Erratum to: Characterization of Generalized Gradient Young Measures Generated by Sequences in W 1,1 and BV

Abstract

We thank Emil Wiedemann for pointing out that in the proof of the sufficiency part of Theorem 9 (p. 584) in [2] the reduction to the case with underlying deformation zero does not work. This is so, because in the definition of μ it might happen that μ ∞ x is not a positive measure. The current proof works only if one restricts attention to the subclass of Young measures μ with underlying deformation u ∈ BV( ; R m ) that admit a generating sequence of the special form (u + ϕ j ), where (ϕ j ) ⊂ W 1,1 ( ; R m ) with ϕ j * 0. Instead, to prove Theorem 9 in the stated generality, we extend Step 1 in the proof to the case of Young measures with arbitrary underlying deformation.For u ∈ BV( ; R m ) we letthat is, the set of gradient Young measures that can be generated by sequences converging weakly* to u in BV( ; R m ). First we need the following approximation lemma in BV( ; R m ).Lemma 1 Let u ∈ BV( ; R m ) and ε > 0. Then, there exists a countable family Q of (rotated) rectangles Q ⊂ and a mapping v ∈ BV( ; R m ) with the following properties:The online version of the original article can be found under doi:10.1007/s00205-009-0287-9.1 Here and in the following the numbers refer to those in [2], and we also use the notation introduced there.

Lower Semicontinuity for Integral Functionals in the Space of Functions of Bounded Deformation Via Rigidity and Young Measures

“…we define q -strict convergence in M(Ω; R N ) to comprise µ j * µ and µ j → µ , see [KR10b,KR10a] and also the Reshetnyak Continuity Theorem 2.4 for a discussion why q -strict convergence is important here. It can be shown (by mollification) that smooth measures are dense in M(Ω; R N ) with respect to the q -strict convergence.…”

“…Hence, the so extended f is still a Carathéodory function, f ∞ is jointly continuous and f (x, 0) = 0 for all x ∈ R d \ Ω. The extended representation result for generalized Young measures (2.11) in Section 2.7 (the original result is in Proposition 2 (i) of [KR10a]), together with Theorem 5.1 yields lim inf j→∞ F(u j ) = f (x, q ),…”

Lower semicontinuity for an integral functional in BV

“…These results relied mostly on blow-up techniques. The result in [15] was generalized to x-dependent integrands in [14,Theorem 10], relying on the theory of generalized Young measures, which were first introduced by DiPerna and Majda in [8]. With a general measure µ, the problem was studied in the case p > 1 in [3], and also in [13].…”

“…Here we require the integrand F : R N ×n → R be nonnegative and quasiconvex, with linear growth m|A| ≤ F (A) ≤ M(1 + |A|) for some 0 < m ≤ M and all A ∈ R N ×n , and with a continuous recession function F ∞ . Our proof will rely heavily on the theory of generalized Young measures, particularly results derived in [14]. Once we have the above integral representation, we can derive Jensen's inequalities for generalized Young measures with respect to µ, as was done in [14,Theorem 9] with respect to the Lebesgue measure.…”

“…Our proof will rely heavily on the theory of generalized Young measures, particularly results derived in [14]. Once we have the above integral representation, we can derive Jensen's inequalities for generalized Young measures with respect to µ, as was done in [14,Theorem 9] with respect to the Lebesgue measure. By using these inequalities, we can then prove the following lower semicontinuity theorem (Theorem 4.4) which is the main result of this work: Theorem 1.1.…”