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

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Abstract

We define a class of deformations in \({W^{1,p}(\Omega,\mathbb{R}^n)}\), \({p > n-1}\), with a 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 the existence of minimizers in some models for nematic elastomers and magnetoelasticity.

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Authors and Affiliations

  1. Dipartimento di Matematica ed Applicazioni, Università degli Studi di Napoli “Federico II”, Via Cintia, 80126, Napoli, Italy

    Marco Barchiesi

  2. Faculty of Mathematics, Pontificia Universidad Católica de Chile, Vicuña Mackenna, 4860, Macul, Santiago, Chile

    Duvan Henao

  3. Faculty of Sciences, Universidad Autónoma de Madrid, C/ Tomás y Valiente 7, E-28049, Madrid, Spain

    Carlos Mora-Corral

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  1. Marco Barchiesi
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  2. Duvan Henao
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  3. Carlos Mora-Corral
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Correspondence to Carlos Mora-Corral.

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Communicated by I. Fonseca

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Barchiesi, M., Henao, D. & Mora-Corral, C. Local Invertibility in Sobolev Spaces with Applications to Nematic Elastomers and Magnetoelasticity. Arch Rational Mech Anal 224, 743–816 (2017). https://doi.org/10.1007/s00205-017-1088-1

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