Equivalence between rank-one convexity and polyconvexity for isotropic sets of \({\mathbb R}^{2{\times}2}\). II
type: Item
instance of: scholarly article
title: Equivalence between rank-one convexity and polyconvexity for isotropic sets of \({\mathbb R}^{2{\times}2}\). II
zbMATH Open document ID: 1004.49008
DOI: 10.1016/S0362-546X(01)00808-2
author: Pierre Cardaliaguet
author: Rabah Tahraoui
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2002-08-25T00:00:00Z
review text: As in the Part I of the paper [Nonlinear Analysis, Theory Methods Appl. 50A, No. 8, 1179-1199 (2002; Zbl 1004.49007, preceding review)], the authors consider closed subsets \(K\) of \(2\times 2\) matrices, classes of functions \[ {\mathcal E}(K)= \{u\in W^{1,\infty}(\Omega; \mathbb{R}^2):\nabla u(x)\in K\text{ for a.e. }x\in\Omega\} \] and deal with the problem of characterizing all sets \(K\) for which \({\mathcal E}(K)\) is weakly\(^*\) closed. This leads to study the notions of polyconvexity, quasiconvexity, rank-one convexity for subsets \(K\) of \(2\times 2\) matrices. The sets \(K\) considered in the Part I of the paper were of the form \[ K(\Lambda)= \{A\in \mathbb{R}^{2\times 2}: (\lambda_1(A), \lambda_2(A))\in \Lambda\} \] being \(\lambda_1(A)\) and \(\lambda_2(A)\) the singular values of the matrix \(A\) (i.e., the eigenvalues of \((AA^*)^{1/2}\)) and where \(\Lambda\subset E= \{(x,y)\in \mathbb{R}^2: 0\leq x\leq y\}\). In the present Part II they add the constraint \(\text{det }A\geq 0\), which gives the sets \[ K_+(\Lambda)= \{A\in K(\Lambda): \text{det }A\geq 0\}. \] The main result (Theorem 7.1) is that when \(\Lambda\subset E\) is compact and connected, then \(K_+(\Lambda)\) is polyconvex if and only if it is rank-one convex. Though formally similar to the results of Part I (except for the connectedness assumption that was not required in the unconstrained case), this constrained case seems more difficult and several new tools are developed in the paper.
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 74G65
Mathematics Subject Classification ID: 74B20
zbMATH DE Number: 1788065
zbMATH Keywords: polyconvexity
zbMATH Keywords: quasiconvexity
zbMATH Keywords: rank-one convexity
zbMATH Keywords: singular values
reviewed by: Giuseppe Buttazzo
MaRDI profile type: Publication
cites work: An example of a quasiconvex function that is not polyconvex in two dimensions
cites work: Existence of minimizers for non-quasiconvex functionals arising in optimal design
cites work: Quelques remarques sur les notions de $1-$rang convexité et de polyconvexité en dimensions 2 et 3
cites work: Conditions necessaires de faible fermeture et de 1-rang convexity en dimension 3. (Necessary conditions for weak closure and rank-one convexity in three-dimensional elasticity)
cites work: Q3872840
cites work: Sur la faible fermeture de certains ensembles de contraintes en élasticité non-linéaire plane. (Weak closure of certain sets of constraints in plane nonlinear elasticity)
cites work: Differentiability properties of symmetric and isotropic functions
cites work: Fine phase mixtures as minimizers of energy
cites work: Proposed experimental tests of a theory of fine microstructure and the two-well problem
cites work: Restrictions on microstructure
cites work: Q4296480
cites work: Equivalence between rank-one convexity and polyconvexity for isotropic sets of \({\mathbb R}^{2{\times}2}\). I
cites work: Sur l'équivalence de la 1-rang convexité et de la polyconvexité des ensembles isotropiques de
cites work: Functions with prescribed singular values of the gradient
cites work: Q4856861
cites work: Equilibrium configurations of crystals
cites work: Q4039741
cites work: Q3782314
cites work: Implicit partial differential equations
cites work: On the different convex hulls of sets involving singular values
cites work: Microstructures with finite surface energy: The two-well problem
cites work: Some phase transitions in crystals
cites work: Constitutive theory for some constrained elastic crystals
cites work: Constitutive inequalities for isotropic elastic solids under finite strain
cites work: On the failure of ellipticity of the equations for finite elastostatic plane strain
cites work: Optimal design and relaxation of variational problems, III
cites work: On the non-locality of quasiconvexity
cites work: Q4289223
cites work: Q5690371
cites work: Q4260840
cites work: ELASTIC MATERIALS WITH TWO PREFERRED STATES
cites work: Q4502693
cites work: Rank-one convexity does not imply quasiconvexity
cites work: Q4295485
cites work: Sur une Classe de Fonctionnelles non Convexes et Applications
cites work: Q3878033
cites work: Q4295486
cites work: On various semiconvex hulls in the calculus of variations
cites work: On the structure of quasiconvex hulls
full work available at URL: s0362-546x(01)00808-2
OpenAlex ID: W3029250936
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description: scientific article; zbMATH DE number 1788065