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Sobolev regularity of the inverse for minimizers of the neo-Hookean energy satisfying condition INV

Part of: Manifolds Existence theories Elastic materials Equilibrium (steady-state) problems

Published online by Cambridge University Press:  11 April 2025

Panas Kalayanamit Open the ORCID record for Panas Kalayanamit [Opens in a new window]
Panas Kalayanamit*
Affiliation:
Instituto de Ciencias de la Ingeniería, Universidad de O’Higgins, Avenida Libertador Bernardo O’Higgins, Rancagua, Chile ([email protected])
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Abstract

We study the existence and regularity of minimizers of the neo-Hookean energy in the closure of classes of deformations without cavitation. The exclusion of cavitation is imposed in the form of the divergence identities, which is equivalent to the well-known condition (INV) with $\operatorname{Det} = \operatorname{det}$. We show that the neo-Hookean energy admits minimizers in classes of maps that are one-to-one a.e. with positive Jacobians, provided that these maps are the weak limits of sequences of maps that satisfy the divergence identities. In particular, these classes include the weak closure of diffeomorphisms and the weak closure of homeomorphisms satisfying Lusin’s condition N. Moreover, if the minimizers satisfy condition (INV), then their inverses have Sobolev regularity. This extends a recent result by Doležalová, Hencl, and Molchanova by showing that the minimizers they obtained enjoy extra regularity properties and that the existence of minimizers can still be obtained even when their coercivity assumption is relaxed.

Keywords

Cavitationcondition INVdivergence identitieselastic deformationneo-Hookeanregular minimizersobolev regularity

MSC classification

Primary: 74B20: Nonlinear elasticity
Secondary: 49J45: Methods involving semicontinuity and convergence; relaxation 49Q15: Geometric measure and integration theory, integral and normal currents 74G65: Energy minimization
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 21
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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