Local invertibility in Sobolev spaces with applications to nematic elastomers and magnetoelasticity
The authors define a class of deformations in \(W^{1,p}(\Omega,\mathbb{R}^n), p > n-1\), with a positive Jacobian, that do not exhibit cavitation, characterizing this class in terms of the non-negativity of the topological degree and the equality \(\mathrm{Det} = \det\) that the distributional determinant coincides with the pointwise determinant of the gradient. Condition \(\mathrm{Det }D\mathbf{u} = \det D\mathbf{u}\) can be expressed as \[ \det D\mathbf{u(x)} = \frac{1}{n} \mathrm{Div} [\mathrm{adj} D\mathbf{u(x)} \mathbf{u(x)}], \eqno{(1)} \] where \(\mathrm{Div}\) in the right-hand side denotes the distributional divergence. For smooth maps \(\mathbf{u}\), equality~(1) is a consequence of Piola's identity, and so is the following variant: \[ \mathrm{div} \mathbf{g(u(x))} \det D\mathbf{u(x)} = \mathrm{Div }[\mathrm{adj} D\mathbf{u(x) g(u(x))}] \eqno{(2)} \] for all \(\mathbf{g} \in C^1(\mathbb{R}^n,\mathbb{R}^n)\bigcap W^{1,\infty}(\mathbb{R}^n,\mathbb{R}^n)\). Let be the energy \[ \mathcal{E}(\mathbf{u}):= \sup _{\mathbf{f}\in C^1(\Omega\times \mathbb{R}^n,\mathbb{R}^n),{\|\mathbf{f}\|_{\infty}\leq 1}} \int_{\Omega} [\mathrm{cof} \nabla \mathbf{u(x)} \cdot D\mathbf{f(x, u(x))}+\det \nabla \mathbf{u(x)}\mathrm{div} \mathbf{f(x, u(x))}] d\mathbf{x},\eqno{(3)} \] measuring the new surface in the deformed configuration created by \(\mathbf{u}\) (for example, by the process of cavitation), and quantifies the failure of the equality~(2), where \( \mathcal{E}(\mathbf{u})= 0\) if and only if equality~(2) holds for all \(\mathbf{g}\in C_c^1(\mathbb{R}^n,\mathbb{R}^n)\). As a matter of fact, in our setting of \(W^{1,p}\) maps with \(p > n-1\), condition \( \mathcal{E}(\mathbf{u})= 0\) also corresponds to saying that the graph of \(\mathbf{u}\) is a current with no boundary in \(\Omega\times \mathbb{R}^n\) [\textit{S. Conti} and \textit{C. De Lellis}, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 2, No. 3, 521--549 (2003; Zbl 1114.74004); \textit{M. Giaquinta} et al., Cartesian currents in the calculus of variations I. Cartesian currents. Berlin: Springer (1998; Zbl 0914.49001); Cartesian currents in the calculus of variations II. Variational integrals. Berlin: Springer (1998; Zbl 0914.49002)]. One of the main results of this article is the Theorem~1 characterizing a class of functions that are orientation preserving and do not exhibit cavitation. As said before, the example of the second and the third author [Adv. Calc. Var. 5, No. 4, 355--409 (2012; Zbl 1252.49016)] shows that conditions \(\mathcal{E}(\mathbf{u})= 0\) and \(\mathrm{Det}\, D\mathbf{u} = \det D\mathbf{u}\) are not equivalent, nor does condition \(\det D\mathbf{u}>0\) a.e. imply \(\mathrm{deg}(\mathbf{u},U,\cdot)\geq 0.\) Nevertheless, when they are put together, and under some complementary assumptions, they are actually equivalent. Theorem 1. Let \(p > n-1\) and suppose that \(\mathbf{u}\in W^{1,p}(\Omega\times \mathbb{R}^n)\) satisfies \(\det D \mathbf{u}\in L^1_{\mathrm{loc}}(\Omega).\) The following conditions are equivalent: \((a)\) \( \mathcal{E}(\mathbf{u})= 0\) and \(\det D\mathbf{u} > 0\), a.e.; \((b)\) \((\mathrm{adj} D\mathbf{u}) \mathbf{u} \in L^1_{\mathrm{loc}}(\Omega,\mathbb{R}^n), \det D\mathbf{u(x)} = 0\) for a.e. \(\mathbf{x} \in \Omega, \mathrm{Det} D\mathbf{u} = \det D\mathbf{u}\), and \(\mathrm{deg}(\mathbf{u}, B, \cdot)\geq 0\) for all balls \(B\) for which \(\mathrm{deg}(\mathbf{u}, B, \cdot)\) is defined. Throughout this article, \(p > n-1\) is fixed (the borderline case \(p = n-1\) is yet to be explored, see [\textit{L. D'Onofrio} et al., J. Math. Anal. Appl. 439, No. 1, 171--182 (2016; Zbl 1351.26024)] for a partial result). Let be \(\mathcal{A}_p\) is the class of functions satisfying the conditions of the Theorem~1. The second main theorem of the article is a local invertibility result for functions in \(\mathcal{A}_p\): given \(\mathbf{u} \in \mathcal{A}_p\), for a.e. \(\mathbf{x} \in \Omega\) there exists \(r_{\mathbf{x}}> 0\) such that the ball \(B := B(\mathbf{x}, r_{\mathbf{x}})\) satisfies \[ \mathbf{u}|_B \;\;\mathrm{satisfies}\;\; \mathrm{INV},\; \mathbf{u}(B) = \mathrm{im}_T(\mathbf{u}, B)\; \mathrm{a.e. \;and} \;(\mathbf{u}|B)^{-1}\in W^{1,1}(\mathrm{im}_T(\mathbf{u}, B),\mathbb{R}^n). \eqno{(4)} \] The set \(\mathrm{im}_T(\mathbf{u}, B)\) is the topological image of \(B\) under the mapping \(\mathbf{u}\). The plan of the article is the following: In Section~2, notations are introduced and previous results used in the proofing of the theorems are recalled. In Section~3, also some results are recalled for the degree for \(W^{1,p}\) maps. In Section 4, Theorem 1 is proved. In Section 5, fine properties for functions in \(\mathcal{A}_p\) are explained, notably, weak monotonicity and its consequences, and the Sobolev regularity of the local inverse stated in (4). In Section 6, the stability of the property~(4) under weak convergence in \(W^{1,p}\) is proved. In Section~7, lower semicontinuity results are established for integrals of the form \(\int_{\mathbf{u}(\Omega)}W(D\mathbf{w(y)})d\mathbf{y}\) and \(\int_{\Omega}W_0(D\mathbf{u(x)},\mathbf{w(u(x))})d\mathbf{x},\) and also for \(\mathrm{Div}\)-quasiconvex integrands under incompressibility. In Section~8, the obtained results are applied to show the well-posedness of variational models for nematic elastomers and magnetoelasticity.
applications to nematic elastomers and magnetoelasticity
scientific article; zbMATH DE number 6710298
scientific article