Local Invertibility in Sobolev Spaces with Applications to Nematic Elastomers and Magnetoelasticity

Abstract

We define a class of deformations in W 1,p (Ω, R n), p > n−1, with positive Jacobian that do not exhibit cavitation. We characterize that class in terms of the non-negativity of the topological degree and the equality Det = det (that the distributional determinant coincides with the pointwise determinant of the gradient). Maps in this class are shown to satisfy a property of weak monotonicity, and, as a consequence, they enjoy an extra degree of regularity. We also prove that these deformations are locally invertible; moreover, the neighbourhood of invertibility is stable along a weak convergent sequence in W 1,p , and the sequence of local inverses converges to the local inverse. We use those features to show weak lower semicontinuity of functionals defined in the deformed configuration and functionals involving composition of maps. We apply those results to prove existence of minimizers in some models for nematic elastomers and magnetoelasticity.

Orlicz–Sobolev nematic elastomers

“…such t there exists a subsequence for which (cof Du j )ν t ⇀ (cof Du)ν in L 1 (∂U t , R n ), where ν t is the unit exterior normal to U t . That K ⊂ im T (u j , U t ) ⊂ im T (u j , Ω) then follows by Lemma 2.11 and the homotopy-invariance of the degree (as in [5,Lemma 3.6]).…”

“…From this point onwards the a.e. differentiability can be obtained exactly as in the proof of [5,Prop. 5.9].…”

“…It is closely related to the well-known equation Det Du = det Du, satisfied by all W 1,n maps. In fact, for maps in W 1,p with p > n − 1 it was proved in [5,Corollary 4.7] that det Du > 0 a.e. and E(u) = 0 if and only if det Du(x) = 0 for a.e.…”

“…where Ω ⊂ R 3 , u ∈ W 1,p (Ω, R 3 ) for some p > 3, n ∈ H 1 (u(Ω), R 2 ), and W mec (F, n) = W (α −1 n ⊗ n + √ α(I − n ⊗ n))F (1.2) for a certain α > 0 and some polyconvex energy function W . Functionals with a similar structure appear also in models describing the nematic mesogens with the Landau-de Gennes theory, and in magnetoelasticity and plasticity, see, e.g., [6,12,18,28,5]. The major difficulties are that I depends on the composition of the two unknowns and that the nematic director n is defined in the domain u(Ω) which is also determined only as a part of the solution of the variational problem.…”

“…This article explains the new ideas and the results in the literature of Orlicz-Sobolev spaces that are required to generalize the analysis of [5] (full detail of the proofs is not given since that would render the article unnecessarily long, given the technical difficulties). Section 2 is for notation and preliminaries.…”