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Additional resources for Algebraic Topology Aarhus 1982. Proc. conf. Aarhus, 1982
But since minimizer y ∗ of I in A. 18), it is immediate that y ∗ is not C 1 . 15), see Bauman, Owen and Phillips [1991a, 1992]. 7). e. 4). e. x ∈ Ω. 7) is assumed, or if y∗ is not assumed in advance to be in W 1,∞ (Ω; R3 ), then it is not obvious how to pass to the limit. 16 John M. Ball Problem 5. 4) of the Euler–Lagrange equation. Problem 6. 19). 19) does not hold, then W (Dy∗ ) is essentially unbounded. This is at ﬁrst sight inconsistent with y∗ being a minimizer, but we know from the one-dimensional examples in Ball and Mizel  and from the example of cavitation that it can pay to have the integrand inﬁnite somewhere so that it is smaller somewhere else.
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.
Let U ⊂ A be open with respect to d, with closure U ¯ By the direct method, I attains a minimum yˆ on U . Suppose now that we 24 John M. Ball can prove that inf I > inf I > inf I. ∂U U A Then yˆ ∈ U and is a local, but not global, minimizer with respect to d. I believe that it should be possible to implement this method in some realistic examples, but have not seen it done. 6) for which the domain Ω has nontrivial topology by global minimization in a weakly closed homotopy class, and to Taheri [2001a], who generalizes the results in Post and Sivaloganathan  to a wider class of domains.