A minimization problem involving variation of the domain

Dacorogna, Bernard; Fonseca, Irene

doi:10.1002/cpa.3160450704

Cited by 11 publications

(10 citation statements)

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“…The essence of that difficulty is the lack of compactness. Indeed, while bounds on I imply bounds on Dw under certain coercivity conditions on W , in no way do they imply bounds on Du (see the discussion in [18,32]). We will see in Section 8 that the necessary compactness will be achieved by adding a suitable term to the energy, which, as a matter of fact, will be analyzed in Subsection 7.…”

Section: Quasiconvex Functionals In the Deformed Configurationmentioning

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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Section: Quasiconvex Functionals In the Deformed Configurationmentioning

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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“…where Ω ⊂ R 3 , u ∈ W 1,p (Ω, R 3 ) for some p > 3, n ∈ H 1 (u(Ω), R 2 ), and W mec (F, n) = W (α −1 n ⊗ n + √ α(I − n ⊗ n))F (1.2) for a certain α > 0 and some polyconvex energy function W . Functionals with a similar structure appear also in models describing the nematic mesogens with the Landau-de Gennes theory, and in magnetoelasticity and plasticity, see, e.g., [6,12,18,28,5]. The major difficulties are that I depends on the composition of the two unknowns and that the nematic director n is defined in the domain u(Ω) which is also determined only as a part of the solution of the variational problem.…”

Section: Introductionmentioning

confidence: 99%

Orlicz–Sobolev nematic elastomers

Henao

Stroffolini

2020

Nonlinear Analysis

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We extend the existence theorems in [Barchiesi, ARMA 224], for models of nematic elastomers and magnetoelasticity, to a larger class in the scale of Orlicz spaces. These models consider both an elastic term where a polyconvex energy density is composed with an unknown state variable defined in the deformed configuration, and a functional corresponding to the nematic energy (or the exchange and magnetostatic energies in magnetoelasticity) where the energy density is integrated over the deformed configuration. In order to obtain the desired compactness and lower semicontinuity, we show that the regularity requirement that maps create no new surface can still be imposed when the gradients are in an Orlicz class with an integrability just above the space dimension minus one. We prove that the fine properties of orientation-preserving maps satisfying that regularity requirement (namely, being weakly 1-pseudomonotone, H 1 -continuous, a.e. differentiable, and a.e. locally invertible) are still valid in the Orlicz-Sobolev setting.

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“…Thus, we consider the class of admissible pairs In DAVINI & PARRY [10] various properties of smooth minimizers were formally derived and convexity was assumed tacitly. Also, in DACOROGNA & FONSECA [7] existence and smoothness of minimizers for functionals of the type (1.2) were discussed. Here, existence of minimizers is not the issue.…”

Section: Introductionmentioning

confidence: 99%