Stochastic two-scale convergence of an integral functional

In this paper we discuss the concept of stochastic two-scale convergence, which is appropriate to solve coupled -periodic and stochastichomogenization problems. This concept is a combination of both well-known two-scale convergence and stochastic two-scale convergence in the mean schemes, and is a generalization of the said previous methods. By way of illustration we apply it to solve a homogenization problem related to an integral functional with convex integrand. This problematic relies on the notion of dynamical system which is our basic tool. 1. Introduction The two-scale convergence method by Nguetseng [23] has proved very e¢ cient to handle periodic homogenization problems. It then generated a great number of research activities that increases over time as shown by the vast existing literature to date; see e.g., [2, 3, 5, 6, 20, 37, 38, 39] and the references therein. However, being strictly limited to periodic structures, it quickly showed its inadequacy as far as the non periodic phenomena are concerned, since in nature, few physical phenomena have in fact a periodic behaviour. Furthermore these phenomena are often subjected to randomness. To overcome this inadequacy, Bourgeat et al. [10] introduced the stochastic two-scale convergence in the mean’s method. It has helped to go from deterministic periodic homogenization theory to stochastic homogenization theory, much closer to the reality of natural and physical phenomena. It is worth pointing out that in between the deterministic periodic and stochastic homogenization theories, there exists a recent general deterministic homogenization theory which was built up recently in [24], and which has just been improved in [26]. We also draw the attention of the reader to the papers [41] and [7]. In the …rst paper [41] a systematic treatment of the stochastic two-scale convergence with respect to invariant probability measures is done. It is to be noted that in the said-paper the authors consider probability spaces de…ned on compact metric spaces. In [7] the authors propose an alternative approach to stochastic homogenization by using di¤eomorphisms to perturb the microscale. In this work we rely on the previous two convergence methods (see [23] and [10]) to propose a general method of solving coupled periodic and stochastic homogenization problems. Our method, the stochastic two-scale convergence method combining both two-scale convergence and stochastic two-scale convergence in the mean schemes is a straightforward generalization of the previous method and seems to be appropriate for natural phenomena since most of these phenomena behave 2000 Mathematics Subject Classi…cation. 35B27, 35B40, 37A05, 37A55. Key words and phrases. Dynamical system, homogenization, stochastic two-scale convergence. 1 2 MAMADOU SANGO AND JEAN LOUIS WOUKENG randomly in some scales, and deterministically in other scales. Our multiscale approach is motivated by the fact that usual monoscale approach has proven to be inadequate because of prohibitively large number of variables involved in each physical problem. One can also give at least two reasons quite natural. Firstly, a scale can not be at the same time deterministic and random. Secondly, the application of our results to natural phenomena. This second reason is the most important; in fact let us be more precise and give an extra motivation for the choice in this case, which motivation arises from the following quite simple example. It is known that the human body is an example of medium that presents both a random and deterministic behaviour. In particular it contains an exciting class of nonlinear materials presenting microstructures like myocardium, arterials walls, cartilage, muscles etc. These biological materials are characterized by hierarchical ordering of microstructures, ranging randomly from nanoscale to macroscale. The understanding of these materials requires a proper study, which can be provided by means of deterministic-stochastic homogenization theory. That is why it becomes urgent to develop systematic method and approach for multiscale problems. This is the aim of the present work. Returning to our original issue, we aim at providing a framework suitable to handle both deterministic and stochastic homogenization problems. Our approach is based on the well-known theory of dynamical systems. The organization of the paper is as follows. In Section 2, we give some preliminaries about the dynamical systems. In Section 3, we state the concept of stochastic two-scale convergence and we prove some compactness results. In Section 4, we apply the results of Section 3 to solve a mixed periodic-stochastic homogenization problem related to an integral functional with convex integrand. Unless otherwise speci…ed, vector spaces throughout are assumed to be real vector spaces, and scalar functions are assumed to take real values. We shall always assume that the numerical spaces R and their open sets are each equipped with the Lebesgue measure. 2. Preliminaries on dynamical systems We begin by recalling the de…nition of the notion of a dynamical system. Let ( ;M; ) denote a probability space. An N -dimensional dynamical system on is a family of invertible mappings T (x) : ! , x 2 R , such that the following conditions hold: (i) (Group property) T (0) = id and T (x+ y) = T (x) T (y) for all x; y 2 R ; (ii) (Invariance) The mappings T (x) : ! are measurable and -measure preserving, i.e., (T (x)F ) = (F ) for each x 2 R and every F 2M; (iii) (Measurability) For each F 2 M, the set (x; !) 2 R : T (x)! 2 F is measurable with respect to the product -algebra L M, where L is the -algebra of Lebesgue measurable sets. If is a compact topological space, by a continuous N -dimensional dynamical system on we mean any family of mappings T (x) : ! , x 2 R , satisfying the above group property (i) and the following condition: The mapping (x; !) 7! T (x)! is continuous from R to . STOCHASTIC HOMOGENIZATION 3 Let 1 p 1. An N -dimensional dynamical system T (x) : ! induces a N -parameter group of isometries U (x) : L( )! L( ) de…ned by (U (x) f) (!) = f (T (x)!) , f 2 L( ) which is strongly continuous, i.e., U (x) f ! f in L( ) as x ! 0; see [18, p. 223] or [28, p. 131]. We denote by Di;p (1 i N) the generator of U(x) along the ith coordinate direction, and by Di;p its domain. Thus, for f 2 L( ), f is in Di;p if and only if the limit Di;pf de…ned by Di;pf(!) = lim !0 f (T ( ei)!) f(!) exists strongly in L( ), where ei denotes the vector ( ij)1 j N , ij being the Kronecker symbol. One can naturally de…ne higher order derivatives by setting D p = D 1 1;p D N N;p for = ( 1; :::; N ) 2 N . Now we need to de…ne the stochastic analog of the smooth functions on R . To this end, we set Dp( ) = \i=1Di;p and de…ne D1 p ( ) = f 2 L ( ) : D p f 2 Dp( ) for all 2 N : It is a fact that each element of D1 1( ) possesses stochastic derivatives of any order that are bounded. So as in [4] we denote it by the suggestive symbol C1( ), and it can be shown that C1( ) is dense in L( ), 1 p < 1. At this level, one can naturally de…ne the concept of stochastic distribution: by a stochastic distribution on is meant any continuous linear mapping from C1( ) to the real …eld R. We recall that C1( ) is endowed with its natural topology de…ned by the family of seminorms Nn(f) = supj j n sup!2 jD 1f(!)j (where j j = 1+ :::+ N for = ( 1; :::; N ) 2 N ). We denote the space of stochastic distributions by (C1( )). One can also de…ne the stochastic weak derivative of f 2 (C1( )) as follows: For any 2 N , D f stands for the stochastic distribution de…ned by (D f) ( ) = ( 1)j jf (D ) 8 2 C1( ): As C1( ) is dense in L( ) (1 p < 1), it is immediate that L( ) (C1( )) so that one may de…ne the stochastic weak derivative of any f 2 L( ), and it veri…es the following functional equation: (D f) ( ) = ( 1)j j R fD d for all 2 C1( ). In particular, for f 2 Di;p we have R fDi;1 d = R Di;pfd for all 2 C1( ) so that we may identify Di;pf with D f , where i = ( ij)1 j N . Conversely, if f 2 L ( ) is such that there exists fi 2 L ( ) with (D f) ( ) = R fi d for all 2 C1( ), then f 2 Di;p and Di;pf = fi. Therefore, endowing Dp( ) with the natural graph norm kfkpDp( ) = kfk p Lp( ) + N X i=1 kDi;pfkpLp( ) (f 2 Dp( )) we obtain a Banach space representing the stochastic generalization of the Sobolev spaces W 1;p R , and so, we denote it by W ( ). Now, returning to the general setting of dynamical systems, we recall that a function f 2 L ( ) is said to be invariant for T (relative to ) if for any x 2 R , f T (x) = f -a.e. on . We denote by I nv ( ) the set of functions in L ( ) that are invariant for T . The set I nv ( ) is a closed vector subspace of L p ( ). The dynamical system T is said to be ergodic if every invariant function is -equivalent to a constant. We have the following very useful properties for functions in L ( ). 4 MAMADOU SANGO AND JEAN LOUIS WOUKENG (P1) For f 2 D1 1 ( ), and for -a.e. ! 2 , the function x 7! f(T (x)!) is in C1(RN ) and further D xf(T (x)!) = (D 1 f) (T (x)!) for any 2 N . (P2) For f 2 L ( ), we have f 2 I nv ( ) if and only if Di;1f = 0 for each 1 i N . Let 1 < p <1. Thanks to (P2) above, one can easily check that, for f 2 L ( ), f is in I nv ( ) if and only if Di;pf = 0 for all 1 i N . So if we endow C1( ) with the seminorm