STRICTLY CONTINUOUS EXTENSION OF FUNCTIONALS WITH LINEAR GROWTH TO THE SPACE BV
Filip Rindler, Giles Shaw · The Quarterly Journal of Mathematics · 2015
Having obtained the existence of candidate minimizers in |${\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)$|, the next step in the application of the Direct Method is to examine when |$\mathcal {F}$| satisfies the lower semicontinuity property |$\mathcal {F}[u]\leq \liminf _{j}\mathcal {F}[u_j]$| for every sequence |$(u_j)\subset {\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)$| such that |$u_j{{\mathop {\rightharpoonup }\limits ^{* }}}u$|. In order to do this, a suitable extension of |$\mathcal {F}$| to |${\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)$| must be identified so that a value can be assigned to |$\mathcal {F}[u]$| for |$u\in ({\mathrm {BV}}\setminus {{W}}^{1,1})(\Omega ;{\mathbb {R}}^m)$|. There is no unique extension of |$\mathcal {F}$| from |${{W}}^{1,1}(\Omega ;{\mathbb {R}}^m)$| to |${\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)$|, and so a criterion is needed to identify the ‘right’ extension in this context for as wide a class of integrands |$f$| as possible. In general, we cannot hope to obtain |$\mathcal {F}$| as the weakly* continuous extension to |${\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)$| of |$u\mapsto \int _\Omega f(x,u(x), abla u(x))\,{\mathrm {d}}x$|: Example 2.13 demonstrates a weakly* convergent sequence in |${{W}}^{1,1}(\Omega ;{\mathbb {R}}^m)$|, under which the map |$u\mapsto \int _\Omega \sqrt {1+ |u'(x)|^2}\,{\mathrm {d}}x$| fails to converge. A priori, it is far from clear how one might extend |$\mathcal {F}$| in such a way that every |$u\in {\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)$| has at least one recovery sequence |$(u_j)\subset {{W}}^{1,1}(\Omega ;{\mathbb {R}}^m)$| (i.e. a sequence |$(u_j)$| such that |$u_j{{\mathop {\rightharpoonup }\limits ^{* }}}u$| and |$\mathcal {F}[u_j]\to \mathcal {F}[u]$|): the derivative |$Du$| of a function |$u\in {\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)$| is defined only as a (potentially Lebesgue-singular) matrix-valued measure, in which case the expression |$\int _\Omega f(x,u(x),Du)\,{\mathrm {d}}x$| is not well-defined. As defined in (2), |$\mathcal {F}[u]$| is equal to our original |$\mathcal {F}[u]$| for |$u\in {{W}}^{1,1}(\Omega ;{\mathbb {R}}^m)$|, and is therefore an extension of the original |$\mathcal {F}$| to |${\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)$|. It follows from (1) that, at least in the scalar or |$u$|-independent case, each |$u\in {\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)$| admits a recovery sequence |$(u_j)\subset ({{W}}^{1,1}\cap {{C}}^\infty )(\Omega ;{\mathbb {R}}^m)$|, which implies that (2) meets the minimum criteria for a suitable extension of |${\mathcal F}$|. In general, however, one is unable to say anything about the properties of such a recovery sequence, or if a better extension of |${\mathcal F}$| exists which admits strictly more recovery sequences. Since the restriction of |${\mathcal F}_{\ast \ast }$| to |${{W}}^{1,1}(\Omega ;{\mathbb {R}}^m)$| is lower semicontinuous with respect to weak convergence in |${{W}}^{1,1}(\Omega ;{\mathbb {R}}^m)$|, it can only be used to extend |$\mathcal {F}$| in situations where |$\mathcal {F}$| is also lower semicontinuous over |${{W}}^{1,1}(\Omega ;{\mathbb {R}}^m)$| (i.e. when |$f(x,y,{\,{\scriptsize {\bullet }}\,})$| is convex/quasiconvex) and so the relaxation method cannot be used to extend |$\mathcal {F}$| for general integrands. As defined in (2), however, the restriction of |$\mathcal {F}[u]$| to |${{W}}^{1,1}(\Omega ;{\mathbb {R}}^m)$| is always equal to |$\int _\Omega f(x,u(x), abla u(x))\,{\mathrm {d}}x$|, regardless of the convexity properties of |$f$|. This suggests that, if the extension given by (2) still admits |${{W}}^{1,1}(\Omega ;{\mathbb {R}}^m)$|-recovery sequences, it can be taken as a candidate functional for the extension of |$\mathcal {F}$| to |${\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)$| even when |$f$| is not convex. In order to justify this position, we must find a way of showing that the extension given by (2) always admits |${{W}}^{1,1}(\Omega ;{\mathbb {R}}^m)$|-recovery sequences and argue that no better extension is to be found. This paper is primarily devoted to proving the following theorem, which establishes that (2) defines an extension of |$\mathcal {F}$| valid for general integrands |$f$| in a way that satisfies all of the requirements above. Here, area-strict convergence (defined in Section 2) is a notion of convergence with respect to which |$({{W}}^{1,1}\cap {{C}}^\infty )(\Omega ;{\mathbb {R}}^m)$| is dense in |${\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)$| and which implies weak* convergence. Every area-strictly convergent sequence is thus a recovery sequence and, by area-strict density of |$({{W}}^{1,1}\cap {{C}}^\infty )(\Omega ;{\mathbb {R}}^m)$| in |${\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)$|, (2) is the unique extension of |$\mathcal {F}$| to |${\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)$| for which this holds. Related results can be found in [14; 19, Theorem 3]. A surprising implication of Theorem 1.1 and the failure of the representation (2) for the case |$f=f(x,y,A)$|, |$m>1$|, is that, in contrast to the situation where |$f=f(x,A)$| or |$m=1$|, the relaxation |$\mathcal {F}_{\ast \ast }$|, cannot be area-strictly continuous in general, not even when |$f(x,y,{\,{\scriptsize {\bullet }}\,})$| is convex. Conversely, it must also be the case that, even when |$f(x,y,{\,{\scriptsize {\bullet }}\,})$| is convex or quasiconvex, |$\mathcal {F}$| is not always weakly* lower semicontinuous over |${\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)$|, despite being area-strictly continuous over |${\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)$| and weakly* lower semicontinuous over |${{W}}^{1,1}(\Omega ;{\mathbb {R}}^m)$|. The Rellich–Kondrachov Theorem for |${\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)$| states that |${\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)\hookrightarrow {{L}}^p(\Omega ;{\mathbb {R}}^m)$| for |$p\in [1,1^* ]$| and it is known that this embedding result is sharp, so that |${\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)$| cannot be em-bedded into any higher |${{L}}^p$| space. Hence, the growth hypothesis |$|f(x,y,A)|\leq C(1+ |y|^{1^* }+ |A|)$| in Theorem 1.1 is optimal, in that it represents the weakest natural condition necessary to ensure that |$\mathcal {F}$| is finite on |${\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)$|. It might seem natural that the result holds for |$f$| satisfying |$|f(x,y,A)|\leq C(1+ |y|^p+ |A|)$| when |$p\lt 1^* $|, as a consequence of the fact that in this case the embedding |${\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)\hookrightarrow {{L}}^p(\Omega ;{\mathbb {R}}^m)$| is compact. That (for |$d>1$|) this result is true even when |$p=1^* $| is surprising, and the proof in this case makes use of Lions’ concentration-compactness principle. For the case |$d=1$| where |$1^* =\infty$|, Example 5.1 demonstrates that the theorem does not hold. We will also show that this result holds true for Carathéodory |$f$|, so long as the recession function |$f^\infty$| remains continuous on |$(\Omega \setminus N)\times {\mathbb {R}}^m\times {\mathbb {R}}^{m\times d}$|, where |$N$| is |$\mathcal {H}^{d-1}$| negligible, see Theorem 5.2. We recall here some technical facts about weak and norm convergence in |${{L}}^p$| spaces which will be used in the sequel. A proof of this result can be found in [8] and also in [15]. Let|$f\colon \Omega \times {\mathbb {R}}^m\times {\mathbb {R}}^{m\times d}\to {\mathbb {R}}$|be Carathéodory. Then for every|$\varepsilon >0,$|there exists a compact set|$K_\varepsilon \subset \Omega$|such that|$\mathcal {L}^d(\Omega \setminus K_\varepsilon )\lt \varepsilon$|and|$f|\mkern -2mu{\underline {\mkern 16mu}}(K_\varepsilon \times {\mathbb {R}}^m\times {\mathbb {R}}^{m\times d})$|is continuous. For a proof, see [10, p. 74]. We now define the recession function |$f^\infty$| of an integrand |$f$|, whose purpose is to capture information about the behaviour of |$f(x,y,A)$| as |$|A|\to \infty$|. Note that we require our definition of the recession function to be more restrictive than what is usually found in the literature (where only the existence of |$\lim _{t\to \infty }f(x,y,tA)/t$| or |$\limsup _{t\to \infty }f(x,y,tA)/t$| is assumed). The definition of the recession function implies that, whenever it exists, it must be continuous. This property is necessary in order for the function |$\tilde {f}$| defined in the proof of Lemma 4.3 to be continuous, which in turn is necessary for Reshetnyak's Continuity Theorem to hold. For further intricacies related to the definition of the recession function, we refer the reader to [21]. Note that the recession function is positively|$1$|-homogeneous in the final variable, that is, |$f^\infty (x,y,\lambda A)=\lambda f^\infty (x,y,A)$| for each |$\lambda \geq 0$|. We will denote by |$\mathcal {B}(X)$| the Borel |$\sigma$|-algebra on a normed space |$X$| and the space of |${\mathbb {R}}^{m\times d}$|-valued Radon measures acting on |$X$| (we will always take |$X=\Omega$|, |$X={\mathbb {R}}^m$| or |$X=\Omega \times {\mathbb {R}}^m$|) by |${\mathbf {M}}(X;{\mathbb {R}}^{m\times d})$|. The spaces of scalar-valued and positive measures on |$X$| will be denoted by |${\mathbf {M}}(X)$| and |${\mathbf {M}}^+ (X),$| respectively. A sequence of measures |$(\mu _j)$| is said to converge strictly to |$\mu$| if |$\mu _j{{\mathop {\rightharpoonup }\limits ^{* }}}\mu$| and |$|\mu _j|(X)\to |\mu |(X)$|, where |$|\mu |$| is the total variation measure of |$\mu$|. We will denote the Radon–Nikodym derivative (or polar function) of a measure |$\mu$| with respect to its total variation by |${{\mathrm {d}}\mu }/{{\mathrm {d}}|\mu |}$|. The following theorems concerning the convergence of nonlinear functionals of measures will be of great importance. for every lower semicontinuous function|$f\colon \Omega \times {\mathbb {R}}^m\times {\mathbb {R}}^{m\times d}\to [0,\infty ]$|which is positively|$1$|-homogenous and convex in the last variable. Given a measure |$\mu \in {\mathbf {M}}^+ (\Omega )$|, we say that |$ u$| is a |$\mu$|-measurable |${\mathbf {M}}({\mathbb {R}}^m;{\mathbb {R}}^{m\times d})$|-valued map or parametrized measure if |$ u \colon x\mapsto u _x$| is a function |$ u \colon \Omega \to {\mathbf {M}}({\mathbb {R}}^m;{\mathbb {R}}^{m\times d})$| such that the map |$x\mapsto u _x(A)$| is |$\mu$|-measurable for every fixed |$A\in {\mathcal B}({\mathbb {R}}^m)$|. Let|$\eta \in {\mathbf {M}}(\Omega \times {\mathbb {R}}^m;{\mathbb {R}}^{m\times d})$|and let|$\pi \colon \Omega \times {\mathbb {R}}^m\to \Omega$|be the projection operator defined by|$\pi (x,y)=x$|for|$(x,y)\in \Omega \times {\mathbb {R}}^m$|. Then there exists a|$\pi _\sharp |\eta |$|-almost everywhere unique measure-valued map|$ u \colon \Omega \to {\mathbf {M}}({\mathbb {R}}^m;{\mathbb {R}}^{m\times d})$|such that each|$| u _x|$|is a probability measure and|$\eta =(\pi _\sharp |\eta |)\otimes u ,$|where|$\pi _\sharp |\eta |\in {\mathbf {M}}(\Omega \times {\mathbb {R}}^m;{\mathbb {R}}^{m\times d})$|is uniquely defined by|$\pi _\sharp |\eta |(A\times B):=|\eta |(A)$|. Furthermore,|$|\eta |=(\pi _\sharp |\eta |)\otimes | u |$||$($|where|$| u |$|is defined by|$| u |_x=| u _x|)$|and, up to scaling, this is the only way of factoring|$ u$|over|$\Omega$|and|${\mathbb {R}}^m$|: if|$\eta =\xi \otimes u$|where|$\xi \in {\mathbf {M}}^+ (\Omega )$|and|$| u |\colon F\to {\mathbf {M}}^1({\mathbb {R}}^m),$|then|$\pi _\sharp |\eta |=\xi$|. The standard reference for all of the above is [4], although we note that new proofs of Reshetnyak's theorems can be found in [24]. When both spaces are endowed with their norm topology, the embedding|${\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)\hookrightarrow {{L}}^p(\Omega ;{\mathbb {R}}^m)$|for|$p\in [1,1^* ]$|is continuous. For|$p\in [1,1^* ),$|the embedding is compact. We will now introduce two metrics on |${\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)$|, the strict metric and the area-strict metric. Our interest in these two metrics stems from the fact that they induce a topology which is stronger than the weak* topology on |${\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)$|, yet with respect to which |${{W}}^{1,1}(\Omega ;{\mathbb {R}}^m)$| and |${{C}}^\infty (\Omega ;{\mathbb {R}}^m)$| functions are dense. Strictly convergent sequences are norm-bounded in |${\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)$|, which implies that they have weakly* convergent subsequences. Using the sequential weak* compactness of bounded sets in |${\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)$|, we deduce that strict convergence of a sequence |$(u_j)$| in |${\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)$| implies strict convergence of |$(Du_j)$| in |${\mathbf {M}}(\Omega ;{\mathbb {R}}^m)$|, i.e. |$Du_j{{\mathop {\rightharpoonup }\limits ^{* }}}Du$| and |$|Du_j|(\Omega )\to |Du|(\Omega )$|. It is an immediate consequence of Theorem 1.1 (with the function |$f=|A|$|) that area-strict convergence implies strict convergence. The following example shows that the converse is not true. Under the topology induced by area-strict convergence,|${{C}}^\infty (\Omega ;{\mathbb {R}}^m)$||$($|and hence also|${{W}}^{1,1}(\Omega ;{\mathbb {R}}^m))$|is dense in|${\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)$|. A proof can be found in [6]. We note that area-strict convergence can be interpreted as requiring strict convergence of the graph |$(x,u(x))$| of |$u$|. Although area-strict convergence is necessary for Theorem 1.1 to hold, it is only used in the proof of Lemma 4.3. For every other argument in this paper, strict convergence suffices. In this section, we will first define a map |$\mu \colon {\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)\to {\mathbf {M}}(\Omega \times {\mathbb {R}}^m;{\mathbb {R}}^{m\times d})$| assigning a lifting |$\mu [u]$| to each |$u\in {\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)$|. Our interest in liftings stems from the fact that, for positively |$1$|-homogeneous integrands, they can be used to compute |$\mathcal {F}$| and hence, after an application of Reshetnyak's Continuity Theorem, reduce the question of the strict continuity of |$\mathcal {F}$| to that of the strict continuity of the map |$u\mapsto \mu [u]$|. In this context, liftings were first defined and studied in [18], where the authors also note that strict continuity of the map |$u\mapsto \mu [u]$| implies strict convergence of |$\mathcal {F}$| for positively |$1$|-homogeneous integrands. We will define liftings in a different (although equivalent) way and, as a consequence, provide a cleaner derivation of the properties of liftings that we require. Second, we will prove an embedding result for |${\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)$| equipped with the strict topology which will be needed to prove Theorem 1.1 for the critical case |$p=1^* $|. The following continuity lemma is crucial to our work. It was originally established in [18] using results from [9], but we provide a streamlined, more direct and shorter proof here. If|$u_j\to u$|strictly in|${\mathrm {BV}}(\Omega ;{\mathbb {R}}^m),$|then|$\mu [u_j]\to \mu [u]$|strictly in|${\mathbf {M}}(\Omega \times {\mathbb {R}}^m;{\mathbb {R}}^{m\times })$|. We have that |$|\mu [u_j]|(\Omega \times {\mathbb {R}}^m)=\pi _\sharp |\mu [u_j]|(\Omega )=|Du_j|(\Omega )$| and so the sequence |$(\mu [u_j])$| is bounded in |${\mathbf {M}}(\Omega \times {\mathbb {R}}^m;{\mathbb {R}}^{m\times d})$|. By the sequential Banach–Alaoglu Theorem, |$(\mu [u_j])$| admits a weakly* convergent subsequence, which we do not relabel. Denote the limit of this sequence by |$\eta$|. We will show that |$\eta =\mu [u]$| and, since the argument will apply to any weakly* convergent subsequence of |$(\mu [u_j])$|, it must follow that |$\mu [u_j]\to \mu [u]$| strictly as required. Case 1. |$x\in \Omega \setminus \mathcal {J}_u$|: Case 2. |$x\in \mathcal {J}_u$|: The following lemma is a special case of [9, Theorem D.1]. The authors of [18] define the minimal lifting of |$u$| to be the measure |$\mu [u]$| satisfying the equation |$Q_\varphi (u,\mu [u])=0$| with the additional property that |$\pi _\sharp |\mu [u]|(\Omega )=|Du|(\Omega )$|. This is equivalent to our definition of a lifting. The following corollary is now a direct consequence of Reshetnyak's Continuity Theorem and Lemma 3.2. Simply combine Corollary 3.2 with Reshetnyak's Continuity Theorem 2.5, the discussion following Definition 3.1, and the fact that |$|f(x,y,A)|\leq C|A|$| implies that the restriction of |$f$| to |$\Omega \times {\mathbb {R}}^m\times \partial \mathbb {B}^{m\times d}$| is bounded. Next, we prove an embedding result for the space |${\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)$| equipped with the metric of strict convergence, which will be of use in Section 5. This result is of interest since it yields an extension of the continuous embedding |${\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)\hookrightarrow {{L}}^p(\Omega ;{\mathbb {R}}^m)$| for all |$p\lt 1^* $|, when |${\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)$| is equipped with the usual weak* topology to the critical case |$p=1^* $|. Note that Proposition 3.7 does not hold when |$d=1$|, as Example 5.1 in Section 5 demonstrates. Let |$u_j\to u$| strictly. Since (|$u_j)$| converges to |$u$| in measure (as a consequence of strong |${{L}}^1$| convergence), this then implies, via Vitali's Convergence Theorem, that |$u_j\to u$| in |${{L}}^{1^* }$| if and only if |$(u_j)$| is |$1^* $|-uniformly integrable. In this situation, assuming that |$(u_j)$| is not |$1^* $|-uniformly integrable, we can apply Lions’ concentration-compactness principle [20, Lemma I.1] to arrive at a contradiction. For reasons of clarity, however, we will carry out the derivation here in full. The fact that |$u_j\to u$| in |${{L}}^{1^* }(\Omega ;{\mathbb {R}}^m)$| whenever |$u_j\to u$| area-strictly in |${\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)$| is a necessary consequence of Theorem 1.1 in the case |$p=1^* $|. Letting |$f(x,y,A)=|y|^{1^* }$|, we see that Theorem 1.1 implies |${\|{{u_j}}\|}_{1^* }\to {\|{{u}}\|}_{1^* }$| whenever |$u_j\to u$| area-strictly. Since |${{L}}^{1^* }$| is a uniformly convex space and |$u_j\rightharpoonup u$| in |${{L}}^{1^* }$| (a consequence of the fact that |$(u_j)$| is bounded in |${{L}}^{1^* }$| and that |$u_j$| converges to |$u$| in measure), we therefore have that |$u_j\to u$| in |${{L}}^{1^* }$| (see, for example, [7, Proposition 3.32]). The purpose of this section is to remove the |$1$|-homogeneity assumption which appears in Corollary 3.6. This is achieved by introducing a perspective function|$\tilde {f}$| for the integrand |$f$| and exchanging strict convergence for area-strict convergence. We note here that a similar approach applying Reshetnyak's theorems combined with perspective functions to integral functionals on |${\mathrm {BV}}(\Omega ;{\mathbb {R}})$| can be found in [12]. For a discussion of generalized perspective functions and their relevance to different notions of convexity, the reader is referred to [11]. Strictly speaking, the perspective function of |$f$| is only unique as an element of |${{C}}(\Omega \times ({\mathbb {R}}\times {\mathbb {R}}^m)\times ({\mathbb {R}}^d\times {\mathbb {R}}^{m\times d}))$| where, for realization as an element of |${{C}}(\Omega \times {\mathbb {R}}^{1+m}\times {\mathbb {R}}^{(1+m)\times d})$| the canonical identifications |${\mathbb {R}}\times {\mathbb {R}}^m\cong {\mathbb {R}}^{1+m}$|, |${\mathbb {R}}^d\times {\mathbb {R}}^{m\times d}\cong {\mathbb {R}}^{(1+m)\times d}$|. We will tacitly assume that such a choice has been made and will speak simply of ‘the’ perspective function. It follows immediately from Definition 4.1 that |$\tilde {f}$| is always positively |$1$|-homogeneous in the |$(t,A)$| argument. The following lemma shows that |$\tilde {f}$| inherits the continuity properties of |$f$|. Let|$f\in {{C}}(\Omega \times {\mathbb {R}}^m\times {\mathbb {R}}^{m\times d})$|be such that|$f^\infty$|exists. Then|$\tilde {f}\in {{C}}(\Omega \times {\mathbb {R}}^{1+m}\times {\mathbb {R}}^{(1+m)\times d})$|. That |$\tilde {f}$| is continuous away from where |$|t|=0$| is an immediate consequence of the continuity of |$f$|. Continuity of |$\tilde {f}$| when |$|t|=0$| follows directly from the definition of the recession function. The following construction, which essentially replaces |$u(x)$| with its graph |$(x,u(x))$|, combined with Lemma 4.2 allows us to remove the |$1$|-homogeneity assumption from Corollary 3.6. This section contains the final step in the proof of Theorem 1.1 and also Theorem 5.2, an extension of Theorem 1.1 to Carathéodory integrands, as well as a counterexample to show that the hypotheses of Theorem 5.2 are optimal. Finally, we will weaken our assumptions on the regularity of |$f$| and |$f^\infty$|. The proof of Theorem 5.2 proceeds by using the Scorza Dragoni theorem to determine the result when |$f$| is bounded. An approximation argument is then used to extend this result to the case where |$f^\infty \equiv 0$| (i.e. when |$f$| has ‘negligible growth at |$\infty$|’). Applying this result to |$f-f^\infty ,$| lets us deduce the general result. Let|$u_j\to u$|area-strictly in|${\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)$|and let|$f\colon \Omega \times {\mathbb {R}}^m\times {\mathbb {R}}^{m\times d}\to {\mathbb {R}}$|be a Carathéodory integrand satisfying (3) whose recession function|$f^\infty$|exists on the set|$(\Omega \setminus N)\times {\mathbb {R}}^m\times {\mathbb {R}}^{m\times d},$|where|$N$|is some Borel set satisfying|$(\mathcal {L}^d+ |Du|)(N)=0$|. Then it holds that|$\mathcal {F}[u_j]\to \mathcal {F}[u]$|. In particular, this theorem implies that |$\mathcal {F}$| is area-strictly continuous for any Carathéodory |$f$| where |$f^\infty$| exists on |$(\Omega \setminus N)\times {\mathbb {R}}^m\times {\mathbb {R}}^{m\times d}$| for some Borel set |$N$| with |$\mathcal {H}^{d-1}(N)=0$|. Finally, we finish with an example which shows that Theorem 5.2 is optimal, in the sense that |$f^\infty$| cannot be discontinuous on a set that is charged by |$|Du|$| if we are to expect area-strict continuity from |$\mathcal {F}$|. This work was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) [EP/H023348/1 for the University of Cambridge Centre for Doctoral Training, the Cambridge Centre for Analysis, to G.S., EP/L018934/1 to F.R.]; and the University of Warwick. The authors would like to thank Jan Kristensen for many insightful conversations related to this paper, as well as Helge Dietert, Tom Holding and Marcus Webb for comments and remarks.