Rank-(n – 1) convexity and quasiconvexity for divergence free fields

Palombaro, Mariapia

doi:10.1515/acv.2010.010

Cited by 6 publications

(4 citation statements)

References 10 publications

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“…We refer to Fonseca & Müller [34] for the theory of A-quasiconvexity, of which Div-quasiconvexity is a particular case, and to [3,66,65] for further results on Div-quasiconvexity.…”

Section: Div-quasiconvex Functionals Under Incompressibilitymentioning

confidence: 99%

Local Invertibility in Sobolev Spaces with Applications to Nematic Elastomers and Magnetoelasticity

Barchiesi

Henao

Mora-Corral

2017

Arch Rational Mech Anal

127

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We define a class of deformations in W 1,p (Ω, R n), p > n−1, with positive Jacobian that do not exhibit cavitation. We characterize that class in terms of the non-negativity of the topological degree and the equality Det = det (that the distributional determinant coincides with the pointwise determinant of the gradient). Maps in this class are shown to satisfy a property of weak monotonicity, and, as a consequence, they enjoy an extra degree of regularity. We also prove that these deformations are locally invertible; moreover, the neighbourhood of invertibility is stable along a weak convergent sequence in W 1,p , and the sequence of local inverses converges to the local inverse. We use those features to show weak lower semicontinuity of functionals defined in the deformed configuration and functionals involving composition of maps. We apply those results to prove existence of minimizers in some models for nematic elastomers and magnetoelasticity.

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“…We refer to Fonseca & Müller [34] for the theory of A-quasiconvexity, of which Div-quasiconvexity is a particular case, and to [3,66,65] for further results on Div-quasiconvexity.…”

Section: Div-quasiconvex Functionals Under Incompressibilitymentioning

confidence: 99%

Local Invertibility in Sobolev Spaces with Applications to Nematic Elastomers and Magnetoelasticity

Barchiesi

Henao

Mora-Corral

2017

Arch Rational Mech Anal

127

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“…Here, C ∞ per ((0, 1) n ; M n×n sym ) is the space of all M n×n sym -valued functions that are smooth and periodic on the unit torus, i.e., (0, 1) n with opposite edges identified. It is well known that (symmetric) div-quasiconvexity implies Λ divconvexity (see [16,Lemma 2.4] and also [25,Proposition 3.4]) but that the converse implication is false in general (see [43]). However, both (symmetric) div-quasiconvexity and Λ div -convexity are equivalent for quadratic forms.…”

Section: Compensated Compactnessmentioning

confidence: 99%

Shape optimization of light structures and the vanishing mass conjecture

2023

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This work proves rigorous results about the vanishing-mass limit of the classical problem to find a shape with minimal elastic compliance. Contrary to all previous results in the mathematical literature, which utilize a soft mass constraint by introducing a Lagrange multiplier, we here consider the hard mass constraint. Our results are the first to establish the convergence of approximately optimal shapes of (exact) size ε ↓ 0 to a limit generalized shape represented by a (possibly diffuse) probability measure. This limit generalized shape is a minimizer of the limit compliance, which involves a new integrand, namely the one conjectured by Bouchitté in 2001 and predicted heuristically before in works of Allaire & Kohn and Kohn & Strang from the 1980s and 1990s. This integrand gives the energy of the limit generalized shape understood as a fine oscillation of (optimal) lower-dimensional structures. Its appearance is surprising since the integrand in the original compliance is just a quadratic form and the non-convexity of the problem is not immediately obvious. In fact, it is the interaction of the mass constraint with the requirement of attaining the loading (in the form of a divergence-constraint) that gives rise to this new integrand. We also present connections to the theory of Michell trusses, first formulated in 1904, and show how our results can be interpreted as a rigorous justification of that theory on the level of functionals in both two and three dimensions, settling this open problem. Our proofs rest on compensated compactness arguments applied to an explicit family of (symmetric) div-quasiconvex quadratic forms, computations involving the Hashin-Shtrikman bounds for the Kohn-Strang integrand, and the characterization of limit minimizers due to Bouchitté & Buttazzo.

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“…Here, C ∞ per ((0, 1) n ; M n×n sym ) is the space of all M n×n sym -valued functions that are smooth and periodic on the unit torus, i.e., (0, 1) n with opposite edges identified. It is well known that (symmetric) div-quasiconvexity implies Λ div -convexity (see [15,Lemma 2.4] and also [23,Proposition 3.4]) but that the converse implication is false in general (see [41]). However, both (symmetric) div-quasiconvexity and Λ div -convexity are equivalent for quadratic forms.…”

Section: Compensated Compactnessmentioning

confidence: 99%

Shape optimization of light structures and the vanishing mass conjecture

Babadjian,

Iurlano,

Rindler

2021

Preprint

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This work proves rigorous results about the vanishing-mass limit of the classical problem to find a shape with minimal elastic compliance. Contrary to all previous results in the mathematical literature, which utilize a soft mass constraint by introducing a Lagrange multiplier, we here consider the hard mass constraint. Our results are the first to establish the convergence of approximately optimal shapes of (exact) size ε ↓ 0 to a limit generalized shape represented by a (possibly diffuse) probability measure. This limit generalized shape is a minimizer of the limit compliance, which involves a new integrand, namely the one conjectured by Bouchitté in 2001 and predicted heuristically before in works of Allaire & Kohn and Kohn & Strang from the 1980s and 1990s. This integrand gives the energy of the limit generalized shape understood as a fine oscillation of (optimal) lower-dimensional structures. Its appearance is surprising since the integrand in the original compliance is just a quadratic form and the non-convexity of the problem is not immediately obvious. In fact, it is the interaction of the mass constraint with the requirement of attaining the loading (in the form of a divergence-constraint) that gives rise to this new integrand. We also present connections to the theory of Michell trusses, first formulated in 1904, and show how our results can be interpreted as a rigorous justification of that theory on the level of functionals in both two and three dimensions, finally settling this long-standing open problem. Our proofs rest on compensated compactness arguments applied to an explicit family of (symmetric) div-quasiconvex quadratic forms, computations involving the Hashin-Shtrikman bounds for the Kohn-Strang integrand, and the characterization of limit minimizers due to Bouchitté & Buttazzo.

show abstract

Rank-(n – 1) convexity and quasiconvexity for divergence free fields

Cited by 6 publications

References 10 publications

Local Invertibility in Sobolev Spaces with Applications to Nematic Elastomers and Magnetoelasticity

Local Invertibility in Sobolev Spaces with Applications to Nematic Elastomers and Magnetoelasticity

Shape optimization of light structures and the vanishing mass conjecture

Shape optimization of light structures and the vanishing mass conjecture

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