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

TL;DR

This paper proves the existence of minimizers for the neo-Hookean energy without cavitation and shows that their inverses have Sobolev regularity, extending previous results to broader classes of deformations.

Contribution

It establishes the existence of minimizers satisfying divergence identities and demonstrates Sobolev regularity of their inverses under the INV condition, relaxing previous coercivity assumptions.

Findings

Minimizers exist in classes of maps without cavitation satisfying divergence identities.

Minimizers satisfying INV have inverses with Sobolev regularity.

Existence results extend to broader classes including weak closures of diffeomorphisms and homeomorphisms.

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 $\text{Det} = \det$. 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 N condition. Moreover, if the minimizers satisfy condition INV, then their inverses have Sobolev regularity. This extends a recent result by Dole\v{z}alov\'{a}, 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.