By Allaire G., Braides A., Buttazzo G.
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Additional resources for School on Homogenization
4) which turns out not to be lower semicontinuous. 1. e. n on ∂Ω. j* , χ* (x ) = 0 if not. 5). Proof. j (x ) dx. 4), it is perfectly legitimate to interchange the order of minimiza→ → tions in χ and j . 5). 8), the interrelation between the minimizers χ* and → j* is now obvious. 1. 5) is clearly not convex if λ > 0. g. Chapter X in ), that won’t be detailed here. 4) may have no solution ; thus, we need to introduce its relaxation, but we don’t want to compute it with the general arguments of the relaxation theory, rather we use the homogenization theory (see section 2).
2. 11) F (u, A) = ϕ(x, Du) dx A for all A ∈ An and u ∈ W1,p (A; IRN ). 3. 1) (see Acerbi & Fusco ). Convex functions are quasiconvex; the two notions coincide only in the case n = 1 or N = 1. Examples of quasiconvex non convex functions are polyconvex functions: we say that f : Mn×N → IR is polyconvex if f (ξ) is a convex function of the vector of all minors of the matrix ξ. In the case n = N = 2 this means that f (ξ) = g(ξ, det ξ), with g convex. 2. We will just give an idea of the proof. First of all one can obtain a representation for F (u, A) when u = ξx is linear (or affine, which is the same because of the translation invariance): since F (ξx, ·) is a measure (absolutely continuous with respect to the Lebesgue measure), then, by Riesz Theorem, there exists a function gξ such that F (ξx, A) = gξ (x) dx A for all A ∈ An .
We begin with a plane bounded domain Ω, occupied by a linearly elastic material with isotropic Hooke’s law A , and loaded on its boundary by → some known force f . Admissible designs are obtained by removing a subset H ⊂ Ω, consisting of one or more holes (the new boundaries created this way are tractionfree). The holes H are actually the degenerate limit of a second material whose Hooke’s law is going to zero. We recall that a Hooke’s law is a fourth-order tensor acting on symmetric matrices (it plays the role of conductivity in this problem).