Encyclopedie des sciences mathematiques. Geometrie by Molk J. (ed.)

By Molk J. (ed.)

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30 Three-dimensional differential geometry [Ch. 1 satisfy the following Cauchy problem for a linear system of three ordinary differential equations with respect to three unknowns: dζj dγ i (t) = Γpij (γ(t)) (t)ζp (t), 0 ≤ t ≤ 1, dt dt ζj (0) = ζj0 , where the initial values ζj0 are given by ζj0 := F 0j . Note in passing that the three Cauchy problems obtained by letting = 1, 2, or 3 only differ by their initial values ζj0 . 1, p. 388]). Hence each one of the three Cauchy problems has one and only one solution.

Then there exist x1 ∈ Ω, a path γ ∈ C 1 ([0, 1]; R3 ) joining x0 to x1 , τ ∈ ]0, 1[, and an open interval I ⊂ [0, 1] containing τ such that γ(t) = x + (t − τ )ei for t ∈ I, where ei is the i-th basis vector in R3 . Since each function ζj is continuously difdγ i dζj (t) = Γpij (γ(t)) (t)ζp (t) for all 0 ≤ t ≤ 1, ferentiable in [0, 1] and satisfies dt dt we have dζj (τ ) + o(t − τ ) dt = ζj (τ ) + (t − τ )Γpij (γ(τ ))ζp (τ ) + o(t − τ ) ζj (t) = ζj (τ ) + (t − τ ) for all t ∈ I. Equivalently, F j (x + (t − τ )ei ) = F j (x) + (t − τ )Γpij (x)F p (x) + o(t − x).

Then there exist a constant C(Θ) and orientation-presk erving mappings Θ ∈ H 1 (Ω; Ed ), k ≥ 1, that are isometrically equivalent to Θk such that k Θ −Θ H 1 (Ω;Ed ) k ≤ C(Θ) (∇Θk )T ∇Θk − ∇ΘT ∇Θ 1/2 . L1 (Ω;Sd ) 1 d Hence the sequence (Θ )∞ k=1 converges to Θ in H (Ω; E ) as k → ∞ if the k T k ∞ T 1 sequence ((∇Θ ) ∇Θ )k=1 converges to ∇Θ ∇Θ in L (Ω; Sd ) as k → ∞ . Should the Cauchy-Green strain tensor be viewed as the primary unknown (as suggested above), such a sequential continuity could thus prove to be useful when considering infimizing sequences of the total energy, in particular for handling the part of the energy that takes into account the applied forces and the boundary conditions, which are both naturally expressed in terms of the deformation itself.

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