Reiterated Homogenization in BV via Multiscale Convergence

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1 Reiterated Homogenization in BV via Multiscale Convergence Feb 25, 20 Rita Ferreira I.C.T.I. - Carnegie Mellon Portugal, F.C.T./C.M.A. da U.N.L. Quinta da Torre, Caparica, Portugal rferreir@andrew.cmu.edu, ragf@fct.unl.pt Irene Fonseca Department of Mathematical Sciences Carnegie Mellon University, Pittsburgh, PA 523, USA fonseca@andrew.cmu.edu Abstract Multiple-scale homogenization problems are treated in the space BV of functions of bounded variation, using the notion of multiple-scale convergence developed in [30]. In the case of one microscale Amar s result [3] is recovered under more general conditions; for two or more microscales new results are obtained. AMS Classification Numbers (200): 49J45, 35B27, 26A45 Key Words: periodic homogenization, multiscale convergence, linear growth, space BV of functions of bounded variation, BV -valued measures, -convergence. Introduction and Main Results Here we are concerned with the description of the macroscopic behavior of a microscopically heterogeneous system. Several approaches have been proposed to handle the minimization of oscillating functionals, such as the method of asymptotic expansions, G-convergence, H-convergence, -convergence and two-scale convergence (we refer to [] and references therein). In the case in which the microscopic properties of the system are periodic, the method of two-scale convergence has proven to be particularly successful. It was introduced by Nguetseng [37], and further developed by Allaire [] and by Allaire and Briane [2], and it provides a mathematical rigorous justification for the formal asymptotic expansions that were commonly used in the study of homogenization problems (see [0], [34] and [40]). In [3] Amar extended the notion of two-scale convergence to the case of bounded sequences of Radon measures with finite total variation, which was then used to study the asymptotic behavior of sequences of positively - homogeneous and periodically oscillating functionals with linear growth, defined in the space BV of functions of bounded variation. Precisely, the following result is given in [3]. Theorem A (cf. [3, Thm. 4.]). Let R N be an open and bounded set Lipschitz, let Q := [0, ] N, and let f : R N R N! [0, ) be a function such that (A) for all 2 R N, f(, ) is continuous and Q-periodic; (A2) for all y 2 Q, f(y, ) is convex, positively -homogeneous, and of class C (R N \{0}); (A3) there exists a constant C > 0 such that for all y 2 Y, 2 R N, C 6 f(y, ) 6 C.

2 For each " > 0, let I " : BV ()! R be the functional defined by x I " (u) := f ", ddu dkduk (x) dkduk(x) + v(x) u(x) p dx, where v 2 L N/(N ) (), p 2 (, N/(N )] if N >, and p 2 (, ) if N =, and ddu/dkduk represents the Radon Nikodym derivative of Du with respect to its total variation kduk. Then for each " > 0, there exists a unique u " 2 BV () such that x I " (u " ) = min I "(w) = inf f w2bv () w2w, () ", rw(x) dx + v(x) w(x) p dx. Moreover, there exist u 2 BV () and µ 2 M(; BV # (Q)) such that {u " } ">0 weakly-? converges to u in BV () as "! 0 + and, up to a subsequence, {Du " } ">0 two-scale converges to the measure on Q, u,µ := Dudy + D y µ as "! 0 +. Furthermore, lim I "(u " ) = inf I sc (w, ) = I sc (u, µ), "!0 + w2bv () 2M(;BV # (Y )) where I sc is the two-scaled homogenized functional defined for w 2 BV () and 2 M(; BV # (Q)) by I sc d w, (w, ) := f y, dk w, k (x, y) dk w, k(x, y) + v(x) w(x) p dx. Q Finally, in the minimizing pair (u, µ) the function u 2 BV () is uniquely determined. The proof of Theorem A is based on the so-called two-scale convergence method, which has the virtue of taking full advantage of the periodic microscopic properties of the media, enabling the explicit characterization of the local behavior of the system: The asymptotic behavior as "! 0 + of the energies F " and of the respective minimizers u " is given with regard to both macroscopic and microscopic levels, through the two space variables x (the macroscopic one) and y (the microscopic one), and through the two unknowns u and µ. The next step of the two-scale convergence method is to obtain the e ective or homogenized problem, that is, the limit problem only involving the macroscopic space variable x, and which has as solution the function ū(x) := R u(x, y) dy. This is usually done via an average process with respect Q to the fast variable y of the two-scale homogenized problem. For the class of functions f considered by Amar [3], Theorem A provides an alternative characterization of the homogenized problem previously obtained by Bouchitté [2], [3], and summarizes as follows: Theorem B (cf. [2, Thm. 2.]). Let R N be an open and bounded set, let Y := (0, ) N, and let f : R N R N! R be a function such that (B) for all 2 R N, f(, ) is measurable and Y -periodic; (B2) for all y 2 Y, f(y, ) is convex; (B3) there exists a constant C > 0 such that for all y 2 Y, 2 R N, C For each " > 0, let F " : L ()! (, ] be the functional defined by ( x F " (u) := f ", ru(x) dx if u 2 W, (), otherwise. C 6 f(y, ) 6 C( + ). Here, and in the sequel, the subscript # stands for Q Q n-periodic functions (or measures), n 2 N, with respect to the variables (y,, y n), where each Q i, i 2 N, is a copy of Q. We refer the reader to Section 2 for the notations used throughout this paper. We will give a precise meaning for this statement further below. 2

3 Then, the sequence of functionals {F " } ">0 -converges as "! 0 + with respect to the strong topology of L () to the functional F 0 : L ()! (, ] given by F 0 (u) := F h (u) if u 2 BV (), otherwise, where, for u 2 BV (), F h (u) := with f hom (ru(x)) dx + (f hom ) dd s u dkd s uk (x) dkd s uk(x), f hom ( ) := inf f(y, + r (y)) dy : 2 W, # (Y ), (f hom) f hom (t ) ( ) := lim, Y t! t and Du = rul N b + D s u is the Radon-Nikodym decomposition of Du with respect to the N-dimensional Lebesgue measure L N. We recall (see [24]) that {F " } ">0 -converges, as "! 0 + and with respect to the strong topology of L (), to the functional F 0 if for all u 2 L (), where F 0 (u) = n lim inf F "(u) := inf "!0 + lim sup "!0 + F " (u) := inf lim inf F "(u) = lim sup F " (u), "!0 + "!0 + F " (u " ): u " 2 L (), u "! u in L () o lim inf "!0 n + F "(u " ): u " 2 L (), u "! u in L (), o lim sup. "!0 + Moreover, under the coercivity condition in (B3), if we consider the analogous functional I " of [3], i.e., the functional I " (u) := F " (u) + R v u p dx, for u 2 L (), where F " is as in Theorem B, and v and p are as in Theorem A, then, Lipschitz and using the continuous injection of BV () in L p (), lim inf I "(w) = lim inf I "(w) = "!0 + w2l () "!0 + w2w, () min I 0(w) = w2l () min w2bv () Ih (w), where I 0 (w) := F 0 (w) + R v w p dx, I h (w) := F h (w) + R v w p dx, and F 0 and F h were introduced in Theorem B. In particular, if f satisfies conditions (A), (A2) and (A3), then I h (u) = I sc (u, µ), where I sc and (u, µ) 2 BV () M(; BV # (Y )) are as in the statement of Theorem A. The proof of Theorem B relies on integral functionals of measures and their formulation by duality, while, as we mentioned before, the proof of Theorem A is based on the two-scale convergence method and is very similar to that of [, Thm. 3.3] in which the subdi erentiability of f and the regularity and boundedness of r f play a crucial role. In particular, the arguments used in [3] do not apply neither under weaker regularity hypotheses than those in (A2) nor under more general linear estimates from above and from below than those in (A3). Some questions then naturally arise: Is it possible to derive the two-scale homogenized functional under weaker hypotheses than those considered in [3]? May we establish the relation between the two-scale homogenized functional I sc and the homogenized functional I hom in a systematic and direct way? How to generalize this analysis to the case of multiple microscales? And to the vectorial case? The goal of this paper is precisely to give answers to these questions. We start by recalling the notion of (n + )-convergence for sequences of Radon measures introduced in [3] for n =, and generalized in [30] for any n 2 N. Let d, m, n, N 2 N, let R N be an open set, and set Y := (0, ) N. Let %,..., % n be positive functions on (0, ) such that for all i 2 {,, n} and for all j 2 {2,, n}, lim % % j (") i(") = 0, lim = 0. (.) "!0 + "!0 + % j (") 3

4 Definition.. Let {µ " } ">0 M(; R m ) be a sequence of Radon measures with finite total variation on. We say that {µ " } ">0 (n + )-scale converges to a Radon measure µ 0 2 C 0 (; C # (Y Y n ; R m )) 0 ' M y# ( Y Y n ; R m ) with finite total variation in the product space Y Y n, where each Y i is a copy of Y, if for all ' 2 C 0 (; C # (Y Y n ; R m )) we have x lim ' x, "!0 + % ("),, x dµ " (x) = '(x, y,, y n ) dµ 0 (x, y,, y n ), % n (") Y Y n in which case we write µ " (n+)-sc " *µ 0. This notion of convergence is justified by a compactness result, which asserts that every bounded sequence in M(; R m ) admits a (n + )-scale converging subsequence (see [30, Thm 3.2]). The (usual) weak-? limit in M(; R m ) is the projection on of the (n + )-scale limit, and so the latter captures more information on the oscillatory behavior of a bounded sequence in M(; R m ) than the former (see [30, Prop. 3.3]). This leads us to the study of the asymptotic behavior with respect to the (n + )-scale convergence of first order derivatives functionals with linear growth of the form F " (u) := f for u 2 BV (; R d ), where x % ("),, x % n ("), ru(x) dx + f (y,, y n, ) := lim sup t! f x % ("),, x % n ("), dd s u dkd s uk (x) dkd s uk(x) (.2) f(y,, y n, t ) t is the recession function of a real valued function f : R nn R d N! R, separately periodic in the first n variables. We start by characterizing the (n + )-scale limits of u " L N b, Du "b ">0 M(; R d ) M(; R d N ), whenever {u " } ">0 is a bounded sequence in BV ; R d. Definition.2. For i 2 N, define the space M? Y Y i ; BV # Y i ; R d of all BV # Y i ; R d -valued Radon measures µ 2 M Y Y i ; BV # Y i ; R d with finite total variation, for which there exists a R d N -valued Radon measure 2 M y# Y Y i ; R d N, with finite total variation in the product space Y Y i, such that for all B 2 B( Y Y i ), E 2 B(Y i ), D yi (µ(b)) (E) = (B E). We say that is the measure associated with D yi µ. We refer the reader to [30] for more detailed considerations on the space M? Y Y i ; BV # Y i ; R d, i 2 N. The following result holds (see [30, Thm..0]). Theorem.3. Let {u " } ">0 BV (; R d? ) be such that u " * u weakly-? in BV (; R d ) as "! 0 +, for some u 2 BV (; R d ). Assume that, in addition to satisfying (.), the length scales %,..., % n are well separated, i.e., there exists m 2 N such that for all i 2 {2,, n}, Then m %i (") lim = 0. (.3) "!0 + % i (") % i (") a) u " L N b (n+)-sc " * u, where u 2 M y# Y Y n ; R d is the measure defined by u := u L N b L nn y,,y n, 4

5 i.e., if ' 2 C 0 ; C # Y Y n ; R d then h u, 'i = '(x, y,, y n ) u(x) dxdy dy n. Y Y n b) there exist a subsequence {Du " 0} of {Du " } and, for all i 2 {,, n}, measures µ i 2 M? Y Y i ; BV # (Y i ; R d such that Du " 0 (n+)-sc " 0 * u,µ,,µ n, where u,µ,,µ n 2 M y# Y Y n ; R d N is the measure given by nx u,µ,,µ n := Du b L nn y,,y n + i= i L (n y i)n i+,,y n + n, (.4) i.e., if ' 2 C 0 ; C # Y Y n ; R d N then h u,µ,,µ n, 'i = '(x, y,, y n ) : ddu(x)dy dy n Y Y n nx + '(x, y,, y n ) : d i (x, y,, y i )dy i+ dy n i= Y Y n + '(x, y,, y n ) : d n (x, y,, y n ), Y Y n and each i 2 M y# Y Y i ; R d N is the measure associated with D yi µ i, i 2 {,, n}. Using Theorem.3, we seek to characterize and relate the functionals n o F sc (u, µ,, µ n ) := inf lim inf F "(u " ): u " 2 BV ; R d (n+)-sc, Du " "!0 + " * u,µ,...,µ n (.5) and n o F hom (u) := inf lim inf F "(u " ): u " 2 BV ; R d?, u " *" u weakly-? in BV ; R d "!0 + for u 2 BV ; R d and µ i 2 M? Y Y i ; BV # (Y i ; R d, i 2 {,, n}, where F " is given by (.2). Before we state our main result, we introduce some notation. Fix k 2 N and let g : R kn R d N! R be a Borel function. We recall that the e ective domain of g, dom e g, is the set dom e g := (y,, y k, ) 2 R kn R d N : g(y,, y k, ) <, while the conjugate function of g is the function g : R kn R d N! R defined by g (y,, y k, ) := (.6) sup : g(y,, y k, ), y,..., y k 2 R N, 2 R d N, (.7) 2R d N and the biconjugate function of g is the function g : R kn R d N! R defined by g (y,, y k, ) := sup : g (y,, y k, ), y,..., y k 2 R N, 2 R d N. (.8) 2R d N 5

6 We define a function g homk : R (k )N R d N! R by setting g homk (y,, y k for y,..., y k 2 R N, 2 R d N. Feb 25, 20, ) := inf g(y,, y k Y k, y k, + r k (y k )) dy k : k 2 W, # Y k; R d (.9) Let f : R nn R d N! R be a Borel function. If n =, we set f hom := f hom, where f hom is given by (.9) for k = and with g replaced by f, that is, f hom ( ) := inf f(y, + r (y )) dy : 2 W, # Y ; R d. Y If n = 2, we define f hom := (f hom2 ) hom, which is the function given by (.9) for k = and with g replaced by f hom2, where the latter is the function given by (.9) for k = 2 and with g replaced by f. Precisely, where f hom ( ) := inf f hom2 (y, + r (y )) dy : 2 W, # Y Y ; R d, f hom2 (y, ) := inf f(y, y 2, + r 2 (y 2 )) dy 2 : 2 2 W, # Y 2 Y 2; R d. Similarly, if n = 3 we define f hom := (f hom3 ) hom2 hom, i.e., f hom ( ) := inf (f hom3 ) hom2 (y, + r (y )) dy : 2 W, # Y Y ; R d, where with (f hom3 ) hom2 (y, ) := inf f hom3 (y, y 2, + r 2 (y 2 )) dy 2 : 2 2 W, # Y 2 Y 2; R d, f hom3 (y, y 2, ) := inf f(y, y 2, y 3, + r 3 (y 3 )) dy 3 : 3 2 W, # Y 3 Y 3; R d. Recursively, for n 2 N we set f hom := (f homn ) homn. (.0)... hom Consider the following conditions: (F) for all 2 R d N, f(, ) is Y Y n -periodic; (F2) for all y,..., y n 2 R N, f(y,, y n, ) is convex; (F3) there exists C > 0 such that for all y,..., y n 2 R N, 2 R d N, f(y,, y n, ) 6 C( + ); (F4) for all > 0 there exist c 2 R N, b 2 R, such that c! 0 as! 0 +, and for all y,..., y n 2 R N, 2 R d N, f(y,, y n, ) + c + b > 0; (F4) there exists C > 0 such that for all y,..., y n 2 R N, 2 R d N, f(y,, y n, ) > C; C 6

7 (F5) for every y, 0..., yn 0 2 R N, > 0, there exists = (y, 0, yn, 0 ) such that for all y,..., y n 2 R N with (y, 0, yn) 0 (y,, y n ) 6, and for all 2 R d N, f(y, 0, yn, 0 ) f(y,, y n, ) 6 ( + ); (F6) for all > 0 there exists ã 2 L # (Y Y n ) such that kã k L # (Y Yn)! 0 as! 0 +, and there exits > 0 such that for all y,..., y n, y, 0..., yn 0 2 R N with (y,, y n ) (y, 0, yn 0 ) 6, and for all y n, 2 R d N, f(y,, y n, y n, ) > ã (y 0,, y 0 n, y n ) + ( + o())f(y 0,, y 0 n, y n, ) (as! 0 + ). If n > 3, then we assume in addition that for a.e. y n, y n 2 R N we have ã (, y n, y n ) 2 C # (Y Y n 2 ) with kã (, y n, y n )k C# (Y Y n 2) 2 L (Y n Y n ); (F7) there exist 2 (0, ) and L, C > 0, such that for all y,..., y n 2 R N, for all 2 R d N with =, and for all t > L, f f(y,, y n, t ) (y,, y n, ) 6 C t t ; (F8) the conjugate function f of f is a bounded function on its e ective domain, dom e f. The next proposition will be used to establish integral representations for the multiple-scale functional F sc in (.5) and for the homogenized functional F hom in (.6). Proposition.4. Let f : R nn R d N! R be a Borel function satisfying hypotheses (F), (F3) and (F4). For > 0, let f be the function defined by f (y,, y n, ) := f(y,, y n, ) +. Then, (i) For all y,..., y n 2 R N, 2 R d N, the limit lim!0 +((f ) ) (y,..., y n, ) =: ((f 0 +) ) (y,..., y n, ) (.) exists, ((f 0 +) ) : R nn R N! R is positively -homogeneous and convex in the last variable, and (f ) 6 ((f 0 +) ) 6 (f ). Furthermore, if in addition a) f also satisfies (F2), then ((f 0 +) ) f ; b) d = and f also satisfies (F7), then ((f 0 +) ) (f ). (ii) For all 2 R N, the limit lim ((f ) ) hom ( ) =: ((f0 +) ) hom ( ) (.2)!0 + exists, with ((f 0 +) ) hom : R N! R positively -homogeneous, convex, and such that (f ) hom 6 ((f0 +) ) hom 6 ((f0 +) ) hom 6 (f ) hom. Furthermore, if in addition a) f also satisfies (F2) and (F8), then ((f 0 +) ) hom (fhom ) = (f ) hom ; b) f also satisfies (F2) and (F7), then ((f 0 +) ) hom (f ) hom ; c) d = and f also satisfies (F7), then ((f 0 +) ) hom (f ) hom. 7

8 Remark.5. Hypothesis (F7) is common within variational problems with linear growth conditions (see, for example, [4, Sect. 4], [9]). We will prove (see Lemma 3.2 below) that under hypotheses (F), (F3), (F4) and (F7), we have (f hom ) = (f ) hom ; in the scalar case, these conditions also ensure the equality (f ) = (f ). Other su cient conditions to guarantee that (f hom ) = (f ) hom are (F) (F4) and (F8) (see Lemma 3. below), which is an hypothesis on f that is often considered when dealing with duality problems (see, for example, [42, Ch. II.4]). Unless stated otherwise, we will always assume that the length scales %,..., % n satisfy (.) and (.3). A simple example of such functions is the case in which for all i 2 {,, n}, % i = " i. Our main result is the following. Theorem.6. Let R N be an open, bounded set Lipschitz, let Y i := (0, ) N, i 2 {,, n}, and let f : R nn R d N! R be a Borel function satisfying (F) (F4), (F5) and (F6). Then, for all (u, µ,, µ n ) 2 BV ; R d M? ; BV # Y ; R d M? Y Y n ; BV # Y n ; R d, F sc (u, µ,, µ n ) = + f Y Y n Y Y n f Moreover, for all u 2 BV ; R d, y,, y n, y,, y n, d ac u,µ,...,µ n dl (x, y,, y (n+)n n ) dxdy dy n d s u,µ,...,µ n dk s u,µ,...,µ n k (x, y,, y n ) dk s u,µ,...,µ n k(x, y,, y n ). (.3) F hom (u) = inf µ 2M?(;BV # (Y ;R d )),..., µ n 2M?( Y Y n ;BV # (Yn;R d )) F sc (u, µ,, µ n ) = f hom (ru(x)) dx + (f 0+,hom) dd s u dkd s uk (x) dkd s uk(x), (.4) where (f 0+,hom) := ((f 0 +) ) hom is the function defined by (.2) (note that in view of (F2), (f ) f ). Furthermore, if in addition (i) f satisfies one of the two conditions (F4) or (F8), then (f 0+,hom) (f hom ) ; (ii) f satisfies (F7), then (f 0+,hom) (f ) hom. We remark that in Theorem.6 we do not assume coercivity nor boundedness from below of f. The main ingredients of the proof are the unfolding operator (see [9], [2]; see also [3]) and Reshetnyak s continuityand lower semicontinuity-type results. The approach via the unfolding operator, in connection with the notion of two-scale convergence and in the framework of homogenization problems, sometimes referred as periodic unfolding method, has already been adopted by other authors in the Sobolev setting (see, for example, [9], [20], [3]). We use the convexity hypothesis (F2) when establishing the lower bound for the infimum defining F sc, which is based on a sequential lower semicontinuity argument. We start by proving that the (n+)-scale convergence of a sequence of measures absolutely continuous with respect to the Lebesgue measure is equivalent to the weak-? convergence on the product space Y Y n in the sense of measures of the unfolded sequence, i.e., the image through the unfolding operator of the original sequence (see Lemma 3.4). Then we prove that the energy F " does not increase by means of the unfolding operator (see Lemma 3.2). In order to conclude we need sequential lower semicontinuity of the functional F ( ) := + Y Y n f Y Y n f d ac y,, y n, dl (x, y,..., y (n+)n n ) dxdy dy n y,, y n, d s dk s k (x, y,..., y n ) 8 dk s k(x, y,..., y n )

9 for 2 M y# ( Y Y n ; R d N ), with respect to weak-? convergence in the sense of measures, which requires convexity of f in the last variable (see, for example, [4]). In the scalar case d = we can overcome this di culty by a relaxation argument with respect to the weak topology of W, (), which cannot be applied in the vectorial case since quasiconvexity is a weaker condition than convexity (see, for example, [22]). As a corollary of Theorem.6, we obtain the following result concerning the scalar case d =. Corollary.7. Let R N be an open and bounded set Lipschitz, let Y i := (0, ) N, i 2 {,, n}, and let f : R nn R N! R be a Borel function satisfying conditions (F), (F3), (F4), (F5) and (F6) with d = and with o() replaced by o() in (F6). Then, for all (u, µ,, µ n ) 2 BV () M? ; BV # (Y )) M? ( Y Y n ; BV # (Y n )), F sc (u, µ,, µ n ) = y,, y n, d ac u,µ,...,µ n dl (x, y,, y (n+)n n ) dxdy dy n d s u,µ y,, y n,,...,µ n dk s u,µ,...,µ n k (x, y,, y n ) dk s u,µ,...,µ n k(x, y,, y n ), Y Y n f + ((f 0 +) ) Y Y 2 where ((f 0 +) ) is the function defined by (.). Moreover, for all u 2 BV (), (.5) F hom (u) = inf µ 2M?(;BV # (Y )),..., µ n 2M?( Y Y n ;BV # (Yn)) = (f ) hom (ru(x)) dx + F sc (u, µ,, µ n ) ((f 0 +) ) hom dd s u dkd s uk (x) dkd s uk(x), (.6) where ((f 0 +) ) hom is the function defined by (.2). Furthermore, if in addition (i) f satisfies the coercivity condition (F4), then ((f 0 +) ) (f ) and ((f 0 +) ) hom (f ) hom ; (ii) f satisfies (F7), then ((f 0 +) ) (f ) and ((f 0 +) ) hom (f ) hom. Remark.8. (Comments on the hypotheses) (i) If f is bounded from below, then (F4) is satisfied: it su ces to take c 0 and b b, where b := inf f 2 R. Hypothesis (F4) may be regarded as a stronger version of the condition (F4)? for all > 0 there exists b 2 R such that for all y,..., y n, 2 R N, f(y,, y n, ) + + b > 0, so f cannot decrease as but it can decrease as with 2 (0, ): If f : R nn R d N! [0, ) is a nonnegative function, and b 2 R, c > 0, then for all 2 (0, ), f(y,, y n, ) := f(y,, y n, ) c + b is a function satisfying (F4)?. We do not assume (F4)? in place of (F4) in Theorem.6 and Corollary.7 because in general the former is not inherit neither by f hom nor by f from f, whereas the latter is. We observe that if f is lower semicontinuous and independent of (y,, y n ), then f satisfies (F4)? if, and only if, it satisfies f( ) lim inf > 0. (.7)! Moreover, if f is in addition convex, then (.7) is a necessary and su cient condition for the sequentially lower semicontinuity with respect to weak-? convergence in the sense of measures of the functional u 2 L ; R d N 7! f(u(x)) dx. 9

10 Furthermore, (.7) yields d ac lim inf f(u " (x)) dx > f "!0 + dl N (x) dx + f d s dk s k (x) dk s k(x) whenever u " L N? b * weakly-? in M ; R d N (see [32, Thm. 5.2]). This fact will be used when establishing (.4) and (.6). (ii) If f satisfies a growth condition of the form f(y,, y n, ) 6 C( + ) and is convex in the last variable, then (see []) (F5) holds if, and only if, the function f : R nn R d N [0, )! R defined by f(y,, y n,, t) := tf y,, y n, t if t > 0, f (y,, y n, ) if t = 0, is continuous. In particular, if f is continuous, positively -homogeneous in the last variable, and satisfies (F2), (F3), and (F4)?, then it also satisfies (F5) since in this setting f is continuous. The continuity of f will be crucial in our analysis in order to apply Reshetnyak s continuity- and lower semicontinuity-type results (see Lemmas 3.5 and 3.6 below). (iii) Hypothesis (F6) is a weaker version of the hypothesis (F6) there exist a continuous, positive function! satisfying!(0) = 0, and a function a 2 L # (Y n) such that for all y,..., y n, y 0,..., y 0 n, y n, 2 R d N, we f(y,, y n, y n, ) f(y 0,, y 0 n, y n, ) 6!( (y,, y n ) (y 0, y 0 n ) ) a(y n ) + f(y,, y n, ), which often appears in the literature (see, for example, [6], [4]). If f is of the form f(y,, y n, ) := g(y,, y n )h(y n, ), where g is a continuous and Y Y n - periodic function, and h is a function satisfying (F) (F5), then f satisfies (F) (F6); in particular, we may consider g, which corresponds to the case of one microscale (i.e., n = ) and so, in this situation, (F6) is trivially satisfied. Other simple examples of functions satisfying (F) (F6) are functions of the form f(y,, y n, ) := g(y,, y n )h( ), where g is continuous and Y Y n -periodic, and h satisfies (F2) (F4). Remark.9. (i) Equalities (.3) and the first one in (.4) are valid under the more general growth condition from below (F4)? (introduced in Remark.8 (i)). The reason why this condition is not enough in order to conclude the second equality in (.4) is that in general it is not inherited by f hom, while (F4) is and this ensures that f hom satisfies (.7), which, as we will see, will play a crucial role in the proof. (ii) In Theorem.6 and Corollary.7, we need the length scales to satisfy condition (.3) only to establish the equalities (.4) and (.6) involving F hom. In the case in which n = and d =, we recover Amar s integral representation [3] of the two-scale homogenized functional F sc under more general conditions (see Remark.8 (ii) and (iii)). Furthermore, if we assume a priori compactness of a diagonal infimizing sequence for the sequence of functionals {F " } ">0, we recover Amar s result [3] under more general conditions. We observe that even if a priori compactness of a diagonal infimizing sequence is assumed in Theorem A, the coercivity condition is still needed to validate the arguments in [3]. We also recover Bouchitté s integral representation [2] of the e ective energy F hom without assuming coercivity of f and without assuming convexity of f in the second variable, but assuming continuity in the first one in order to apply Reshetnyak Continuity Theorem, while in [2] f is assumed to be convex in the second variable and coercive, but only measurable and Y -periodic in the first variable. If n = and d > in Theorem.6, then we recover De Arcangelis and Gargiulo s integral representation [26] of the e ective energy F hom without assuming f to be bounded from below, but assuming f to be 0

11 continuous in the first variable and convex in the second one, while in [26] f is only required to be nonnegative, measurable and Y -periodic in the first variable and continuous in the second one. As we mentioned before, our hypotheses are related to the periodic unfolding method and Reshetnyak Continuity Theorem s hypotheses. In the case in which n > 2, Theorem.6 and Corollary.7 provide new results in the literature in that, to the best of our knowledge, the homogenization of nonlinear periodically oscillating functionals with linear growth and characterized by n > 2 microscales has not yet been carried out. Finally, in the framework of homogenization by -convergence in the BV setting and for n = we also mention the works by Braides and Chiatò Piat [5] and Carbone, Cioranescu, De Arcangelis and Gaudiello [7] concerning the convex case; and Bouchitté, Fonseca and Mascarenhas [4, Sect. 4.3], Attouch, Buttazzo and Michaille [7, Sect. 2.3] and Babadjian and Millot [8] regarding the nonconvex case. This paper is organized as follows. In Section 2, we collect the necessary notation and we recall some basic properties of (R m -valued) Radon measures and of functions of bounded variation. We also recall some results established in [30] that will be used in the subsequent sections. In Section 3 we prove Proposition.4 and Theorem.6, and in Section 4 we prove Corollary Notation and preliminaries 2.. Notation In the sequel is a -compact separable metric space, is an open subset of R N, N 2 N, and Y := (0, ) N is the reference cell. For each i 2 N, Y i stands for a copy of Y. Given x 2 R N, we write [x] and hxi to denote the integer and the fractional part of x componentwise, respectively, so that x = [x] + hxi and [x] 2 N, hxi 2 Y. Let n, m 2 N. If x, y 2 R m, then x y stands for the Euclidean inner product of x and y, and x := p x x for the Euclidean norm of x. The space of (m n)-dimensional matrices will be identified with R mn, and we write R m n. If = ( ij ) 6i6m,6j6n, = ( ij ) 6i6m,6j6n 2 R m n, then : := mx i= j= nx ij ij represents the inner product of and, while := p : denotes the norm of. If a 2 R m and b 2 R n, then a b stands for the (m n)-dimensional rank-one matrix defined by a b := (a i b j ) 6i6m,6j6n. Let g : R nn! R m be a function. We denote the Lipschitz constant of g on a set D R nn by Lip(g; D); if D coincides with the domain of g we omit its dependence. We say that g is Y Y n -periodic if for all i 2 {,, n}, apple 2 N, y,..., y n 2 R N, one has g(y,, y i + apple,, y n ) = g(y,, y i,, y n ). We will consider the Banach spaces C # (Y Y n ; R m ) := g 2 C(R nn ; R m ): g is Y Y n -periodic endowed with the supremum norm k k, and C 0 (; C # (Y Y n ; R m )), which is the closure with respect to the supremum norm k k of C c (; C # (Y Y n ; R m )). The latter is the space of all functions g : R nn! R m such that for all z 2, g(z, ) 2 C # (Y Y n ; R m ) and for all y,..., y n 2 R N, g(, y,..., y n ) 2 C c (; R m ). The spaces C k # (Y Y n ; R m ), C # (Y Y n ; R m ), C k c (; C k # (Y Y n ; R m )), C c (; C # (Y Y n ; R m )), C k 0 (; C k # (Y Y n ; R m )) and C 0 (; C # (Y Y n ; R m )) are now defined in an obvious way. If m = the co-domain will often be omitted (e.g., we write C 0 () instead of C 0 (; R)). C represents a generic positive constant, whose value may change from expression to expression, and " stands for a positive small parameter, often considered as taking its values on a positive sequence converging to zero; in this case, " 0 represents a subsequence of ", and we write " 0 ".

12 2.2. Measure theory For m 2 N, the m-dimensional Lebesgue measure is denoted by L m. Feb 25, 20 The Borel -algebra on is denoted by B(), and M(; R m ) is the Banach space of all Radon measures : B()! R m endowed with the total variation norm k k. If ' 2 C 0 () and = (,, m) 2 M(; R m ), then we set '(z) d (z) := '(z) d (z),, If ' = (',, ' m ) 2 C 0 (; R m ) and 2 M(; R), then we define '(z) d (z) := ' (z) d (z),, '(z) d m (z). ' m (z) d (z). We write M # (Y Y n ; R m ) and M y# ( Y Y n ; R m ) to denote the duals of C # (Y Y n ; R m ) and C 0 (; C # (Y Y n ; R m )), respectively. Let E R n be a Borel set and let µ : B(E)! [0, ] be a positive Radon measure. If 2 M(E; R m ), then (see for example, [32]) by Lebesgue Decomposition Theorem we can decompose as = ac + s = d ac dµ µ be + s, where ac is absolutely continuous with respect to µ, s and µ are mutually singular The space of functions of bounded variation A function u :! R d, d 2 N, is said to be a function of bounded variation, and we write u 2 BV ; R d, if u 2 L ; R d and its distributional derivative Du belongs to M ; R d N, that is, if there exists a measure Du 2 M ; R d N such that for all 2 C c (), j 2 {,, d} and i 2 {,, N} one has u j (x) dx = (x) dd i u j i where u = (u,, u d ) and Du j = (D u j,, D N u j ). The space BV ; R d is a Banach space when endowed with the norm kuk BV (;Rd ) := kuk L (;R d ) + kduk(). We will also consider the space BV # Y ; R d := u 2 BV loc R N ; R d : u is Y -periodic, endowed with the norm of BV Y ; R d. Notice that if u 2 BV # Y ; R d, then Du 2 M # Y ; R d N. We will consider the weak-? convergence in BV ; R d. We recall that {u j } j2n BV ; R d is said to weakly-? converge in BV ; R d to some u 2 BV ; R d if u j! u (strongly) in L ; R d? ) and Du j * Du weakly-? in M ; R d N. If u 2 BV ; R d, then Du = rul N b + D s u is the Radon-Nikodym decomposition of Du with respect to L N b Some preliminary results We start this subsection by providing a simple example of a measure in the space M? Y Y i ; BV # Y i ; R d, i 2 N, introduced in Section. For simplicity, assume i =, and let 2 M(; R) be a real-valued Radon measure with finite total variation, let v 2 BV # Y ; R d, and consider the mapping µ : B 2 B() 7! (B)v 2 BV # Y ; R d. Then µ 2 M ; BV # Y ; R d, and := Dv 2 M y# Y ; R d N is the measure associated with D y µ : B 2 B() 7! D y (µ(b)) = (B)Dv in the sense of Definition.2, that is, h, 'i := R Y '(x, y) d (x)ddv(y), ' 2 C 0 ; C # Y ; R d N. We refer the reader to [30] for more detailed considerations on the space M? Y Y i ; BV # Y i ; R d, i 2 N. The next result shows that Theorem.3 fully characterizes the (n + )-scale limit of bounded sequences in BV ; R d (see [30, Prop..]). 2

13 Proposition 2.. Let u 2 BV ; R d and for i 2 {,, n}, let µ i 2 M? Y Y i ; BV # Y i ; R d. Then there exists a bounded sequence {u " } ">0 BV ; R d for which a) and b) of Theorem.3 hold (with " 0 replaced by "). Remark 2.2. Since every (n + )-scale convergent sequence in M(; R m ) is also a weakly-? convergent sequence in the sense of measures (see [30, Prop. 3.3]), it follows that any such sequence is bounded in M(; R m ). We now recall a density type result proved in [30, Prop. 3.4], which will play an important role in the proof of our main results. Proposition 2.3. Let R N be an open and bounded set such is Lipschitz. Let u 2 BV ; R d, and for each i 2 {,, n}, let µ i 2 M? Y Y i ; BV # Y i ; R d. Then there exist sequences {u j } j2n C ; R d and { (i) j } j2n Cc ; C# Y Y i ; R d satisfying? u j *j u weakly-? in BV (; R d ), lim j! ru j (x) dx = kduk(), ru j + nx i= r (i) yi j L (n+)n? b Y Y n *j? i= * j u,µ,...,µ n weakly-? in M y# Y Y n ; R d N, nx lim ru j (x) + r (i) yi j! j (x, y,, y i ) dxdy dy n Y Y n where u,µ,...,µ n is the measure defined in (.4), and = k u,µ,...,µ n k( Y Y n ), (2.) j? * u,µ,,µ n weakly-? in M y# Y Y n ; R d N R, lim k jk( Y Y n ) = k u,µ j!,,µ n k( Y Y n ), (2.2) where, for any B 2 B( Y Y n ), j(b) := B ru j (x) + nx i= u,µ,,µ n (B) := u,µ,,µ n (B), L (n+)n (B). r (i) yi j (x, y,, y i ) dxdy dy n, L (n+)n (B), Finally, we recall that in view of Riemann-Lebesgue s Lemma, if ' 2 C(; C # (Y Y n ; R m )) then ', % ("),,?* '(, y,, y n ) dy dy n (2.3) % n (") Y Y n weakly-? in L loc (; Rm ). In particular, if ' 2 C 0 ; C # Y Y n ; R d L (; R m ). then (2.3) holds weakly-? in Also, if a : R nn! R is a Y Y n -periodic function such that for some 6 p 6 and for a.e. y n 2 Y n we have a(, y n ) 2 C # (Y Y n ) and ka(, y n )k C# (Y Y n ) 2 L p (Y n ), then (see [27]) ( a %,, (") % n(") * ā weakly in L p loc (RN ) if 6 p <, a %,,? (") % n(") * ā weakly-? in L loc (RN ) if p =, (2.4) where ā := R Y Y n a(y,, y n ) dy... dy n. 3

14 3. Proof of Theorem.6 Throughout this section we will assume that n = 2. The cases in which n = or n > 3 do not bring any additional technical di culties. For n = 2 the energies F " in (.2) take the form x F " (u) := f % ("), x % 2 ("), ru(x) dx + f x % ("), x % 2 ("), dd s u dkd s uk (x) dkd s uk(x) (3.) for u 2 BV ; R d, where, we recall, %, % 2 : (0, )! (0, ) are functions satisfying (.) (with n = 2) and f is the recession function associated with f. Due to the convexity hypothesis (F2), the limit superior defining f is actually a limit (see, for example, [32]), so that f : R N R N R d N! R is given by by f f(y, y 2, t ) (y, y 2, ) := lim t! t Moreover, under hypotheses (F) (F3) and (F4)? on f, we have that f is a Borel function satisfying (F), (F2), and the growth condition 0 6 f (y, y 2, ) 6 C. (3.2) Notice that in view of (F3), (F4)? and (3.2), the functional F " is well defined (in R) for every u 2 BV ; R d. In Theorem 3. below we will establish (.3). We will use the unfolding operator (see [9], [2]; see also [3]): For % > 0, T % : L (; R m )! L (R N ; L # (Y 2; R m )) is defined by h x i T % (g)(x, y 2 ) := g % + %(y 2 [y 2 ]) % for x, y 2 2 R N, g 2 L (; R m ), where g is the extension by zero of g to R N. Clearly T % is linear, and for every g 2 L (; R m ) and (see [3, Prop. A.]). kt % (g)k L ( Y 2;R m ) 6 kt % (g)k L (R N Y 2;R m ) = k gk L (R N ;R m ) = kgk L (;R m ), (3.3) lim %!0 + g(x) R N Y 2 T % (g)(x, y 2 ) dxdy 2 = 0 (3.4) Similarly, we define the operator A % : L ( Y 2 ; R m )! L (R N ; L # (Y ; L (Y 2 ; R m ))) by A % (h)(x, y, y 2 ) := h h x i % + %(y [y ]), y 2 = T % (h(, y 2 ))(x, y ) for x, y 2 R N, y 2 2 Y 2, h 2 L ( Y 2 ; R m ), % where h is the extension by zero of h to R N Y 2. A % is linear, and for all h 2 L ( Y 2 ; R m ), ka % (h)k L ( Y Y 2;R m ) 6 ka % (h)k L (R N Y Y 2;R m ) = k hk L (R N Y 2;R m ) = khk L ( Y 2;R m ) (3.5) by (3.3) and Fubini s Theorem. Moreover, we notice that for a.e. y 2 2 Y 2, we have lim h(x, y2 ) T % (h(, y 2 ))(x, y ) dxdy = 0 %!0 + R N Y by (3.4), and R N Y R N h(x, y2 ) T % (h(, y 2 ))(x, y ) dxdy 6 2 h(x, y2 ) dx 2 L (Y 2 ), 4

15 where we used (3.3) to obtain T % (h(, y 2 ))(x, y ) dxdy = R N Y R N h(x, y2 ) dx. Thus, Lebesgue Dominated Convergence Theorem yields lim %!0 + R N Y Y 2 h(x, y2 ) A % (h)(x, y, y 2 ) dxdy dy 2 = lim %!0 + R N Y Y 2 h(x, y2 ) T % (h(, y 2 ))(x, y ) dxdy dy 2 = 0. Theorem 3.. Let R N be an open, bounded set Lipschitz, let Y = Y 2 := (0, ) N, and let f : R N R N R d N! R be a Borel function satisfying conditions (F) (F3), (F4)?, (F5), (F6) for n = 2. Then (.3) holds (with n = 2). The proof of Theorem 3. is hinged on some lemmas. The first lemma unfolds the rapidly oscillating sequence. Lemma 3.2. Under the same hypotheses of Theorem 3., if {v " } ">0 L ; R d N is a bounded sequence then, for all > 0, x lim inf f "!0 + % ("), x % 2 ("), v "(x) dx > lim inf f y, y 2, A %(") T %2(")(v " ) (x, y, y 2 ) dxdy dy 2, "!0 + Y Y 2 (3.6) where f (y, y 2, ) := f(y, y 2, ) +. Proof. Fix > 0 and > 0. Let b 2 R be given by (F4)? (see Remark.8), and let ã 2 L # (Y Y 2 ) and > 0 be given by (F6). Then f (,, ) > b, (3.7) and, for all y, y 0, y 2 2 R N, 2 R d N such that y y 0 6, f (y, y 2, ) > ã (y 0, y 2 ) + ( + o())f (y 0, y 2, ) o() (as! 0 + ). (3.8) Set c := sup " kv " k L (;R d N ), " := % (") and " 2 := % 2 ("). Define Notice that "2 and, by (3.3), [ "2 := apple 2 N : " 2 (apple + Y 2 ) \ 6= ;, "2 := int " 2 (apple + Y 2 ). (3.9) apple2 "2 sup kt "2 (v " )k L (R N Y 2;R d N ) 6 c. (3.0) ">0 Recalling that ṽ " stands for the extension by zero to the whole R N of v ", using (F3), a change of variables and (F), in this order, we obtain x f, x, v " (x) dx = " " 2 "2 f > X apple2 "2 = X apple2 "2 x ", x " 2, ṽ " (x) dx " 2(apple+Y 2) Y 2 f "2 \ x f, x, 0 dx " " 2 x f, x, ṽ " (x) dx CL N "2 \ " " 2 "2 apple + " 2 y 2, y 2, ṽ " (" 2 apple + " 2 y 2 ) " N 2 dy 2 CL N "2 \. " " 5 (3.)

16 Since x " 2 = apple whenever x 2 "2 (apple + Y 2 ), L N (" 2 (apple + Y 2 )) = " N 2 and [y 2 ] = 0 for all y 2 2 Y 2, in view of the definition of T "2 (v " ), by Fubini s Theorem, and from (3.) we get x f, x, v " (x) dx " " 2 > X = > apple2 "2 "2 Y 2 f Y 2 f " 2(apple+Y 2) Y 2 f "2 " h x " 2 i + " 2 " y 2, y 2, ṽ " " 2 h x " 2 i + " 2 y 2 dy 2 dx CL N "2 \ "2 h x i + " 2 y 2, y 2, T "2 (v " )(x, y 2 ) dxdy 2 CL N "2 \ " " 2 " "2 h x i + " 2 y 2, y 2, T "2 (v " )(x, y 2 ) dxdy 2 (b + C)L N "2 \, " " 2 " where in the last inequality we used (3.7). By (.) there exists " > 0 such that for all 0 < " 6 " one has 0 < " 2 /" < /2 p N. For any such ", " h 2 x i sup + " 2 x " D 2 x E y 2 = sup + " 2 y 2 <, x2,y 22Y 2 " " 2 " " x2,y 22Y 2 " " 2 " thus (3.8) and (3.0) yield "2 h x i f + " 2 y 2, y 2, T "2 v " )(x, y 2 ) dxdy 2 Y 2 " " 2 " x >, y 2 dxdy o() " Y 2 ã Y 2 f x ", y 2, T "2 (v " )(x, y 2 ) dxdy 2 o() c. (3.2) (3.3) Defining " and " as in (3.9) (with " 2 and Y 2 replaced by " and Y, respectively), and reasoning as in (3.) (3.2), we conclude that x f, y 2, T "2 (v " )(x, y 2 ) dxdy 2 Y 2 " (3.4) > f y, y 2, A " T "2 (v " ) (x, y, y 2 ) dxdy dy 2 (b + C)L N " \. Y Y 2 By the Riemann Lebesgue Lemma we have that for a.e. y 2 2 Y 2, ã ( /", y 2 ) * R Y ã (y, y 2 ) dy weakly in L loc (RN ). Hence, x lim inf ã, y 2 dxdy 2 > L N () ã (y, y 2 ) dy dy 2, (3.5) "!0 + Y 2 " Y Y 2 where we have also used Fatou s Lemma and Fubini s Theorem. In view of (3.2) (3.5), we obtain x lim inf f, x, v " (x) dx "!0 + " " 2 > + o() lim inf f y, y 2, A " T "2 (v " ) (x, y, y 2 ) dxdy dy 2 "!0 + Y Y 2 + L N () ã (y, y 2 ) dy dy 2 o() c, Y Y 2 (3.6) where we also used the convergences L N " \, L N "2 \! 0 as "! 0 +, is Lipschitz and so L N (@) = 0. Finally, recalling that kã k L # (Y Y2)! 0 as! 0 +, passing (3.6) to the limit as! 0 + we get (3.6). 6

17 Remark 3.3. The previous proof can be easily generalized to the case in which n > 3 by using (2.4) in place of Riemann Lebesgue Lemma (see (3.5)). We now show that, similarly to what happens in the L p -case with p 2 (, ) (see [2, Prop. 2.4]), 3- scale convergence of a sequence of measures absolutely continuous with respect to the Lebesgue measure is equivalent to a weak-? convergence in the sense of measures in a product space of the unfolded sequence. Lemma 3.4. Let R N be open and bounded, let {v " } ">0 L ; R d N be a bounded sequence and let 2 M y# Y Y 2 ; R d N. Then v " L N b 3-sc " * if, and only if, A % (") T %2(")(v " ) L 3N? b Y Y 2 * weakly-? in M y# Y Y 2 ; R d N as "! 0 +. Proof. For > 0, define the sets [ W := apple 2 N : (apple + Y ), := int apple2w (apple + Y ). Take 2 C c (), 2 C # (Y ) and 2 2 C # Y 2; R d N, and let ' := 2. Set " := % (") and " 2 := % 2 ("). By (.) we can find " > 0 such that for all 0 < " 6 " one has dist(supp, \ " ) > 2" p N, dist(supp, \"2 ) > 2" p N. (3.7) Fix any such ". Using (3.7), the definition of A ", Fubini s Theorem, and the equalities x " = apple if x 2 " (apple + Y ) and [y ] = 0 if y 2 Y, in this order, we get '(x, y, y 2 ) : A " T "2 (v " ) (x, y, y 2 ) dxdy dy 2 Y Y 2 h x i = '(x, y, y 2 ) : T "2 (v " ) " + " (y [y ]), y 2 dxdy dy 2 " Y Y 2 " X = '(x, y, y 2 ) : T "2 (v " )(" apple + " y, y 2 ) dx dy dy 2. Y Y 2 apple2w " " (apple+y ) Performing the change of variables x = " apple + ", by Fubini s Theorem the last integral in (3.8) becomes Y Y 2 X apple2w " Considering now the change of variables y = x " Y Y 2 = = X Y Y 2 apple2w " (3.8) '(" apple + ", y, y 2 ) : T "2 (v " )(" apple + " y, y 2 ) " N dy ddy 2. (3.9) Y " (apple+y ) " ' " Y Y 2 ' apple, and using again Fubini s Theorem, (3.9) reduces to ' " apple + ", x apple, y 2 : T "2 (v " )(x, y 2 ) dx ddy 2 " " h x " i + ", x ", y 2 : T "2 (v " )(x, y 2 ) dx ddy 2 h x i " + " y, x, y 2 : T "2 (v " )(x, y 2 ) dxdy dy 2, " " where in the first equality we used the Y -periodicity of. We claim that if x 2 \ " [ \ "2 then (3.20) " h x " i + " Y \ supp = ;. (3.2) 7

18 In fact, if there was z 2 (" [ x " ] + " Y ) \ supp, then z = " [ x " ] + " y for some y 2 Y and, by (3.7), 2" p N < dist(supp, x) 6 z x = " h x " i + " y x = " D x " E + " y 6 2" p N, which is a contradiction. Hence, (3.2) holds. Consequently, h x i " + " y, x, y 2 : T "2 (v " )(x, y 2 ) dxdy dy 2 " " " Y Y 2 ' = "2 Y Y 2 ' h x i " + " y, x, y 2 : T "2 (v " )(x, y 2 ) dxdy dy 2. " " (3.22) Arguing as in (3.8) (3.20), we have "2 Y Y 2 ' = = = = = Y Y 2 Y Y 2 Y Y 2 Y Y 2 h x i " + " y, x, y 2 : T "2 (v " )(x, y 2 ) dxdy dy 2 " " X apple2w "2 " 2(apple+Y 2) h x i ' " + " y, x, y 2 : v " (" 2 apple + " 2 y 2 ) dx dy dy 2 " " X h "2 ' " apple + " i 2 + " y, " 2 apple + " 2, y 2 : v " (" 2 apple + " 2 y 2 ) " N 2 dy 2 dy d apple2w Y 2 " " " " "2 X h "2 ' " apple + " i 2 + " y, " 2 apple + " 2, x apple : v " (x) dx dy d apple2w " "2 2(apple+Y 2) " " " " " 2 apple "2 h x i ' " + " 2 + " y, " h 2 x i + " 2, x : v " (x) dx dy d "2 " " 2 " " " 2 " " 2 apple "2 h x i " + " 2 y 2 + " y, " h 2 x i + " 2 y 2, x : v " (x) dxdy dy 2, " " 2 " " " 2 " " 2 "2 Y Y 2 ' (3.23) where in the fourth equality we used the Y 2 -periodicity of 2. In view of (3.8) (3.20) and (3.22) (3.23), we conclude that '(x, y, y 2 ) : A " T "2 (v " ) (x, y, y 2 ) dxdy dy 2 Y Y 2 x = (a " (x, y, y 2 )) (b " (x, y 2 )) 2 : v " (x) dxdy dy 2, "2 Y Y 2 " 2 (3.24) where a " (x, y, y 2 ) := " apple "2 " h x " 2 i + " 2 " y 2 + " y, b " (x, y 2 ) := " 2 " h x " 2 i + " 2 " y 2, x, y, y 2 2 R N. Notice that for all x 2, y 2 Y and y 2 2 Y 2, a " (x, y, y 2 ) x 6 2 p N(" + " 2 ), b " (x, y 2 ) x " 6 2 p N " 2 " (3.25) 8

19 Using (3.24) and (3.7), we obtain '(x, y, y 2 ) : A " T "2 (v " ) (x, y, y 2 ) dxdy dy 2 ' x, x, x : v " (x) dx Y Y 2 " " 2 x = (a " (x, y, y 2 )) (b " (x, y 2 )) 2 : v " (x) dxdy dy 2 "2 Y Y 2 " 2 x x (x) 2 : v " (x) dxdy dy 2 "2 Y Y 2 " " 2 x 6 k 2 k L # (Y2;Rd N ) (a " (x, y, y 2 )) (b " (x, y 2 )) (x) v " (x) dxdy dy 2 Y Y 2 " x 6 k 2 k L # (Y2;Rd N ) k k L ()Lip( ) b " (x, y 2 ) 6 C " + " 2 + " 2, " Y Y 2 " + k k L # (Y) Lip( ) a " (x, y, y 2 ) x v " (x) dxdy dy 2 (3.26) where in the last inequality we used (3.25) and the fact that sup " kv " k L (;R d N ) <. Since functions of the form ' = 2 are dense in C 0 ; C # Y Y 2 ; R d N, and since A %(") T %2(")(v " ) L Y Y 2 ; R d N, {v " } L ; R d N are bounded sequences (see (3.3) and (3.5)), using a density argument, (.), and passing (3.26) to the limit as "! 0 +, we conclude that v " L N b 3-sc "* if, and only if, A %(") T %2(")(v " ) L 3N? b Y Y 2 * weakly-? in My# Y Y 2 ; R d N as "! 0 +. The next lemma is a Reshetnyak continuity type result for functions not necessarily positively -homogeneous, and similar to [35, Thm. 5] (see [25] for related results). Lemma 3.5. Let U R l be an open set such that L l (U) <. Let g : U R m! R be a function such that ḡ : U R m [0, )! R defined by ḡ(z,, t) := tg z, t if t > 0, g (z, ) if t = 0, (3.27) is continuous and bounded on U S m, where g (z, ) := lim sup t! g(z, t )/t is the recession function of g and S m is the unit sphere in R m R. If 2 M(U; R m ), let 2 M(U; R m R) denote the measure defined by ( ) := ( ), L l ( ). Assume that j, 2 M(U; R m ) are such that Then j? * j weakly-? in M(U; R m R), lim j! k jk(u) = k k(u). (3.28) lim g z, d ac j j! U dl l (z) = g z, d ac dl l (z) U dz + U g z, dz + g z, U d s j dk s j k(z) dk s jk(z) d s dk s k (z) dk s k(z). (3.29) Proof. Since ḡ is a continuous and bounded function on U S m, in view of (3.28) Reshetnyak Continuity Theorem (see [38], and also [5, Thm. 2.39]) yields lim ḡ z, j! U d j dk jk (z) dk jk(z) = ḡ z, U 9 d dk k (z) dk k(z). (3.30)

20 We claim that (3.30) reduces to (3.29). µ 2 M(U; R m ) with respect to L l as then µ = Feb 25, 20 In fact, writing the Lebesgue decomposition of an arbitrary µ = dµac dl l Ll bu + µ s, dµ ac dl l, L l bu + (µ s, 0), k µk = are the Lebesgue decomposition of µ and k µk with respect to L l, respectively. In view of the Besicovitch Derivation Theorem, for L l -a.e. z 2 U, we have and for kµ s k-a.e. z 2 U, we have ac dµ ac dl l, L l bu + kµ s k, (3.3) dµ d µ dk µk (z) = (z), dl l dµ ac (z),, (3.32) dl l d µ dµ s dk µk (z) = dkµ s k (z), 0. (3.33) From (3.3) (3.33), and taking into account the positive -homogeneity of (, t) 2 R m [0, ) 7! ḡ(z,, t), we deduce that d µ ḡ z, U dk µk (z) dk µk(z) = ḡ z, dµac U dl l (z), dµ s dz + ḡ z, U dkµ s k (z), 0 dkµ s k(z) = g z, dµac dl l (z) dz + g dµ s (3.34) z, dkµ s k (z) dkµ s k(z), U where in the last equality we used the definition of ḡ. By (3.34) we conclude that (3.30) reduces to (3.29). Next we prove a Reshetnyak lower semicontinuity type result for functions not necessarily positively - homogeneous (see also [23], [33]). Lemma 3.6. Let U R l be an open set such that L l (U) <. Let g : U R m! R be a function satisfying g(z, ) 6 C( + ), for some C > 0 and for every (z, ) 2 U R m, and such that for all z 2 U, g(z, ) is convex. Assume further that for all z 2 U and > 0, there exists = ( z, ) > 0 such that for all z 2 U with z z <, and 2 R m, we have g( z, ) g(z, ) 6 ( + ). If j, 2 M(U; R m ) are such that? j * j weakly-? in M(U; R m ) as j!, then lim inf g z, d ac j j! U dl l (z) dz + g d s j z, U dk s j k(z) dk s jk(z) > g z, d ac U dl l (z) dz + g d s (3.35) z, U dk s k (z) dk s k(z).? Proof. Let j, 2 M(U; R m ) be such that j * j weakly-? in M(U; R m R). Defining j, 2 M(U; R m R) as in Lemma 3.5, we see that j? * j weakly-? in M(U; R m R). Let ḡ : U R m R! R be the function introduced in (3.27). Then (see Remark.8 (ii)) ḡ is a continuous function, and ḡ(z,, t) 6 2C (, t) for all (z,, t) 2 U R m [0, ). Moreover, since for each i 2 N there exist functions a i : U! R and b i : U! R m such that U g(z, ) = sup i2n a i (z) + b i (z), g (z, ) = sup i2n b i (z), (see [32, Prop. 2.77]), we have that for all (z,, t) 2 U R m [0, ), ḡ(z,, t) = sup i2n a i (z)t + b i (z). 20

21 Thus for all z 2 U, (, t) 2 R m [0, ) 7! ḡ(z,, t) is convex and positively -homogeneous. So, Reshetnyak Lower Semicontinuity Theorem (see [38], and also [5, Thm. 2.38]) yields lim inf ḡ z, j! U d j dk jk (z) dk jk(z) > ḡ z, U Finally, we observe that by (3.34), (3.36) reduces to (3.35). d dk k (z) dk k(z). (3.36) Proof of Theorem 3.. Fix (u, µ, µ 2 ) 2 BV ; R d M? ; BV # Y ; R d M? Y ; BV # Y 2 ; R d, and set G(u, µ, µ 2 ) := y, y 2, d ac u,µ,µ 2 dl 3N (x, y, y 2 ) dxdy dy 2 We will proceed in two steps. Y Y 2 f + Step. We start by proving that Y Y 2 f d s u,µ y, y 2,,µ 2 dk s u,µ,µ 2 k (x, y, y 2 ) dk s u,µ,µ 2 k(x, y, y 2 ). F sc (u, µ, µ 2 ) > G(u, µ, µ 2 ). Let {" h } h2n be an arbitrary sequence of positive numbers converging to zero as h!, and by Proposition 2. let {u h } h2n BV ; R d 3-sc be a bounded sequence such that Du h " h * u,µ,µ 2. We claim that lim inf F " h (u h ) > G(u, µ, µ 2 ). (3.37) h! Since {Du h } h2n is bounded in M ; R d N (see Remark 2.2), in view of (F3), (F4)? and (3.2), we have that {F "h (u h )} h2n is bounded. Therefore, we may assume without loss of generality that the limit inferior in (3.37) is actually a limit and that this limit is finite (which is true up to a subsequence). By Proposition 2.3 (with µ i = 0), for each h 2 N we can find a sequence u (h) j (h) j where, for B 2 B(),? * j u h weakly-? in BV ; R d, u (h) j j2n W, ; R d such that? * j h weakly-? in M(; R d N R), lim j! k (h) j k() = k hk(), (3.38) (h) j (B) := B ru (h) j (x) dx, L N (B), h(b) := Du h (B), L N (B). Under hypotheses (F) (F3), (F4)?, (F5) (see also Remark.8 (ii)), it can be shown that for fixed h 2 N, x Lemma 3.5 applies to U := and g(x, ) := f( % (" h ), x % 2(" h ), ), which ensures the continuity of the functional with respect to the convergence (3.38), that is, lim j! F "h u (h) = F "h (u h ). Consequently, F "h lim lim F " h h! j! u (h) j j = lim h! F " h (u h ). (3.39) Moreover, given ' 2 C 0 ; C # Y Y 2 ; R d N lim lim ' x, h! j! x % (" h ), x % 2 (" h ) : ru (h) j we have x (x) dx = lim ' x, h! % (" h ), x : ddu h (x) % 2 (" h ) = '(x, y, y 2 ) : d u,µ,µ 2 (x, y, y 2 ), Y Y 2 (3.40) 2

22 where we have used the weak-? convergence ru (h) j 3-sc Du h " h * u,µ,µ 2. In addition, in view of (3.38), sup sup h2n j2n L N b Feb 25, 20? * j Du h in M ; R d N, and the 3-scale convergence ru (h) j (x) dx <. (3.4) Using the separability of C 0 ; C # Y Y 2 ; R d N and a diagonalization argument, from (3.39), (3.40) and (3.4), we can find a sequence {j h } such that j h! as h!, and such that w h := u (h) j h satisfies w h 2 W, ; R d, rw h L N b 3-sc " h * u,µ,µ 2, lim h! F " h (w h ) = lim h! F " h (u h ). (3.42) Set c := sup h krw h k L (;R d N ) < and fix > 0. Then by Lemmas 3.2 and 3.4, and by Lemma 3.6 applied to U := Y Y 2 and g(x, y, y 2, ) := f (y, y 2, ), where f (y, y 2, ) := f(y, y 2, ) +, we conclude that x lim F " h (u h ) + c = lim F " h (w h ) + c > lim inf f h! h! h! % (" h ), x % 2 (" h ), rw h(x) dx (3.43) > lim inf h! Y Y 2 f y, y 2, A %(" h ) T %2(" h ) rw h (x, y, y 2 ) dxdy dy 2 > F sc (u, µ, µ 2 ), where Since f F sc (u, µ, µ 2 ) := Y Y 2 f + Y Y 2 f y, y 2, d ac u,µ,µ 2 y, y 2, dl 3N (x, y, y 2 ) dxdy dy 2 d s u,µ,µ 2 dk s u,µ,µ 2 k (x, y, y 2 ) (y, y 2, ) = f (y, y 2, ) +, from (3.43) we deduce that dk s u,µ,µ 2 k(x, y, y 2 ). lim F " h (u h ) + c > G(u, µ, µ 2 ) + k u,µ,µ h! 2 k( Y Y 2 ). (3.44) Finally, letting! 0 + we obtain (3.37). Step 2. We prove that F sc (u, µ, µ 2 ) 6 G(u, µ, µ 2 ). (3.45) Let {" h } h2n be a sequence of positive numbers converging to zero as h!, and let {u j } j2n C ; R d, () j j2n C c ; C# Y ; R d and (2) j j2n C c ; C# Y Y 2 ; R d be the sequences given by Proposition 2.3. For each h, j 2 N define u h,j 2 C ; R d by u h,j (x) := u j (x) + % (" h ) () j x, x % (" h ) + % 2 (" h ) (2) j x, x % (" h ), x. (3.46) % 2 (" h ) Using (.), (2.3), and (2.), in this order, we have that for all ' 2 C 0 ; C # Y Y 2 ; R d N lim lim x ' x, j! h! % (" h ), x : ru h,j (x) dx = '(x, y, y 2 ) : d u,µ,µ % 2 (" h ) 2 (x, y, y 2 ). (3.47) Y Y 2 Moreover, F "h (u h,j ) = = f x f % (" h ), x % 2 (" h ), ru h,j(x) dx x % (" h ), x % 2 (" h ), ru j(x) + r () y j x x, + r (2) y2 j % (" h ) 22 x, x % (" h ), x + # h,j (x) dx, % 2 (" h )