Local invertibility in Sobolev spaces with applications to nematic elastomers and magnetoelasticity
type: Item
instance of: scholarly article
title: Local invertibility in Sobolev spaces with applications to nematic elastomers and magnetoelasticity
zbMATH Open document ID: 1371.35284
author: M. Barchiesi
author: Duvan Henao
author: Carlos Mora-Corral
published in: Archive for Rational Mechanics and Analysis
publication date: 2017-05-02T00:00:00Z
full work available at URL: http://hdl.handle.net/11368/2955323
review text: The authors 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, characterizing this class in terms of the non-negativity of the topological degree and the equality \(\mathrm{Det} = \det\) that the distributional determinant coincides with the pointwise determinant of the gradient. Condition \(\mathrm{Det }D\mathbf{u} = \det D\mathbf{u}\) can be expressed as \[ \det D\mathbf{u(x)} = \frac{1}{n} \mathrm{Div} [\mathrm{adj} D\mathbf{u(x)} \mathbf{u(x)}], \eqno{(1)} \] where \(\mathrm{Div}\) in the right-hand side denotes the distributional divergence. For smooth maps \(\mathbf{u}\), equality~(1) is a consequence of Piola's identity, and so is the following variant: \[ \mathrm{div} \mathbf{g(u(x))} \det D\mathbf{u(x)} = \mathrm{Div }[\mathrm{adj} D\mathbf{u(x) g(u(x))}] \eqno{(2)} \] for all \(\mathbf{g} \in C^1(\mathbb{R}^n,\mathbb{R}^n)\bigcap W^{1,\infty}(\mathbb{R}^n,\mathbb{R}^n)\). Let be the energy \[ \mathcal{E}(\mathbf{u}):= \sup _{\mathbf{f}\in C^1(\Omega\times \mathbb{R}^n,\mathbb{R}^n),{\|\mathbf{f}\|_{\infty}\leq 1}} \int_{\Omega} [\mathrm{cof} \nabla \mathbf{u(x)} \cdot D\mathbf{f(x, u(x))}+\det \nabla \mathbf{u(x)}\mathrm{div} \mathbf{f(x, u(x))}] d\mathbf{x},\eqno{(3)} \] measuring the new surface in the deformed configuration created by \(\mathbf{u}\) (for example, by the process of cavitation), and quantifies the failure of the equality~(2), where \( \mathcal{E}(\mathbf{u})= 0\) if and only if equality~(2) holds for all \(\mathbf{g}\in C_c^1(\mathbb{R}^n,\mathbb{R}^n)\). As a matter of fact, in our setting of \(W^{1,p}\) maps with \(p > n-1\), condition \( \mathcal{E}(\mathbf{u})= 0\) also corresponds to saying that the graph of \(\mathbf{u}\) is a current with no boundary in \(\Omega\times \mathbb{R}^n\) [\textit{S. Conti} and \textit{C. De Lellis}, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 2, No. 3, 521--549 (2003; Zbl 1114.74004); \textit{M. Giaquinta} et al., Cartesian…
Mathematics Subject Classification ID: 35Q74
Mathematics Subject Classification ID: 74B20
Mathematics Subject Classification ID: 74F15
Mathematics Subject Classification ID: 35R05
zbMATH DE Number: 6710298
zbMATH Keywords: Sobolev spaces
zbMATH Keywords: degree for Sobolev maps
zbMATH Keywords: applications to nematic elastomers and magnetoelasticity
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
OpenAlex ID: W2587210832
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Local invertibility in Sobolev spaces with applications to nematic elastomers and magnetoelasticity (DOI)
DOI: 10.1007/S00205-017-1088-1