Relaxation of bulk and interfacial energies

Optimal Flux Densities for Linear Mappings and the Multiscale Geometry of Structured Deformations

“…The dependence of ψ on x was not included in [10] but can be handled, thanks to assumption (H 2), as in Barroso et al [5] again taking into account Remark 3.3 of [10].…”

Second-Order Structured Deformations: Relaxation, Integral Representation and Applications

“…(2) The hypotheses listed above are similar to the ones in [12] and [7] where there is no explicit dependence on x , and with the hypotheses in [9] where the density functions depended explicitly on the variable x . (3) It is well known that the bulk energy may have potential wells and for this reason it is desirable to consider…”

“…This interpretation of I dis (g, G) is justified by considering a sequence {u n } in SBV 2 (Ω, R 3 ) with u n → g and ∇u n → G both in L 1 and by writing 9) showing that M L 3 := (∇g − G) L 3 is the absolutely continuous part of the limit of the singular measures [u n ] ⊗ ν un H 2 that capture the submacroscopic disarrangements associated with (g, G) . Moreover, the energy density (A, L) −→ W 2 (A, L) of [7] provides the remaining portion…”

Quasiconvexification in W1,1 and Optimal Jump Microstructure in BV Relaxation

“…Section 5 deals with the proof of the lower bound inequality I > /*, which is a modified version of the corresponding argument in (a draft of) [6]. The changes include choosing the rescaling factors so that the weak * limit measure /i does not see the boundary of the rescaled unit cube, and so that as the rescaled variation measures converge weakly * on a cube, they do not lose any mass (see Lemma 5.1).…”

“…As mentioned in the introduction, we rely heavily on [6], and we use the blow-up method introduced by Fonseca …”