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Some developments of the homogenization theory and related questions

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(1):. DOCTORA L T H E S I S. Some Developments of the Homogenization Theory and Related Questions. Jonas Engström. Luleå University of Technology Department of Mathematics :|: -|: - -- ⁄ -- .

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(52) 49 (2004). APPLICATIONS OF MATHEMATICS. No. 2, 111–122. BOUNDS AND NUMERICAL RESULTS FOR HOMOGENIZED DEGENERATED p-POISSON EQUATIONS ,. , and. , Lule˚ a. (Received November 13, 2001). Abstract. In this paper we derive upper and lower bounds on the homogenized energy density functional corresponding to degenerated p-Poisson equations. Moreover, we give some non-trivial examples where the bounds are tight and thus can be used as good approximations of the homogenized properties. We even present some cases where the bounds coincide and also compare them with some numerical results. Keywords: homogenization, bounds, degenerated, p-Poisson equation MSC 2000 : 35B27, 35J60, 74Q20. 1. Introduction In many types of materials, e.g. composites, the physical properties can be modelled by a Y -periodic function λ. For small values of ε the function λ(x/ε) will oscillate rapidly which means that the material is strongly heterogeneous on a local scale. Nevertheless, the material will globally act as a homogeneous medium. It is extremely difficult to find the effective properties λhom which describe this homogeneous medium. The field of mathematics that rigorously defines the notion of effective properties is known as homogenization. Consider a class of physical problems described by a minimum energy principle of the form    1 x p (1) Eε = min λ |Du| dx − f u dx , u ε Ω p Ω where u belongs to some subset of W 1,1 (Ω) and represents the state of the material. It is known that the energy Eε → Ehom as ε → 0, where Ehom is of the type    Ehom = min λhom (Du) dx − f u dx , u. Ω. Ω. 111.

(53) and λhom is defined as (2).  λhom (ξ) = min v. Y. 1 λ(y)|ξ + Dv|p dy. p. For a proof of these homogenization results when λ is bounded between two positive constants see e.g. [13]. The degenerated case, i.e. when λ is allowed to approach zero or infinity, was studied in [6] (see also [2]). By using numerical methods it is possible to compute λhom by formula (2), see e.g. [3]. Another approach is to find bounds on λhom . Bounds for the case when λ is bounded between two positive constants were presented in [10] (see also [9]). These bounds, combined with reiterated homogenization (i.e. introducing λ of the form λ(x/ε, . . . , x/εm )), have played a central role in the development of new optimal structures, see e.g. [1], [7], [8], [11], [12], [14] and [16]. Moreover, these bounds were used in an extension of the Ponte Castaneda variational principle ([4] and [5]) to obtain bounds for a class of more general nonlinear problems than those described above, see [15]. The main results of this paper are that we prove lower and upper bounds on λhom for the degenerated case (see Theorem 1 and Theorem 2), i.e. we find functions λlower and λupper such that λlower (ξ)  λhom (ξ)  λupper (ξ). Moreover, we present some illustrative examples where the bounds are tight and thus can be used as a good approximation of λhom . 2. Notation and preliminary results Let Ω be an open bounded subset of n , Y the unit cube in n and ·, · the Euclidean inner product. Let 1 < p < ∞, 1/p + 1/q = 1, and let λ be a Y -periodic (weight) function such that λ > 0 a.e., and λ, λ−1/(1−p) are in L1loc (. n. ).. The set of all real-valued functions u in L1loc (Ω) such that uλ1/p is in Lp (Ω) is 1,1 denoted by Lp (Ω, λ). The set of functions u in Wloc (Ω) such that u and |Du| are in 1,p p 1,p L (Ω, λ) is denoted by W (Ω, λ). Moreover, by W0 (Ω, λ) we mean the closure of 1,1 1,p C01 (Ω) in W 1,p (Ω, λ) and Wper (Y, λ) is the set of real functions u in Wloc ( n ) such that u is Y -periodic and u ∈ W 1,p (Y, λ). Define the family (uε ) as the set of solutions of the variational problems    1 x p λ |Du| dx − min f u dx . Eε = ε u∈W01,p (Ω,λ(x/ε)) Ω p Ω 112.

(54) Under the additional assumption that λ belongs to the Muckenhoupt class Ap it was proved in [6] that (uε ) converges weakly in W01,1 (Ω) to the unique solution uhom of the homogenized problem    Ehom = min λ (Du) dx − f u dx hom 1,p u∈W0. (Ω). Ω. where λhom is defined as (3). Ω. . λhom (ξ) =. min 1,p. v∈Wper (Y,λ). Y. 1 λ(y)|ξ + Dv|p dy. p. It was also proved that Eε → Ehom . We remark that the family of solutions (uε ) of the minimization problems described above are also solutions to the weighted weak formulations of the p-Poisson equations, namely  ⎧  

(55) x ⎪ ⎨ λ |Duε |p−2 Duε , Dϕ dx = f ϕ dx, ε Ω Ω ⎪ ⎩ uε ∈ W01,p (Ω, λ(x/ε)) for every ϕ in W01,p (Ω, λ(x/ε)). The homogenized solution uhom satisfies the homogenized problem  ⎧ ⎨ b(Duhom ), Dϕ dx = f ϕ dx, ⎩ for every ϕ in. Ω. Ω. uhom ∈. W01,p (Ω),. W01,p (Ω),. where b is given by  b(ξ) = λ(y)|ξ + Dwξ |p−2 (ξ + Dwξ ) dy Y. and wξ is the solution of the local problem ⎧ ⎨ λ(y)|ξ + Dwξ |p−2 (ξ + Dwξ ), Dϕ dy = 0, Y (4) ⎩ 1,p wξ ∈ Wper (Y, λ), 1,p (Y, λ). for every ϕ in Wper. 1. The solution wξ of the local problem (4) is also the minimizer in the local minimization problem (3) and

(56) 1 b(ξ), ξ . λhom (ξ) = p. 113.

(57) 3. Bounds In this section we present upper and lower bounds on the homogenized energy density functional λhom defined in (3). The bounds are given in the following two theorems: Theorem 1. Let λhom be defined as in (3). Then we have the upper bound def. λhom (kei )  λupper(kei ) = |k|. p1. where {e1 , . . . , en } is the canonical basis in  λi =. . 1. p n. 1. 0. 1/(1−p) λi. 1−p dyi. ,. and. 1. ... 0. . λ dy1 . . . dyi−1 dyi+1 . . . dyn . 0. Theorem 2. Let λhom be defined as in (3). Then we have the lower bound λhom (kei )  λlower (kei )  1  1 1 def = |k|p ... λ1/(1−p) 1−p dy1 . . . dyi−1 dyi+1 . . . dyn , i p 0 0 where {e1 , . . . , en } is the canonical basis in  λ. 1/(1−p). i =. n 1. and. λ1/(1−p) dyi .. 0. 2. By linearity, we get lower and upper bounds on λhom for all ξ ∈. n. when p = 2. For the proofs of Theorem 1 and Theorem 2 we need the following two lemmata, which themselves are of independent interest: Lemma 3. Let D be a measurable set in d such that |D| = 1. Moreover, let a  0 be a weight function such that a ∈ L1 (D) and a1/(1−r) ∈ L1 (D), where 1 < r < ∞. Then  1−r  r 1/(1−r) min a(x)|1 + u(x)| dx = a(x) dx , u∈U. D. D. where (5). U=.    u dx = 0 . u ∈ Lr (D, a) : D. 114.

(58) Moreover, the minimum is attained for . −1 a1/(1−r) dx a1/(1−r) − 1.. u = D. . The reversed Hölder inequality and (5) imply that .  a(x)|1 + u(x)|r dx  D. D.  =. r 1−r     1 + u(x) dx a(x)1/(1−r) dx   1−r 1/(1−r) a(x) dx .. D. D. Equality holds in Hölder’s inequality when ca1/(1−r) = |1 + u| = 1 + u. The zero average constraint implies that  c=. a. 1/(1−r). −1 dx .. D. Since a ∈ L1 (D) and a1/(1−r) ∈ L1 (D), it follows that 0 < c < ∞. Moreover, u  ∈ Lr (D, a) since .  | u|r a dx =. |ca1/(1−r) − 1|r a dx    1/(1−r) C a dx + a dx < ∞,. D. D. D. D. . where C is a constant. Lemma 4. Let λhom be defined as in (3). Then λ∗hom (ξ).  =. inf. σ∈V σ dy=0 Y. Y. 1 1−q λ |ξ + σ|q dy, q. where λ∗hom is the Legendre transform of λhom and V is defined as V =.    1,p σ ∈ Lq (Y, λ1−q ) : σ, Dv dy = 0 for all v ∈ Wper (Y, λ) . Y. 115.

(59) . Let f :. n. →. be defined as f (ξ) =. . The Legendre transform f ∗ of f is. n. where s is a fixed vector in. 1 λ|ξ + s|p , p. f ∗ (σ) = sup {σ, ξ − f (ξ)} = def. ξ∈. n. 1 1−q q λ |σ| − σ, s. q. This implies Young’s inequality σ, ξ  f (ξ) + f ∗ (σ). (6). for all σ, ξ ∈. n. ,. with equality for σ = λ|ξ + s|p−2 (ξ + s).. (7). Inequality (6) implies that for any measurable function σ we have  λhom (ξ) =. min 1,p. v∈Wper (Y,λ). . 1 λ(y)|ξ + Dv|p dy p. Y. . min 1,p. v∈Wper (Y,λ). 1 σ, ξ − λ1−q |σ|q + σ, Dv dy. q Y. This implies  (8). λhom (ξ)  sup σ∈V. 1 σ, ξ − λ1−q |σ|q dy. q Y. Actually we have equality in (8). This fact will be clear if we prove that  (9). λhom (ξ)  sup σ∈V. 1 σ, ξ − λ1−q |σ|q dy. q Y. Let wξ be the minimizer in (3) and let σ1 be defined as (10). σ1 = λ|ξ + Dwξ |p−2 (ξ + Dwξ ).. Then it follows by (6) and (7) that  λhom (ξ) = 116. 1 σ1 , ξ − λ1−q |σ1 |q + σ1 , Dwξ  dy. q Y.

(60) Next we note that σ1 ∈ V and thus (9) holds. Indeed, by (10) and Remark 1 we have   σ1 , Dϕ dy = λ|ξ + Dwξ |p−2 (ξ + Dwξ ), Dϕ dy = 0 Y. for every ϕ ∈. Y. 1,p Wper (Y, λ). . and (10) implies that  q 1−q |σ1 | λ dy = |ξ + Dwξ |p λ dy < ∞.. Y. Y. We now proceed as follows: . 1 λhom (ξ) = sup σ, ξ − λ1−q |σ|q dy q σ∈V Y    1 1−q q λ |σ| dy . = sup η, ξ − inf σ∈V η Y q. (11). Y. Let F : (12). n. σ dy=η. →. be defined as  1 1−q q λ |σ| dy = F (η) = inf σ∈V Y q Y. σ dy=η.  inf. σ∈V σ dy=0 Y. Y. 1 1−q λ |η + σ|q dy. q. In view of (11) and (12) it follows that λhom (ξ) = sup[η, ξ − F (η)] = F ∗ (ξ). η. Since F is convex and lower semicontinuous we have λ∗hom (ξ) = F ∗∗ (ξ) = F (ξ) and the proof is complete.. . of Theorem 1. Without loss of generality we prove the result for k = 1. 1,p (Y, λ) : v = v(yi )}. Lemma 3 then gives Let Mi = {v ∈ Wper  1 λhom (ei ) = λ(y)|ei + Dv(y)|p dy min 1,p v∈Wper (Y,λ) Y p  1 λ(y)|1 + Di v(yi )|p dy  min v∈Mi Y p  1 1 λi |1 + Di v(yi )|p dyi = min v∈Mi 0 p 1−p  1 1 1/(1−p) = λi dyi . p 0  117.

(61) of Theorem 2. Without loss of generality we prove the result for k = 1. Let    Si = σ ∈ V : σ = (0, . . . , σi (y1 , . . . , yi−1 , yi+1 , . . . , yn ), . . . , 0) and σ dy = 0 . Y. By using Lemma 4 and Lemma 3 we obtain λ∗hom (ei ).  =. inf. σ∈V σ dy=0 Y. Y. 1 1−q λ |ei + σ|q dy q. . 1 1−q λ |1 + σi (y1 , . . . , yi−1 , yi+1 , . . . , yn )|q dy Y q  1  1 1 1−q = inf λ i |1 + σi |q dy1 . . . dyi−1 dyi+1 . . . dyn ... σ∈Si 0 0 q 1−q  1  1 1 = ... λ1−q 1−p dy . . . dy dy . . . dy . 1 i−1 i+1 n i q 0 0  inf. σ∈Si. This implies the following lower bound on λhom (ei ): λhom (ei ) = sup {ei , ξ − λ∗hom (ξ)} ξ∈. n.  sup {t − λ∗hom (tei )} tei ∈. n. = sup{t − |t|q λ∗hom (ei )} t∈.  1−q   1  1 |t|q 1−q 1−p  sup t − ... λ i dy1 . . . dyi−1 dyi+1 . . . dyn q t∈ 0 0  1  1 1 = ... λ1/(1−p) 1−p dy1 . . . dyi−1 dyi+1 . . . dyn . i p 0 0 . 4. Some examples In this section we apply the bounds from Theorem 1 and Theorem 2 in two illustrative examples. The examples are presented in 2 for simplicity. Let us first remark that when the upper and lower bounds are equal we know the effective energy density functional exactly. For instance, this is the case when λ is of the type λ(y) = f (y1 )g(y2 ), 118. λ(y) is Y -periodic..

(62) Then it follows from Theorem 1 and Theorem 2 that 1 p. λhom (e1 ) =. 1 λhom (e2 ) = p. . 1. 1−p  f (y1 )1/(1−p) dy1. 0. . 1. g(y2 )1/(1−p) dy2. 0. 1. g(y2 ) dy2 ,. 0 1−p  1. f (y1 ) dy1 . 0. We now give one example of this situation where the conductivity degenerates on the unit cell boundary. We note that it is easy to make the mistake of believing that we have zero conductivity in one direction and infinitely high conductivity in the other direction. However, as we show below, the homogenized energy density functional is nonzero and finite in both directions. 5. Consider the special case when p = 2 and let λ : Y -periodic and defined as   λ(y) = |y1 (1 − y1 )|−1/2 y1 −. 2. →. be. 1/2 1  on Y. 2. Then, by Theorem 1 and Theorem 2, we have λhom (e1 ) =. 1 2. 1 λhom (e2 ) = 2.  . 1. λ−1 (y1 ) dy1. 0 1. λ(y1 ) dy1 = 0. √ −1 1 2 2  1  K √ =√ , 3 2 2 π  1 ,. −1. 2K. √. 2. where K(·) is the complete elliptic integral of the first kind. This means that λhom (e1 ) ≈ 0.40451,. λhom (e2 ) ≈ 0.84721,. that is, the effective conductivity in the y2 -direction is only about twice as high as the effective conductivity in the y1 -direction. We now consider an example where the upper and lower bounds are very tight. This means that we have a good explicit estimate of the effective energy density functional. This fact can be used to obtain error estimates for numerical computations. We demonstrate this by comparing the bounds with numerical computations done in MATLAB using the FEMLAB toolbox. 6. Let Dr =.     1 1  1   , y ∈ Y : y −   r, 0  r  2 2 2 119.

(63) and let λ :. 2. →. be Y -periodic and defined as ⎧   α ⎨ 1 y − 1 , 1  , r 2 2 λ(y) = ⎩ 1,. y ∈ Dr , y ∈ Y \ Dr ,. where −2 < α < 2p − 2. By symmetry, the homogenized energy density functional λhom (e1 ) = λhom (e2 ). Let r = 0.4 and let λ− , λ+ denote the lower and upper bounds, respectively.. Figure 1. This picture shows 9 unit cells for each of the values α = −1/2, α = 1/2, α = 1 and α = 3/2. The radius is r = 0.4.. (a) The linear case, p = 2: We present the results rounded to five digits. α −3/2 −1 −1/2 1/2 1 3/2 120. λhom (ei ) 0.70147 0.63470 0.56597 0.44172 0.39389 0.35639. λ− 0.67425 0.62059 0.56203 0.43797 0.38207 0.33773. λ+ 0.74023 0.65433 0.57082 0.44482 0.40284 0.37078.

(64) (b) The nonlinear case, p = 3: We present the results rounded to four digits. α −1 1 2 3. λhom (ei ) 0.4325 0.2649 0.2219 0.1952. λ− 0.4260 0.2608 0.2121 0.1821. λ+ 0.4482 0.2708 0.2353 0.2132. As we see from these tables, the lower and upper bounds are very tight, which means that they can be used as a good approximation of the homogenized energy density functional. Acknowledgement. We thank the referee and Professor Lars-Erik Persson for valuable comments which have improved the final version of this paper. References [1] A. Braides, D. Lukkassen: Reiterated homogenization of integral functionals. Math. Models Methods Appl. Sci. 10 (2000), 47–71. [2] J. Byström, J. Engström, and P. Wall: Reiterated homogenization of degenerated nonlinear elliptic equations. Chinese Ann. Math. Ser. B 23 (2002), 325–334. [3] J. Byström, J. Helsing, and A. Meidell: Some computational aspects of iterated structures. Compos-B: Engineering 32 (2001), 485–490. [4] P. Ponte Castaneda: Bounds and estimates for the properties of nonlinear heterogeneous systems. Philos. Trans. Roy. Soc. London Ser. A 340 (1992), 531–567. [5] P. Ponte Castaneda: A new variational principle and its application to nonlinear heterogeneous systems. SIAM J. Appl. Math. 52 (1992), 1321–1341. [6] R. De Arcangelis, F. Serra Cassano: On the homogenization of degenerate elliptic equations in divergence form. J. Math. Pures Appl. 71 (1992), 119–138. [7] J.-L. Lions, D. Lukkassen, L.-E. Persson, and P. Wall: Reiterated homogenization of monotone operators. C. R. Acad. Sci. Paris Ser. I Math. 330 (2000), 675–680. [8] J.-L. Lions, D. Lukkassen, L.-E. Persson, and P. Wall: Reiterated homogenization of nonlinear monotone operators. Chinese Ann. Math. Ser. B 22 (2001), 1–12. [9] D. Lukkassen: On some sharp bounds for the off-diagonal elements of the homogenized tensor. Appl. Math. 40 (1995), 401–406. [10] D. Lukkassen, L.-E. Persson, and P. Wall: On some sharp bounds for the homogenized p-Poisson equation. Appl. Anal. 58 (1995), 123–135. [11] D. Lukkassen: Formulae and bounds connected to optimal design and homogenization of partial differential operators and integral functionals. Ph.D. thesis. Dept. of Math., Tromsö University, Norway, 1996. [12] D. Lukkassen: Bounds and homogenization of integral functionals. Acta Sci. Math. 64 (1998), 121–141. [13] P. Marcellini: Periodic solutions and homogenization of nonlinear variational problems. Ann. Mat. Pura Appl. 117 (1978), 139–152. [14] P. Wall: Homogenization of some partial differential operators and integral functionals. Ph.D. thesis. Dept. of Math., Lule˚ a University of Technology, Sweden, 1998.. 121.

(65) [15] P. Wall: Bounds and estimates on the effective properties for nonlinear composites. Appl. Math. 45 (2000), 419–437. [16] P. Wall: Optimal bounds on the effective shear moduli for some nonlinear and reiterated problems. Acta Sci. Math. 65 (2000), 553–566. Authors’ address: J. Byström, J. Engström, and P. Wall, Dept. of Mathematics, Lule˚ a University of Technology, S-97187 Lule˚ a, Sweden, e-mail: johanb@sm.luth.se, jonase @sm.luth.se, wall@sm.luth.se.. 122.

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(67)

(68) Homogenization of random degenerated nonlinear monotone operators 1. J. Engstr¨om1 , L.-E. Persson1 , A. Piatnitski2 and P. Wall1 Department of Mathematics, Lule˚ a University of Technology, SE-971 87 Lule˚ a, Sweden. 2. Narvik University College, N-8505 Narvik, Norway and. P.N.Lebedev Physical Institute of RAS, Leninski pr., 53, Moscow 117924, Russia.. Abstract This paper deals with homogenization of random nonlinear monotone operators in divergence form. We assume that the structure conditions (strict monotonicity and continuity conditions) degenerate and are given in terms of a weight function. Under proper integrability assumptions on the weight function we construct the effective operator and prove the homogenization result.. AMS Subject Classification (2000): 35B27, 35B40. Keywords: Stochastic homogenization, random operators, degenerated monotone operators.. 1. Introduction. Mathematical description of microscopically heterogeneous media usually involves rapidly oscillating functions of the form a = a (x/ε) where ε is a small positive parameter characterizing the microscopic length scale of the media. The aim of homogenization theory is to provide the macroscopic rigorous description of the studied media. Homogenization is at present a well developed area and there is a vast literature on the topic, see e.g. [1]-[8] and [12]-[19]. The homogenization problems for various random structures are widely discussed in the physical and mathematical literature, see e.g. [13] and its bibliography. The first rigorous results for random elliptic operators in divergence form with stochastically homogeneous coefficients were obtained by Kozlov in [14] and independently by Papanicolaou and Varadhan in [18]. Later on, many other random models were investigated, among them are random porous media (see for instance [12]), convection-diffusion problems (see e.g. [3]), nonlinear models (see e.g. [17]) etc. In [6] Bourgeat et al. developed the stochastic version of the two-scale convergence approach. 1.

(69) We study random nonlinear monotone operators in divergence form, which satisfy weighted structure conditions with weight Λ (x) > 0 being a statistically homogeneous random field. Concerning this weight we assume that Λ ∈ L1loc (Rn ) and Λ−1/(p−1) ∈ L1loc (Rn ) and also that some uniform integrability condition of Muckenhoupt type (see Definition 2 below) holds. The corresponding Dirichlet problem takes the form (f ∈ L∞ (Q), Q ⊂ Rn ): ⎧ ⎨ −div(A(x/ε, Duε )) = f in Q, ⎩ u ∈ W 1,p (Q, Λ (x/ε)), ε 0 where A(x/ε, ·) is a statistically homogeneous field which satisfies the degenerated structure conditions (17) and (18) below, and Q is a regular domain in Rn . In the paper we prove the a.s. convergence uε.  u weakly in W01,1 (Q),. A(x/ε, Duε )  b(Du) weakly in L1 (Q)n , and show that the limit function u is the unique solution of the following effective equation ⎧ ⎨ −div(b(Du)) = f in Q, (1) ⎩ u ∈ W 1,p (Q). 0 The coefficients b(ξ) here are expressed in terms of solutions of an auxiliary problem involving random variables a(ξ) = A(0, ξ). For details see (9) and (13) In periodic case similar homogenization results were obtained in [2] where the framework of weighted Sobolev spaces was used. We believe that one can make use of the singular measure approach developed in [5], [16], [20] and [21], to investigate the problems of this type. Notice that for non-degenerated random operators stronger convergence holds, namely (see e.g. [11] or [17]): uε. . A(x/ε, Duε ) . u weakly in W01,p (Q), b(Du) weakly in Lp (Q)n .. The paper is organized as follows: Section 2 contains the setup and some technical statements. In section 3 we define the class of potential vectorfunctions in a weighted space; then in section 4 we introduce an auxiliary stochastic problem and construct the formal homogenized operator; Finally in the last section the homogenization result is proved.. 2. Notation and preliminaries. First we recall the notion of random dynamical system. Let (Ω, F, μ) be a probability space. A family of measurable mappings Tx : Ω → Ω (x ∈ Rn ) 2.

(70) is called a n-dimensional random dynamical system if it satisfy the following properties: 1. T0 = I (i.e. T0 is the identity mapping) and Tx+y = Tx Ty (for every x, y ∈ Rn ). 2. The map Tx : Ω → Ω preserves the measure μ i.e. for every x ∈ Rn and every U ∈ F μ (U) = μ (Tx (U)) . 3. For any measurable function f on Ω, the function f (Tx ω) defined on Rn ×Ω is measurable (Rn ×Ω is endowed with the product σ-algebra B ×F, where B stands for the Borel σ-algebra). Given such a dynamical system we can introduce a wide class of statistically homogeneous random fields. Indeed, let f : Ω → Rn be a random function then the function F (x) = f (Tx ω) is a statistically homogeneous random field. If ω ∈ Ω is fixed, the function F (x) is called a realization of f . We say that f = f (ω) is invariant if f (ω) = f (Tx ω) a.e. in Ω, for every x ∈ Rn . A dynamical system is ergodic if every invariant function is constant a.s. We assume in the rest of this work that the dynamical system Tx is ergodic. The following result will be useful later (for a proof see [13]). Lemma 1 Let Ω0 be a measurable subset of Ω such that μ(Ω0 ) = 1. Then there exists a measurable subset Ω1 ⊂ Ω0 such that μ(Ω1 ) = 1 and for any ω ∈ Ω1 we have Tx ω ∈ Ω0 for a.e. x ∈ Rn . Now we proceed by introducing weight functions and weighted spaces. Assume that λ : Ω → R is a measurable function such that λ > 0 a.s. and λ ∈ L1 (Ω),. λ−1/(p−1) ∈ L1 (Ω),. (2). for some p, 1 < p < ∞. Then by the Fubini theorem almost all realizations satisfies Λ (x) = λ (Tx ω) > 0 a.e. and Λ ∈ L1loc (Rn ) ,. Λ−1/(p−1) ∈ L1loc (Rn ) .. (3). We denote by Lp (Ω, λ) the set of functions u in L1 (Ω) such that uλ1/p ∈ Lp (Ω), and by Lploc (Rn , Λ) the set of functions u ∈ L1loc (Rn ) such that uΛ1/p ∈ Lploc (Rn ) . Let Q be a regular bounded domain in Rn , then W 1,p (Q, Λ) stands 1,1 for the space of functions u in Wloc (Q) such that u ∈ Lp (Q, Λ) and Du ∈ n 1,p Lp (Q, Λ) . Denote by W0 (Q, Λ) the completion of C01 (Q) in W 1,p (Q, Λ) with respect to the norm 1/p  p p (|u| + |Du| ) Λ dx . Q. 3.

(71) The conditions (3) are rather natural but not sufficient for our purposes. We will impose a stronger version of these conditions, namely the so-called Muckenhoupt condition. For the reader convenience we formulate it below. Definition 2 Let K ≥ 1 and let Λ be a positive function on Rn . Then Λ belongs to the class Ap (K) if for every cube Q ⊂ Rn with faces parallel to the coordinate planes the following condition is satisfied: . 1   Q . . . Λ dx Q. 1   Q . p−1. . 1 − p−1. Λ. ≤ K.. dx. Q. here and in what follows |B| stands for the Lebesgue measure of a Borel set B. We also define Ap = ∪K≥1 Ap (K). Now we define the set of weights used in this paper. p is defined as the set of positive functions λ : Ω → R Definition 3 The class NK whose realizations belong to Ap (K) a.s.. Some properties of the weight functions satisfying the Muckenhoupt condition, are given by the following statement proved in [9]. Lemma 4 Let K ≥ 1. Then there exist two positive constants δ = δ (n, p, K) and C = C (n, p, K) such that for every cube Q ⊂ Rn with faces parallel to the coordinate planes and every Λ ∈ Ap (K) . . 1   Q . 1   Q . . 1  1+δ.  Λ. 1+δ. ≤ C . dy. Q. 1  Q . 1  1+δ. −(1+δ)/(p−1). Λ. ≤ C . dy. Q. 1  Q .  Λdy,. (4). Λ−1/(p−1) dy.. (5). Q.  Q. We end this section by formulating a version of compensated compactness lemma, adapted to the framework of weighted spaces. For the proof see [2]. Lemma 5 (Compensated compactness) Let ν ∈ Ap , K ≥ 1, and let Q be an open bounded subset of Rn . Given a family of weights {Λε : Λε ∈ Ap (K), ε > 0}, suppose that {uε } is a family of functions such that  p 1. Q |Duε | Λε dy ≤ C1 < ∞ for all ε > 0, 2. There is u ∈ W 1,p (Q, ν) such that uε → u in L1 (Q). and (Aε ) is a family of vector functions in Rn such that  q −1/(p−1) dy ≤ C2 < ∞ for all ε > 0, 3. Q |Aε | Λε 4.

(72) 4. there exists g ∈ L∞ (Q) such that div(Aε ) = g on C01 (Q) for every ε > 0,. n n 5. there is A ∈ Lq Q, ν −1/(p−1) such that Aε  A weakly in L1 (Q) . . Then. Q.  Aε , Duε φ dy →. Q. A, Du φ dy,. for every φ ∈ C0∞ (Q) .. 3. Potential functions in a weighted space. n. Recall that a vector field f ∈ Lploc (Rn ) is said to be potential if there exists 1,p a function u ∈ Wloc (Rn ) such that f = Du. A vector field v is said to be solenoidal if div v = 0 in the weak sense, i.e.  v, Dφ dx = 0 for all φ ∈ C0∞ (Rn ) . Rn. Now we turn to random vector fields. Let us first recall the definition of potential and solenoidal random fields in the non-weighted case. A random function n f ∈ Lp (Ω) is said to be potential if almost all its realizations are potential. Notice that by the Fubini theorem the realizations of f are a.s. elements of n Lploc (Rn ) . Solenoidal random vector field is defined similarly. In order to define potential vector fields in a weighted probability space, notice that due (2) and H¨older inequality we have .  |f (ω)| dμ ≤ Ω. 1/p    p−1 p 1 − p−1 |f (ω)| λ(ω) dμ (λ(ω)) dμ < ∞, p. Ω. Ω. for any f ∈ Lp (Ω, λ)n . Therefore, any element of Lp (Ω, λ)n belongs to L1 (Ω) and its realizations belong to L1loc (Rn ). It is then natural to say that a vector field f ∈ Lp (Ω, λ)n is potential if n almost all its realizations are potential vector fields in L1loc (Rn ) . We denote p n this space by Lpot (Ω, λ) . We also define p Vpot.

(73)  p (Ω, λ) = f ∈ Lpot (Ω, λ) : f dμ = 0 . Ω. Note that since convergence in Lp (Ω, λ) implies convergence for a subsequence of p (Ω, λ) is closed in Lp (Ω, λ)n . almost all realizations in L1loc (Rn ), the space Vpot Lemma 6 Let f ∈ Lppot (Ω, λ). Then there is σ = σ(n, p, K) > 0 such that the n n realizations of f a.s. belong to L1+σ loc (R ) .. 5.

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