Calculus of Variations and Geometric Measure Theory
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M. Barchiesi - D. Henao - C. Mora-Corral

Local invertibility in Sobolev spaces with applications to nematic elastomers and magnetoelasticity

created by barchiesi on 01 Aug 2015
modified on 02 Apr 2017


Published Paper

Inserted: 1 aug 2015
Last Updated: 2 apr 2017

Journal: Arch. Ration. Mech. Anal.
Volume: 224
Number: 2
Pages: 743--816
Year: 2017
Doi: 10.1007/s00205-017-1088-1
Links: official version


We define a class of deformations in $W^{1,p}(\Omega,\mathbb{R}^n)$, $p>n-1$, 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 between the distributional determinant and 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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