Relaxation of certain integral functionals depending on strain and chemical composition

Abstract

We provide a relaxation result in BV × L q , 1 ≤ q < +∞ as a first step towards the analysis of thermochemical equilibria.

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

“…, and this latter property is inherited by CQf and CQf ∞ as it can be easily verified arguing as in [10,Proposition 2.3]. Thus Proposition 2.14 holds when p = ∞ just requiring that f satisfies (2.6) and (2.7).…”

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

“…Since the proof develops along the lines as in [31,Proposition 2.2], in turn inspired by [19], we omit it. (iv) A proof entirely similar to [10,Proposition 3.4] (see also [31,Proposition 2.6]) ensures that for every (q, z) ∈ R m × R d×N , Q(f ∞ )(q, z) = (Qf ) ∞ (q, z), hence we will adopt the notation Qf ∞ . In particular if f satisfies (F 1 ) − (F 3 ), Proposition 3.1 guarantees that Qf ∞ is continuous in both variables.…”

Lower semicontinuous envelopes in W1,1×Lp

“…Since ∇u n (x) = (∇u * ̺ n )(x), (25) Passing to the limit on the right hand side of (24), exploiting (25) in the third line and applying [18,Lemma 2.5], in the fourth line, we get Moreover by virtue of (21) and arguing as in the estimate of formula (5.11) of [18] we can conclude that lim sup Then we can exploit (22) and argue again as done for (5.11) in [18] in order to evaluate lim sup Taking into account that |Du|(∂Q(x 0 , ε)) = 0 for a.e. ε one obtains from (21)…”

“…The following results will be exploited in the sequel. We omit the proofs since they are very similar to [21,Proposition 2.2], in turn inspired by [10].…”