Abstract
We obtain a measure representation for a functional arising in the context of optimal design problems under linear growth conditions. The functional in question corresponds to the relaxation with respect to a pair$(\chi,u)$, where$\chi$is the characteristic function of a set of finite perimeter and$u$is a function of bounded deformation, of an energy with a bulk term depending on the symmetrized gradient as well as a perimeter term.
Highlights
In optimal design one aims to find an optimal shape which minimizes a cost functional
The optimal shape is a subset E of a bounded, open set Ω ⊂ RN which is described by its characteristic function χ : Ω → {0, 1}, E = {χ = 1}, and, in the linear elasticity framework, the cost functional is usually a quadratic energy, so we are lead to the problem min χ(x)W1(Eu(x)) + (1 − χ(x))W0(Eu(x)) dx, (χ,u) Ω
As a simple consequence of the density of smooth functions in LD(Ω) we show in remark 3.5 that, under the above growth conditions on W0, W1, F (χ, u; A) = FLD (χ, u; A), for every χ ∈ BV (A; {0, 1}), u ∈ BD(Ω), A ∈ O(Ω)
Summary
In optimal design one aims to find an optimal shape which minimizes a cost functional. For general energy densities f , Barroso, Fonseca and Toader [12] studied the relaxation of (1.2) for u ∈ SBD(Ω) under linear growth conditions but placing no convexity assumptions on f They showed that the relaxed functional admits an integral representation where a surface energy term arises naturally. The invariance of the studied functional with respect to rigid motions, required in [29], is replaced by a weaker condition stating continuity with respect to infinitesimal rigid motions Their result relies, as in papers mentioned above, on the global method for relaxation, as well as on the characterization of the Cantor part of the measure Eu, due to De Philippis and Rindler [25], which extends to the BD case the result of Alberti’s rank-one theorem in BV. The fact that our functionals have an explicit dependence on the χ field prevented us from applying existing results (such as [5, 19]) directly and required us to obtain direct proofs
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