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Invertibility of Sobolev maps through approximate invertibility at the boundary and tangential polyconvexity

Carlos Mora-Corral, David Mur-Callizo

TL;DR

The paper addresses invertibility of Sobolev deformations in nonlinear elasticity when $u\in W^{1,p}$ with $p> d-1$, positive Jacobian, and boundary data that are approximately invertible. It develops a variational framework with interior polyconvex energy and a boundary energy that is tangentially polyconvex, proving injectivity a.e. and existence of minimizers for a broad class of boundary conditions. A new notion of tangential polyconvexity is introduced, linked to an extension polyconvexity concept, and complemented by a weak continuity result for tangential minors to secure lower semicontinuity of the boundary term. The work also provides a counterexample showing the necessity of boundary regularity, clarifies the relationship with interface polyconvexity, and yields a robust existence theory for nonlinear elasticity problems with non-Dirichlet boundary data. Overall, this advances the mathematical foundation for preventing interpenetration and obtaining minimizers in elasticity models with surface interactions and approximate boundary invertibility.

Abstract

We work in a class of Sobolev $W^{1,p}$ maps, with $p > d-1$, from a bounded open set $Ω\subset \mathbb{R}^{d}$ to $\mathbb{R}^{d}$ that do not exhibit cavitation and whose trace on $\partial Ω$ is also $W^{1,p}$. Under the assumptions that the Jacobian is positive and the deformation can be approximated on the boundary by injective maps, we show that the deformation is injective. We prove the existence of minimizers in this class for functionals accounting for a nonlinear elastic energy and a boundary energy. The energy density in $Ω$ is assumed to be polyconvex, while the energy density in $\partial Ω$ is assumed to be tangentially polyconvex, a new type of polyconvexity on $\partial Ω$.

Invertibility of Sobolev maps through approximate invertibility at the boundary and tangential polyconvexity

TL;DR

The paper addresses invertibility of Sobolev deformations in nonlinear elasticity when with , positive Jacobian, and boundary data that are approximately invertible. It develops a variational framework with interior polyconvex energy and a boundary energy that is tangentially polyconvex, proving injectivity a.e. and existence of minimizers for a broad class of boundary conditions. A new notion of tangential polyconvexity is introduced, linked to an extension polyconvexity concept, and complemented by a weak continuity result for tangential minors to secure lower semicontinuity of the boundary term. The work also provides a counterexample showing the necessity of boundary regularity, clarifies the relationship with interface polyconvexity, and yields a robust existence theory for nonlinear elasticity problems with non-Dirichlet boundary data. Overall, this advances the mathematical foundation for preventing interpenetration and obtaining minimizers in elasticity models with surface interactions and approximate boundary invertibility.

Abstract

We work in a class of Sobolev maps, with , from a bounded open set to that do not exhibit cavitation and whose trace on is also . Under the assumptions that the Jacobian is positive and the deformation can be approximated on the boundary by injective maps, we show that the deformation is injective. We prove the existence of minimizers in this class for functionals accounting for a nonlinear elastic energy and a boundary energy. The energy density in is assumed to be polyconvex, while the energy density in is assumed to be tangentially polyconvex, a new type of polyconvexity on .

Paper Structure

This paper contains 12 sections, 27 theorems, 75 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Class $\overline{\mathcal{A}}_{p}\left(\Omega\right)$
  4. Injectivity of maps in $\overline{\mathcal{A}}_{p} \cap \mathop{\mathrm{AIB}}\nolimits$
  5. Counterexample to global injectivity in $\mathcal{A}_{p} \cap \mathop{\mathrm{AIB}}\nolimits$
  6. Weak continuity of minors of tangential derivatives
  7. Minors of linear maps
  8. Convergence of minors of tangential derivatives
  9. Tangential polyconvexity and quasiconvexity
  10. Interface polyconvexity
  11. Tangential polyconvexity in surface potentials
  12. Existence of minimizers

Key Result

Lemma 2.2

Let $\Omega$ be a Lipschitz domain. Let $p>d-1$, let $u\in W^{1,p}\left(\partial\Omega;\mathbb{R}^{d}\right)$ and let $\left\{u_{k}\right\}_{k\in\mathbb{N}}\subset W^{1,p}(\partial\Omega;\mathbb{R}^{d}) \cap \mathop{\mathrm{AIB}}\nolimits$ be such that $u_{k}\rightharpoonup u$ in $W^{1,p}(\partial\O

Figures (1)

  • Figure 1: Counterexample to a.e. injectivity in $\mathcal{A}_{p}((-1,1)\times(0,1))\cap\mathop{\mathrm{AIB}}\nolimits$.

Theorems & Definitions (60)

  • Definition 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Theorem 2.7
  • Definition 3.1
  • Proposition 3.2
  • ...and 50 more