Invertibility of Orlicz-Sobolev maps
In this very interesting article, the authors made contributions to an important problem in nonlinear elasticity: the global invertibility of deformations. Let \(\Omega\) be a bounded open set in \(\mathbb{R}^n\) and let \(\mathbf{u}: \Omega \to \mathbb{R}^n\) be the deformation of an elastic body whose reference configuration is \(\Omega\). The global invertibility of \(\mathbf{u}\) is related to the interpenetration of matter, and is relevant to the minimisation of elastic energies. A classical work by \textit{J. M. Ball} [Proc. R. Soc. Edinb., Sect. A, Math. 88, 315--328 (1981; Zbl 0478.46032)] shows that if \(\mathbf{u}\) is \(W^{1,p}\) for \(p>n\), \(\det\,D\mathbf{u}>0\), and the trace of \(\mathbf{u}\) on \(\partial\Omega\) is an invertible map, then \(\mathbf{u}\) is globally invertible. On the other hand, things are more complicated when \(n-1 \leq p \leq n\) and are pertaining to the formation of cavitation in elastic bodies. For example, ``injectivity + \(\det D\mathbf{u}>0\)'' is not preserved under weak limits. In the case \(p>n-1\), one may recover \(\det\,D\mathbf{u}>0\) under suitable geometric conditions on \(\mathbf{ u}\) formulated in terms of topological degree. See [\textit{S. Müller} and \textit{S. J. Spector}, Arch. Ration. Mech. Anal. 131, No. 1, 1--66 (1995; Zbl 0836.73025)]. Motivated by a recent work [\textit{D. Henao} et al., Adv. Calc. Var. 14, No. 2, 207--230 (2021; Zbl 1475.74011)], the authors extend the global invertibility result from the Sobolev scale to the Orlicz scale of maps. More precisely, they establish global invertibility among a new admissible class of deformations, which consists of orientation-preserving Orlicz-Sobolev maps with an integrability just above \(n-1\), whose traces on the boundary are also Orlicz-Sobolev which, in connection with [Müller and Spector, loc. cit.], do not present cavitation either in the interior or on the boundary. For the entire collection see [Zbl 1495.74001].
orientation-preserving Orlicz-Sobolev map
scientific article; zbMATH DE number 7598424
scientific article