using a modified finite difference approach. These adjusted finite differences involve inner variations in conjunction with a Piola-type transformation to preserve the curl-structure within the matrix Maxwell system. The proposed method is further applied to the linear relaxed micromorphic model. As a result, for a physically nonlinear version of the relaxed micromorphic model, we demonstrate that for arbitrary <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2024_2915_Article_IEq1.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$\epsilon > 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϵ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, the displacement vector u belongs to <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2024_2915_Article_IEq2.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">$W^{\frac{3}{2}-\epsilon ,2}(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>W</mi> <mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> <mo>-</mo> <mi>ϵ</mi> <mo>,</mo> <mn>2</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, and the microdistortion tensor P belongs to <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2024_2915_Article_IEq3.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">$W^{\frac{1}{2}-\epsilon ,2}(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>W</mi> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>-</mo> <mi>ϵ</mi> <mo>,</mo> <mn>2</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> while <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2024_2915_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">${{\,\textrm{Curl}\,}}P$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mspace width="0.166667em" /> <mtext>Curl</mtext> <mspace width="0.166667em" /> </mrow> <mi>P</mi> </mrow> </math></EquationSource> </InlineEquation> belongs to <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2024_2915_Article_IEq5.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">$W^{\frac{1}{2}-\epsilon ,2}(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>W</mi> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>-</mo> <mi>ϵ</mi> <mo>,</mo> <mn>2</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.

Global regularity in a nonlinear relaxed micromorphic continuum on Lipschitz domains

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发表日期：

2025年1月9日

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Dorothee Knees,
Sebastian Owczarek,
Patrizio Neff

摘要

In this paper, we investigate the global higher regularity properties of weak solutions for a linear elliptic system coupled with a nonlinear Maxwell-type system defined on Lipschitz domains. The regularity result is established using a modified finite difference approach. These adjusted finite differences involve inner variations in conjunction with a Piola-type transformation to preserve the curl-structure within the matrix Maxwell system. The proposed method is further applied to the linear relaxed micromorphic model. As a result, for a physically nonlinear version of the relaxed micromorphic model, we demonstrate that for arbitrary $\epsilon > 0$ $ϵ > 0$ , the displacement vector u belongs to $W^{\frac{3}{2}-\epsilon ,2}(\Omega )$ $W^{\frac{3}{2} - ϵ, 2} (Ω)$ , and the microdistortion tensor P belongs to $W^{\frac{1}{2}-\epsilon ,2}(\Omega )$ $W^{\frac{1}{2} - ϵ, 2} (Ω)$ while ${{\,\textrm{Curl}\,}}P$ $Curl P$ belongs to $W^{\frac{1}{2}-\epsilon ,2}(\Omega )$ $W^{\frac{1}{2} - ϵ, 2} (Ω)$ .