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<p>We provide a structural analysis of the space of functions of bounded deviatoric deformation, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\textrm{BD}_{\textrm{dev}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>BD</mtext> <mtext>dev</mtext> </msub> </math></EquationSource> </InlineEquation>, which arises in models of plasticity and fluid mechanics. The main result is the identification of the annihilator and a rigidity theorem for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\textrm{BD}_{\textrm{dev}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>BD</mtext> <mtext>dev</mtext> </msub> </math></EquationSource> </InlineEquation>-maps with constant polar vector in the wave cone characterizing the structure of singularities for such maps. This result, together with an explicit kernel projection operator, enables an iterative blow-up procedure for relaxation, homogenization, and integral representation problems, allowing for integrands with explicit dependence on <i>u</i> as well as <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\({{\mathcal {E}}}_d u\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">E</mi> <mi>d</mi> </msub> <mi>u</mi> </mrow> </math></EquationSource> </InlineEquation>. Our approach overcomes several difficulties as compared to the <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\textrm{BD}\)</EquationSource> <EquationSource Format="MATHML"><math> <mtext>BD</mtext> </math></EquationSource> </InlineEquation> case, in particular due to the lack of invariance of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\({{\mathcal {E}}}_d\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">E</mi> <mi>d</mi> </msub> </math></EquationSource> </InlineEquation> under orthogonalization of the polar directions.</p>

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Rigidity and Functional Properties of \(\textrm{BD}_{\textrm{dev}}(\Omega )\)

  • 期刊论文

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  • 发表日期：

    2026年6月21日

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  • Marco Caroccia,
  • Nicolas Van Goethem

摘要

We provide a structural analysis of the space of functions of bounded deviatoric deformation, \(\textrm{BD}_{\textrm{dev}}\) BD dev , which arises in models of plasticity and fluid mechanics. The main result is the identification of the annihilator and a rigidity theorem for \(\textrm{BD}_{\textrm{dev}}\) BD dev -maps with constant polar vector in the wave cone characterizing the structure of singularities for such maps. This result, together with an explicit kernel projection operator, enables an iterative blow-up procedure for relaxation, homogenization, and integral representation problems, allowing for integrands with explicit dependence on u as well as \({{\mathcal {E}}}_d u\) E d u . Our approach overcomes several difficulties as compared to the \(\textrm{BD}\) BD case, in particular due to the lack of invariance of \({{\mathcal {E}}}_d\) E d under orthogonalization of the polar directions.