Limits of discrete systems without convexity ... - Semantic Scholar

Limits of discrete systems without convexity hypotheses Andrea Braides and Maria Stella Gelli SISSA, via Beirut 4, 34013 Trieste, Italy Email: [email protected], [email protected]

1 Introduction The use of non-convex discrete energies to derive a continuum theory by using a variational approach was recently considered by Truskinovsky [11] and by Braides, Dal Maso and Garroni [5]. In both those papers the authors consider an array of material points linked by nearest-neighbour interactions and an energy of the form   X E" (u) = " " ui ?"ui?1 ; (1.1) i where " is the distance between neighbouring points in the reference con guration, ui denotes the position of the i-th material point (i = 1; : : :; L=") in the deformed con guration and " is a Lennard Jones type potential characterized by being convex until a threshold T" and then concave. In both cases, though with some technical dierences, the authors highlight that the eect of the convex part is to give rise to a bulk (elastic) energy, while the concave part contributes to a penalization of fracture. Minimizers, local minimizers and stationary points under suitable boundary conditions and body forces are characterized as having only a nite number of interactions lying in the concave region of " which give rise to fracture in the limit description, while all the other interactions contribute to the bulk. The limit continuum energy can be explicitly described by using the terminology of ?-convergence (which is a variational convergence that ensures the convergence of minimum problems, see [8] or [6]), and the theory of freediscontinuity functionals, of the form

E (u) =

Z

;L)nS (u)

(0

f(u) _ dt +

X

t2S (u)

g([u]);

(1.2)

de ned on piecewise-Sobolev functions on the segment (0; L) (S(u) denotes the set of discontinuities of u and [u] stands for the jump of u), and can be carried to dimension higher than one by using the framework of special functions of bounded 1

variation (see [3]). In the case treated in [5] the limit energy densities f and g are de ned as the limit of the convex part of " and of a suitable scaling of its concave part, respectively. In this paper we deal with the case when the energy " does not have a concave-convex behaviour. Our only assumption is that we can nd p > 1, a constant C > 0 and an interval I" = [T"? ; T"+ ] (possibly degenerating to a point or a half line) such that " (z)  C(jz jp ?1) if z 2 I" and " (z)  C" if z 62 I" . In this case a `homogenization' process takes place in parallel with the passage from a discrete to a continuum theory. As a consequence, periodic microstructure may appear in minimizers, underlined by a convexi cation of "(z) for `small' values of z, as well as a fragmentation of fracture, mathematically translated into a `subadditive envelope' on " (z) for `large' values of z. This `regularization' produces convex functions f" and subadditive functions g" whose asymptotic behaviour as " ! 0 describes the limit energy densities f and g, respectively. The precise statement of this convergence result can be found in Theorem 3.1.

gε

fε Tn

+

Tn

Figure 1: `regularization of " '. We underline that, contrary to the case of a convex-concave behaviour, a sequence of minimizers for E" may not converge in L1 (0; L) to minimizers of the continuum limit,but only in measure, due to the possible fragmentation of fracture. Moreover, the limit g is not described anymore by simply taking the pointwise limit of g" , but by a more complex limit procedure. In order to simplify the presentation we treat the case when T" ! 1 only. In this case the representation of the limit energy by a functional E as in (1.2) is complete. When one or both T" remain bounded the domain of the limit functional is not a space of piecewise Sobolev functions, but that of functions of bounded variation on (0; L), and the representation must take the possible interaction between f and g into account. As this aspect of the problem has been studied in detail in [5] we refer to that paper for the complete form of the limit energy. Furthermore, we refer to [7] for an extension to the N-dimensional setting. 2

2 Notation and preliminaries

For a set A of R we denote intA the interior of A. We write sgn t and [t] to denote the sign of t and the integer part of t, respectively. We write L1(A) or jAj to denote the Lebesgue measure of A  R. We write # to denote the counting measure and t the Dirac mass at t. We use standard R notation for Sobolev and Lebesgue spaces. If  is a measurable function then ?B  dx is its mean value on the set B. If  is a Borel measure and B is a Borel set, then the measure  B is de ned as  B(A) := (A \ B). The letter c will denote a strictly positive constant whose value may vary from line to line. 2.1

Special functions of bounded variation

We recall that the space SBV (a; b) is de ned as the space of functions u 2 L1 (a; b) P whose distributional derivative Du can be written as Du = f L1 + t2S (u) att for some f 2 L1 (a; b), a (at most P countable) set S(u)  (a; b) and a sequence of real numbers (at)t2S (u) with t jatj < +1. It can be easily seen that for such functions the left hand-side and right hand-side approximate limits u?(t), u+ (t) exist at every point, and that S(u) = ft 2 R : u? (t) 6= u+ (t)g and at = u+ (t) ? u? (t). We will write u_ = f, which is an approximate gradient of u. This notation describes a particular case of a SBV -functions space as introduced by De Giorgi and Ambrosio [9]. We will mainly deal with functionals whose natural domain is that of piecewise-W 1;p functions, which is a particular sub-class of SBV (a; b) corresponding to the conditions u_ 2 Lp (a; b) and #(S(u)) < +1, but we nevertheless use the more general SBV notation for future reference and for further generalization to higher dimensions (see [2]). For an introduction to SBV functions we refer to the book by Ambrosio, Fusco and Pallara [3], while approximation methods for free-discontinuity problems are discussed by Braides [4]. A class of energies on SBV (a; b) are those of the form (1.2) with f; g : R ! [0; +1]. Lower semicontinuity conditions on E are equivalent to requiring that f is lower semicontinuous and convex and g is lower semicontinuous and subadditive; i.e., g(x + y)  g(x) + g(y). The latter can be interpreted as a condition penalizing fracture fragmentation, whereas convexity penalizes oscillations. If ' is not lower semicontinuous and convex (respectively, subadditive) then we may consider its lower semicontinuous and convex (respectively, subadditive) envelope; i.e., the greatest lower semicontinuous and convex (respectively, subadditive) function not greater than ', that we denote by ' (respectively, sub? '). For a discussion on the role of these conditions for the lower semicontinuity of E we refer to [4] Section 2.2.

3

?-convergence We recall the de nition of De Giorgi's ?-convergence in a metric space space (X; d): given a family of functionals Fn : X ! [0; +1], n 2 N, for u 2 X we de ne 2.2

n

o

F 0(u) = ?(d)-limninf Fn (u) := inf limninf Fn(un ) : lim d(un; u) = 0 ; (2.1) n and n o F 00(u) = ?(d)-lim sup Fn (u) := inf lim sup Fn(un) : lim d(u ; u) = 0 : (2.2) n n n

n

Note that the functions F 0 and F 00 are lower semicontinuous. If these two quantities coincide then their common value is called the ?-limit of the sequence (Fn) at u, and is denoted by ?- limn Fn(u) or ?(d)-limn Fn (u). Equivalently, F(u) = ?limn Fn(u) if and only if the two following conditions are satis ed: (i) (lower semicontinuity inequality) for all sequences (un ) converging to u in X we have F(u)  lim infn Fn(un ); (ii) (existence of a recovery sequence) there exists a sequence (un) converging to u in X such that F(u)  lim supn Fn(un). We will use as d the L1 -metric or a metric giving convergence in measure. For a comprehensive study of ?-convergence we refer to the book of Dal Maso [8] (see also [6] Part II). The reason for the introduction of this notion is explained by the following fundamental theorem. Theorem 2.1 Let F = ?-limn Fn, and let a compact set K  X exist such that inf X Fn = inf K Fn for all n. Then 9 min F = lim inf F : n X n X Moreover, if (un) is a converging sequence such that limn Fn(un) = limn inf X Fn then its limit is a minimum point for F .

We recall a simple 1-dimensional ?-convergence result. Proposition 2.2 For any n 2 N let hn : R ! [0; +1] be convex and lower semicontinuous functions such that the limit limn hn(x) =: h(x) exists for all x 2 R. Assume in addition that h is lower semicontinuous and int (fx : h(x) 6= +1g) 6= ;. Then for all n;  2 Lp (a; b) such that n *  weakly in Lp (a; b), we have limninf

Z

b

a

hn (n) dt 

Z

b

h() dt:

a

Proof. Let (Bi )i2I be a Borel partition of (a; b). Then, for n;  as above, by convexity we have Z

b

a

hn(n ) dt 

X

I

Z



jBi jhn ? n dt : Bi

4

Under the hypotheses above we have h(z)  lim infn hn(zn ) if zn ! z. Hence, limninf

Z

b a

hn(n ) dt 

X



Z

jBi j limninf hn ? n dt = Bi

I




P ?R

X

I

Z



jBi jh ?  dt : Bi

Let wB be de ned as wB (x) = i ?B u dy B (x). It suces to consider a sequence of Borel partitions (Bn ) such that wB ! u pointwise a.e. and use Fatou's Lemma. i

i

n

3 The main result For future reference, we state and prove the convergence result allowing for a more general dependence on the underlying lattice than in the Introduction, at the expense of a slightly more complex notation. We begin by identifying the functions de ned on a lattice with a subset of measurable functions. Consider an open interval (a; b) of R and two sequences (n ), (an ) of positive real numbers with an 2 [a; a + n ) and n ! 0. For n 2 N let a  x1n < : : : < xNn < b be the partition of (a; b) induced by the intersection of (a; b) with the set an + n Z. We de ne An (a; b) the set of the restrictions to (a; b) of functions constant on each [a + kn; a + (k + 1)n ), k 2 Z. A function u 2 An (a; b) will be identi ed by Nn + 1 real numbers c0n ; : : :; cNn such that 8 < cin if x 2 [xin; xin+1), i = 1; : : :; Nn ? 1 u(x) = : c0n if x 2 (a; x1n) (3.1) cNn if x 2 [xNn ; b). For n 2 N let n : R ! [0; +1] be a given Borel function and de ne En : L1 (a; b) ! [0; +1] as n

n

n

8 Nn ?1 > > X
i=1 > : +1 otherwise in L1 (a; b). We will prove the following ?-convergence result. Theorem 3.1 For all n 2 N let Tn 2 R exist with lim T  = 1; lim  T  ! 0; n n n n n

(3.2)

(3.3)

and such that, if we de ne Fn; Gn : R ! [0; +1] as

Fn(z) =

8 < :

n(z)

+1

Tn?  z  Tn+ z 2 R n [Tn? ; Tn+ ] 5

(3.4)

8 > n n n n : 0 if z = 0 the following conditions are satis ed: there exists p > 1 such that

(3.5)

Fn(z)  jz jp

8z 2 R sup zinf Fn (z) < +1 n 2R Gn(z)  c > 0 8z = 6 0 and, moreover, there exist F; G : R ! [0; +1] such that

(3.6) (3.7) (3.8)

?- lim F  = F; n n

(3.9)

?- lim sub? Gn = G: (3.10) n Then, (En)n ?-converges to E with respect to the convergence in measure on L1 (a; b), where 8Z > >
a t2S (u) > : +1 otherwise in L1 (a; b). Remark 3.2 Note that the hypotheses (3.9,3.10) are not restrictive up to passing to a subsequence by a compactness argument. F(u) _ dt +

Proof. For simplicity of notation we deal with the case Tn = ?Tn? =: Tn , the general case following by simple modi cations. With xed u 2 L (a; b) and a sequence (un)  An (a; b) such that un ! u in measure and supn En (un) < +1. Up to a subsequence, we can suppose in addition that un converges to u pointwise a.e. We now construct for each n 2 N a function vn 2 SBV (a; b) and a free-discontinuity energy such that vn still converges to u +

1

and we can use that energy to give a lower estimate for En(un). Set 

In := i 2 f1; : : :; Nn ? 1g and

i+1 : un (xn )

? un(xin ) > T

n

8 > < ci

(cin+1 ? cin ) (x ? xi ) n n vn (x) := > : un (x) n+

6





n

x 2 [xin; xin+1); i 2= In x elsewhere in (a; b).

(3.11)

(3.12)

We have that, for " > 0 xed,

fx : jvn(x) ? un(x)j > "g  fx 2 [xin; xin ); i 2= In ; jun(xin ) ? un(xin )j > "g: (3.13) Since, for i 2= In we have jun(xin ) ? un (xin)j  n Tn , then fx : jvn(x) ? un(x)j > "g = ; if n is large enough. Hence, the sequence (vn )n converges to u in measure +1

+1

+1

and pointwise a.e. Moreover, by (3.8)

c#In  En (un)  M;

(3.14)

with M = supn En (un). By the equiboundedness of #In, we can suppose that S(vn ) = fxin+1gi2I tends to a nite set. For the local nature of the arguments in the following reasoning, we can also assume that S consists of only one point x0 2 (a; b). Now, consider the sequence (wn )n de ned by n

wn (x) =

Z 8 > > v (a) + v_ n (t) dt > < n (a;x) Z > > v (a) + v_ n (t) dt + n > : (a;x)

if x < x0 X

(3.15)

[vn](t) if x  x0.

t2S (vn )

P

Note that wn (a) = vn (a), w_ n = v_ n, S(wn ) = fx0g and [wn](x0) = t2S (v ) [vn](t). Such a sequence still converges to u a.e. Indeed, since x0 is the limit point of the sets S(vn ), for any  > 0 xed we can nd n0() 2 N such that for any n  n0() and for any i 2 In jx0 ? xin+1 j < . Hence, by construction, for any n  n0() and for any x 2 (a; b) n [x0 ? ; x0 + ], wn(x) = vn(x), that is, the two sequences (vn ) and (wn) have the same pointwise limit. Since w_ n = v_ n on (a; b), by (3.6) we have that kw_ nkL (a;b)  M. Then, using Poincare's inequality on each interval, it can be easily seen that (wn )n is equibounded in W 1;p ((a; b) n fx0g). Since it also converges to u pointwise a.e., by using a compactness argument, we get that u 2 W 1;p((a; b) n fx0g) and, up to subsequences, n

p

w_ n * u_ weakly in Lp (a; b): Moreover, since for any two points a < x1 < x0 < x2 < b we have wn(x2) = wn(x1 ) + u(x2) = u(x1) +

Z x2

x1 Z x2 x1

w_ n dt + [wn](x0) u_ dt + [u](x0);

taking points x1; x2 in which wn converges to u and passing to the limit as n ! +1, we have [wn](x0) ! [u](x0): (3.16) 7

We can now rewrite our functionals in terms of vn : X X n n (_vn ) + Gn([vn](xin+1) ? sgn([vn ](xin+1))n Tn ) En(un ) = =

i=2In b

i2In

a

t2S (vn )

Z

Fn (_vn) dt +

X

Gn ([vn](t) ? sgn([vn](t))n Tn ):

>From (3.14) we also have En(un ) 



Z

b

a

Z

a

b

Fn(_vn ) dt + sub? Gn

? X

[vn](t) ? sgn([vn ](t))nTn




t2S (vn )

Fn (w_ n) dt + sub? Gn([wn](x0) + o(1))

as n ! +1. Passing to the liminf as n ! +1, using (3.16) and Proposition 2.2, we have limninf En(un)  limninf

 as desired.

Z

b

a

Z

b

a

Fn (w_ n) dt + limninf sub? Gn([wn](x0) + o(1))

F(u) _ dt + G([u](x0))

We now turn our attention to the construction of recovery sequences for the ?-limsup. We may assume in what follows that inf z2 Fn(z) = Fn(0). Step 1 We rst prove the limsup inequality for u ane on (a; b). Set  = u;_ for each n in N we can nd n1 ; n2 2 R, tn 2 [0; 1] such that p 1 2 jtnn + (1 ? tn)n ?  j  2(b ?na) tn Fn(n1 ) + (1 ? tn)Fn (n2 )  Fn () + o(1) (3.17) i jnj  c = c(): Note that in the last inequality the choice of the constant c can be chosen independent of n thanks to (3.6) and (3.7). It can be easily seen that it is not restrictive to make the following assumptions on ni :

R

p

n1 > ; Fn (n1 )  Fn(n2 ); (jn1 j + jn2 j) n  1: (3.18) We de ne a piecewise ane function vn 2 L1 (a; b) with the following properties: vn (x) = u(x) on (a; x1n]; v_ n j [xi ; xi+1) := vni 2 fn1 ; n2 g; n n

8

and vni is de ned recursively by 8 1 vn = n18 > > p > i < p P > < vi if 2n  vn(an ) + vnj n + vni n ? u(xin+1)  n n i +1 v => > j =1 > > n : 1 : 2 i n + n ? vn otherwise. (3.19) p Since 0  vn ? u  n by de nition, (vn )n converges to u uniformly, and hence in measure and, moreover,  (3.20) n1 := # i 2 f0; : : :; Nng : vni = n1  tnNn : Indeed, from (3.17), (3.18) and (3.19) we deduce p ?
1 2 n Nn tn(n ? ) + (1 ? tn)(n ? )  2 n  vn (xNn ) ? u(xNn ) = n1 (n1 ? )n + (Nn ? n1 )(n2 ? )n ; so that ( n1 ? tn Nn )(n1 ? n2 )  0: Now, consider the sequence (un)  An(a; b) de ned by un(xin ) = vn (xin) for i = 1; : : :; Nn ; un (a) = vn (a) and un(b) = vn(b): Since (3.13) still holds with un; vn as above, it can be easily checked that (un)n converges to u in measure. Hence, recalling (3.17), (3.18) and (3.20), En (un) = nFn (n1 ) n1 + (Nn ? n1 )n Fn (n2 )  tnn Nn Fn(n1 ) + (1 ? tn)Nn n Fn(n2 )  Nnn (Fn () + o(1))  (b ? a)Fn () + o(1): Taking the limsup as n ! +1 we get lim sup En(un )  F()(b ? a) = E(u): n

n

n

The same construction as above works also in the case of a piecewise ane function: S let [a; b] = [aj ; bj ] with a1 = a; bj = aj +1 and u_ constant on each (aj ; bj ), then it suces to repeat the procedure above on each (aj ; bj ) to provide functions vnj in An (aj ; bj ) such that vnj ! u j (a ; b ) in measure j j

lim sup n

X

j

fi: x 2(a ;b )g i n

j

 j i+1 j i  Zb n n vn(xn )? vn (xn)  F(u) _ dx: n

j

aj

9

With j xed de ne ynj := maxfxin 2 (aj ; bj )g. Then, the recovery sequence un is de ned in (aj ; bj ) as un(x) = vnj (x) ?

X

` > lim n

n

> > > >
j (i?in) > > N ?
P > > > znj + sgn(znj )n Tn : j =1

n

if in < i  in + N if i > in + N.

Clearly (wn)n ! u in measure and N X

N i X z n i En(wn ) = n n  + sgn(zn )Tn = Gn (zni ) + (b ? a)Fn (0) ; n i=1 i=1 the estimate follows from (3.21) by passing to the limit as n ! +1. 



10

(3.22)

Step 3 Let u 2 SBV (a; b) be such that E(u) < +1, then

u = v + w with v(x) =

Z

x

a

u_ dt + c and w(x) =

m X j =1

zj [x ;b): j

For j = 1; : : :m let wnj be de ned as in Step 2 with jumps in j fxin+i gNi=1 and let vn be a recovery sequence for v such that it is constant on each [xin ; xin +n Nj ). m P The sequence un = vn + wnj converges in measure to u and S

j

n;j

n;j

n;j

j =1



lim sup En(un ) = lim sup En (vn) + n

n

m X j +1



En(wnj )  E(v) + E(w) = E(u);

as desired.

Corollary 3.3 Let

n : R ! [0; +1] satisfy the hypotheses of Theorem 3.1. Assume that, in addition, for all n 2 N, Fn is lower semicontinuous and convex and Gn is lower semicontinuous and subadditive. Then, for any u 2 L1 (a; b), En (u) ?-converges to E(u) with respect to the strong topology of L1 (a; b).

Proof. It suces to produce a recovery sequence converging strongly in L (a; b). 1

Note that in Step 1, by the convexity of Fn , we can choose n1 = n2 = n in (3.17). Then vn = u and un turns out to be the piecewise-constant interpolation of u at points fxing. It is easy to check that un ! u strongly in L1 (a; b). It remains to show that also for functions of the form z[x0 ;b) it is possible to exhibit a sequence that converges strongly in L1 (a; b). To this end it suces to note that in Step 2, since Gn = sub? Gn, we can nd a sequence (zn ) such that (3.21) is replaced by limn zn = z and limn Gn(zn ) = G(z). Hence, the sequence wn de ned by (3.22) converges to u strongly in L1 (a; b) and it is a recovery sequence.

4 Examples

Example 4.1 (i) The typical example of a sequence of functions which satisfy the hypotheses of Theorem 3.1 (and indeed of Corollary 3.3) is given ( xed (n ) converging to 0 and C > 0) by 1? n(z) =  (n z 2 ) ^ C); n p with p = 2, Tn = 1= C=n, 

p

2 jz j  C=n Fn(z) = z +1 otherwise,

11



6 0 Gn(z) = C0 jzz j== 0,

so that

E(u) =

Z

b

a

ju_ j dt + C#(S(u)) 2

on SBV (a; b). (ii) In many cases, we can prove a convergence result without the hypothesis that limn nTn = 0 by using Theorem 3.1 and a comparison argument. As an

C/ λ

-

-1/2

λn

C/ λ

n

-1/2

λn

-

-1/2

λn

n

-1/2

λn

Figure 2: the potentials n in Example 4.1 (i) and (ii). example, let

8
0

z2 if z  0. In this case the ?-limit (with respect to both the convergence in measure and L1 convergence) is given by 8Z >
a ju_ j dt + C#(S(u)) if u 2 SBV (a; b) and u > u on S(u) : +1 otherwise. In fact, clearly, the construction of a recovery sequence can be achieved in the same way, while the opposite inequality can be obtained by applying Theorem 3.1 to 8 1 ? (n z 2) ^ C) if z > 0 < j (z) =  n : 1 ? 2  (n z ) ^ j) if z  0, noting that n  nj for all n and j, and using the arbitrariness of j. We now give an example which illustrates the eect of the operation of the subadditive envelope. Example 4.2 If we take   1 p n (z) = z 2 ^  + (jz j n ? 1)2 n n

n

12

with n converging to 0, then we obtain F(z) = z 2 and o n 2 G(z) = sub? (1 + z 2 ) = min k + zk : k = 1; 2; : : : :

-1

λn

-

-1/2

-1/2

λn

λn

Figure 3: the functions n and G in Example 4.2. We now give an example which shows that the ?-convergence result stated in Theorem 3.1 is sharp, in the sense that it cannot be improved to a result also with respect to the strong convergence in L1 (a; b), unless by adding further hypotheses as in Corollary 3.3.

Example 4.3 Consider a lattice of step size n = n2 and the functions [0; +1) de ned as 8 if jxj  n

a : +1 otherwise in L1(a; b), u_ 2 dt + 3#(S(u)) if u 2 SBV (a; b) E 1 (u) := > a : +1 otherwise in L1 (a; b), 13

and with E (u), E (u) the ?-liminf and the ?-limsup of En with respect to the strong topology of L1 (a; b) de ned as in (2.1) and (2.2), respectively. We will see that the sequence En has dierent ?-limits in L1 (a; b) depending on the choice of rn . Case 1 If rn = n, then E (u) = E (u) = E(u). In particular the ?-limit in L1 coincides with the ?-limit in measure. Proof. Since the strong convergence in L1 (a; b) implies the convergence in measure, we easily get E (u)  E(u): We now prove that E(u)  E (u) for a u 2 SBV (a; b) with u_ 2 L1 (a; b), u(a+) = 0 and only a positive jump z in x0 . If in is the index such that x0 2 [xin ; xin +1 ), we de ne un 2 An(a; b) as 0

00

0

00

0

00

n

n

8 Z xi+1 n > > > u_ dt > > > a > > < Z xin+1

un(xin ) = >

if i < in

if i > in .

a

We have lim n = lim n

Z

b

a

(4.1)

u_ dt + n + 1 + z if i = in

> a > > > Z xi+1 > n > > : u_ dt + z

jun ? uj dt

XZ

i

xin+1 Z x

xin

xin

ju_ j ds dt + (n + 1)(xin ? x ) + (n + 1 + z)(x ? xin ) n +1

0

0

n

 lim c 1 = 0: n n Hence, un ! u strongly in L (a; b) and, moreover, 1

En (un) 

XZ

i6=in

xin+1

xin

u_ 2 dt + Gn (n + z + cn ) + Gn(?n ? cn )  E(u);

where cn is de nitively positive. Case 2 If rn = n2 , then E (u) = E (u) = E 1 (u). In particular the ?-limits in measure and in the L1 convergence are dierent. Proof. Note that in this case the sequence un de ned in (4.1) is a recovery sequence for E 1 (u). Indeed, de nitively, cn + z < n2 , hence E(u) < En(un) = E 1 (u) + o(1). 0

00

14

Note also that the sequence vn de ned by 8 Z xi+1 n > > > u_ dt > > > a > > < Z xin+1

vn (xin ) = >

if i < in

u_ dt + z + 1 + n2 if i = in

> a > > > Z xin+1 > > > : u_ dt + z

if i > in .

a

is such that En(vn ) = E(u) + o(1) but it converges to u only in measure, that is, (vn )n is a recovery sequence for the convergence in measure but not for the strong convergence in L1 (a; b) . It remains only to prove the liminf inequality. With xed a function u 2 L1 (a; b) and a sequence (un )n  An (a; b) such that un ! u strongly in L1 (a; b) and supn En(un ) < +1. Proceeding as in the proof of Proposition 3.1 we get that u 2 SBV (a; b) with a nite set of jumps that we assume to be non-empty (otherwise there is nothing to prove). For any t 2 S(u), let In (t) be the set of points of In whose limit point is t (this set is de nitively non-empty). Taking notations (3.1) and (3.11) into account, we claim that there exists at least an index i 2 In(t) such that jcin+1 ? cin j < n2 . Indeed, for any x 2 [xin; xin+1), we have jcin+1 ? cin j  jcin+1 ? u(x + n )j + ju(x + n ) ? u(x)j + ju(x) ? cin j: Integrating on [xin; xin+1) and summing on i 2 In (t), we get 1 X jci+1 ? ci j  2 Z b ju (x) ? u(x)j dx + Z b? ju(x +  ) ? u(x)j dx: n n n n2 i2I (t) n a a n

n

If jcin+1 ? cin j  n2 for each i 2 In (t), the left hand side term remains bounded from below. Indeed, X jcin+1 ? cinj: 1  #(In (t))  n12 i2I (t) n

Since the right term goes to 0 as n goes to +1, we get a contradiction. Hence, there exists at least one point i(t) 2 In (t) such that n < i(t) < n + n4. Now, it can be easily checked that En(un)  E(u). Case 3 If rn = n for n even and rn = n2 for n odd, then E (u) = E 1(u), E (u) = E(u). Hence, the ?-limit with respect to the L1 -convergence may not exist. 0

00

5 A remark on second-neighbour interactions In [7] the eect of long-range interaction in discrete systems has been inverstigated. In particular, [7] Theorem 5.3 can be applied to second-neighbour interactions, by 15

considering functionals of the form   i+2 ) ? u(xi )  i+1 i  X X n : (5.1) En(u) = n n1 u(xn )? u(xn) + 2n n2 u(xn 2 n n i i If both sequences of functions ( ni )n satisfy conditions of Corollary 3.3 and some additional growth conditions from above, then the conclusions of Theorem 3.1 hold with   1 2 F(z) = lim (z) + 2 (z) ; n n n and   z   z  2 G(z) = lim  + 4 n n n n n 2n : This means that En can be decomposed as the sum of three `nearest-neighbour type' functionals, with underlying lattices n Z, 2n Zand n (2Z+1), respectively, whose ?-convergence can be studied separately. We now show that a similar conclusion does not hold if we remove the convexity/concavity hypothesis on ni .

Example 5.1 Let (n) be a sequence of positive numbers converging to 0, and let M > 2 be xed. Let En be given by (5.1) with k n (z) =

(

z2

p



kn z ? kn

1 k kn G



p

if jz j  pkn?1 if jz j > kn?1

(k = 1; 2), where 



if jz j < 8 G1(z) = M 1 if jz j  8

1 G2(z) = 1M ifif jjzz jj  > 1.

Neither Gi is subadditive and we have   1 2 if j z j < 8 ? ? 2 1 sub G (z) = 21 ifif jjzz jj > sub G (z) = 1 if jz j  8 1. We can view En as the sum of a rst-neighbour interaction functional and two second-neighbour interaction functionals, to whom we can apply separately Theorem 3.1, obtaining the limit functionals E (u) = 1

for the rst, and

E (u) = 2

Z

b

a Z

a

b

ju_ j dt + 2

ju_ j dt + 2

16

X

S (u) X

S (u)

sub? G1([u]) sub? G2([u])

for each of the second ones. We will show that the ?-limit of En is strictly greater than E 1 (u) + 2E 2(u) at some u 2 SBV (a; b). Let u be given simply by u = (t0 ;b) with t0 2 (a; b). In this case E 1(u) + 2 2E (u) = 4. Suppose that there exist un 2 An(a; b) converging to u and such that lim supn En(un )  4. In this case it can be easily seen that for n large enough there must exist in such that un(xi ) ? un(xi ?1 ) > 4; un(xi +1 ) ? un(xi ) < ?4; but jun(xi ?1 ) ? un(xi ?2 )j < 1; jun(xi +2 ) ? un(xi +1 )j < 1: This implies that un(xi ) ? un(xi ?2 ) > 3; un(xi +2 ) ? un(xi ) < ?3; so that lim supn En(un )  2M, which gives a contradiction. n

n

n

n

n

n

n

n

n

n

n

n

References [1] L. Ambrosio, Existence theory for a new class of variational problems, Arch. Rational Mech. Anal. 111 (1990), 291-322. [2] L. Ambrosio & A. Braides. Energies in SBV and variational models in fracture mechanics. In Homogenization and Applications to Material Sciences, (D. Cioranescu, A. Damlamian, P. Donato eds.), GAKUTO, Gakkotosho, Tokio, Japan, 1997, p. 1-22. [3] L. Ambrosio, N. Fusco and D. Pallara, Special Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, Oxford, to appear. [4] A. Braides, Approximation of Free-Discontinuity Problems, Lecture Notes in Mathematics 1694, Springer Verlag, Berlin, 1998. [5] A. Braides, G. Dal Maso and A. Garroni, Variational formulation of softening phenomena in fracture mechanics: the one-dimensional case, Arch. Rational Mech. Anal. (1999), to appear. [6] A. Braides and A. Defranceschi, Homogenization of Multiple Integrals, Oxford University Press, Oxford, 1998. [7] A. Braides and M.S. Gelli, Limits of discrete systems with long-range interactions. Preprint SISSA, 1999. [8] G. Dal Maso, An Introduction to ?-convergence, Birkhauser, Boston, 1993. [9] 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), 199-210. 17

[10] D. Mumford and J. Shah, Optimal approximation by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math. 17 (1989), 577-685. [11] L. Truskinovsky, Fracture as a phase transition, Contemporary research in the mechanics and mathematics of materials (R.C. Batra and M.F. Beatty eds.) CIMNE, Barcelona, 1996, 322-332.

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