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Archive for Rational Mechanics and Analysis

, Volume 224, Issue 2, pp 743–816 | Cite as

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

  • Marco Barchiesi
  • Duvan Henao
  • Carlos Mora-Corral
Article

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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© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Dipartimento di Matematica ed ApplicazioniUniversità degli Studi di Napoli “Federico II”NapoliItaly
  2. 2.Faculty of MathematicsPontificia Universidad Católica de ChileMaculChile
  3. 3.Faculty of SciencesUniversidad Autónoma de MadridMadridSpain

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