Global-Local subadditive ergodic theorems and application to homogenization in elasticity

We establish a global-local ergodic theorem about subadditive processes which seems to be a flexible tool to identify some limit problems in homogenization involving several small parameters. When the subadditive process is parametrized in a separable space, we show that the convergence takes place in the variational sense of the epiconvergence (or r-convergence). Some applications are given in the setting of nonlinear elasticity.


Introduction
The Ackoglu-Krengel subadditive ergodic theorem asserts, for a subadditive process A -SA, the existence of a pointwise limit for the sequence SAn/meas (An) where (An)n is a family of cubes in Rd whose size tends to infinity. This result seems to be firstly used in the setting of the calculus of variation by G. Dal Maso-L. Modica [10]. In this context, we would like to generalize this theorem to sequences indexed by convex sets. Indeed, homogenization of nonconvex integral functionals with linear growth seems to require this generalisation (see Y. Abddaimi-C. Licht-G. Michaille [2]). In these applications, the limit density (or its regular part in a nonreflexive case) appears to be the limit of a suitable subadditive process and it is of interest to study, from a variational point of view, the "stability" of the limit with respect to perturbations. This is the reason why we study the variational property of the previous convergence when the process depends on a parameter in a metric space. On the other hand many mathematical modelings in homogenization involve several small parameters and the limit problem, in the sense of epiconvergence, depends on their relative behavior. The previous (global) subadditive theorem or the local and more generally the global-local version, according to the various relative behaviors, seems to be an eflicient mathematical tool to identify the limit problem. Consequently, we study the pointwise limit of SAn Qr/meas (An)rq when the "size " p(An ) tends to infinity and that of the cube Qr tends to zero where S is defined on the product Bb(Rd) x Bb(Rq) of bounded Borel sets of Rd and Rq.
The paper is organized as follows. In section 2, we investigate the invariant case : the subadditive set function is invariant when the set index is translated in Zd in the global version, when the set index is translated in Rq in the local version. The result obtained in the global version is well known when the indices are [0, n[d. We give a complete proof of the generalization to a suitable family of convex indices (An)n through some arguments of Nguyen Xuhan Xanh-H. Zessin [18] and various ideas explained in M.A. Ackoglu-U. Krengel [3] and U. Krengel [11]. After giving the local theorem, we mix the two versions to obtain a global-local subadditive theorem and a complete description of the limit.
In view of some applications (see G. Bouchitté-I. Fonseca-L. Mascarenhas [7]), we generalize, in section 3, the previous global result to the quasiperiodic case. Section 4 is devoted to the random case. The subadditive set function takes its values in ~1 (S~, T, P) where (S~, T, P) is a probability space and the translation of the index in Zd modifies the function through a group of P-preserving transformations in the global version. When the family (An)n is constituted of suitable intervals of Rd, we recover the Ackoglu-Krengel ergodic theorem. Our generalisation is perhaps known (see for instance various remarks in U. Krengel ~11~, chapter 7) but we give an exhaustive proof which is a natural extension of the proof of the invariant case and a complete description of the limit in the nonergodic case. We recall without proof the local version due to M.A. Ackoglu-U. Krengel [3] and we give a global-local subadditive theorem.
In section 5, when the subbaditive process depends on a parameter varying through a separable metric space and when the set valued maps 03C9 ẽ pi are random sets, where epi SA (cv, . ) denotes the epigraph of .), we establish, in the global case, a variational almost sure convergence of previous sequences with respect to the parameter : the limit is obtained in the sense of epiconvergence (also called r-convergence). The method consists in applying the previous results to the Baire approximate of -SAn/meas (An) which is a superadditive process. The conclusion then follows thanks to a characterization of epiconvergence by the pointwise convergence of the Baire approximate. We do not give the local or global-local version which are easy adaptations of the previous method.
In the last section, we first recall some results about stochastic homogenization of nonconvex integral functionals and particularly those with linear growth, and give three applications. In the two first one, using Theorem 5.2 about almost sure epiconvergence of parametrized subadditive processes, we establish the continuity of homogenized energy or homogenized density energy with respect to some parameters. The last application concerns a modeling of elastic adhesive bonded joints. At least three parameters appears : the stiffness of the adhesive, the thickness c of the layer filled by the adhesive and the size A of heterogenities. Using the global or the local subadditive ergodic theorem, we give the limit problems corresponding to the cases A « ~ or c « A.     then, /or every sequence (rn)n~N of positive reals tending to zero, lim S A nQ r n ( x 0 ) | A n | r q n n~+ẽ xists and is equal to this common value.
The conclusion will follow if we prove In addition, if every set E in T such that Tz(E) = E for every z E Zd has a probability equal to 0 or 1, (Tz)zEZd is said to be ergodic. A sufficient condition to ensure ergodicity of (z)z~Zd is the following mixing condition : for every E and F in T lim P(TzE n F) = P(E)P(F) Izl+oo which expresses an asymptotic independance. In the sequel, F (resp. Fm, me N*) will denote the 03C3-algebra of invariant sets of T for (z)z~Zd) (resp. for (z)z~mZd, and EF (resp. EFm ) will denote the conditional expectation operator with respect to F (resp. to j~).
A subadditive process for (z)z~Zd is a set function S : Bb(Rd) L 1 (S~, T, P) such that i) VA, B E Bb(Rd) with A n B = 0, SA~B _ sA + SB ii) VA E Bb(Rd), , Vz E Zd, Sz+A = SA o Tz (covariance).
The following result generalizes Theorem 2.1 in a stochastic framework and gives an explicit formula for the limit in the non ergodic case. For the study of the speed of convergence in the ergodic case (more precisely in the independent case), we refer the reader to G. Michaille-J. Michel-L. Piccinini [15]      According to the previous inequality, letting 6  Then as a corollary of Theorems 4.1 and 4.2 we obtain the following global-local and local-global subadditive ergodic theorems, the proof of which being an easy extension of the proof of Theorem 2.2.

Parametric subadditive processes
In what follows, we assume that T is P-complete and that (z)z~Zd is ergodic.
This section is concerned with the variational property of the almost sure convergence studied in section 4, when the subadditive process depends on a parameter which belongs to a separable metric space. For convenience and in view to use some usual concepts of the calculus of variations, the process S will be assumed to be superadditive that is -S is subadditive.
In this context, under (i), (ii) and (iii) every set valued map w e epi sA(., c~), where epi sA(., w) denotes the epigraph of x ~ SA(x,03C9), is a random set and with the terminology of R.T. Rockafellar [17] or H. Attouch-R J.B. Wets [5], every map (x, w) ~ SA(x, cv) is a random lsc function.
We recall that for f, , For any g : X ~ R, and k E N*, we define the Baire approximate of g by gk ( x) := inf ~g(y) + y)~. yEX If g is lsc in X, non identically equal to +oo and satisfies : 03B1 > 0, ~03B2 > 0 ~x0 E X such that b'x E X, g(x) + 03B1d(x, x0) + ,Q > 0 then g~ is lipschitzian with Lipschitz constant k and g = sup g~. Attouch-R J.B. Wets [5] and for a complete study of epiconvergence see H. Attouch E4~ .
In these conditions we state in the theorem below that the almost sure convergence in Theorem 4.1 is variational in the sense of epiconvergence. When S is additive, we recover the law of large numbers for random lsc functions firstly established by H. Attouch-R.J.B. Wets [5]. For more details and an upper bound of the tail probabilities of the law, we refer the reader to G. Michaille-J. Michel-L. Piccinini [15]. Theorem 5.2: If S satisfies (i) -(iv) and -S satisfies the hypotheses of  [4] for stability properties of epiconvergence). We adopt the same notation for this new process. It is easily seen that, for every fixed x, A ~ -infy~X{SA(y, .)+kd(x,y)|A|} is a subadditive process satisfying all the hypothesis of Theorem 2.1 (the measurability comes from the measurability of epi , see H. Attouch-R J.B Wets [5] or C. Hess [11] ). Therefore, D denoting a dense countable subset of X, there exists S21 E T, P(S21) = 1 such that ~03C9 E Hi and 'd~ e D n~+~ m~N* 0 3 A 9 (S[0,m[d md(.,03C9))k (x)dP(03C9) dm E N*.
By equi-lipschitz property of the Baire approximations, the above inequality is satisfyed for every (w, x) in 521 x X. Going to the limit on k, we obtain finally epilimin fSAn |An| ( where 1 p +00. We equip 9 with the trace a-field a(9) of the product a-field of RRdxMmxN and define the group of transformation (Tz)zEZd in G, by zg(x, a) = g(x + z, a).
We finally consider a map f from S2 x Rd x MmxN into R, which is T0 B(Rd) 0 B(Mm d) measurable and such that, for every 03C9 in SZ, f (w, ., .) belongs to 9. In the sequel, to shorten the notations, f will also denote the partial map w H f (c~, ., .) from 52 into ~.
It is clear that the maps Tz f from H into p are (T, Q(~)) measurable. The process f is said to be stationary if, for every z in Zd,  [2] when p = 1 and f is assumed to be stationary only. It is precisely about this last extension that some encountered technical difficulties ( [2], pp. 195-199) motivate us to generalize subadditive theorems to sequences indexed by convex sets. We now give two new applications respectively using Theorem 5, 4.1 and 2.2.

Application to optimization of integral functionals in stochastic homogenization
Let (X, d) be a separable metric space. According to the probabilistic setting stated above, we consider a map f from X x H x Rd x Mmxd into R which is B(X) 0 T ® B(Rd) 8) measurable and which fullfils, for every fixed 8 in X conditions (i) and (ii) below.
(i) w f (8, W, ., .) is a stationary and ergodic process.  (Fen (., cv, A) )nEN is non increasing. Note also, that when p > 1, Then, according to conditions fulfiled by the process W ~ f(03B8,03C9,.,.), and to hypotheses (i), (ii), S is a parametrized subadditive process. Thanks to (ii), we actually have continuity of 0 H Applying Theorem 5.2, we deduce that 03C9 a.s., -I~(03C9,.) epiconverges to -I = -meas(O) infnEN* . Therefore, if the set > 0} of ~-minimizers of is relatively compact in X, we have lim I~ (03C9, 03B8) = sup I(03B8).~0 BEX So, roughly speaking, for maximizing the random energy with respect to a (physical) parameter 0, it suffices, for the small values of c, to maximize the deterministic homogenized energy I. 6.2 Application to the continuity of an homogenized density with respect to a geometrical parameter Let us onsider Di CC]O,l[2, i = 1, 2 and A = ~D1, D2~ equipped with the probability presence pi and p2 of Di and D2. We set S1 = AZ2, define the classical Bernoulli product probability space (H, T, P) and the random chessboard = ~z~Z2((03C9z + z} in R2. Let now f : R2 -~ R be a given function satisfying the growth conditions 6.23 and a, b two numbers in R. We denote by I (R2)  Bouchitté [6]. In our case, above result is an essential tool for describing this problem in a probabilistic setting. 6.3 Application to a modeling of elastic adhesive bonded joints Here, we extend or give more direct proofs of some results of [12], [13] to where we refer for a detailled presentation of the problem (see also [1]). This problems devoted to the modelling of elastic adhesive bonded joints.
Let 0 be a domain with lipschitz boundary in R3 whose intersection S with the plane x3 = 0 is assumed to have a positive two dimensional Hausdorff measure In the sequel x = (x, x3) denotes a current point of R3. If e is a small positive parameter intended to tend to zero, BE :_ e} (respectively O~ :== (9 B B~) denotes the interior of the part of the reference configuration filled by the adhesive (respectively by the adherents). The adhesive and the adherents are assumed to be perfectly stuck together along := {x E O : : ±x3 == e}. They are modeled as hyperelastic. The small positive parameters ~, and A are associated respectively with the low stiffness and the size of heterogenities of the adhesive. We will denote by s the 3-uplet ( , e, A), and s tends to zero means that there exists a sequence En, 03BBn))n going to (0, 0, 0). Moreover, we assume that lims~0 2~ = l with l E [0, +~[. The stored strain energy associated with a displacement field v is then given by the following functional where w denotes a random parameter The structure made of the elastic bodies and the adhesive is clamped on a part fo of 80 with H2(03930) > 0, and is subjected to applied body forces f and applied surface forces g on ri :== 80 B ro. We shall make precisely the following assumptions on the exterior loading and  50 We study the behavior of Us when s tends to zero. Due to the small stiff ness in the layer BE:, the limit displacement field vs can at the limit develop discontinuities along S to which Be shrinks, and converges in R3) to a solution of the limit problem : Qh is the quasiconvex envelope of h, the density of the surface energy defined below and [v] is the jump of the displacement field v through S. Actually, arguing as in [13], it suffices to exhibit the almost sure epilimit of Fs.
The limit problem describes the equilibrium of deformable bodies filling the closure of C~~ = C~ n {~x3 > 0} as reference configurations, made of hyperelastic materials with energy density Qh, subjected to the loading ( f, g), clamped on ro and constrained along S to which Be shrinks. The density b is assumed to be a stationary and ergodic process, that is satisfies (ST) and (ER) with d = 2 and m = N = 3, with, more precisely, It is easily seen that the process w e also satisfies (ST) and (ER). Moreover we assume that the deterministic density h satisfies (H2). In the sequel, to shorten notations, we omit the random variable w. In order to work in a fixed space, we extend Fs by +00 in L2 (C~, R3 ) ~ V and we define the limit energy by R3) : v E W1,2(OBS,R3), v = 0 on ro} and where is defined in Theorems 6.1, 6.4 below and depends on the relative behavior of A and 6.
The limit problem is defined in term of epiconvergence in the space L2 ( C~, R3 ) equipped with its strong topology. More precisely we want to prove that, almost surely, F = epi lims~0 Fs, that is, the sequence of random functions (Fs)s fulfils the two following conditions for every w in a set ~2' of full probability and every u in L2 ( C~, R3 ) : (E1) for every us converging to u F ( u ) liminfs-7 (E2 ) there exists vs in L2 (C~, R3 ) converging to u in L2 (C~, R3 ) such that F(u) > lim sups~0 Fs(us). tends weakly to a bounded Borel measure v. Our method consists in analyzing the limit measure v. More precisely, if v = 03BDa + 03BDsing where va is absolutely continuous with respect to the Lebesgue measure on O and 03BDsing is the singular part of v, we prove a > Qh(.,   Moreover using the Poincaré inequality, it can be easily proved (see [1]) that ~ strongly tends to 16  for the first term of the right hand side and the integral representation (see for instance Dacorogna [9]) of the quasiconvex envelope of the second term. But (see Attouch [4] for the first equality) Third step. If u is not smooth, for (El) we approximate u by ua strongly in W1,2(O, R3) and consider = us -REu + R~u03B4 and conclude as in [13]. For (E2), we reason by density and a diagonalization argument. r-i Remark: It is straightforward to establish (cf [13]) where, for every Q E M3x3 dx: 03C6 E Q.x + R3)}. k~+T his new expression of conform to physical intuition : since A is lower than c, we begin to homogenize the layer, then we let the thickness of the layer tends to zero.
Remark: In the non ergodic case, according to Theorem 4.1, we obtain the following expression of the density of the surface energy : The end of the proof is then identical to that of the previous case. Third step. In the case when u is not smooth, we reason by a density and a diagonalization argument. where the last integral in the right hand side is obviously equal to zero.