aa r X i v : . [ m a t h . A P ] J a n Derivation of Darcy’s law in randomly perforated domains
A. GIUNTI
Abstract:
We consider the homogenization of a Poisson problem or a Stokes system in a randomlypunctured domain with Dirichlet boundary conditions. We assume that the holes are spherical andhave random centres and radii. We impose that the average distance between the balls is of size ε and their average radius is ε α , α ∈ (1; 3). We prove that, as in the periodic case [2], the solutionsconverge to the solution of Darcy’s law (or its scalar analogue in the case of Poisson). In the samespirit of [12, 14], we work under minimal conditions on the integrability of the random radii. Theseensure that the problem is well-defined but do not rule out the onset of clusters of holes. We are interested in the effective behaviour of a Stokes system or a Poisson equation in a boundeddomain D ε ⊆ R , perforated by many random small holes H ε . We impose Dirichlet boundary conditionson the boundary of the holes and of the domain. Problems like the one studied in this paper arisemostly in fluid-dynamics where a Stokes system in a punctured domain models the flow of a viscous andincompressible fluid through many disjoint obstacles. We focus on the regime where the effective equationis given by Darcy’s law or its scalar analogue in the case of the Poisson problem. For the latter, thiscorresponds to the case where the average density of harmonic capacity of the holes H ε goes to infinityin the limit ε ↓
0. In the case of Stokes the same is true, this time with the harmonic capacity beingreplaced by the so-called
Stokes capacity . This is a vectorial version of the harmonic capacity where theclass of minimizers further satisfies the incompressibility constraint (see (4.8)).We construct the randomly punctured domain D ε as follows: Given α ∈ (1 ,
3) and a bounded C , -domain D ⊆ R , we define D ε := D \ H ε , H ε := [ z ∈ Φ ∩ ε D B ε α ρ z ( εz ) . (0.1)Here, the set of centres Φ is a Poisson point process of intensity λ > ε D := { x ∈ R : εx ∈ D } .The radii R = { ρ z } z ∈ Φ ⊆ [1; + ∞ ) are independent and identically distributed random variables satisfyingfor a constant C < + ∞ E (cid:2) ρ α (cid:3) C. (0.2)This condition is minimal in order to ensure that, P -almost surely and when ε is small, the set H ε doesnot fully cover the domain D , hence implying that D ε = ∅ (see Lemma 1.1). However, condition (0.2)does not prevent that, with high probability, the balls in H ε do overlap.For ε > D ε as above, we consider the (weak) solution to either − ∆ u ε = f in D ε u ε = 0 on ∂D ε (0.3)or to − ∆ u ε + ∇ p ε = f in D ε ∇ · u ε = 0 in D ε u ε = 0 on ∂D ε (0.4)1n the case of the Stokes system, we further assume that E (cid:2) ρ α + β (cid:3) C, for some β >
0. (0.5)We refer to the next section for a more detailed discussion on what conditions (0.2) and (0.5) entail interms of the geometric properties of the set H ε .It is easy to see that in the case of spherical periodic holes having distance ε and radius ε α , α ∈ (1 , H ε is asymptotically of order ε − α ; The same is true in the case ofthe Stokes capacity. When α = 3 these limits are thus finite. In the case of the Poisson problem, thesolutions to (0.3) thus converge to the solution u ∈ H ( D ) to − ∆ u + µu = f in D , where the constant µ > Brinkmannsystem , namely a Stokes system in D with no-slip boundary conditions and with the additional term ˜ µu in the system of equations [1]. The term ˜ µ > µ is apositive-definite matrix. For α ∈ (1; 3) as in the present paper, the solutions to (0.3) or (0.4) need to berescaled by the factor ε − α in order to converge to a non-trivial limit. The effective equations, in thiscase, are either u = kf in D or Darcy’s law u = K ( f − ∇ p ) in D [2]. Here, k, K are related to the rescaledlimit of the density of capacity and admit a representation in terms of a corrector problem solved in theexterior domain R \ B (0).When α = 1, namely when the distance between holes and their size have the same order ε , theeffective equations for (0.3) and (0.4) are as in the case α ∈ (1 , k, K obtained inthe limit, however, are determined by a corrector problem of different nature. In this case indeed, there isonly one microscopic scale ε and the relative distance between the connected components of the holes H ε does not tends to infinity for ε →
0. This yields that the corrector equations are solved in the periodiccell and not in the exterior domain R \ B (0) [3].For holes that are not periodic, the extremal regimes α ∈ { , } have been rigorously studied bothin deterministic and random settings. For α = 3 we mention, for instance [6, 9, 16, 17, 18, 21, 22, 23]and refer to the introductions in [12] and [14] for a detailed overview of these results. We stress thatthe homogenization of (0.3) and (0.4) when H ε is as in (0.1) with α = 3 has been studied in the seriesof papers [12, 13, 14]. These works prove the convergence to the effective equation under the minimalassumption that H ε has finite averaged capacity density. There is no additional condition on the minimaldistance between the balls in the set of H ε .There are many works devoted also to the regime α = 1. We refer, in particular, to [4] where (0.3)and (0.4) are studied for a very general class of stationary and ergodic punctured domains. For thesedomains, the formulation of the corrector equation for the the effective quantities k, K is solved in theprobability space (Ω , F , P ) generating the holes.There is fewer mathematical literature concerning the homogenization of (0.3) or (0.4) in the regime α ∈ (1; 3). For periodic holes, this has been studied in [2]. These results have been extended for certainregimes to compressible Navier-Stokes systems [15] or to elliptic systems in the context of linear elasticity[19]. We are not aware of analogous results when the holes H ε are not periodic. The present paperconsiders this problem when H ε is random and, in the same spirit of [12, 14], allows that the balls in H ε overlap and cluster.The main result of this paper is the following: 2 heorem 0.1. Let σ ε := ε − − α and let H ε and D ε be the random sets defined in (0.1) .(a) Let u ε ∈ H ( D ε ) solve (0.3) with f ∈ L q ( D ) for q ∈ (2; + ∞ ] . Then, if the marked point process (Φ , R ) satisfies (0.2) , for every p ∈ [1; 2) we have that lim ε ↓ E (cid:2) ˆ D | σ ε u ε − kf | p (cid:3) = 0 , with k := (4 πλ E (cid:2) ρ (cid:3) ) − .Here, and in the rest of the paper, E (cid:2) · (cid:3) denotes the expectation under the probability measure for (Φ , R ) .(b) Let u ε ∈ H ( D ε ; R ) solve (0.4) with f ∈ L q ( D ; R ) for q ∈ (2; + ∞ ] . If (Φ , R ) satisfies (0.2) , thenfor every p ∈ [1; 2) we have lim ε ↓ E (cid:2) ˆ D | σ ε u ε − K ( f − ∇ p ∗ ) | p (cid:3) = 0 , with K := (6 πλ E (cid:2) ρ (cid:3) ) − and p ∗ ∈ H ( D ) (weakly) solving −∇ · ( ∇ p ∗ − f ) = 0 in D ( ∇ p ∗ − f ) · ν = 0 on ∂D D p ∗ = 0 . As mentioned above, condition (0.2) is minimal in order to ensure that the set D ε is non-empty for P -almost every realization. A lower stochastic integrability assumption for the radii, indeed, yields that,in the limit ε ↓ P -almost surely H ε , covers the full set D (see Lemma 1.1 in the next section).By the Strong Law of the Large Numbers, condition (0.2) implies that the density of capacity is almostsurely of order ε − α as in the periodic case. As already remarked in [12] in the case α = 3, with (0.4) werequire that the radii satisfy the slightly stronger assumption (0.5). While (0.2) seems to be the optimalcondition in order to control the density of harmonic capacity, the lack of subadditivity of the Stokescapacity calls for a better control on the geometry of the set H ε .The ideas used in the proof of Theorem 0.1 are an adaptation of the techniques used in [2, 7] for theperiodic case. They are combined with the tools developed in [12, 14] to tackle the case of domains havingholes that may overlap. As shown in [2], the uniform bounds on the sequences { σ ε u ε } ε> , { σ ε ∇ u ε } ε> are obtained by means of a Poincaré’s inequality for functions that vanish on ∂D ε . If v ∈ H ( D ε ), sincethe function vanishes on the holes H ε , the constant in the Poincaré’ s inequality is of order σ − ε << v ∈ H ( D ), this would instead be of order 1 (dependent on the domain D ). Note that, as for α = 3we have σ ε = 1, there is no gain in using a Poincar’e’s inequality in H ( D ε ) instead of in H ( D ) inthis regime. In the case of centres of H ε that are distributed like a Poisson point process, the is a lowprobability that some regions of D ε have few holes, thus leading to a worse Poincaré’s constant. Thiscauses the lack of uniform bounds for the family { σ ε u ε } ε> in L ( D ).Equipped with uniform bounds for the rescaled solutions of (0.3), one may prove Theorem 0.1, ( a ) byconstructing suitable oscillating test functions { w ε } ε> . These allow to pass to the limit in the equationand identify the effective problem. We stress that a crucial ingredient in these arguments is given by thequantitative bounds obtained in [11] in the case α = 3. These bounds may indeed also be extended to thecurrent setting sot that the rate of convergence of the measures − σ − ε ∆ w ε ∈ H − ( D ) is quantified. Thisallows to control the convergence of the duality term h− ∆ w ε ; u ε i H − ( D ); H ( D ) . There is a fine balance the3onvergence of − σ ε ∆ w ε with the right space where we have uniform bounds for { σ ε u ε } ε> . In contrastwith the periodic case, the unboundedness of { σ ε u ε } ε> in L ( D ) requires for a careful study of the dualityterm above. For the precise statements, we refer to (3.3) in Lemma 3.1 and Lemma 3.3. The same ideassketched here apply also to the case of solutions to (0.4). This time, the oscillating test functions { w ε } ε> are replaced by the reduction operator R ε of Lemma 4.1. Remark . We comment below on some variations and corollaries of Theorem 0.1:( i ) If Φ = Z d or is a stationary point process satisfying for a finite constant C < + ∞ max z i ,z j ∈ Φ | z i − z j | < C P -almost surely,then the convergence of Theorem 0.1 holds also with p = 2. In this case, indeed, we may drop thelogarithmic factor in the bounds of Lemma 2.1.The assumption R ⊆ [1; + ∞ ) may be also weakened to R ⊆ [0; + ∞ ), provided that E (cid:2) ρ − γ (cid:3) < + ∞ , for an exponent γ ∈ (1; + ∞ ]. In this case, the convergence of Theorem 0.1 holds in L p ( D ) for p ∈ [1; ¯ p ) with ¯ p = ¯ p ( γ ) ∈ [1; 2) such that ¯ p ( γ ) → γ → + ∞ .( ii ) A careful inspection of the proof of Theorem 0.1 yields that, under assumption (0.5) and for a source f ∈ W , ∞ , the convergences in both ( a ) and ( b ) may be upgraded to E (cid:2) ˆ D | σ ε u ε − u | p (cid:3) . ε κ , for an exponent κ > α, β .( iii ) The quenched version of Theorem 0.1, namely the P -almost sure convergence of the families in L p ( D ), holds as well provided that we restrict to any vanishing sequence { ε j } j ∈ N that convergesfast enough. For instance, it suffices that j + ǫ ε j → ǫ >
0. It is a technical but easy argument toobserve that, under this assumption, limits (3.3) of Lemma 3.1 and (4.1)-(4.2) of Lemma 4.1 vanishalso P -almost surely. From these, the quenched version of Theorem 0.1 may be shown as done inthe annealed case. To control the limits in (3.3), (4.1) and (4.2) without taking the expectation,one may follow the same lines of the current proof and control most of the terms by the Strong Lawof Large Numbers. Condition j + ǫ ε j → { ε j } j ∈ N is needed inorder to obtain quenched bounds for the term in (3.31) by means of Borel-Cantelli’s Lemma.( iv ) The analogue of Theorem 0.1 holds also for a general dimension d > α ∈ (1; dd − ) and rescale the solutions by σ ε = ε − dd − + α . In this case, (0.2) and (0.5) hold with theexponent α replaced by dα .The paper is structured as follows: In the next section we describe the setting and introduce thenotation that we use throughout the proofs. Subsection 1.2 is devoted to discussing the minimality ofassumption (0.2) and what condition (0.5) implies on the geometry of the holes H ε . In Section 2, we showthe uniform bounds on the family { σ ε u ε } ε> , with u ε solving (0.3) or (0.4). In Section 3 we argue Theorem0.1 in case ( a ), while in Section 4 we adapt it to case ( b ). The proof of case ( b ) is conceptually similar tothe one for ( a ), but it is technically more challenging. It heavily relies on the geometric properties of theholes implied by condition (0.5). Finally, Section 5 contains the proof of the main auxiliary results usedthroughout the paper. 4 Setting and notation
Let D ⊆ R be an open set having C , -boundary. We assume that D is star-shaped with respect to apoint x ∈ R . This assumption is purely technical and allows us to give an easier formulation for the setof holes H ε . With no loss of generality we assume that x = 0.The process (Φ; R ) is a stationary marked point process on R having identically and independentdistributed marks on [1 + ∞ ). In other words, (Φ; R ) may be seen as a Poisson point process on thespace R × [1; + ∞ ), having intensity ˜ λ ( x, ρ ) = λf ( ρ ). The expectation in (0.2) or (0.5) is thereforetaken with respect to the measure f ( ρ )d ρ . We denote by (Ω; F , P ) the probability space associated to(Φ , R ), so that the random sets in (0.1) and the random fields solving (0.3) or (0.4) may be written as H ε = H ε ( ω ), D ε = D ε ( ω ) and u ε ( ω ; · ), respectively. The set of realizations Ω may be seen as the setof atomic measures P n ∈ N δ ( z n ,ρ n ) in R × [1; + ∞ ) or, equivalently, as the set of (unordered) collections { ( z n , ρ n ) } n ∈ N ⊆ R × [1; + ∞ ).We choose as F the smallest σ -algebra such that the random variables N ( B ) : Ω → N , ω { ω ∩ B } are measurable for every set B ⊆ R the Borel σ -algebra B R . Here and throughout the paper, p ∈ [1; + ∞ ) we define the space L p (Ω) as the space of( F -measurable) random variables F : Ω → R endowed with the norm E (cid:2) | F ( ω ) | p (cid:3) p . For p = + ∞ , we set L ∞ (Ω) as the space of P -essentially bounded random variables. We denote by L p (Ω × D ), p ∈ [1; + ∞ ),the space of random fields F : Ω × R → R that are measurable with respect to the product σ -algebraand such that E (cid:2) ´ D | F ( ω, x ) | p d x (cid:3) p < + ∞ . The spaces L p (Ω) , L p (Ω × R ) are separable for p ∈ [1 , + ∞ )and reflexive for p ∈ (1 , + ∞ ) (see e.g. [5][Section 13,4]). The same definition, with obvious modifications,holds in the case of the target space R replaced by R .We often appeal to the Strong Law of Large Numbers (SSLN) for averaged sums of the form ∩ B R ) − X z ∈ Φ ∩ B R X z , where { X z } z ∈ Φ ε ( D ) are identically distributed random variables that have sufficiently decaying correlations.Here, we send the radius of the ball B R to infinity. It is well-known that such results hold and we referto [14][Section 5] for a detailed proof of the result that is tailored to the current setting. We use the notation . or & for C or > C where the constant depends only on α , λ , D and, in case ( b ),also on β in (0.5). Given a parameter p ∈ R , we use the notation . p if the implicit constant also dependson the value p . For r >
0, we write B r for the ball of radius r centred in the origin of R . We denote by h · ; · i the duality bracket between the spaces H − ( D ) and H ( D ).When no ambiguity occurs, we skip the argument ω ∈ Ω in all the random objects considered in thepaper. If (Φ; R ) is as in the previous subsection, for a set A ⊆ R d , we defineΦ ε ( A ) := (cid:8) z ∈ Φ : εz ∈ A (cid:9) , N ε ( A ) := ε ( A ) . For x ∈ R , we define the random variables d x := 12 min z ∈ Φ z = x | z − x | , R x := min (cid:8) d x , (cid:9) , d x,ε := εd x , R ε,x := εR x . (1.1)5 .2 On the assumptions on the radii In this subsection we discuss the choice of assumptions (0.2) and (0.5) in Theorem 0.1. We postpone tothe Appendix the proofs of the statements. The next result states that assumption (0.2) is sufficient tohave only microscopic holes whose size vanishes in the limit ε ↓
0. Moreover, it is also necessary in orderto have that holes H ε do not cover the full domain D . Lemma 1.1.
The following conditions are equivalent:(i) The process satisfies (0.2) ;(ii) For P -almost every realization and for every ε small enough the set D ε = ∅ .Furthermore, ( i ) ( or ( ii ) ) implies that for P -almost realization lim ε ↓ | D ε | = | D | . In the following result we provide the geometric information on H ε that may be inferred by strength-ening condition (0.2) to (0.5). Roughly speaking, the next lemma tells that, under condition (0.5), wehave a control on the maximum number of holes of comparable size that intersect. More precisely, we maydiscretize the range of the size of the radii { ρ z } z ∈ Φ ε ( D ) and partition the set of centres Φ ε ( D ) accordingto the order of magnitude of the associated radii. The next statement says that there exists an M ∈ N (that is independent from the realization ω ∈ Ω) such that, provided that the step-size of the previousdiscretization is small enough, each sub-collection contains at most M holes that overlap when dilated bya factor 4. This result allows to treat also the case of the Stokes system in Theorem 0.1, (b) and motivatesthe need of the stronger assumption (0.5) in that setting. Lemma 1.2.
Let (Φ , R ) satisfy (0.5) . Then:(i) There exists κ = κ ( α, β ) > , k max = k max ( α, β ) , M = M ( α, β ) ∈ N such that for P -almost everyrealization and for every ε small enough it holds sup z ∈ Φ ε ( D ) ε α ρ z ε κ (1.2) and we may rewrite H ε = k max [ i =1 [ z ∈ I i,ε B ε α ρ z ( εz ) , inf z ∈ I ε,i ε α ρ z > ε κ sup z ∈ I ε,i − ε α ρ z for i = 1 , · · · , k max (1.3) such that for every i = 1 , · · · k max { B ε α ρ z ( εz ) } z ∈ I i,ε ∪ I i − ,ε , contains at most M elements that intersect. (1.4) (ii) For every δ > there exists ε = ε ( δ ) > and a set B ∈ F such that P ( B ) > − δ and for every ω ∈ B and ε ε inequality (1.2) holds and there exists a partition of H ε satisfying (1.3) - (1.4) . In this section we provide uniform bounds for the family { σ ε u ε } ε> and { σ ε ∇ u ε } ε> . We stress that, asin [2], this is done by relying on a Poincaré’s inequality for functions that vanish in the holes H ε . Theorder of magnitude of the typical size (i.e. ε α ) and distance (i.e. ε ) of the holes yields that the Poincaré’s6onstant scales as the factor σ ε introduced in Theorem 0.1. This, combined with the energy estimate for(0.3) or (0.4), allows to obtain the bounds on the rescaled solutions. We mention that the next resultscontain both annealed and quenched uniform bounds. The quenched versions are not needed to proveTheorem 0.1, but may be used to prove the quenched analogue described in Remark 0.2, (iii). Lemma 2.1.
Let u ε be is as in Theorem 0.1. Then for every p ∈ [1; 2)lim sup ε ↓ E (cid:2) ˆ D | σ ε ∇ u ε | + | log ε | − | σ ε u ε | + ˆ D | σ ε u ε | p (cid:3) . p . (2.1) Furthermore, for P -almost every realization, the sequences { σ ε u ε } ε> and { σ ε ∇ u ε } ε> are bounded in L p ( D ) , p ∈ (1; 2) , and in L ( D ) , respectively. This, in turn, is a consequence of
Lemma 2.2.
For every p ∈ [1; 2] and for every v ∈ H ( D ε ) we have (cid:0) ˆ D | σ ε v | p (cid:1) p . C ε ( p ) (cid:0) ˆ D |∇ v | (cid:1) × for p ∈ [1; 2) | log ε | if p = 2 , (2.2) where the random variables { C ε ( p ) } ε> satisfy lim sup ε ↓ C ε ( p ) . p P -almost surely, lim sup ε ↓ E (cid:2) C qε ( p ) (cid:3) . p for every q ∈ [1; + ∞ ) . (2.3) Proof of Lemma 2.2.
As first step, we argue that the following Poincaré’s inequality holds: Let V bea convex domain. Assume that V ⊆ B r for some r >
0. Let s < r . Then, for every q ∈ [1; 2] and u ∈ H ( V \ B s ) such that u = 0 on ∂B s it holds (cid:0) ˆ V \ B s | u | q (cid:1) q . r q s (cid:0) ˆ V \ B s |∇ u | (cid:1) . The proof of this result is standard and may be easily proven by writing the integrals in sphericalcoordinates. We stress that the assumptions on V allows to write the domain V \ B s as { ( ω, r ) ∈ S n − × R + , s ∧ R ( ω ) r < R ( ω ) } for some function R : S → R satisfying k R k L ∞ ( S ) r .As second step, we construct an appropriate random tesselation for D : We consider the Voronoitesselation { V z } z ∈ Φ associated to the point process Φ, namely the sets V z := (cid:8) y ∈ R : | y − z | = min z ∈ Φ | z − y | (cid:9) , for every z ∈ Φ.We define V ε,z := (cid:8) y ∈ R : 1 ε y ∈ V z (cid:9) , A ε := (cid:8) z ∈ Φ α : V z,ε ∩ D = ∅ (cid:9) . Note that, by the previous rescaling, we have that, if diam ( V z ) := r z , then diam ( V ε,z ) = εr z .It is immediate to see that, for every realization ω ∈ Ω, the sets { V ε,z } z ∈ A ε are essentially dis-joint, convex and cover the set D . Since Φ is stationary, the random variables { r z } z ∈ Φ are identically7istributed. Furthermore, they are distributed as a generalized Gamma distribution having intensity g ( r ) = C ( λ ) r exp − c ( d,λ ) r [20][Proposition 4.3.1.]. From this, it is a standard computation to show thatlim ε ↓ ε E (cid:2) | A ε | q (cid:3) q = | D | for every q ∈ [1 , + ∞ ) (2.4)and that there exists a constant c = c ( λ ) > F : R + → R (that is integrablewith respect to the measure g ( r ) dr ) E (cid:2) exp ( cr ) (cid:3) . , | E (cid:2) F ( r z ) F ( r y ) (cid:3) − E (cid:2) F ( r ) (cid:3) | . E (cid:2) F ( r ) (cid:3) ε − c | x − y | . (2.5)Equipped with { V ε,z } z ∈ A ε , we argue that for every realization of H ε and all p ∈ [1; 2) it holds ˆ D | v | p σ − pε C ε ( p ) (cid:0) ˆ D |∇ v | (cid:1) p (2.6)with C ε ( p ) p := (cid:0) ε P z ∈ A ε r − p z (cid:1) − p . Note that by (2.4), (2.5) and the Law of Large Numbers the family { C ε ( p ) } ε> satisfies (2.3). We show (2.6) as follows: For every v ∈ H ( D ε ), we rewrite ˆ D | v | p = X z ∈ A ε ˆ V εz | v | p . Since ρ z >
1, we have that B ε α ( εz ) ⊆ B ε α ρ z ( εz ) so that the function v ∈ H ( D ε ) vanishes on B ε α ( εz ).Hence, thanks to the choice of { V ε,z } z ∈ A ε , we apply Lemma 2 in each set V εz with B s = B ε α ( εz ) and B r = B εr z ( εz ) and infer that ˆ D | v | p . ε ε − p α X z ∈ A ε r z (cid:0) ˆ V εz |∇ v | (cid:1) p . (2.7)Since p ∈ [1 , ˆ D | v | p . σ − pε (cid:0) ε X z ∈ A ε r − p z (cid:1) − p (cid:0) X z ∈ A ε ˆ V εz ∩ D |∇ v | (cid:1) p , i.e. inequality (2.6). This concludes the proof of (2.2) in the case p ∈ [1; 2).To tackle the case p = 2 we need a further manipulation: we distinguish between points z ∈ A ε having r z > − log ε or r z − log ε : ˆ D | v | = X z ∈ Aεrz − log ε ˆ V εz | v | + X z ∈ Aεrz> − log ε ˆ V εz | v | . (2.8)We apply Poincaré’s inequality in H ( D ) on every integral of the second sum above. This implies that X z ∈ Aεrz> − log ε ˆ V εz ∩ D | v | . σ − ε ˆ D |∇ v | (cid:0) ε X z ∈ A ε ε − σ ε r z > − log ε (cid:1) , so that Chebyschev’s inequality and (2.5) yield X z ∈ Φ ε ( D ) dz> − log ε ˆ V εz ∩ D | v | . σ − ε C ε (2) ˆ D |∇ v | , C ε (2) := (cid:0) ε P z ∈ A ε exp (cid:0) r z (cid:1)(cid:1) . Note that, again by (2.4)-(2.5) and the Law of Large Numbers,this definition of C ε (2) satisfies (2.3). Inserting the previous display into (2.10) implies that ˆ D | v | . X z ∈ Aεrz − log ε ˆ V εz ∩ D | v | + σ ε C ε (2) ˆ D |∇ v | . (2.9)We now apply Lemma 2 in the remaining sum and obtain (2.7) with p = 2, where the sum is restrictedto the points z ∈ A ε such that r z − log ε . From this, we infer that ˆ D | v | . σ ε ( | log ε | + C ε (2) ) ˆ D |∇ v | . (2.10)By redefining C ε (2) = min (cid:0) ε P z ∈ A ε exp (cid:0) r z (cid:1) ; 1 (cid:1) , the above inequality immediately implies (2.2) for p = 2.The proof of Lemma 2.2 is complete. Proof of Lemma 2.1.
We prove Lemma 2.1 for u ε solving (0.3). The case (0.4) is analogous. Since f ∈ L q ( D ) with q ∈ (2; + ∞ ], we may test (0.3) with u ε and use Hölder’s inequality to control ˆ D |∇ u ε | (cid:0) ˆ D | f | q (cid:1) q (cid:0) ˆ D | u ε | qq − (cid:1) q − q . We thus appeal to (2.2) with p = qq − and obtain that (cid:0) ˆ D | σ ε ∇ u ε | (cid:1) . C ε ( qq − − q (cid:0) ˆ D | f | q (cid:1) q . (2.11)Thanks to (2.3) of Lemma 2.2, this yields that the sequence { σ ε ∇ u ε } ε> is bounded in L ( D ) for P -almostevery realization. Similarly, we infer (2.1) by taking the expectation and applying Hölder’s inequality.We argue the remaining bounds for the terms of u ε in a similar way: We combine Lemma 2.2 withthe same calculation above for (2.11) and apply Hölder’s inequality. This establishes Lemma 2.1. ( a ) Lemma 3.1.
There exists an ε = ε ( d ) such that for every ε < ε and P -almost every realization thereexists a family { w ε } ε>ε ⊆ W , + ∞ ( R ) such that k w ε k L ∞ ( R ) = 1 , w ε = 0 in H ε and lim sup ε ↓ ˆ D | σ − ε ∇ w ε | . , lim ε ↓ ˆ D | w ε − | = 0 . (3.1) In addition, lim sup ε ↓ E (cid:2) ˆ D | σ − ε ∇ w ε | (cid:3) . , lim ε ↓ E (cid:2) ˆ D | w ε − | (cid:3) = 0 , (3.2) and for every φ ∈ C ∞ ( D ) and v ε ∈ H ( D ε ) satisfying the bounds of Lemma 2.1 and such that σ ε v ε ⇀ v in L (Ω × D ) , it holds E (cid:2) |h− ∆ w ε ; v ε φ i − k − ˆ D vφ | (cid:3) → . (3.3) Here, the constant k is as in Theorem 0.1, ( a ) . roof of Theorem 0.1, ( a ) . The proof is similar to the one in [2]. We first show that σ ε u ε ⇀ u in L p ( D × Ω), p ∈ [1 , u ∗ ∈ L p (Ω × R d ), p ∈ [1 , P -almost surely, the function u ∗ = kf in D . This, in particular, also implies that the full family { σ ε u ε } ε> weakly converges to u ∗ .We restrict to the converging subsequence { σ ε j u ε j } j ∈ N . However, for the sake of a lean notation, weforget about the subsequence { ε j } j ∈ N and continue using the notation u ε and ε ↓
0. Let ε and { w ε } ε> be as in Lemma 3.1. For every ε < ε , χ ∈ L ∞ (Ω) and φ ∈ C ∞ ( D ) we test equation (0.3) with χw ε φ andtake the expectation: E (cid:2) χ ˆ D ∇ ( w ε φ ) · ∇ u ε (cid:3) = E (cid:2) χ ˆ D f w ε φ (cid:3) . Using Leibniz’s rule, integration by parts and the bounds for u ε and w ε in Lemma 2.1 and 3.1 we reduceto lim ε ↓ E (cid:2) χ h− ∆ w ε ; u ε φ i (cid:3) = E (cid:2) χ ˆ D f φ (cid:3) . We now appeal to (3.3) in Lemma 3.1 applied to the converging subsequence { u ε } ε> and conclude that E (cid:2) χ ˆ D φ ( k − u ∗ − f ) (cid:3) = 0 . Since χ ∈ L ∞ (Ω) and φ ∈ C ∞ ( D ) are arbitrary, we infer that for P -almost every realization u ∗ = kf for(Lebesgue-)almost every x ∈ D . We stress that in this last statement we used the separability of L p ( D ), p ∈ [1 , ∞ ). This establishes that the full family σ ε u ε ⇀ kf in L p (Ω × D ), p ∈ [1 , a ) it remains to upgrade the previous convergence from weak to strong.We fix p ∈ [1 , f , the function u ∗ ∈ L q ( D ), for some q ∈ (2; + ∞ ]. Let { u n } n ∈ N ⊆ C ∞ ( D ) be an approximating sequence for u ∗ in L q ( D ).Since w ε ∈ W , ∞ ( D ), the function w ε u n ∈ H ( D ). Hence, by Lemma 2.2 applied to u ε − w ε u n weobtain E (cid:2) ˆ D | σ ε u ε − w ε u n | p (cid:3) σ − pε E (cid:2) C ( p ) p (cid:0) ˆ D |∇ ( σ ε u ε − w ε u n ) | (cid:1) p (cid:3) and, since p < C ( p ) satisfies (2.3) of Lemma 2.2, also E (cid:2) ˆ D | σ ε u ε − w ε u n | p (cid:3) (cid:0) σ − ε E (cid:2) ˆ D |∇ ( σ ε u ε − w ε u n ) | (cid:3)(cid:1) p . (3.4)We claim that lim ε ↓ σ − ε E (cid:2) ˆ D |∇ ( σ ε u ε − w ε u n ) | (cid:3) = k − ˆ D | u n − u ∗ | , (3.5)so that lim sup ε ↓ E (cid:2) ˆ D | σ ε u ε − w ε u n | p (cid:3) . ˆ D | u n − u ∗ | . (3.6)Provided this holds, we establish Theorem 0.1, ( a ), as follows: By the triangle inequality we have that ˆ D | σ − ε u ε − u | p ˆ D | u n − u ∗ | p + ˆ D | σ − ε u ε − w ε u n | p + ˆ D | w ε − | p | u n | . u ∗ and u n ∈ C ∞ ( D ) are deterministic, we take the expectation and use Lemma 3.1 with (3.6) toget lim sup ε ↓ E (cid:2) ˆ D | σ ε u ε − u | p (cid:3) . ˆ D | u n − u ∗ | p + ( ˆ D | u n − u ∗ | ) p . This implies the statement of Theorem 0.1, ( a ), since p < { u n } n ∈ N converges to u in L ( D ).We thus turn to (3.5): We skip the lower index n ∈ N and write u instead of u n . If we expand theinner square, we write σ − ε E (cid:2) ˆ D |∇ ( σ ε u ε − w ε u ) | (cid:3) = σ ε E (cid:2) ˆ D |∇ u ε | (cid:3) − E (cid:2) ˆ D ∇ u ε · ∇ ( w ε u ) (cid:3) + σ − ε E (cid:2) ˆ D |∇ ( w ε u ) | (cid:3) . (3.7)For first term in the right-hand we use (0.3) and the fact that σ ε u ε ⇀ u ∗ in L p (Ω × D ) with p ∈ [1 , ε ↓ σ ε E (cid:2) ˆ D |∇ u ε | (cid:3) = ˆ D f u ∗ . (3.8)We focus on the remaining two terms in (3.7): Using Leibniz’s rule and an integration by parts we havethat E (cid:2) ˆ D ∇ u ε · ∇ ( w ε u ) (cid:3) = E (cid:2) ˆ D w ε ∇ u ε · ∇ u (cid:3) + E (cid:2) h− ∆ w ε : u ε u i (cid:3) − E (cid:2) ˆ D u ε ∇ w ε · ∇ u (cid:3) . Thanks to Lemma 2.1, Lemma 3.1 and since u ∈ C ∞ ( D ), the first and second term vanish in the limit ε ↓
0. Hence, lim ε ↓ E (cid:2) ˆ D ∇ u ε · ∇ ( w ε u ) (cid:3) = lim ε ↓ E (cid:2) h− ∆ w ε ; u ε u i (cid:3) . (3.9)By Lemma 2.1 and since u ε ⇀ u ∗ , we may apply (3.3) of Lemma 3.1 with φ = u and v ε = u ε to the limiton the right-hand side above. This yieldslim ε ↓ E (cid:2) ˆ D ∇ u ε · ∇ ( w ε u ) (cid:3) = ˆ k − u ∗ u. (3.10)We now turn to the last term in (3.7). Also here, we use Leibniz rule to compute σ − ε E (cid:2) ˆ D |∇ ( w ε u ) | (cid:3) = σ − ε (cid:18) E (cid:2) ˆ D |∇ w ε | u (cid:3) + E (cid:2) ˆ D |∇ u | w ε (cid:3) + 2 E (cid:2) ˆ D u w ε ∇ w ε · ∇ u (cid:3)(cid:19) . By an argument similar to the one for (3.10), we reduce tolim ε ↓ σ − ε E (cid:2) ˆ D |∇ ( w ε u ) | (cid:3) = lim ε ↓ σ − ε E (cid:2) h− ∆ w ε ; w ε u i . We now apply (3.3) of Lemma 3.1 to v ε = w ε u and φ = u . This implies thatlim ε ↓ σ − ε E (cid:2) ˆ D |∇ ( w ε u ) | (cid:3) = ˆ k − u . (3.11)Inserting (3.8), (3.10) and (3.11) into (3.7) we have thatlim ε ↓ E (cid:0) ˆ D | σ ε u ε − w ε u | q (cid:1) q = ˆ D f u ∗ + ˆ D k − u − k − ˆ D u ∗ u. (3.12)Since u ∗ = kf , it is easy to see the the right-hand side above equals the right-hand side of (3.5). Thisestablishes (3.5) and concludes the proof of Theorem 0.1, case ( a ).11 .1 Proof of Lemma 3.1 Lemma 3.1 may be proven in a way that is similar to [14][Lemma 3.1]. The first crucial ingredient is thefollowing lemma, that allows to find a suitable partition of the holes H ε by dividing this set into a partcontaining well separated holes and another one containing the clusters. The next result is the analogueof [14][Lemma 4.2] with the different rescaling of the radii of the balls generating the set H ε .For every x ∈ R , we recall the definition of R ε,x in (1.1). We have: Lemma 3.2.
Let γ ∈ (0 , α − . Then there exists a partition H ε := H εg ∪ H εb , with the followingproperties: • There exists a subset of centres n ε ( D ) ⊆ Φ ε ( D ) such that H εg := [ z ∈ n ε ( D ) B ε α ρ z ( εz ) , min z ∈ n ε ( D ) R ε,z > ε γ , max z ∈ n ε ( D ) ε α ρ z ε γ . (3.13)• There exists a set D εb ( ω ) ⊆ R satisfying H εb ⊆ D εb , Cap( H εb , D εb ) . C ( γ ) ε α X z ∈ Φ ε ( D ) \ n ε ( D ) ρ z and for which B Rε,z ( εz ) ∩ D εb = ∅ , for every z ∈ n ε ( D ) .Finally, we have that lim ε ↓ ε X z ∈ Φ ε ( D ) \ n ε ( D ) ρ α z = 0 , P -almost surely , lim ε ↓ E (cid:2) ε X z ∈ Φ ε ( D ) \ n ε ( D ) ρ α z (cid:3) = 0 . (3.14)Let γ in Lemma 3.2 be fixed. We construct w ε as done in [14, ]: we set w ε = w gε ∧ w bε with w εb := − argmin Cap( H εb ; D εb ) in D εb R \ D εb w εg = w ε,z in B R ε,z ( εz ) , z ∈ n ε ( D )1 in R \ S z ∈ n ε ( D ) B R ε,z ( εz ) (3.15)where for each z ∈ n ε ( D ), the function w ε,z vanishes in the hole B ε α ρ z ( εz ) and solves w εg = − ∆ w ε,z = 0 in B R ε,z ( εz ) \ B ε α ρ z ( εz )0 on ∂B ε α ρ z ( εz )1 on ∂B R ε,z ( εz ) (3.16)We also define of the measure µ ε = X z ∈ n ε ( D ) ∂ n w ε,z δ ∂B Rε,z ( εz ) ∈ H − ( D ) . (3.17)We stress that all the previous objects depend on the choice of the parameter γ in Lemma 3.2. The nextresult states that this parameter may be chosen in so that the norm k µ ε − πλ E (cid:2) ρ (cid:3) k H − ( D ) is suitablysmall. This, together with Lemma 3.2, provides the crucial tool to show Lemma 3.1:12 emma 3.3. There exists γ ∈ (0 , α − such that if µ ε is as in (3.17) there exists κ > such that forevery random field v ∈ H ( D ) E (cid:2) h ( σ − ε µ ε − π E (cid:2) ρ (cid:3) ); v i (cid:3) . ε κ (cid:0) σ − ε E (cid:2) ˆ D |∇ v | (cid:3) + E (cid:2) ˆ D | v | (cid:3) (cid:1) . Proof of Lemma 3.1.
By construction, it is clear that, for P -almost every realization, the functions w ε ∈ W , ∞ ( R ) ∩ H ( R ), vanish in H ε and are such that k w ε k L ∞ ( R ) = 1.We now turn to (3.1). Using the definitions of w εg and w εb and Lemma 3.2 we have that k w ε − k L ( D ) = k w gε − k L ( D ) + k w bε − k L ( D ) . (3.18)By Poincaré’s inequality in each ball { B R ε,z ( εz ) } z ∈ n ε ( D ) we bound k w gε − k L ( D ) X z ∈ n ε ( D ) ε k∇ w εg k L ( D ) . ε α − ε X z ∈ n ε ( D ) ρ z . (3.19)Thanks to (0.2) and the Strong law of Large numbers, for P -a.e. realization the right-hand side vanishesin the limit ε ↓ | w bε − |
1, we may use thedefinition of D εb to bound k w bε − k L ( D ) | D bε ∩ D | X z ∈ Φ ε ( D ) \ n ε ε α ( ρ z ∧ ε − α ) . ε X z ∈ Φ ε ( D ) \ n ε ρ α z . Thanks to (3.14) in Lemma 3.2, the right-hand side vanishes in the limit ε ↓ P -almost every realization.Combining this with (3.19) and (3.18) yields (3.1) for w ε −
1. Inequality (3.1) for σ − ε ∇ w ε follows byLemma 3.2 and the definition (3.15) of w ε as done in [14][Lemma 3.1]. Limit (3.2) may be argued as doneabove for (3.1), this time appealing to the bound (0.2) and the stationarity of (Φ , R ).It thus remains to show (3.3). Using (3.15), (3.17) and the fact that φu ε ∈ H ( D ε ), we may decompose h− ∆ w ε ; φv ε i = h µ ε ; φv ε i + ˆ D ∇ w bε · ∇ ( φv ε ) . (3.20)Since v ε is assumed to satisfy the bounds in Lemma 2.1, Hölder’s inequality, Lemma 2.1 , definition (3.15)and (3.14) of Lemma 3.2 imply thatlim ε ↓ E (cid:2) | ˆ D ∇ w bε · ∇ ( φv ε ) | (cid:3) lim ε ↓ E (cid:2) Cap( H εb ; D εb ) (cid:3) = 0 . This and (3.20) thus yield thatlim sup ε ↓ E (cid:2) |h− ∆ w ε ; φv ε i − k − ˆ D vφ | (cid:3) = lim sup ε ↓ E (cid:2) |h µ ε ; φv ε i − k − ˆ D vφ | (cid:3) . Using the triangle inequality and the assumption v ε ⇀ v in L (Ω × D ), we further reduce tolim sup ε ↓ E (cid:2) |h− ∆ w ε ; φv ε i − k − ˆ D vφ | (cid:3) = lim sup ε ↓ E (cid:2) |h ( − σ ε ∆ w ε − k − ; φσ − ε v ε i| (cid:3) (3.21)By Lemma 3.3, there exists κ > ε ↓ E (cid:2) |h ( − σ ε ∆ w ε − πλ E (cid:2) ρ (cid:3) ); φσ − ε v ε i| (cid:3) lim sup ε ↓ ε κ (cid:0) σ − ε E (cid:2) ˆ D |∇ ( φσ ε v ε ) | (cid:3) + E (cid:2) ˆ D ( φσ ε v ε ) (cid:3) (cid:1) . Thanks to the assumptions on v ε , we infer that the right-hand side is zero. This, together with (3.21),yields (3.3). The proof of Lemma 3.1 is thus complete.13 roof of Lemma 3.3. We divide the proof into steps. The strategy of this proof is similar to the one for[11][Theorem 2.1, (b)].
Step 1: (Construction of a partition for D)
Let Q := [ − ; ] ; for k ∈ N and x ∈ R we define Q ε,k,x := εz + kεQ, Q ε,x := Q ε, ,x Let N k,ε ⊆ Z be a collection of points such that | N k,ε | . ε − and D ⊆ S x ∈ N k,ε Q ε,k,x . For each x ∈ N k,ε we consider the collection of points N ε,k,x := { z ∈ n ε ( D ) : εz ∈ Q ε,k,x } ⊆ Φ ε ( D ) and define the set K ε,k,x := (cid:0) Q ε,k,x [ z ∈ N ε,k,x Q ε,z (cid:1) \ [ z ∈ ˜Φ ε ( D ) \ N ε,k,x Q ε,z . Since by definition of n ε ( D ) in Lemma 3.2 the cubes { Q ε,z } z ∈ ˜Φ ε ( D ) are all disjoint, we have that D ⊆ [ x ∈ N ε,k K ε,k,x , sup x ∈ N ε,k | diam ( K ε,k,x ) | . kε, ( k − ε | K k,x | ( k + 1) ε for every x ∈ N ε,k . (3.22)Note that the previous properties hold for every realization ω ∈ Ω. Step 2.
For k ∈ N fixed, let { K ε,x,k } x ∈ N k,ε be the covering of D constructed in the previous step. Wedefine the random variables S ε,k,x := 4 π | K ε,x,k | X z ∈ N ε,k,x Y ε,z Y ε,z := ε ρ z R ε,z R ε,z − ε α ρ z . (3.23)and construct the random step function m ε ( k ) = 4 π X x ∈ N ε,k S ε,k,x K ε,k,x . Let v be as in the statement of the lemma and m ε ( k ) as above. The triangle and Cauchy-Schwarzinequalities imply that E (cid:2) h σ − ε µ ε − πλ E (cid:2) ρ (cid:3) ; v i (cid:3) (3.24) E (cid:2) k σ − ε µ ε − m ε ( k ) k H − (cid:3) E (cid:2) k∇ v k L ( D ) (cid:3) + E (cid:2) k m ε ( k ) − πλ E (cid:2) ρ (cid:3) k L (cid:3) E (cid:2) k v k L ( D ) (cid:3) , so that the proof of the lemma reduces to estimating the norms E (cid:2) k σ − ε µ ε − m ε ( k ) k H − (cid:3) , E (cid:2) k m ε ( k ) − πλ E (cid:2) ρ (cid:3) k L (cid:3) . We now claim that there exists a γ > k ∈ NE (cid:2) k σ − ε µ ε − m ε ( k ) k H − ( D ) (cid:3) . ε κ σ − ε , E (cid:2) k m ε ( k ) − πλ E (cid:2) ρ (cid:3) k L ( D ) (cid:3) . ε κ (3.25)for a positive exponent κ >
0. Combining these two inequalities with (3.24) establishes Lemma 3.3.In the remaining part of the proof we tackle inequalities (3.25). We follow the same lines of [11][Theorem1.1, (b)]. and thus only sketch the main steps for the argument.14 tep 3.
We claim that E (cid:2) k σ − ε µ ε − m ε ( k ) k H − ( D ) (cid:3) . ( kε ) | log ε | ε − ( α − − γ )(2 − α ) + . (3.26)We first argue that that k σ − ε µ ε − m ε ( k ) k H − ( D ) . ( εk ) ε X z ∈ n ε ( D ) ρ z ( εd z ) − , (3.27)This follows by Lemma 5.1 applied to the measure σ − ε µ ε : In this case, the random set of centres is Z = ˜Φ ε ( D ), the random radii R = { R ε,z } z ∈ ˜Φ( D ) , the functions g i = σ − ε ∇ ν w ε,z , z ∈ ˜Φ ε ( D ) and thepartition { K ε,k,x } x ∈ N ε,k of the previous step. Note that, by construction, this partition satisfies theassumptions of Lemma 5.1. The explicit formulation of the harmonic functions { w ε,z } z ∈ n ε ( D ) defined in(3.16) (c.f. also [11][(2.24)]) implies that for every z ∈ n ε ( D ) ˆ ∂B Rε,z ( εz ) | σ − ε ∂ ν w ε,z | . ε ρ z d − z , ˆ ∂B ε,z σ − ε ∂ ν w ε,z (3.23) = Y ε,z . (3.28)Therefore, Lemma 5.1 and the bounds (3.28) yield that k σ − ε µ ε − m ε ( k ) k H − ( D ) . sup x ∈ N k,ε diam( K ε,k,x ) X z ∈ ˜Φ ε ( D ) ρ z ( εd z ) − , which implies (3.27) thanks to (3.22).It thus remains to pass from (3.27) to (3.26): We do this by taking the expectation and arguing as for[11][Inequality (4.22)]. We rely on the stationarity of ( φ, R ), the properties of the Poisson point processand the fact that z ∈ n ε implies that ε α ρ z ε γ and R ε,z > ε γ . Step 4.
We now turn to the left-hand side in the second inequality of (3.25) and show that E (cid:20) k m ε ( k ) − πλ E (cid:2) ρ (cid:3) k L ( D ) (cid:21) k − ε − ( α − − γ ) + k − + ε γ (3.29)+ ε ( α − − γ )( α − + ε γ ) − α + εkε − ( α − − γ )(2 − α ) + . The proof of this step is similar to [11][Theorem 2.1, (b)]: Using the explicit formulation of m ε ( k ) wereduce to E (cid:20) k m ε ( k ) − πλ E (cid:2) ρ (cid:3) k L ( D ) (cid:21) . X x ∈ N k,ε E (cid:2) ( S k,ε,x − λ E (cid:2) ρ (cid:3) ) (cid:3) If ˚ N ε,k := { x ∈ N ε,k : dist( Q ε,k,x ; ∂D ) > ε } , we split X x ∈ N k,ε E (cid:2) ( S k,ε,x − λ E (cid:2) ρ (cid:3) ) (cid:3) . ( εk ) X x ∈ N k,ε \ ˚ N ε,k E (cid:2) ( S k,ε,x − λ E (cid:2) ρ (cid:3) ) (cid:3) + X x ∈ ˚ N k,ε E (cid:2) ( S k,ε,x − λ E (cid:2) ρ (cid:3) ) (cid:3) . (3.30)Since ∂D is C and compact, for ε small enough (depending on D ) we have( εk ) X x ∈ N k,ε \ ˚ N ε,k E (cid:2) ( S k,ε,x − λ E (cid:2) ρ (cid:3) ) (cid:3) . εk E (cid:2) ρ ε α ρ<ε γ (cid:3) (0.2) . εkε − ( α − − γ )(2 − α ) + .
15y stationarity, the second term in (3.30) is controlled by E (cid:2) ( S k,ε, − λ E (cid:2) ρ (cid:3) ) (cid:3) . (3.31)Hence, X x ∈ N k,ε E (cid:2) ( S k,ε,x − λ E (cid:2) ρ (cid:3) ) (cid:3) . E (cid:2) ( S k,ε, − λ E (cid:2) ρ (cid:3) ) (cid:3) + εkε − ( α − − γ )(2 − α ) + . The remaining term on the righ-hand side may be controlled by the right-hand side in (3.29) by meansof standard CLT arguments as done in [11][Inequality (4.23)] for the analogous term. We stress that thecrucial observation is that the random variables S k,ε,x − λ E (cid:2) ρ (cid:3) are centred up to an error term. Wemention that in this case the set K ε,k,x has been defined in a different way from [11] and we use properties(3.22) instead of [11][(4.13)]. This yields (3.29) Step 5.
We show that, given (3.26) and (3.29) of the previous two steps, we may pick γ and k ∈ N such that inequalities (3.25) hold: Thanks to the definition of σ ε and since α ∈ (1; 3), we may find γ closeenough to α −
1, e.g. γ = ( α − k ∈ N , e.g. k = − ( α − εk ) | log ε | ε − ( α − − γ )(2 − α ) + σ − ε k ε ( α − − γ ) ε α − . This, thanks to (3.26), implies that the first inequality in (3.25) holds with the choice κ = α − > γ and κ yield that also the right hand side of (3.29) is bounded by ε − ( α − . Thisyields also the remaining inequality in (3.25) and thus concludes the proof of Lemma 3.3. Proof of Lemma 3.2.
The proof of this lemma follows the same construction implemented in the proof of[11][Lemma 4.1] with d = 3, δ = γ and with the radii { ρ z } z ∈ Φ ε ( D ) rescaled by ε α instead of ε . Note thatthe constraint for γ is due to this different rescaling. In the current setting, we replace ε by ε γ in thedefinition of the set K εb in [11][(4.7)]. Estimate (3.14) may be argued as [11][Lemma 4.4] by relying on(0.2). ( b ) The next lemma is the analogue of Lemma 3.1:
Lemma 4.1.
For every δ > , there exists an ε > and a set A δ ∈ F , having P ( A δ ) > − δ , such thatfor every ω ∈ A δ and ε ε there exists a linear map R ε : { φ ∈ C ∞ ( D, R d ) : ∇ · φ = 0 } → H ( D, R d ) satisfying R ε φ = 0 in H ε , ∇ · R ε φ = 0 in D and such that lim sup ε ↓ E (cid:2) A δ ˆ D | σ − ε ∇ R ε ( φ ) | (cid:3) . k v k C ( D ) , E (cid:2) A δ ˆ D | R ε ( φ ) − φ | (cid:3) → . (4.1) Furthermore, if v ε satisfies the bounds of Lemma 2.1 and σ ε v ε ⇀ v in L (Ω × D ) , then E (cid:2) A δ | ˆ ∇ R ε ( φ ) · ∇ v ε − K − ˆ ρv | (cid:3) → . (4.2)16 roof of Theorem 0.1, ( b ) . The proof of this statement is very similar to the one for case ( a ) and weonly emphasize the few technical differences: Using the bounds of Lemma 2.1, we have that, up to asubsequence, u ε j ⇀ u ∗ in L p (Ω × D ), 1 p <
2. We prove that u ∗ = K ( f − ∇ p ) , (4.3)where p ∈ H ( D ) is the unique weak solution to − ∆ p = −∇ · f in D ( ∇ p − f ) · n = 0 on ∂D . , ˆ D p = 0 . (4.4)Identity (4.3) also implies that the full { u ε } ε> converges to u ∗ .As for the proof of Theorem 0.1, case (a), we restrict to the converging subsequence { u ε j } j ∈ N but weskip the index j ∈ N in the notation. We start by noting that, using the divergence-free condition for u ε and that u ε vanishes on ∂D , we have that for every φ ∈ C ∞ ( D ) and χ ∈ L ∞ (Ω) E (cid:2) χ ˆ D ∇ φ · u ∗ (cid:3) = 0 . (4.5)Let χ ∈ L ∞ (Ω) and φ ∈ C ∞ ( D ) with ∇ · φ = 0 in D be fixed. For every δ >
0, we appeal to Lemma4.1 to infer that there exists an ε δ > A δ ∈ F , having P ( A δ ) > − δ , such that for every ω ∈ A δ and for every ε ε δ we may consider the function R ε φ ∈ H ( D ε ) of Lemma 4.1. Testing equation (0.4)with R ε ( ρ ), and using that the vector field R ε v is divergence-free, we infer that E (cid:2) A δ χ ˆ D ∇ u ε : ∇ ( R ε φ ) (cid:3) = E (cid:2) χ A δ ˆ D ( R ε φ ) f (cid:3) . Using Lemma 4.1 and the bounds of Lemma 2.1 this implies that in the limit ε ↓ E (cid:2) A δ χ ˆ D ( u ∗ − Kf ) φ (cid:3) = 0 . We now send δ ↓ E (cid:2) χ ˆ D ( u ∗ − Kf ) v (cid:3) = 0 . (4.6)Since D has C , -boundary and is simply connected, the spaces L p ( D ), p ∈ (1 , + ∞ ) admit an L p -Helmoltz decomposition L p ( D ) = L p div ( D ) ⊕ L p curl ( D ) [10][Section III.1]. This, the separability of L p ( D ), p ∈ [1 , + ∞ ), and the arbitrariness of χ and φ in (4.6), allows us to infer that for P -almost realization thefunction u ∗ satisfies u ∗ = Kf + ∇ p ( ω ; · ) for p ( ω ; · ) ∈ W ,p ( D ), p ∈ [1; 2). By a similar argument, we mayuse (4.5) to infer that for P -almost every realization and for every v ∈ W .q ( D ), q > ˆ D ( ∇ p ( · ; ω ) + Kf ) · ∇ v = 0 . Since (4.4) admits a unique mean-zero solution, we conclude that p ( ω, · ) does not depend on ω . Finally,since D is regular enough and f ∈ L q ( D ), standard elliptic regularity yields that p ∈ H ( D ). Thisconcludes the proof of (4.3).We now upgrade the convergence of the family { u ε } ε> to u ∗ from weak to strong: We claim that forevery δ > A δ ⊆ Ω with P ( A δ ) > − δ such thatlim ε ↓ E (cid:2) A δ ˆ D | σ ε u ε − u ∗ | q (cid:3) = 0 . (4.7)17ere, q ∈ [1 , a ):In this case, we rely on Lemma 4.1 instead of Lemma 3.1 and use that, thanks to the definition (4.4), itholds ˆ D f ( f − ∇ p ) = ˆ D ( f − ∇ p ) . From (4.7), the statement of Theorem 0.1, ( b ) easily follows: Let, indeed, q ∈ [1 ,
2) be fixed. For every δ >
0, let A δ be as above. We rewrite E (cid:2) ˆ | σ ε u ε − u | q (cid:3) = E (cid:2) A δ ˆ | σ ε u ε − u | q (cid:3) + E (cid:2) Ω \ A δ ˆ | σ ε u ε − u | q (cid:3) and, given an exponent p ∈ ( q ; 2), we use Hölder’s inequality and the assumption on A δ to control E (cid:2) ˆ | σ ε u ε − u | q (cid:3) E (cid:2) A δ ˆ | σ ε u ε − u | q (cid:3) + δ − qp E (cid:2) ˆ | σ ε u ε − u | p (cid:3) qp . Since by Lemma 2.1 the family σ ε u ε is uniformly bounded in every L p (Ω × D ) for p ∈ [1 , ε ↓ E (cid:2) ˆ | σ ε u ε − u | q (cid:3) lim sup ε ↓ E (cid:2) A δ ˆ | σ ε u ε − u | q (cid:3) + δ − qp (4.7) . δ − qp . Since δ is arbitrary, we conclude the proof of Theorem 0.1, ( b ). This section is devoted to arguing Lemma 4.1 by leveraging on the geometric information on the clustersof holes H ε contained Lemma 1.2. The idea behind these proof is in spirit very similar to the one forLemma 3.1 in case (a): As in that setting, indeed, we aim at partitioning the holes of H ε into a subset H εg of disjoint and “small enough” holes and H εb where the clustering occurs.The main difference with case ( a ), however, is due to the fact that we need to ensure that the so-calledStokes capacity of the set H εb , namely the vector(St-Cap( H εb )) i = inf (cid:26) ˆ |∇ v | : v ∈ C ∞ ( R ; R ) , ∇ · v = 0 in R , v > e i in H εb (cid:27) , i = 1 , , ε ↓
0. The divergence-free constraint implies that, in contrast with the harmoniccapacity of case (a), the Stokes capacity is not subadditive. This yields that, if H εb is constructed as inLemma 3.2, then we cannot simply control its Stokes-capacity by the sum of the capacity of each ball of H εb .We circumvent this issue by relying on the information on the length of the clusters given by Lemma1.2. We do this by adopting the exact same strategy used to tackle the same issue in the case of theBrinkmann scaling in [12]. The following result is a simple generalization of [12][Lemma 3.2] and upgradesthe partition of Lemma 3.2 in such a way that we may control the Stokes-capacity of the clustering holesin H εb . For a detailed discussion on the main ideas behind this construction, we refer to [12][Subsection2.3]. Lemma 4.2.
Let γ > be as chosen in Lemma 3.3. For every δ > there exists ε > and A δ ⊆ Ω with P ( A δ ) > − δ such that for every ω ∈ Ω and ε ε we may choose H εg , H εb of Lemma 3.2 as follows: There exist Λ( β ) > , a sub-collection J ε ⊆ I ε and constants { λ εl } z l ∈ J ε ⊆ [1 , Λ] such that H εb ⊆ ¯ H εb := [ z j ∈ J ε B λ εj ε α ρ j ( εz j ) , λ εj ε α ρ j Λ ε κ . • There exists k max = k max ( β, d ) > such that we may partition I ε = k max [ k = − I εk , J ε = k max [ i = − J εk , with I εk ⊆ J εk for all k = 1 , · · · , k max and [ z i ∈I εk B ε α ρ i ( εz i ) ⊆ [ z j ∈ J εk B λ εj ε α ρ j ( εz j );• For all k = − , · · · , k max and every z i , z j ∈ J εk , z i = z j B θ λ εi ε α ρ i ( εz i ) ∩ B θ λ εj ε α ρ j ( εz j ) = ∅ ;• For each k = − , · · · , k max and z i ∈ I εk and for all z j ∈ S k − l = − J εl we have B ε α ρ i ( εz i ) ∩ B θλ εj ε α ρ j ( εz j ) = ∅ . (4.9) Finally, the set D εb of Lemma 3.2 may be chosen as D εb = [ z i ∈ J ε B θε α λ εi ρ i ( εz i ) . (4.10) The same statement is true for P -almost every ω ∈ Ω for every ε > ε (with ε depending, in thiscase, also on the realization ω ).Proof of Lemma 4.2. The proof of this result follows the exact same lines of of [12][Lemma 3.2]. We thusrefer to it for the proof and to [12][Subsection 3.1] for a sketch of the ideas behind the quite technicalargument. We stress that the different scaling of the radii does not affect the argument since the necessaryrequirement is that ε α << ε . This holds for every choice of α ∈ (1 , H ε . For every δ >
0, we thus select the set A δ of Lemma 1.2 containing those realizations where the partition of H ε satisfies (1.2) and (1.4). Oncerestricted to the set A δ , the construction of the set H εb is as in [12][Lemma 3.1].Equipped with the previous result, we may now proceed to prove Lemma 4.1: Proof of Lemma 4.1.
The proof of this is similar to the one in [12][Lemma 2.5] for the analogous operatorand we sketch below the main steps and the main differences in the argument. For δ >
0, let ε > A δ ⊆ Ω be the set of Lemma 4.2; From now on, we restrict to the realization ω ∈ A δ . For every ε < ε we appeal to Lemma 3.2 and Lemma 4.2 to partition H ε = H εb ∪ H εg . We recall the definitions of the set n ε ⊆ Φ ε ( D ) in (3.13) in Lemma 3.2 and of the subdomain D εb ⊆ D in (4.10) of Lemma 4.2.19 tep 1. (Construction of R ε ) For every φ ∈ C ∞ ( D ), we define R ε φ as R ε φ := φ εb in D εb φ εg in D \ D εb ,where the functions φ εb and φ εg satisfy φ εb = 0 in H εb , φ εb = φ in D \ D εb , ∇ · φ εb = 0 in D, k φ εb − φ k pL p . p | D εb | for every p > k σ ε ∇ φ bε k L . ε P z ∈ Φ ε ( D ) \ n ε ρ z . (4.11)and φ εg = φ in D εb , φ εg = 0 in H εg , ∇ · φ εg = 0 in D , k∇ ( φ εg − φ ) k L ( D ) . ε α P z ∈ n ε ( D ) ρ z , k φ εg − φ k pL p ( D ) . ε δ +3 P z ∈ n ε ( D ) ρ α + βz . (4.12) Step 2. (Construction of φ bε ) We construct φ εb as done in [12][Proof of Lemma 2.5, Step 2]: For every z ∈ J ε , we define B θ,z := B θλ ε ε α ρ z ( εz ) , B z := B λ ε ε α ρ z ( εz ) . It is clear that the previous quantities also depend on ε . However, in order to keep a leaner notation, weskip it in the notation. We use the same understanding for the function φ εb and the sets { I ε,i } k max i = − and { J ε,i } k max i = − of Lemma 4.2.We define φ b by solving a finite number of boundary value problems in the annuli [ z ∈ I k B θ,z \ B z , for k = − , · · · , k max We stress that, thanks to Lemma 4.2, for every k = − , · · · , k max , each one of the above collections containsonly disjoint annuli. Let φ ( k max +1) = φ . Starting from k = k max , at every iteration step k = k max , · · · , − z ∈ I ε,k the Stokes system − ∆ φ ( k ) + ∇ π ( k ) = − ∆ φ ( k +1) in B θ,z \ B z ∇ · φ ( k ) = 0 in B θ,z \ B z φ ( k ) = 0 on ∂B θ,z φ ( k ) = φ ( k +1) on ∂B z .We then extend φ ( k ) to φ ( k +1) outside S z ∈ I k B θ,z and to zero in S z ∈ I k B z .20he analogue of inequalities of [12][(4.12)-(4.14)], this time with the factor ε d − d replaced by ε α andwith d = 3, is k∇ φ ( k ) k L ( D ) . k∇ φ k L ( D ) + ε d X z ∈∪ k max i = k J i ρ z k φ k L ∞ ( D ) , k φ ( k ) k C ( D ) . k φ k C ( D ) , (4.13)and ∇ · φ ( k ) = 0 in D, φ i = 0 in [ z ∈ S k max i = k I i B ε α ρ z ( εz ) . (4.14)Moreover, φ ( k ) − φ = 0 in D \ [ z ∈∪ k max i = k J i B θ,z , k∇ ( φ ( k ) − φ ) k L ( D ) . X z ∈∪ k max i = k J i (cid:16) k∇ φ k L ( B θ,z ) + ε d ρ z k φ k L ∞ ( D ) (cid:17) . (4.15)These inequalities may be proven exactly as in [12]. We stress that condition (4.9) in Lemma 4.2 is crucialin order to ensure that this construction satisfies the right-boundary conditions. In other words, the mainrole of Lemma 4.2 is to ensure that, if at step k the function φ ( k ) vanishes on a certain subset of H εb , then φ ( k +1) also vanishes in that set (and actually vanishes on a bigger set).We set φ εb = φ ( − obtained by the previous iteration. The first property in (4.11) is an easy conse-quence of (4.14) and the first identity in (4.15). We recall, indeed, that thanks to Lemma 4.2 we havethat H εb = [ z ∈ S k max i = − I i B ε α ρ z ( εz ) , D εb = [ z ∈∪ k max k = − J i B θ,z . The second property in (4.11) follows immediately from (4.14). The third line in (4.11) is an easyconsequence of the first line in (4.11) and the second inequality in (4.13). Finally, the last inequality in(4.11) follows by multiplying the last inequality in (4.15) with the factor σ ε and using that, since φ ∈ C ∞ ,we have that k σ − ε ∇ ( φ εb − φ ) k L ( D ) . k φ k C ( D ) ε − α X z ∈∪ k max k = − J k ( ε α ρ z + ε α ρ z ) . ε X z ∈∪ k max k = − J k (( ε α ρ z ) + 1) ρ z . Thanks to Lemma 4.2 and the definition of the set n ε in Lemma 3.2, the previous inequality yields thelast bound in (4.11). Step 3. (Construction of φ εg ) Equipped with φ εb satisfying (4.11), we now turn to the construction of φ εg . Also in this case, we follow the same lines of [12][Proof of Lemma 2.5, Step 3] and exploit the factthat the set H εg is only made by balls that are disjoint and have radii ε α ρ that are sufficiently small. Wedefine the function φ εg exactly as in [12][Proof of Lemma 2.5, Step 3] with the radius a i,ε in [12][(4.18)]being defined as a ε,z = ε α ρ z instead of ε d − d ρ z . More precisely, for every z ∈ n ε , we write a ε,z := ε α ρ z , d ε,z := min (cid:26) dist( εz, D εb ) ,
12 min ˜ z ∈ nε,z =˜ z (cid:0) ε | z − ˜ z | (cid:1) , ε (cid:27) (4.16)21nd we set T z = B a ε,z ( εz ) , B z := B dε,z ( εz ) , B ,i := B d ε,z ( εz ) , C z := B z \ T z , D z := B ,z \ B z . With this notation, we define the function φ εg as in [12][(4.19)-(4-21)]. Also in this case, identities,[12][(4.22)-(4.23)] hold. By Lemma 3.2 It is immediate to see that this construction satisfies the firsttwo properties in (4.12).We now turn to show the remaining part of (4.12): We remark that, since z ∈ n ε ( D ), Lemma 3.2 anddefinition (4.16) yield that ( a ε,z d ε,z ) ε γ , a ε,z ε γ ρ α + βz , (4.17)where γ > β > z ∈ n ε ( D ) k∇ ( φ εg − φ ) k L ( D i ) . ε γ ε α ρ z , k φ εg − φ k pL p ( D i ) . ε γp d ε,z (4.18) k∇ ( φ εg − φ ) k L ( C i ) . ε α ρ z , k φ εg − φ k pL p ( C i ) . ε γ +3 ρ α + βz k∇ ( φ εg − φ ) k L ( T i ) + k φ εg − φ k pL p ( T i ) . ε γ +3 ρ α + βz . Since B ,z = D z ∪ C z ∪ T z and the function φ εg − φ is supported only on S z ∈ n ε ( D ) B ,z , we infer that forevery z ∈ n ε ( D ), it holds k∇ ( φ εg − φ ) k C ( B z ) . ε α ρ z + ε γ +3 ρ α + βz , k φ εg − φ k pL p ( B z ) . ε δ +3 ρ α + βz . Summing over z ∈ n ε we obtain the last two inequalities in (4.12). We thus established (4.12) andcompleted the proof of Step 1. Step 4. (Properties of R ε ) We now argue that R ε defined in Step 1. satisfies all the propertiesenumerated in Lemma 4.2. It is immediate to see from (4.12) and (4.11) that R ε φ vanishes on H ε andis divergence-free in D . Inequalities (4.1) also follow easily from the inequalities in (4.12) and (4.11) andarguments analogous to the ones in Lemma 3.1. We stress that, in this case, we appeal to condition (0.5)and, in the expectation, we need to restrict to the subset A δ ⊆ Ω of the realizations for which R ε may beconstructed as in Step 1.To conclude the proof, it only remains to tackle (4.2). We do this by relying on the same ideas usedin Lemma 3.1 in the case of the Poisson equation. We use the same notation introduced in Step 2. Webegin by claiming that (4.2) reduces to show that for every i = 1 , · · · , ε ↓ E (cid:2) | X z ∈ n ε ( D ) ˆ ∂B z ( ∂ ν w iε,z − q iε,z ν i ) φ i v ε,i − K − ˆ D v ε,i φ i | (cid:3) = 0 , (4.19)where w iε,z ( x ) := ¯ w i ( x − εzε α ρ z ) , q iε,z ( x ) = ( ε α ρ z ) − ¯ q i ( x − εzε α ρ z ) , x ∈ B z , with ( ¯ w i , ¯ q i ) solving ∆ ¯ w i − ∇ ¯ q i = 0 in R d \ B ∇ · ¯ w i = 0 in R d \ B ¯ w i = e i on ∂B ¯ w i → | x | → + ∞ .
22e use the definition of R ε φ to rewrite for every ω ∈ A δ ˆ D ∇ v ε · ∇ R ε ( φ ) = ˆ D ∇ v ε · ∇ ( φ εg − φ ) + ˆ D ∇ v ε · ∇ ( φ εb − φ ) + ˆ D ∇ v ε · ∇ φ. We claim that, after multiplying by A δ and taking the expectation, the last two integrals on the right-hand side vanish in the limit. In fact, using the triangle and Cauchy-Schwarz’s inequalities and combiningthem with (4.11) and the uniform bounds for { v ε } ε> we have thatlim sup ε ↓ E (cid:2) A δ | ˆ D ∇ v ε · ∇ ( φ εb − φ ) + ˆ D ∇ v ε · ∇ φ | (cid:3) . lim sup ↓ E (cid:2) A δ ε X z ∈ Φ ε ( D ) \ n ε ( D ) ρ z (cid:3) (3.14) = 0 . Hence, we show (4.2) provided thatlim ε ↓ E (cid:2) A δ | ˆ D ∇ v ε · ∇ ( φ εg − φ ) − K − ˆ D v · φ | (cid:3) = 0 . Furthermore, since σ − ε v ε ⇀ v in L p (Ω × D ), p ∈ [1 ,
2) and φ ∈ C ∞ ( D ), it suffices to prove thatlim ε ↓ E (cid:2) A δ | ˆ D ∇ v ε · ∇ ( φ εg − φ ) − K − ˆ D σ − ε v ε · φ | (cid:3) = 0 . We further reduce this to (4.19) iflim ε ↓ E (cid:2) A δ | ˆ D ∇ v ε · ∇ ( φ εg − φ ) − X z ∈ n ε ( D ) ˆ ∂B z ( ∂ ν w iε,z − q iε,z ν i ) φ i v ε,i | (cid:3) = 0 . An argument analogous to the one outlined in [12] to pass from the left-hand side of [12][(4.34)] to theone in [12][(4.39)] yields thatlim ε ↓ E (cid:2) A δ | ˆ D ∇ v ε · ∇ ( φ εg − φ ) − X z ∈ n ε ( D ) φ i ( εz ) ˆ ∂B z ( ∂ ν w iε,z − q iε,z ν i ) v ε,i | (cid:3) = 0 . (4.20)We stress that in the current setting we use again the uniform bounds on the sequence σ − ε ∇ u ε and werely on estimates (4.18) instead of [12][(4.26)-(4.30)]. To pass from (4.20) to (4.19) it suffices to use thesmoothness of φ and, again, the bounds on the family { v ε } ε> . We thus established that (4.2) reduces to(4.19).We finally turn to the proof of (4.19). By the triangle inequality it suffices to show thatlim ε ↓ E (cid:2) |h ˜ µ ε,i ; φ i v ε,i i − K − ˆ v ε,i φ i | (cid:3) = 0 for all i = 1 , , µ ε,i ∈ H − ( D ), i = 1 , ,
3, are defined as˜ µ ε,i := X z ∈ n ε ( D ) g iε,z δ ∂B z , g iε,z := ( ∂ ν w iε,z − q iε,z ν i ) . (4.22)We focus on the limit above in the case i = 1. The other values of i follow analogously. We skip the index i = 1 in all the previous objects. As done in the proof of (3.3) in Lemma 3.1, it suffices to show that thereexists a positive exponent κ > E (cid:2) |h ˜ µ ε ; φv ε i − K − ˆ v ε φ | (cid:3) ε κ (cid:18) ˆ D | σ − ε ∇ v ε | + ˆ D | σ − ε v ε | (cid:19) + r ε , (4.23)23ith lim ε ↓ r ε = 0. From this, (4.21) follows immediately thanks to the bounds assumed for { v ε } ε> .The proof of (4.23) is similar to (3.3): For k ∈ N to be fixed, we apply once Lemma 5.1 to thisnew measure σ − ε ˜ µ ε , with Z = { εz } z ∈ n ε ( D ) , R = { d ε,z } z ∈ ˜Φ ε ( D ) , { g z,ε } z ∈ ˜Ψ ε ( D ) and with the partition { K ε,z,k } z ∈ N k,ε constructed in Step 1 in the proof of Lemma 3.3. This implies that k σ − ε ˜ µ ε − ˜ µ ε ( k ) k H − . kε (cid:0) σ − ε X z ∈ ˜Φ ε ( D ) d − ε,z ˆ ∂B z | g z,ε | (cid:1) (4.24)˜ µ ε ( k ) := X x ∈ N ε,k (cid:0) | K ε,x,k | X z ∈ N k,x,ε σ − ε ˆ ∂B z g ε,z (cid:1) K ε,k,x Appealing to the definition of g ε,z and to the bounds for ( ¯ w, ¯ q ) obtained in [1][Appendix], for each z ∈ n ε ( D ) it holds that σ − ε ˆ ∂B z | g ε,z | . ε ρ z d − z , | σ − ε ˆ ∂B z g ε,z − πε ρ z | . ε ρ z ( ε α ρ z εd z ) (4.17) . ε γ . This, (4.24), (4.22), the triangle inequality and the definition of K − , imply that E (cid:2) |h ˜ µ ε ; φv ε i − K − ˆ v ε φ | (cid:3) . (cid:0) ε X z ∈ n ε ( D ) ρ z d − z (cid:1) (cid:0) ˆ D |∇ ( φv ε ) | (cid:1) + (cid:0) ˆ D | m ε ( k ) − k − | (cid:1) (cid:0) ˆ D | σ − ε v ε | (cid:1) + ε γ , where µ ε ( k ) is as in Step 2 of Lemma 3.3 and k is as in Theorem 0.1, ( a ). From this, we argue (4.2)exactly as done in Step 2-5 of Lemma 3.3. We established Lemma 4.1. Proof of Lemma 1.1. ( i ) ⇒ ( ii ): We prove thatlim ε ↓ | H ε ∩ D | = 0 P -almost surely. (5.1)We do this by bounding | H ε ∩ D | X z ∈ Φ ε ( D ) ( ε α ρ ∧ ε α X z ∈ Φ ε ( D ) ρ ρ<ε − α + X z ∈ Φ ε ( D ) ρ>ε − α X z ∈ Φ ε ( D ) ρ z and, for 0 < δ < α − | H ε ∩ D | ε δ ε ( D )) + ε α X z ∈ Φ ε ( D ) ρ ε − ( α − δ <ρ<ε − α + X z ∈ Φ ε ( D ) ρ>ε − α . ε δ ε ( D )) + X z ∈ Φ ε ( D ) ρ α ρ>ε − ( α − δ + ε X z ∈ Φ ε ( D ) ρ α ρ>ε − α . ε δ ε ( D )) + X z ∈ Φ ε ( D ) ρ α ρ>ε − ( α − δ . Since Φ is a Poisson point process and we assumed (0.2), the right-hand side above vanishes P -almostsurely in the limit ε ↓
0. This concludes the proof of (5.1) and immediately yields ( ii ).24 ii ) ⇒ ( i ): This is equivalent to show that if E (cid:2) ρ α (cid:3) = + ∞ then for P -almost every realization and ε small enough, the set D ε = ∅ . With no loss of generality, let us assume that diam( D ) = 1. We claim thatif ε j := 2 − j , then the events A j := (cid:26) B (0) ⊆ B ε αj ρ z ( ε j z ) for some z ∈ Φ ε j ( D ) \ Φ ε j − ( D ) (cid:27) , j ∈ N satisfy X j ∈ N P ( A j ) = + ∞ . (5.2)Since the events are independent, by Borel-Cantelli’s Lemma we conclude that for P -almost every realiza-tion there exists j ∈ N such that for all j > j we have B (0) ⊆ B ε αj ρ z ( ε j ρ z ), for some z ∈ Φ ε j ( D ).We now argue that this suffices to prove that, for P -almost every realization and all ε < − j , with j ∈ N as above, there is an element z ∈ Φ ε ( D ) such that B (0) ⊆ B ε α ρ z ( εz ). Let, indeed, assume that ε j +1 ε ε j . Then, since Φ ε j ( D ) ⊆ Φ ε ( D ), we may find z ∈ Φ ε ( D ) such that B (0) ⊆ B ε αj ρ z ( ε j ρ z ), i.e. | ε j z | ε αj ρ z −
2. This, in particular, yields that | εz |
14 ( ε j ε ) α − ε α ρ z − εε j α< ε α ρ z − , i.e. B (0) ⊆ B ε α ρ z ( εz ) for z ∈ Φ ε ( D ).We argue (5.2): Let B j = { ( z, ρ z ) ∈ ( 1 ε j D \ ε j − D ) × R + : ε j | z | + 2 < ε αj ρ z } . then, if Ψ = (Φ; R ) denotes the extended point process on R d × [1; + ∞ ) with intensity ˜ λ ( x, ρ ) = λf ( ρ )(c.f. Section 1), we rewrite P ( A j ) = 1 − P (Ψ( B j ) = 0) = 1 − exp (cid:18) − λ ˆ εj D \ εj − D ˆ + ∞ B j ( x ) f ( ρ )d ρ d x (cid:19) . Since ˆ εj D \ εj − D ˆ + ∞ B j ( x ) f ( ρ )d ρ d x = ˆ + ∞ f ( ρ ) ρ>ε − αj ˆ εj D \ εj − D d x & | D | ε − j ˆ + ∞ ρ> ε − αj & ε − j P (12 ε − αj < ρ < ε − αj ) , we bound P ( A j ) > − exp (cid:18) Cε − j P (12 ε − αj < ρ < ε − αj ) (cid:19) . Recalling that ε j = 2 − j , we may sum over j ∈ N in the previous inequality and get that X j ∈ N P ( A j ) > X j ∈ N (1 − exp (cid:8) − Cε − j P (12 ε − αj ρ ε − αj +1 ) (cid:9) )25e may assume that ε − j P (12 ε − αj ρ ε − αj +1 ) →
0. If not, indeed, (5.2) immediately follows. Since ε j = 2 − j , we have that X j ∈ N P ( A j ) & X j ∈ N ε − j P (12 ε − αj ρ ε − αj +1 ) & X j ∈ N E (cid:2) ρ α ε − αj ρ ε − αj +1 (cid:3) ≃ E (cid:2) ρ α (cid:3) . By the assumption E (cid:2) ρ α (cid:3) = + ∞ , this establishes (5.2). The proof of Lemma 1.1 is complete. Proof of Lemma 1.2.
The proof of this lemma relies on an application of Borel-Cantelli’s lemma andfollows the same lines of the one in [12][Lemma 5.1].For κ >
0, let k max = ⌊ κ ⌋ + 1. We partition the set of centres Φ ε ( D ) in terms of magnitude of theassociated radii: We write Φ ε ( D ) = S k max k = − I ε,k with I ε, − := { z ∈ Φ ε ( D ) : ε α ρ z < ε κ } , I ε,k max := (cid:8) z ∈ Φ ε ( D ) : ε α ρ z > ε − k max κ (cid:9) I ε,k := { z ∈ Φ ε ( D ) : ε − kκ ε α ρ z < ε − ( k +1) κ } for − k k max − k = − , · · · , k max , the previous partition satisfies (1.3) ofLemma 1.2.For any set χ ⊆ Φ ε ( D ), we say that A contains a chain of length M ∈ N , M >
2, if there exist z , · · · , z M ∈ χ such that B ε α ρ zi ( εz i ) ∩ B ε α ρ zj ( εz j ) = ∅ , for all i, j = 1 , · · · M . We say that A contains achain of size 1 if and only if A = ∅ .Equipped with this notation, (1.2) follows provided we argue that for κ suitably chosen, there exists k < k max − P -almost surely and for ε small, the sets { I ε,k ∪ I ε,k +1 } k max k = k are empty. This isequivalent to prove that they do not contain any chain of size at least 1. Similarly, (1.4) is obtained if wefind an M ∈ N such that P -almost surely and for ε small enough, all the sets { I ε,k ∪ I ε,k +1 } k k = − containchains of length at most M − M ∈ N and k = − , · · · , k max , we define the events A k,ε,M := (cid:8) I ε,k ∪ I ε,k +1 contains a chain of length at least M (cid:9) . We claim that if κ < min (cid:0) β ; α β αβ (cid:1) , then there exists k ∈ N , k < k max such that for every k ∈{ k , · · · , k max } P ( \ ε > [ ε<ε A k,ε, ) = 0 (5.3)and there exists M = M ( α, β ) ∈ N such that for every k = − , · · · , k −
1, also P ( \ ε > [ ε<ε A k,ε,M ) = 0 . (5.4)These claims immediately yield (1.2) and (1.4) and conclude the proof of Lemma 1.2, ( i ).The argument for (5.3) and (5.4) relies on an application of Borel-Cantelli’s Lemma and is analogousto the one for [12][Lemma 5.1]. We thus only sketch the proof. As shown in [12][Proof of Lemma 5.1,(5.5) to (5.6)], up changing the constant 4 in the definition of chain, we may reduce to prove (5.3)-(5.4)for a sequence { ε j } j ∈ N = { r j } j ∈ N , with r ∈ (0 , , R ), it is easy tosee that P ( A k,ε,M ) . p p M − (5.5)26here p := P ( { There is z ∈ Φ ε ( D ) with ε α ρ z > ε − kκ } ) ,p := P ( { There is z ∈ Φ ε ( B ε − ( k +1) κ (0) ) with ε α ρ z > ε − kκ } ) . Using the moment condition (0.5) and provided κ < β this yields p . ε αβ +( kκ − α + β ) , p . ε αβ . Hence, by (5.5), we have that P ( A k,ε,M ) . ε αβ +( kκ − α + β ) ε ( M − αβ . On the one hand, if κ < min (cid:0) β ; α β αβ (cid:1) , then we may pick k := ⌊ κ α β αβ ⌋ and observe that for every k ∈ { k , · · · , k max } we have that P ( A k,ε, ) . ε αβ . On the other hand, if M ∈ N is chosen big enough, for every k = − , · · · , k also P ( A k,ε,M ) . ε αβ . Using these two bounds, we may apply Borel-Cantelli to the family of events { A k,ε j ,M } and conclude (5.3)and (5.4).We now turn to case (ii). Identity (5.4) may be rewritten as P ( [ ε > \ ε<ε ( k [ k = − A k,ε,M ) c ) = 1 . This implies that for every δ >
0, we may pick ε > P ( T ε<ε ( S k max k = − A k,ε,M ) c ) > − δ .The statement of ( ii ) immediately follows if we set A δ := T ε<ε ( S k max k = − A k,ε,M ) c . The same argumentapplied to (5.3) implies the same statement for (1.2). Lemma 5.1.
Let Z := { z i } i ∈ I ⊆ D be a collection of points and let R := { r i } i ∈ I ⊆ R + such that theballs { B r i ( z i ) } i ∈ I are disjoint. We define the measure M := X i ∈ I g i δ ∂B ri ( z i ) ∈ H − ( D ) , where g i ∈ L ( ∂B r i ( z i )) . Then, there exists a constant C < + ∞ such that for every Lipschitz and(essentially) disjoint covering { K j } j ∈ J of D such that B r i ( z i ) ⊆ K j OR B r i ( z i ) ∩ K j = ∅ for every i ∈ I , j ∈ J we have that k M − m k H − ( D ) C max j ∈ J diam ( K j ) (cid:0)X i ∈ I k g i k L ( ∂B ri ( z i )) r − i (cid:1) , with m := X j ∈ J (cid:0) | K j | X i ∈ I,zi ∈ Kj ˆ ∂B ri ( z i ) g i (cid:1) K j . Proof.
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