By Jacob Palis, Manfredo DoCarmo

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The second approach is to consider the true minimizers of the discrete problems, and to try to understand what functional their limit minimizes. One example of this approach is the interesting study by Friesecke and Theil [2001] of a model two-dimensional problem of a lattice of particles linked by harmonic springs between their nearest and next nearest neighbours. They determine open regions of atomic parameters in which the Cauchy–Born hypothesis holds in the appropriate limit, and open regions in which it does not.

15) For the more general case of a thermoviscoelastic material (of strain-rate type), TR , η, ψ, qR are assumed to be functions of Dy, Dyt , θ, grad θ. By the same method we ﬁnd that ψ = ψ(Dy, θ), and that S · Dyt − η = −Dθ ψ , qR · grad θ ≥ 0, θ where TR = DA ψ + S Dy, Dyt , θ, grad θ . 15). 17) 1 for some matrix-valued function Σ, where U = (DyT Dy) 2 . 2 Existence of Solutions Problem 12. Prove the global existence and uniqueness of solutions to initial boundary-value problems for properly formulated dynamic theories of nonlinear elasticity.

For |C − 1| < γ we have that 1 W (CA) − W (A) = 0 d W (C(t)A) dt dt 1 DA W C(t)A · (C − 1)A dt = 0 1 = DA W C(t)A C(t)A T · (C − 1)C(t)−1 dt 0 1 ≤K W C(t)A + 1 · C − 1 · C(t)−1 dt 0 1 ≤ 2Kγ W C(t)A + 1 dt . 0 Let θ(A) = sup|C−1|<γ W (CA). 24) follows. 24) that DA W (CA)AT = DA W (CA)(CA)T C−T ≤ K W (CA) + 1 C−T ≤ 3K W (A) + 1 , as required. 4. (i) Given ϕ as in the theorem, deﬁne for |τ | sufﬁciently small yτ (x) := y(x) + τ ϕ y(x) . e. x ∈ Ω. 1. e. x ∈ Ω and limτ →0 yτ − y Hence I(yτ ) ≥ I(y) for |τ | suﬃciently small.