Explicit examples of extremal quasiconvex quadratic forms that are not polyconvex

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• 作者：Davit Harutyunyan ; Graeme Walter Milton
• 关键词：26B25 ; 35A23 ; 49J40 ; 74B05
• 刊名：Calculus of Variations and Partial Differential Equations
• 出版年：2015
• 出版时间：October 2015
• 年：2015
• 卷：54
• 期：2
• 页码：1575-1589
• 全文大小：464 KB
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  7.Kang, H., Kim, E., Milton, G.W.: Sharp bounds on the volume fractions of two materials in a two-dimensional body from electrical boundary measurements: the translation method. Calc. Var. Partial Differ. Equ. 45, 367鈥?01 (2012)MathSciNet CrossRef MATH
  8.Kang, H., Milton, G.W.: Bounds on the volume fractions of two materials in a three dimensional body from boundary measurements by the translation method. SIAM J. Appl. Math. 73, 475鈥?92 (2013)MathSciNet CrossRef MATH
  9.Kang, H., Milton, G.W., Wang, J.-N.: Bounds on the volume fraction of the two-phase shallow shell using one measurement. J. Elast. 114, 41鈥?3 (2014)MathSciNet CrossRef MATH
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  11.Kohn, R.V., Lipton, R.: Optimal bounds for the effective energy of a mixture of isotropic, incompressible, elastic materials. Arch. Ration. Mech. Anal. 102, 331鈥?50 (1988)MathSciNet CrossRef MATH
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  25.Serre, D.: Condition de Legendre-Hadamard: Espaces de matrices de rang\(\ne 1\) . (French) [Legendre-Hadamard condition: Space of matrices of rank\(\ne 1\) ]. Comptes rendus de l鈥橝cad茅mie des sciences, Paris 293, 23鈥?6 (1981)
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  28.Van Hove, L.: Sur le signe de la variation seconde des int茅grales multiples 谩 plusieurs functions inconnues. Acad. Roy. Belgique Cl. Sci. M茅m. Coll. 24, 68 (1949)
  
• 作者单位：Davit Harutyunyan (1)
  Graeme Walter Milton (1)
  
  1. Department of Mathematics, The University of Utah, Salt Lake City, USA
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Systems Theory and Control
  Calculus of Variations and Optimal Control
  Mathematical and Computational Physics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0835

文摘

We prove that if the associated fourth order tensor of a quadratic form has a linear elastic cubic symmetry then it is quasiconvex if and only if it is polyconvex, i.e. a sum of convex and null-Lagrangian quadratic forms. We prove that allowing for slightly less symmetry, namely only cyclic and axis-reflection symmetry, gives rise to a class of extremal quasiconvex quadratic forms, that are not polyconvex. Non-affine boundary conditions on the potential are identified which allow one to obtain sharp bounds on the integrals of these extremal quasiconvex quadratic forms of \(\nabla u\) over an arbitrary region. Mathematics Subject Classification 26B25 35A23 49J40 74B05