Lower semicontinuous envelopes in W^1,1×L^p

Abstract

It is studied the lower semicontinuity of functionals of the type Ω f (x, u, v, ∇u)dx with respect to the (W 1,1 × L p )-weak * topology. Moreover in absence of lower semicontinuity, it is also provided an integral representation in W 1,1 × L p for the lower semicontinuous envelope.We are interested in studying the lower semicontinuity and relaxation of (1) with respect to the L 1 -strong ×L p -weak convergence. Clearly, bounded sequences {u n } ⊂ W 1,1 (Ω; R n ) may converge in L 1 , up to a subsequence, to a BV function. In this paper we restrict our analysis to limits u which are in W 1,1 (Ω; R n ). Thus, our results can be considered as a step towards the study of relaxation in BV (Ω; R n ) × L p (Ω; R m ) of functionals (1).We will consider separately the cases 1 < p < ∞ and p = ∞. To this end we introduce for 1 < p < +∞ the functionalfor any pair (u, v) ∈ W 1,1 (Ω; R n ) × L p (Ω; R m ), and for p = ∞ the functional J ∞ (u, v) := inf lim inf J(u n , v n ) :

A relaxation result in B V × L p for integral functionals depending on chemical composition and elastic strain

“…The following result can be deduced in full analogy with [11,Theorem 13], where it has been proven for J ∞ .…”

“…The proof is omitted since it can be performed as in [11,Lemma 8 and Remark 9]. In [11] it is not required that f satisfies (H 2 ) p , (p ∈ (1, ∞]).…”

“…The proof is omitted since it can be performed as in [11,Lemma 8 and Remark 9]. In [11] it is not required that f satisfies (H 2 ) p , (p ∈ (1, ∞]). Indeed, the equality J p = J CQf p holds independently on this assumption on f , but in order to remove hypothesis (H 0 ) from the representation theorem we need to assume that CQf inherits the same properties as f , which is the case as it has been observed in Proposition 2.7.…”

“…ii) A convex-quasiconvex function is separately convex. iii) By [11,Proposition 3], the growth condition from above in (H 1 ) p , ii), entail that there exists γ > 0 such that …”

“…f (x, u, v, ∇u) ≡ W (x, u, ∇u) + ϕ (x, u, v), the analysis in the case 1 < p < ∞ was considered already in [10] in order to describe image decomposition models. In [11] a general f was taken into account when the target u is in W 1,1 ( ; R n ). In this manuscript we consider f ≡ f (b, ξ), (b, ξ ) ∈ R m × R n×N and the target u ∈ BV ( ; R n ).…”