Liftings, Young Measures, and Lower Semicontinuity

Abstract

This work introduces liftings and their associated Young measures as new tools to study the asymptotic behaviour of sequences of pairs (uj , Duj )j for (uj )j ⊂ BV(Ω; R m ) under weak* convergence. These tools are then used to prove an integral representation theorem for the relaxation of the functionalto the space BV(Ω; R m ). Lower semicontinuity results of this type were first obtained by Fonseca & Müller [Arch. Ration. Mech. Anal. 123 (1993), 1-49] and later improved by a number of authors, but our theorem is valid under more natural, essentially optimal, hypotheses than those currently present in the literature, requiring principally that f be Carathéodory and quasiconvex in the final variable. The key idea is that liftings provide the right way of localising F in the x and u variables simultaneously under weak* convergence. As a consequence, we are able to implement an optimal measure-theoretic blow-up procedure.

Relaxation for Partially Coercive Integral Functionals with Linear Growth

“…exists and is equal to the Radon-Nikodým derivative dη dµ (x). In addition to the usual version of the Besicovitch Differentiation Theorem, we shall make use of a new generalised version, first proved in [28], which applies to measures that behave like graphs. The following two results are taken from Theorem 5.1 and Lemma 5.2 in [28].…”

“…Indeed, an example of Dal Maso [13] shows that there exists a continuous (u-independent) integrand f : Ω×R d → [0, ∞) which is both convex and positively one-homogeneous in the final variable, but for which F is not equal to F 1 * * over W 1,1 (Ω; R) (despite the convexity of f !). On the other hand, it was recently shown in [28] that a satisfactory integral formula for the sequential weak* relaxation, F w * * * , of F to BV(Ω; R m ) (i.e., F w * * * [u] is defined analogously to F 1 * * [u] but with L 1 -convergence replaced by weak* convergence in BV) does always exist for essentially any Carathéodory integrand f which is quasiconvex and of linear growth in the final variable. However, integrands arising from limits of the kind of models given by (2) are not coercive, so it need not be the case that every sequence (u j ) j ⊂ W 1,1 (Ω; R m ) with lim sup j F[u j ] < ∞ can be assumed to be weakly* convergent.…”

“…(4) It turns out that partial coercivity implies that F is coercive in small cylinders B d (x, r) × B m (y, R) about every pair (x, y) ⊂ Ω × R m which 'matters from the perspective of computing F and F 1 * * '. In order to derive a formula for F 1 * * which makes minimal assumptions on f , our approach is to find a way of making the preceding statement rigorous in order to reduce the computation of F 1 * * to an application of the weak* technology developed in [28].…”

“…In this paper we build on the weak* relaxation results developed in [28] to establish an improved relaxation theory for F as defined by (1) under natural conditions on the integrand f . In particular, we allow for arbitrarily large growth in the y-variable (but still requiring that f does not become degenerate as |y| → ∞).…”

“…The first two of these inequalities are proved simultaneously in Section 4 via an intricate measure-theoretic truncation argument, which allows us to replace the L 1 (Ω; R m )convergent sequence (u j ) j with a weakly* convergent sequence ( u j ) j and to reduce the problem to an application of the weak* lower semicontinuity theory developed in [28]. We note that the work in [28] features in two distinct ways here: the main weak* relaxation result (quoted below as Theorem 2.18) is used to obtain the final inequality above, but the theory of liftings together with an associated Besicovitch Differentiation Theorem is also used in an essential way to control the error term which arises when exchanging (u j ) j for ( u j ) j . Whilst previous works have only been able to use measure-theoretic techniques to localise at points x ∈ Ω, the key point here is that liftings allow us to use measure theory to localise at points (x, u(x)) ∈ Ω × R m with a single limit.…”