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1 Max-Planck-Institut fur Mathematik in den Naturwissenschaften Leipzig Approximation of non-convex functionals in GBV by Roberto Alicandro, Andrea Braides and Jayant Shah Preprint-Nr.:

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3 Approximation of non-convex functionals in GBV 1 Introduction Roberto Alicandro, Andrea Braides SISSA, via Beirut 4, Trieste, Italy Jayant Shah Northeastern University, Department of Mathematics 360 Huntington Avenue, Boston, MA 02115, USA In many problems of Computer Vision, the unknown is a pair (u; K) with K a union of (suciently smooth) closed curves contained in a xed open set R 2 and u : n K! R belonging to a class of (suciently smooth) functions. A variational formulation of some of these problems was given by Mumford and Shah [14] introducing the functional F (u; K) = jruj 2 dx + c 1 H 1 (K) + c 2 ju? gj 2 dx : (1) nk In this case g is interpreted as the input picture taken from a camera, u is the `cleaned' image, and K is the relevant contour of the objects in the picture; c 1 and c 2 are contrast parameters, and H 1 (K) denotes the total length of K. Problems involving functionals of this form are usually called free-discontinuity problems, after a terminology introduced by De Giorgi. They have been intensively studied in recent times through weak formulations in the framework of the spaces of special functions of bounded variation (see [10], [9], [2], [5]). The presence of the unknown surface K leads to numerical problems, and some kind of approximation of this functional is needed to obtain approximate smooth solutions. The Ambrosio and Tortorelli approach [3] provides a variational approximation of the Mumford and Shah functional (1) via elliptic functionals. The lack of convexity of the limiting functional is overcome by the introduction of an additional function variable which approaches the characteristic of the complement of the set K. The approximating functionals have the form F (u; v) = v 2 jruj 2 dx + c 1 jrvj (1? v)2 dx + c 2 ju? gj 2 dx ; (2) dened on functions u; v such that u; v 2 H 1 () and 0 v 1. The interaction of the terms in the second integral provide an approximate interfacial energy, as in the theory of phase transitions for Cahn - Hilliard uids. This phenomenon had previously been described in analytic terms by Modica and Mortola [13] in the case of phase boundaries. The adaptation of the Ambrosio and Tortorelli approximation to obtain as limits more complex surface energies does not seem to follow easily from their approach. nk 1

4 Motivated by applications in Computer Vision and Fracture Mechanics, in this paper we study a variant of the Ambrosio Tortorelli construction by considering functionals of the form G (u; v) = v 2 jruj dx + c 1 jrvj (1? v)2 dx + c 2 ju? gj dx; (3) where 1. Even though the form of these functionals is quite similar to the previous one, the domain of the limiting functional will be dierent. In fact, as we have G (u; 1) R jruj+c 2ju?gj dx, it is clear that the limit of these functionals will be nite if u 2 BV (). In fact we prove (Theorem 4.1 and Example 4.6) that G converge to functionals related to the function-surface energy ju G(u; K) = jduj( n +? u? j K) + c 1 K 1 + ju +? u? j dh1 + c 2 ju? gj nk dx ; (4) where jduj(a) denotes the total variation on A of the distributional derivative Du, and u are the traces of u on both sides of K. We push this approach further, constructing a variational approximation for a wide class of non-convex functionals dened on spaces of generalized functions of bounded variation. The paper is divided as follows. In Section 2 we introduce the spaces of generalized functions of bounded variation GBV and GSBV, which are needed for a weak formulation of the functionals in (1){(4), and the notion of?-convergence, which precises in which sense the convergence of these functionals is understood. In Section 3 we state the many preliminaries which are needed in the course of the proof. Section 4 is devoted to the statement and proof of the main result, in a slightly more general form than above. The proof of the result lies on a lower bound which is obtained by a new denition of the limit interfacial energy density, taking into account the interaction of the rst two integrals of the approximating energies G, and on an upper bound which is obtained by direct construction and a density result of pairs function-polyhedral surface. Section 5 contains the statement and proof of the approximation result for general isotropic functionals with convex bulk energy density and concave surface energy density dened on GBV. 2 Notation We use standard notation for Sobolev and Lebesgue spaces. L n will denote the Lebesgue measure in R n and H k will denote the k-dimensional Hausdor measure. A() and B() will be the families of open and Borel sets, respectively. If is a Borel measure and E is a Borel set, then the measure B is dened as B(A) = (A \ B). Let A 0 A be open sets. By a cut-o function between A 0 and A we mean a function 2 C 1 0 (A) with 0 1 and = 1 on A 0. 2

5 2.1 Generalized functions of bounded variation Let u 2 L 1 (). We say that u is a function of bounded variation on if its distributional derivative is a measure; i.e., there exist signed measures i such that ud i dx =? d i for all 2 Cc 1 (). The vector measure = ( i ) will be denoted by Du. The space of all functions of bounded variation on will be denoted by BV (). It can be proven that if u 2 BV () then the complement of the set of Lebesgue points, that will be called the jump set of u, is rectiable, i.e. there exists a countable family (? i ) of graphs of Lipschitz functions of (n? 1) variables such that H n?1 ( n S 1? i) = 0. Hence, a normal u can be dened H n?1 -a.e. on, as well as the traces u of u on both sides of as u (x) = lim? u(y) dy ;!0 fy2b + (x):hy?x; u(x)i>0g where? R B u dy = jbj?1 R B u dy. If u 2 BV () we dene the three measures D a u, D j u and D c u as follows. By the Radon Nikodym Theorem we set Du = D a u + D s u where D a u << L n and D s u is the singular part of Du with respect to L n. D a u is the absolutely continuous part of Du with respect to the Lebesgue measure, D j u = Du is the jump part of Du, and D c u = D s u ( n ) is the Cantor part of Du. We can write then Du = D a u + D j u + D c u : It can be seen that D j u = (u +? u? ) u H n?1, and that the Radon Nikodym derivative of Du with respect of L n is the approximate gradient ru of u. A function u 2 L 1 () is a special function of bounded variation on if D c u = 0, or, equivalently, if its distributional derivative can be written as Du = ru L n + (u +? u? ) u H n?1 : The space of special functions of bounded variation on is denoted SBV (). We will also use the auxiliary spaces SBV p () = fu 2 SBV () : jruj 2 L p (); H n?1 ( ) < +1g: We dene the space GBV () of generalized functions of bounded variation as the space of all functions u 2 L 1 () whose truncations u T = (?T ) S _ (u ^ T ) are in BV () for any T > 0. For such functions we can dene = T S >0 u T, and the approximate gradient and the traces u as the limits of the corresponding quantities dened for u T. Moreover, we dene the measure jd c uj : B()! [0; +1] as jd c uj(b) = sup jd c u T j(b) = T >0 3 lim T!+1 jdc u T j(b):

6 If u 2 BV () jd c uj coincides with the usual notion of total variation of D c u. Finally, we set GSBV ( = fu 2 GBV () : jd c uj = 0g = fu 2 L 1 () : u T 2 SBV () for all Tg: For a detailed study of the properties of BV -functions we refer to [2], [11] and [12]. For an introduction to the study of free-discontinuity problems in the BV setting we refer to [2]. 2.2 Relaxation and?-convergence Let (X; d) be a metric space. We rst recall the notion of relaxed functional. Let F : X! R [ f+1g. Then the relaxed functional F of F, or relaxation of F, is the greatest d-lower semicontinuous functional less than or equal to F. We say that a sequence F j : X! [?1; +1]?-converges to F : X! [?1; +1] (as j! +1) if for all u 2 X we have (i) (lower limit inequality) for every sequence (u j ) converging to u F (u) lim inf j F j (u j ); (5) (ii) (existence of a recovery sequence) there exists a sequence (u j ) converging to u such that F (u) lim sup F j (u j ); (6) j or, equivalently by (5), F (u) = lim j F j (u j ): (7) The function F is called the?-limit of (F j ) (with respect to d), and we write F =?-lim j F j. If (F ) is a family of functionals indexed by > 0 then we say that F?-converges to F as! 0 + if F =?-lim j!+1 F j for all ( j ) converging to 0. The reason for the introduction of this notion is explained by the following fundamental theorem. Theorem 2.1 Let F =?-lim j F j, and let a compact set K X exist such that inf X F j = inf K F j for all j. Then 9 min F = lim X j inf X F j: (8) Moreover, if (u j ) is a converging sequence such that lim j F j (u j ) = lim j inf X F j then its limit is a minimum point for F. The denition of?-convergence can be given pointwise on X. It is convenient to introduce also the notion of?-lower and upper limit, as follows: let F : X! [?1; +1] and u 2 X. We dene?- lim inf!0 + F (u) = infflim inf!0 + F (u ) : u! ug; (9) 4

7 ?- lim sup F (u) = infflim sup F (u ) : u! ug: (10)!0 +!0 + If?- lim inf!0 + F (u) =?- lim sup!0 + F (u) then the common value is called the?-limit of (F ) at u, and is denoted by?- lim!0 + F (u). Note that this denition is in accord with the previous one, and that F?-converges to F if and only if F (u) =?- lim!0 + F (u) at all points u 2 X. We recall that: (i) if F =?-lim j F j and G is a continuous function then F + G =?- lim j (F j + G); (ii) the?-lower and upper limits dene lower semicontinuous functions. From (i) we get that in the computation of our?-limits we can drop all d-continuous terms. Remark (ii) will be used in the proofs combined with approximation arguments. For an introduction to?-convergence we refer to [8]. For an overview of?- convergence techniques for the approximation of free-discontinuity problems see [5]. 3 Preliminaries In the following will denote a bounded open set in R n with Lipschitz boundary. We denote by W() the space of all functions w 2 SBV () satisfying the following properties: (i) H n?1 (S w n S w ) = 0; (ii) S w is the intersection of with the union of a nite number of pairwise disjoint (n? 1)-dimensional simplexes; (iii) w 2 W k;1 ( n S w ) for every k 2 N. The following result is due to Cortesani and Toader [7] (see also [6]). Theorem 3.1 (Strong approximation in SBV 2 ) Let u 2 SBV 2 () \ L 1 (). Then there exists a sequence (w j ) in W() such that w j! u strongly in L 1 (), rw j! ru strongly in L 2 (; R n ), lim sup h!+1 kw j k 1 kuk 1 and lim sup j!+1 S wj (w + j ; w? j ; w j ) dh n?1 (u + ; u? ; u ) dh n?1 for every upper semicontinuous function : R R S n?1! [0; +1) such that (a; b; ) = (b; a;?), for every a; b 2 R and 2 S n?1. The next result is a particular case of a theorem by Bouchitte, Braides and Buttazzo [4], and deals with relaxation in BV of isotropic functionals. Theorem 3.2 (Relaxation in BV ) Let g : R! [0; +1] be a lower semicontinuos function with g(t) g(0) = 0; lim = 1; t!0 + t 5

8 and such that the map t! g(jtj) is subadditive and locally bounded. Let F : BV ()! [0; +1] be dened by F (u) := 8 jruj dx + g(ju +? u? j) dh n?1 if u 2 SBV 2 () \ L 1 () +1 otherwise in BV () Then the relaxation of F with respect to the L 1 ()-topology is given on BV () by the functional F (u) = jruj dx + g(ju +? u? j) dh n?1 + jd c uj(): The following lemma is a commonly used tool (see [5]). Lemma 3.3 (Supremum of measures) Let : A()! [0; +1) be an open-set superadditive function, let 2 M + (), let i be positive Borel functions such that (A) R A i d for all A 2 A() and let (x) = sup i i (x). Then (A) R A d for all A 2 A(). We nally include a `slicing' result by Ambrosio (see [1]). We introduce rst some notation. Let 2 S n?1, and let := fy 2 R n : hy; i = 0g be the linear hyperplane orthogonal to. If y 2 and E R n we dene E ;y = ft 2 R : y + t 2 Eg. Moreover, if u :! R we set u ;y : ;y! R by u ;y (t) = u(y + t). Theorem 3.4 (a) Let u 2 BV (). Then, for all 2 S n?1 belongs to BV ( ;y ) for H n?1 -a.a. y 2. For such y we have the function u ;y u 0 ;y(t) = hru(y + t); i for a.a. t 2 ;y (11) ;y = ft 2 R : y + t 2 g; (12) v(t) = u (y + t) or v(t) = u (y + t); (13) according to the cases h u ; i > 0 or h u ; i < 0 (the case h u ; i = 0 being negligible). Moreover, we have jd c u ;y j(a ;y )dh n?1 (y) = jhd c u; ij(a) (14) for all A 2 A(), and for all Borel functions g X t2;y g(t) dh n?1 (y) = g(x)jh u ; ijdh n?1 : (15) (b) Conversely, if u 2 L 1 () and for all 2 fe 1 ; : : : ; e n g and for a.e. y 2 u ;y 2 BV ( ;y ) and jdu ;y j( ;y ) dh n?1 (y) < +1 ; (16) then u 2 BV (). 6

9 4 The main result Using the space GBV dened in the previous section, it is possible to give a weak formulation for problems as in (1) and (4), which has been successfully used to obtain solutions of free-discontinuity problems (see [2]). In what follows we drop the term containing R ju? gj 2 dx, which is of lower order, and does not aect the form of the?-limit, and we generalize the form of the functional (3). Theorem 4.1 Let W : [0; 1]! [0; +1) be a continuous function such that W (x) = 0 if and only if x = 1, and let : [0; 1]! [0; 1] be an increasing lower semicontinuous function with (0) = 0, (1) = 1, and (t) > 0 if t 6= 0. Let G : L 1 () L 1 ()! [0; +1) be dened by G (u; v) = 8 (v)jruj + 1 W (v) + jrvj2 dx if u; v 2 H 1 () and 0 v 1 a.e. +1 otherwise. Then there exists the?- lim!0+ G (u; v) = G(u; v) with respect to the L 1 () L 1 ()-convergence, where G(u; v) = and 8 jruj dx + g(ju +? u? j) dh n?1 + jd c uj() if u 2 GBV () and v = 1 a.e. +1 otherwise, with c W (x) := 2 R 1 x p W (s) ds. g(z) := min f (x)z + 2c W (x) : 0 x 1g; (17) The proof of the theorem above will be a consequence of the propositions in the rest of the section. Before entering into the details of the proof, we dene also a `localized version' of our functionals as follows: and G (u; v; A) = G(u; v; A) = 8 8 A A (v)jruj + 1 W (v) + jrvj2 dx if u; v 2 H 1 () and 0 v 1 a.e. +1 otherwise. jruj dx + g(ju +? u? j) dh n?1 + jd c uj(a) \A if u 2 GBV () and v = 1 a.e. +1 otherwise, 7

10 for any A 2 bounded open set. Remark 4.2 By the assumptions on and W, it can be easily proved that g satises the following properties (i) g is increasing, g(0) = 0 and (ii) g is subadditive, i.e. lim g(z) = 2c W (0) = 4 z! p W (s) ds; g(z 1 + z 2 ) g(z 1 ) + g(z 2 ) 8z 1 ; z 2 2 R + ; (iii) g is Lipschitz-continuous with Lipschitz constant 1; (iv) g(z) z for all z 2 R + and g(z) lim = 1; z!0 + z (v) for any T > 0 there exists a constant c T > 0 such that z c T g(z) for all z 2 [0; T ]. Proposition 4.3 Let n = 1. Then G(u; v)?- lim inf!0 + G (u; v) for all u; v 2 L 1 (). Proof. It suces to consider the case in which the right-hand side is nite. Let j! 0 +, u j! u and v j! v in L 1 () be such that lim j!+1 G j (u j ; v j ) =?- lim inf!0 + G (u; v). Up to passing to subsequences we may suppose We have u j! u; and v j! v a.e. (18) W (v j ) dx < c j ; hence, by the continuity of W, for any > 0 L 1? x 2 : W (v(x)) > = lim j!+1 L 1? x 2 : W (v j (x)) > = 0. We conclude that W (v) = 0 a.e., i.e. v = 1 a.e. By simplicity, suppose that = (a; b) (otherwise we split into its connected components). We now use a discretization argument similar to the one used in the proof of [3]. Let N 2 N and consider the intervals IN k = a + (k? 1) N (b? a); a + k N (b? a) ; k 2 f1; ::; Ng: Up to passing to subsequences we may suppose that lim inf v j j!+1 IN k 8

11 exists for all N 2 N and k 2 f1; ::; Ng. Let z 2 (0; 1) be xed and consider the set n o JN z = k 2 f1; ::; Ng : lim v j z : j!+1 inf Note that for any (; ) interval in R and for any w 2 H 1 (; ) we have, by Young's inequality, 1 W (w) + jw0 j 2 dx 2 I k N p W (w)jw 0 j dx 2 From this inequality we deduce, arguing as in [3], that 1 2 Then z w() p : W (s) ds w() p W (s) ds #J z N lim j!+1 G j (u j ; v j ) < +1: #J z N C with C independent of N. Hence, up to a subsequence, we may suppose J z N = fk N 1 ; ::; k N L g with L independent of N, and up to a further subsequence that there exist S = ft 1 ; ::; t L g [a; b] such that ki N lim N!+1 N = t i for any i 2 f1; ::; Lg. For every > 0 we have I k N S := S + [?; ] for all k 2 J z N and for N large enough. Then lim inf G j!+1 j (u j ; v j ) lim inf G j!+1 j (u j ; v j ; n S ) LX + lim inf G j (u j ; v j ; (t i? ; t i + )) j!+1 lim inf (z) ju 0 j!+1 jj dt ns + LX lim inf G j!+1 j (u j ; v j ; (t i? ; t i + )): (19) With xed i 2 f1; : : : ; Lg, we focus our attention on the term G j (u j ; v j ; (t i? ; t i + )). By denition and by (18), we have that for any > 0 there exist 9

12 x 1 ; x 2 2 (t i? ; t i + ) such that lim u j(x 1 ) = u(x 1 ) < ess- inf u + ; j!+1 (t i?;t i+) lim u j(x 2 ) = u(x 2 ) > ess- sup u? ; j!+1 (t i?;t i+) lim v j(x 1 ) = lim v j(x 2 ) = 1: (20) j!+1 j!+1 Let x i j 2 [x 1 ; x 2 ] be such that v j (x i j ) = inf [x 1;x 2] v j. Then we obtain the following estimate: G j (u j ; v j ; (t i? ; t i + )) G j (u j ; v j ; (x 1 ; x 2 )) (v j (x i j)) x 2 u x2 0 j dx + 2 x 1 (v j (x i j))ju j (x 2 )? u j (x 1 )j +2 vj (x 1) n v j (x i j ) inf t2[0;1] vj (x 1) +2 t x 1 p W (s) ds + 2 vj (x2) (t)ju j (x 2 )? u j (x 1 )j Letting j! +1 and taking into account (20), we get v j (x i j ) p W (s) ds + vj (x 2) t q W (v j )jv 0 jj dx p W (s) ds p W (s) ds o : (21) lim inf G j (u j ; v j ; (t i? ; t i + )) j!+1 inf t2[0;1] n (t) ess- sup (t i?;t i+) u? ess- inf (t i?;t i+) u? t p W (s) ds o : Thus, by the arbitrariness of > 0, lim inf j!+1 G j (u j ; v j ; (t i? ; t i + )) g ess- sup u? ess- inf u (t i?;t i+) (t i?;t i+) (22) Now we turn back to the estimate (19). Since sup j G j (u j ; v j ) < +1, by (19) we get the equiboundness of R ns ju 0 jj dt. Hence u 2 BV ( n S ) and, by (19) and (22), lim inf G j!+1 j (u j ; v j ) (z)jduj( n S ) + LX g ess- sup u? ess- inf u : (23) (t i?;t i+) (t i?;t i+) 10

13 By the arbitrariness of, we deduce that u 2 BV ( n S), i.e., since S is nite, u 2 BV (). Then, letting! 0 in (23), we get lim inf G j!+1 j (u j ; v j ) (z)jduj( n S) + (z)jduj( n ) + X t2 LX g(ju +? u? j(t i )) g(ju +? u? j(t)) ^ (z)ju +? u? j(t) : (24) Finally, letting z! 1 in (24) we obtain the required inequality, since g(t) t. We recover, now, the n-dimensional analogue of the previous inequality, by using Theorem 3.4. Proposition 4.4 Let n 2 N. Then G(u; v)?- lim inf!0 + G (u; v) for all u; v 2 L 1 (). Proof. In the following we will use the notation G 0 =?- lim inf!0 + G. Let 2 S n?1 be xed and let be the hyperplane through 0 orthogonal to. For any u 2 L 1 (), A 2 A(), y 2 we set A y := ft 2 R : y + t 2 Ag; u y (t) := u(y + t): In particular, if u 2 H 1 (), we get u 0 y(t) := hru(y + t); i: For any u; v 2 H 1 (), 0 v 1, we have, by Fubini's Theorem, = = G (u; v; A) Ay (v(y + t))jru(y + t)j + 1 W (v(y + t)) + jrv(y + t)j2 dt dh n?1 (y) Ay (v y (t))ju 0 yj + 1 W (v y(t)) + jv 0 y(t)j 2 dt dh n?1 (y) G (u y ; v y ; A y ) dh n?1 (y); (25) where G is dened by G (u; v; I) = 8 I (v)ju 0 j + 1 W (v) + jv00 j 2 dt if u; v 2 H 1 (I) and 0 v 1 +1 otherwise, 11

14 for any u; v 2 L 1 (I) and I R open and bounded. Let j! 0 and let u j! u, v j! v in L 1 () be such that lim inf G j!+1 j (u j ; v j ) < +1: (26) Then u j ; v j 2 H 1 (); 0 v j 1 a.e. and, as in the proof of Proposition 4.3, v = 1 a.e. Moreover, by Fubini's Theorem, (u j ) y! u y ; (v j ) y! 1 in L 1 ( y ) for H n?1 -a.a. y 2. Thus by Proposition 4.3 and by Fatou's Lemma we get lim inf G j!+1 j (u j ; v j ; A) lim inf G j!+1 j ((u j ) y ; (v j ) y ; A y ) dh n?1 (y) A y ju 0 yj dt + y \A y g(ju + y? u? yj) d# + jd c u y j(a y ) dh n?1 (y): (27) Let T > 0 and set u T = (?T ) _ (u ^ T ): Since g is increasing, it is clear that we decrease the last term in (27) if we substitute u by u T. Moreover, since u T 2 L 1 (), with ku T k 1 T, by Remark 4.2(v), we have ju + T? u? T j c T g(ju + T? u? T j) for a suitable constant c T depending only on T. Then, by (26) and (27), we have jdu T j(a y ) dh n?1 (y) < +1: Thus, applying Theorem 3.4, we get that u T 2 BV () and, by the arbitrariness of (u j ) and(v j ), G 0 (u; 1; A) A jhru T ; ij dx+ g(ju + T?u? T j)jh u ; ij dh n?1 +jhd c u T ; ij(a) \A (28) for all A 2 A() and 2 S n?1. Consider the superadditive increasing function dened on A() by and the Radon measure (A) := G 0 (u; 1; A) := L n + g(ju + T? u? T j)h n?1 T + jd c u T j: 12

15 Fixed a sequence ( i ) i2n, dense in S n?1, we have, by (28), for all i 2 N, where i(x) = 8 (A) jhru T (x); i ij jh u (x); i ij Hence, applying Lemma 3.3, we get G 0 (u; 1; A) A jru T j dx + for all A 2 A(). In particular G 0 (u; 1; ) A i d L n a.e. on jd c u T j a.e. on n T jh u (x); i ij H n?1 a.e. on T. jru T j dx + T \A g(ju + T? u? T j) dh n?1 + jd c u T j(a) (29) T g(ju + T? u? T j) dh n?1 + jd c u T j(): (30) Finally, by the arbitrariness of T > 0, u 2 GBV () and the thesis follows letting T! +1 in (30). Proposition 4.5 We have?-lim sup!0 + G (u; v) G(u; v) for all u; v 2 L 1 (). Proof. It suces to prove the inequality for v = 1 a.e. Since we will use density and relaxation arguments, we divide the proof into ve steps, passing from a particular choice of u to the general one. In the following we will use the notation G 00 =?- lim sup!0 + G. Step 1. Suppose that u 2 W() and = \ K with K a (n? 1)-dimensional simplex. Up to a translation and rotation argument, we can suppose that K is contained in the hyperplane := fx n = 0g. Set h(y) := u + (y)? u? (y); y 2 : By our hypotheses on u, h is regular on ; hence, xed > 0, we can nd a triangulation ft i g N of such that jh(y 1 )? h(y 2 )j < if y 1 ; y 2 2 T i : 13

16 Let h :! R be dened as h (y) := z i y 2 T i ; where z i := min fh(y) : y 2 T i g. Since kh? h k 1 <, by Remark 4.2 (iii), we have that g(h (y)) dh n?1 g(h(y)) dh n?1 + H n?1 ( ): Let x zi realize the minimum in (17) for z = z i. Fixed > 0, there exists T () > 0 such that min n T 0 jv 0 j 2 + W (v) dt : v 2 H 1 (0; T ); v(0) = x zi ; v(t ) = 1o cw (x zi ) + (31) for all T T () and for any i = 1; ::; N. Let v(z i ; ) realize the minimum in (31). For r > 0; > 0 and i 2 f1; ::; Ng, set B r := n (y; t) 2 : y 2 ; jtj < r o and T i := n o y 2 T i : i ) > ; and let i : R (n?1)! R be a cut-o function between T i and T i such that kr i k 1 < C?1. Fix a sequence ( ) such that lim!0+ = 0, set T := T () +, and dene where v (y; t) := 8 < : v i (t) := 1 if (y; t) 2 n B T i (y)v i (t) + (1? i (y)) if y 2 T i ; jtj < T, 8 < : x zi if jtj < v z i ; jtj? if < jtj < T. We have that (v ) 2 H 1 () and v! 1 in L 1 () as! 0+. Hence, we get jrv j W (v ) dx (32) = NX T i NX T 1 NX Ti T int i v 0 z i ; jtj? 2 + W v z i ; jtj?? jr i (y)j 2 jx zi? 1j W (v (y; t)) T jr i (y)j 2 v z i ; jtj? 14? 1 2 dt dh n?1 (y) dt dh n?1 (y)

17 and set NX NX + T i + 1 v ji (y)j 2 0 z i ; jtj? NX T int i T dt dh n?1 (y) T 1 W (v (y; t)) dt dh n?1 (y) jv 0 (z i ; t)j 2 + W (v(z i ; t)) dt dh n?1 (y) +c Hn?1 ( ) + c() 2 NX H n?1 (T i n T i ) T i c W (x zi ) dh n?1 (y) + 2H n?1 ( ) + O(): We now construct a recovery sequence u. Let ~u (z 1 ; z 2 ; t) = u (y; t) = 8 < : 8 z 1?T < t <? z 2?z 1 2 (t + ) + z 1 jtj < z 2 < t < T u(y; t) jtj > T ~u u(y;?t ); u(y; T ); t jtj < T. It can be easily veried that u 2 H 1 () and u! u in L 1 () as! 0 +. Moreover, we have (v )jru j dx = NX T i +? 1 2 jruj dx + (x zi )ju(y; T )? u(y;?t )j dt dh n?1 (y) nb t jruj dx + ch n?1 (T i n T i ) + O() NX Letting, now, tend to 0 +, we obtain, by (32) and (33), T i (x zi )ju +? u? j(y) dh n?1 (y) + O(): (33) G 00 (u; 1) lim sup G (u ; v )!0 + jruj dx + NX T i (ju +? u? j(y) (x zi ) + 2c W (x zi )) dh n?1 (y) + c 15

18 = jruj dx + jruj dx + jruj dx + NX T i (z i (x zi ) + 2c W (x zi )) dh n?1 (y) + c( + ) g(h (y)) dh n?1 (y) + c( + ) g(ju +? u? j(y)) dh n?1 (y) + c( + ): Letting and tend to 0 +, we obtain the required inequality. In order to use the same construction as above in the case = \ SM K i, with M > 1, we now show that we can replace (u ) by a new sequence (^u ) such that ^u 6= u only in a small neighbourhood of K. To this end we again use a cut-o argument. Set K := fy 2 : d(y; K) < g and let : R n?1! R be a cut-o function between K and K with jr j 1 c?1. Dene We have Then nb T jr^u j dx Thus ^u (y; t) := (y)u (y; t) + (1? (y))u(y; t) (y; t) 2 : ^u (y; t) = u (y; t) if (y; t) 2 B T ; ^u (y; t) = u(y; t) if (y; t) 2 n K (?T ; T ): (34) + + and, by (34), we still have nk (?T ;T ) \(K nk) \(K nk) T jruj dx jr (y)jju (y; t)? u(y; t)j dt dh n?1 (y)?t T (y)jru (y; t)j?t +(1? (y))jru(y; t)j dt dh n?1 (y) jruj dx + c T Hn?1 (K n K) + O(): lim sup jr^u j dx =!0 + nb T jruj dx; lim sup!0 + G (^u ; v ) G(u; 1) + c( + ): 16

19 S Step 2. If u 2 W() with = \ M K i, we can generalize in a very natural way the construction of the recovery sequences ^u and v in Step 1, since this construction modies u and v only in a small neighbourhood of each sets K i. Step 3. Let u 2 SBV 2 () \ L 1 (). Then, applying Theorem 3.1 with (a; b; ) = g(ja? bj), there exists a sequence (w j ) 2 W() such that w j! u in L 1 (); and lim sup G(w j ; 1) G(u; 1): j!+1 Then, by the previous steps and by the lower semicontinuity of G 00 G 00 (u; 1) lim inf j!+1 G00 (w j ; 1) lim inf j!+1 G(w j; 1) G(u; 1): Step 4. Since g satises the hypotheses of Theorem 3.2, the relaxation with respect to L 1 ()-topology of the functional is given by F (u) := 8 < : G(u; 1) if u 2 SBV 2 () \ L 1 () +1 otherwise in BV () F (u) = G(u; 1) for all u 2 BV (). Then by the previous steps and by the lower semicontinuity of G 00 we get G 00 (u; 1) F (u) = G(u; 1) for any u 2 BV (). Step 5. We recover the general case by a truncation argument. Let u 2 GBV () and let u j = (?j) _ (u ^ j). Then lim j!+1 G(u j; 1) = G(u; 1): Since u j! u in L 1 () we get the thesis by the lower semicontinuity of G 00. Example 4.6 We illustrate with a few simple examples the behaviour of the function g, given by (17), with dierent choices of.. The terms bulk energy and surface energy refer to the rst and second terms in Equation (17) and what is meant by interaction between the two at each value of z is relative contributions of these terms to g(z) at that z. Let W (v) = (1? v) 2 =4, so that c W (x) = (1? x) 2 =2. We then have (a) if (v) = v 2 then g(z) = jzj=(1 + jzj); (b) if (v) = v then g(z) = jzj? (z2 =4) if jzj 2 (c) if (v) = 0 if v = 0 1 otherwise, 1 if jzj > 2; then g(z) = minfjzj; 1g. 17

20 We see that the `bulk term' and of the `surface term' (i.e. the rst and the second term in (17)) play dierent roles in these examples. Note that in (a) we always have interaction between these two terms (i.e. both terms contribute to the value g(z)) contrary to what happens in the Ambrosio Tortorelli case. The interaction also occurs in (b) for jzj < 2. Note moreover that in the third case the minimal x in the denition of g(z) does not vary with continuity at z = 1. 5 Approximation of general functionals In this section we show how Theorem 4.1 can be used to obtain an approximation of general (isotropic) energies dened on GSBV by a double limit. The set will be a bounded open subset of R n with Lipschitz boundary. Proposition 5.1 Let W and be dened as in Theorem 4.1, let f : [0; +1)! [0; +1) be a convex and increasing function satisfying and let G : L 1 () L 1 ()! [0; +1) be dened by G (u; v) = 8 f(t) lim = 1; (35) t!+1 t (v)f(jruj) + 1 W (v) + jrvj2 dx if u; v 2 H 1 () and 0 v 1 a.e. +1 otherwise. Then there exists the?- lim!0+ G (u; v) = G(u; v) with respect to the L 1 () L 1 ()-convergence, where G(u; v) = 8 f(jruj)dx + and g is dened in (17). g(ju +? u? j)dh n?1 + jd c uj() if u 2 GBV () and v = 1 a.e. +1 otherwise, Proof. The estimate for the?-lim inf can be performed as in Proposition 4.3, noting that in (21) we obtain, by Jensen's inequality, u(x2 )? u(x 1 ) x2 G j (u j ; v j ; IN k?i ) (v j(x i j))jx 2? x 1 jf x 2? x x 1 q W (v j )jv 0 jj dx; from which the lower bound can be easily obtained taking into account (35). The rest of the proof can be obtained following Propositions 4.4 and

21 Remark 5.2 Let K > 0 and N 2, let 0 = a 0 < a 1 < < a N = 1; 0 = b N < b N?1 < b 0 = K; and let f and W be as in the previous proposition. Then there exists satisfying the hypotheses in Theorem 4.1 such that, if G : L 1 () L 1 ()! [0; +1) is dened by G (u; v) = 8 (v)f(jruj) + K W (v) + Kjrvj2 dx if u; v 2 H 1 () and 0 v 1 a.e. +1 otherwise, then the thesis of the previous proposition holds with g : [0; +1)! [0; +1) given by g(z) = minfa i z + b i g: In fact, in this case the formula for g can be easily inverted, obtaining as the piecewise constant function given by (0) = 0 and where c W is dened in Theorem 4.1. () = a i if c?1 W (b i?1=2) < c?1 W l(b i=2); Proposition 5.3 Let W be as in Theorem 4.1. Let '; # : [0; +1)! [0; +1) be functions satisfying (i) ' is convex and increasing, lim t!+1 '(t)=t = +1; (ii) # is concave, lim t!0 + #(t)=t = +1. Then there exist two increasing sequences of functions (' j ) and ( j ), and a sequences of positive real numbers (k j ), converging to sup #, such that if we dene G j (u; v) = 8 j(v)' j (jruj) + k j W (v) + k jjrvj 2 dx if u; v 2 H 1 () and 0 v 1 a.e. +1 otherwise, then for every j 2 N there exist the limits?- lim Gj (u; v) =: G j (u; v)!0 +?- lim j!+1 Gj (u; v) = lim j!+1 Gj (u; v) = G(u; v) with respect to the L 1 () L 1 ()-convergence, where G(u; v) = 8 '(jruj) dx + #(ju +? u? j)dh n?1 if u 2 GSBV () and v = 1 a.e. +1 otherwise. (36) 19

22 Proof. Let # j : [0; +1)! [0; +1) be functions of the form # j (z) = minfa j i z + Bj i g; with 0 = A j 0 < < Aj j = j converging increasingly to #, and let ' j : [0; +1)! [0; +1) be convex increasing functions with ' j (t) lim = j; t!+1 t converging increasingly to '. Let k j = max # j. Set g j = # j =j, K j = k j =j and f j = ' j =j. By the previous remark, applied with g = g j, f = f j and K = K j, we can nd =: j such that if we let G j : L 1 () L 1 ()! [0; +1) be dened by (36) then there exists the?- lim!0+ G j (u; v) = G j (u; v) with respect to the L 1 () L 1 ()-convergence, where 8 ' j (jruj)dx + # j (ju +? u? j)dh n?1 + jjd c uj() G j (u; v) = +1 otherwise. if u 2 GBV () and v = 1 a.e. Since the functionals G j converge increasingly to G, they also?-converge to G as j! +1. Remark 5.4 If ' is convex and even, # is concave and even, and '(t) #(t) lim = lim = M; t!+1 t t!o + t then there exist (' j ), ( j ) and (k j ) such that the functionals G j dened above?-converge with respect to the L 1 () L 1 ()-convergence to G(u; v) = 8 '(jruj)dx + #(ju +? u? j)dh n?1 + MjD c uj() if u 2 GBV () and v = 1 a.e. +1 otherwise. The proof can be obtained directly from Remark 5.2, using the approximation argument of Proposition 5.3. Acknowledgements This paper was written while the second author was visiting the Max-Planck-Institute for Mathematics in the Sciences at Leipzig, on Marie- Curie fellowship ERBFMBICT of the European Union program \Training and Mobility of Researchers. J. Shah is partially supported under NSF Grant DMS and NIH Grant I-R01-NS

23 References [1] L. Ambrosio, A compactness theorem for a new class of functions of bounded variation, Boll. Un. Mat. Ital. 3-B (1989), [2] L. Ambrosio, N. Fusco and D. Pallara, Special Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, Oxford, to appear. [3] L. Ambrosio and V. M. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via?-convergence, Comm. Pure Appl. Math. 43 (1990), [4] G. Bouchitte, A. Braides and G. Buttazzo, Relaxation results for some free discontinuity problems, J. Reine Angew. Math. 458 (1995), 1-18 [5] A. Braides, Approximation of Free-discontinuity Problems, Lecture Notes in Mathematics, Springer Verlag, Berlin, to appear. [6] A. Braides and V. Chiado Piat, Integral representation results for functionals dened on SBV (; R m ), J. Math. Pures Appl. 75 (1996), [7] G. Cortesani and R. Toader, A density result in SBV with respect to nonisotropic energies, Nonlinear Analysis, to appear. [8] G. Dal Maso, An Introduction to?-convergence, Birkhauser, Boston, [9] E. De Giorgi, Free Discontinuity Problems in Calculus of Variations, in: Frontiers in pure and applied Mathematics, a collection of papers dedicated to J.L.Lions on the occasion of his 60 th birthday (R. Dautray ed.), North Holland, [10] E. De Giorgi and L. Ambrosio, Un nuovo funzionale del calcolo delle variazioni, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 82 (1988), [11] L. C. Evans and R. F. Gariepy, Measure theory and ne properties of functions, CRC Press, Boca Raton [12] H. Federer, Geometric Measure Theory, Springer Verlag, New York, [13] L. Modica and S. Mortola, Un esempio di?-convergenza, Boll. Un. Mat. It. 14-B (1977), [14] D. Mumford and J. Shah, Optimal approximation by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math. 17 (1989),