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

TL;DR

This work analyzes the existence and regularity of minimizers for the neo-Hookean energy $E(u)=\int_{\Omega} |Du|^{n-1}+H(\det Du)\,dx$ in cavitation-free deformation classes, enforcing the INV condition to prevent interpenetration of matter. The authors prove a fundamental equivalence between the divergence identities, INV, and invertibility (i.e., $\operatorname{Det}Du=(\det Du)\mathcal{L}^n$ and $u^{-1}\in W^{1,1}$ when appropriate), and establish lower semicontinuity of $E$ on the weak closure of admissible classes, yielding existence of minimizers in the weak closures of diffeomorphism and homeomorphism classes. These minimizers satisfy INV and the determinant identity, and their inverses have Sobolev regularity, extending prior results by showing extra regularity even when the coercivity assumption is relaxed. The results rely on a framework of divergence identities, the geometric image, and degree theory to control invertibility under weak convergence, with significant implications for non-interpenetration and cavitation in nonlinear elasticity. The paper broadens the scope of existence results to more general boundary data and relaxations of coercivity, contributing to the mathematical understanding of physically admissible deformations in elasticity.

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 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.

Theorems & Definitions (14)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Theorem 2.3
• proof