Relaxation for an optimal design problem with linear growth and perimeter penalization

Abstract

The paper is devoted to the relaxation and integral representation in the space of functions of bounded variation for an integral energy arising from optimal design problems. The presence of a perimeter penalization is also considered in order to avoid non existence of admissible solutions, besides this leads to an interaction in the limit energy. Also more general models have been taken into account.

Optimal design of fractured media with prescribed macroscopic strain

“…We observe that the bulk energy density H : BV (Ω; {0, 1})×(L 1 (Ω; R d×N )) 2 → [0, +∞[ depends on the structured deformation on {χ = 0} or {χ = 1} (see (3.5)) and that the interfacial energy density (3.6)) can be further specialized on the various pieces of the decomposition of S(χ, u), as noted in detail in Remark 3.4. We remark also that if we consider the classical deformation setting, that is no jumps in u and G = ∇u, then we recover an optimal design problem studied in [8]; if we consider just one material, then we recover the results in [9].…”

“…We also believe that a similar analysis to the one presented below, can be performed when the range of χ is countable. Such a case is considered, e.g., in [8];…”

Orlicz equi-integrability for scaled gradients

“…Lemma 2 Let U be an Orlicz function satisfying (5) and (6). Let E & R N be a Lebesgue measurable set of finite measure and let ðu n Þ be a uniformly bounded sequence in L U ðE; R m Þ: For any r [ 0 define the standard truncature operators s r : R !…”

“…of the type j Á j p ) for the energy density, by convex functions [satisfying suitable properties, as (5) and (6)]. We refer to the recent works [18,19] aimed to describe thin structures and their bending phenomena, and to the forthcoming paper [16], where optimal design questions are addressed in the same spirit of [5,6]. We believe that our result can have further applications like those to fluid mechanics and multiscale problems (we refer to [21], where homogenization of integral functionals was treated, in a very similar setting to ours).…”

Relaxation for Optimal Design Problems with Non-standard Growth

“…Carita and Zappale in [13] considered a similar functional to the one in [4] and studied the minimum problem inf Ω χ E (x)W 1 (∇u(x)) + (1 − χ E (x))W 2 (∇u(x)) dx + |Dχ E |(Ω) :…”