By Dieudonne J.
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Then Θ is an isometry of Rd . Parts (ii) and (iii) of the above proof thus provide a proof of this theorem under the additional assumption that the mapping Θ is of class C 1 (the extension from R3 to Rd is trivial). 7-1, the special case where Θ is the identity mapping of R3 identiﬁed with E3 is the classical Liouville theorem. This theorem thus asserts that if a mapping Θ ∈ C 1 (Ω; E3 ) is such that ∇Θ(x) ∈ O3 for all x ∈ Ω, where Ω is an open connected subset of R3 , then Θ is an isometry. , such that Θ = J ◦ Θ, where J is an isometry of E3 .
Such that lim d3 (Θn , Θ) = 0. n→∞ Consequently, lim d˙3 (F (Cn ), F (C)) = 0. n→∞ As shown by Ciarlet & C. Mardare [2004b], the above continuity result can be extended “up to the boundary of the set Ω”, as follows. 6). Another extension, again motivated by nonlinear three-dimensional elasticity, is the following: Let Ω be a bounded and connected subset of R3 , and let B be an elastic body with Ω as its reference conﬁguration. e. in Ω as a possible deformation Sect. 8] An immersion as a function of its metric tensor 57 of B when B is subjected to ad hoc applied forces and boundary conditions.
Z∈]x,y[ Since the spectral norm of an orthogonal matrix is one, we thus have |Θ(y) − Θ(x)| ≤ |y − x| for all x, y ∈ B. , Schwartz ) shows that there exist an open neighborhood V of x0 contained in Ω and an open neighborhood V of Θ(x0 ) in E3 such that the restriction of Θ to V is a C 1 -diﬀeomorphism from V onto V . Besides, there is no loss of generality in assuming that V is contained in B and that V is convex (to see this, apply the local inversion theorem ﬁrst to the restriction of Θ to B, thus producing a “ﬁrst” neighborhood V of x0 contained in B, then to the restriction of the inverse mapping obtained in this fashion to an open ball V centered at Θ(x0 ) and contained in Θ(V )).