Existence of the oscillating test functions (Proof of Lemma 5.3.1)

As already mentioned in Subsection 5.1.1, we proceed to prove Lemma 5.3.1 in two steps: We first give an argument in the simplest case of random holes H^ε having periodic centres and i.i.d. radii (cf.

example (a) of Section 5.2). In that case, the crucial role played by assumption (5.1.5) on the random geometry of the set H^ε becomes clear. We then generalize this argument to an arbitrary process (Φ, R) that satisfies the assumptions of Theorem 5.2.1. We observe that, as it becomes apparent in the proofs of this section, the full-probability set of realizations for which the statement of Lemma 5.2.1 holds true is selected by countable repeated applications of Strong Laws of Large Numbers-type of results. The final set Ω⁰ ⊂ Ω in which we prove the existence of the oscillating test functions is thus a countable intersection of full-probability sets and remains of full probability.

Before giving the proof of Lemma 5.3.1, we fix the following notation: For any two open sets A ⊂ B ⊂ R^d, we define

Cap(A, B) := inf

ˆ

|∇v|² : v ∈ C₀^∞(B), v ≥ 1A



. (5.4.1)

For a point process Φ on R^d and any bounded set E ⊂ R^d, we define the random variables

Φ(E) := Φ ∩ E, Φ^ε(E) := Φ ∩ 1

εE



, (5.4.2)

N (E) := #(Φ(E)), N^ε(E) := #(Φ^ε(E)).

For δ > 0, we denote by Φ_δ a thinning for the process Φ obtained as Φ_δ(ω) := {x ∈ Φ(ω) : min

y∈Φ(ω), y6=x

|x − y| ≥ δ}, (5.4.3)

i.e. the points of Φ(ω) whose minimal distance from the other points is at least δ. Given the process Φ_δ, we set Φ_δ(E), Φ^ε_δ(E), N_δ(E) and N_δ^ε(E) for the analogues for Φ_δ of the random variables defined in (5.4.2).

For a fixed M > 0, we define the truncated marks

R^M := {ρj,M}_zj∈Φ, ρj,M := ρj∧ M. (5.4.4) Furthermore, throughout the proofs, we write

a . b

whenever a ≤ Cb for a constant C = C(d) depending only on the dimension d.

Finally, we remark that, under the assumptions of the process (Φ, R) in Theorem 5.2.1, the process (Φ, {ρ^d−2}_z

i∈Z^d) satisfies the assumptions of Section 5.5, and we therefore may apply all the results stated in that section.

5.4.1 Case (a): Periodic centres

In this setting, the holes H^ε are generated by Φ = Z^dand a collection of i.i.d. random variables {ρ_i}_i∈Zd satisfying (5.1.5). It is immediate to check that the marked process

Since the centres of the holes are periodically distributed, the only challenge in the construction of the functions w_ε of Lemma 5.3.1 is due to the random variables {ρ_i}_i∈Zd which might generate very large holes under the mere condition (5.1.5). In fact, in [CM97] the construction of w_ε relies on the assumption that each hole B

ε^d−2d (εzi), zi ∈ Z^d, is strictly contained in the concentric cube of size ε; this allows to explicitly construct w^ε by locally solving a PDE on each of these cubes. In our case, the sole assumption (5.1.5) does not exclude that there are big holes which overlap and where the previous construction breaks down. The main auxiliary result on which Lemma 5.3.1 for the periodic case (a) relies is the following Lemma 5.4.1 on the asymptotic geometry of the set H^ε. Roughly speaking, this lemma ensures that H^ε may be almost surely partitioned into two subsets, a

“good” and a “bad” set of holes which we denote by H_g^ε and H_b^ε, respectively. The set H_g^ε contains most of the holes of H^ε and is made of small balls where the construction of wε may be carried out similarly to [CM97]. The remaining holes, some of which overlap with full probability, are all included in H_b^ε. This set is well separated from H_g^ε and small with respect to the macroscopic size of the domain D: We may indeed enclose H_b^ε into a set D_b^ε⊂ D which is still separated from H_g^ε and such that the harmonic capacity of H_b^ε with respect to this “safety layer” D_b^ε vanishes in the limit ε ↓ 0⁺. This allows us to implicitly define w^εin D_b^ε as the capacitary function of H_b^ε in D^ε_b; this choice ensures that the H¹-norm of wε on D_b^ε converges to zero. Hence, in the verification of (H2) and (H3) of Lemma 5.3.1, we only need to focus on the construction of w_ε on D\D_b^ε.

Lemma 5.4.1. Let δ ∈ (0,_d−2² ) be fixed. Then, there exists ε0 = ε0(δ) > 0 such that for P-almost every ω ∈ Ω and for all ε ≤ ε₀ there exist H_g^ε(ω), H_b^ε(ω), D_b^ε(ω) ⊂ R^d such that

H^ε(ω) = H_g^ε(ω) ∪ H_b^ε(ω), H_b^ε(ω) ⊂ D_b^ε(ω), (5.4.5) dist H_g^ε(ω), D_b^ε(ω)
≥ ε

2, where

lim

ε↓0⁺Cap (H_b^ε(ω), D_b^ε(ω)) = 0. (5.4.6) Moreover, H_g^ε(ω) may be written as the following union of disjoint balls centred in n^ε(ω) ⊂ Z^d∩¹_εD:

H_g^ε(ω) := [

zj∈n^ε

B

ε

d d−2ρj

(εz_j), (5.4.7)

ε^d−2d ρ_j ≤ ε^1+δ < ε

2, lim

ε↓0⁺ε^d#(n^ε) = |D|. (5.4.8) Proof of Lemma 5.4.1. The partition of the set H^ε(ω) in the statement of the lemma clearly depends on the realization ω ∈ Ω; in the sake of a leaner notation, though, in the rest of the proof we omit the argument ω and write H^ε, H_b^ε, H_g^ε instead of H^ε(ω), H_b^ε(ω), H_g^ε(ω). For each z_i∈ Z^d, we denote by Q^ε_i the cube of length ε centered at εzi.

We begin by constructing the set H_b^ε and its “safety layer” D_b^ε. We first include in H_b^εthe particles which are large compared to the size of the cubes Q^ε_i: For δ as in the statement of the lemma,we consider the subset of Z^d given by

J_ε^b :=n

z_i∈ Z^d∩ 1

εD : ε^d−2d ρ_j ≥ ε^1+δo

, (5.4.9)

Case (a): Periodic centres 133

and the corresponding union of balls

H˜_b^ε:= [

zj∈J_b^ε

B2ε^d−2d ρj

(εz_j).

We now extend J_b^ε by including the centres of the balls for which, independently from the size of their radius, the corresponding cell Q^ε_i intersects ˜H_b^ε: We define

I˜_b^ε:=z_i∈ Z^d: Q^ε_i ∩ ˜H_b^ε6= ∅ ⊃ J_b^ε, I_b^ε:= ˜I_b^ε∩1

εD. (5.4.10)

We finally set

H_b^ε := [

zj∈I_b^ε

Bε

d d−2ρj

(εzj), D_b^ε:= [

zj∈ ˜I_b^ε

Q^ε_j. (5.4.11)

By (5.4.11) it is immediate that H_b^ε⊂ D_b^ε. To show (5.4.6) we first argue that provided ε ≤ ε0(δ), with ε₀(δ) such that 2ε^1+δ₀ ≤ ε₀, for every z_j ∈ I_b^ε it holds

B2ε^d−2d (εz_j) ⊂ D_b^ε. (5.4.12)

Indeed, since by definition ˜H_b^ε ⊂ D_b^ε, if zj ∈ J_b^ε, then (5.4.12) follows immediately. If, otherwise, zj ∈ I_b^ε\ J_b^ε, then the assumption ε ≤ ε0 implies B

2ε^d−2d (εzj) ⊂ Q^ε_j ⊂ D_b^ε. By the subadditivity of the capacity (see definition (5.4.1)) we estimate

Cap (H_b^ε(ω), D_b^ε(ω)
^(5.4.11)

≤ X

j∈I^ε_b

Cap

Bε^d−2d ρj

(x_j), D_b^ε(ω)

(5.4.12)

≤ X

j∈I^ε_b

Cap

 Bε

d d−2ρj

(xj), B

2ε

d d−2ρj

(xj)



. X

j∈I_b^ε

ε^dρ^d−2_j .

To conclude (5.4.6), it remains to show that the right-hand side above vanishes almost surely in the limit ε ↓ 0⁺. This follows from Lemma 5.5.3 for the process (Z^d, {ρ^d−2_i }_z

i∈Z^d) provided lim

ε↓0⁺ε^d#(I_b^ε) = 0. (5.4.13)

To show (5.4.13), we first bound by (5.4.10)

ε^d#(I_b^ε) ≤ ε^d#(J_b^ε) + X

zi∈I_b^ε\J_b^ε

|Q^ε_i|.

We note that by (5.4.10) and (5.4.9), there exists a constant c = c(d) such that, provided ε ≤ ε₀(d) (with ε₀(d) possibly smaller than the one above), for any cube Q^ε_i with z_i∈ I_b^ε, there exists z_j ∈ J_b^ε

such that

Q^ε_i ⊂ B

2cε

d d−2ρj

(εzj).

Since all the cubes Q^ε_i are (essentially) disjoint, we use the previous inclusion and the definition of

By Lemma 5.5.2, we have almost surely lim sup

and thus estimate for ε small enough (this time depending on ω) ε^d#(I_b^ε) . ε^d#(J_b^ε) + hρ^d−2i^d−22 X

zj∈J_b^ε



ε^d−2d ρ_jd−2

. (5.4.15)

The first term on right-hand side above tends to zero thanks to ε^d#(J_b^ε) = ε^d X

and the choice δ < _d−2² together with the right-hand side of (5.4.14). By this estimate and Lemma 5.5.3, also the second term on the right-hand side of (5.4.15) vanishes almost surely in the limit ε ↓ 0⁺. We thus established (5.4.13) and therefore also (5.4.6).

We now define H_g^ε := H^ε\H_b^ε, which allows to write H_g^ε as in (5.4.7) with n_ε = (Z^d∩¹_εD)\I_b^ε. The first property in (5.4.8) is immediately implied by J_b^ε⊂ I_b^ε and (5.4.9). The second property in (5.4.8) follows from (5.4.13).

It remains to prove the last inequality in (5.4.5): By the definition of H_g^ε itself, if B

ε This concludes the proof of Lemma 5.4.1.

Proof of Lemma 5.3.1, case (a). Let us fix δ and ε₀(δ) as in the statement of Lemma 5.4.1. Then, we know that we may fix P-almost any event ω ∈ Ω such that we find H_b^ε(ω), H_g^ε(ω) and D^ε_b(ω) as in Lemma 5.4.1. Also in this proof, to keep the notation leaner, we omit the argument ω in the oscillating test functions and in the set of holes and write, for instance, w_ε, H^ε instead of w_ε(ω, ·) and H^ε(ω).

Step 1. We begin by a reduction argument: We claim that we may separately treat the two regions D^ε_b and D\D_b^ε, which contain H_b^ε and H_g^ε respectively, and give an explicit construction for

Case (a): Periodic centres 135

w^ε only in the set D\D_b^ε . We indeed claim that, for P-almost every ω ∈ Ω we may set w^ε= w^ε₁∧ w₂^ε with w1, w2∈ H¹(D) and such that

w₁^ε≡ 1 in D \ D^ε_b, w₁^ε= 0 in H_b^ε, (5.4.16) 0 ≤ w^ε₂ ≤ 1, w₂^ε≡ 1 in D_b^ε, w₂^ε= 0 in H_g^ε, (5.4.17) with, in addition,

w^ε₁ → 1 in H¹(D). (5.4.18)

If true, this decomposition for w^ε yields that (H1) is satisfied and, by (5.4.18), that (H2) needs to be argued only for the sequence {w₂^ε}_ε>0. Finally, since ∇w^ε₁ and ∇w₂^ε have disjoint support, for any sequence {vε}_ε>0⊂ H¹(D) as in (H3) we have that

(−∆w^ε, vε)_H⁻¹_,H¹

0 = ˆ

∇w^ε₁· ∇v_ε+ ˆ

∇w₂^ε· ∇v_ε,

and, by (5.4.18), that the first term on the right-hand side vanishes in the limit ε ↓ 0⁺. Since by an integration by parts, the second term on the right-hand side may be rewritten as (−∆w₂^ε, vε)_H⁻¹_,H¹

0, we deduce that with the previous decomposition we may verify (H3) only for the measures {−∆w^ε₂}_ε>0.

Step 2. Construction of w^ε₁ and w₂^ε. We begin with w^ε₁: Thanks to (5.4.5) of Lemma 5.4.1 for H_b^ε, H_g^εand D^ε_b, together with (5.4.1) and (5.4.6), for every ε ≤ ε0there exists a function ˜w^ε₁ ∈ H₀¹(D^ε_b), such that ˜w₁^ε= 1 in H_b^ε, which satisfies

ˆ

D^ε_b

|∇ ˜w₁^ε|² ≤ 2 Cap(H_b^ε, D^ε_b).

If we now set w^ε₁= 1 − ˜w^ε₁, and trivially extend w^ε₁ by 1 outside D_b^ε, we immediately have that (5.4.16) for w₁^ε is satisfied. In addition, thanks to (5.4.6) and our choice of ˜w^ε₁, also (5.4.18) follows.

We now turn to the construction of w₂^ε: By the properties of H_g^ε, H_b^ε and D^ε_b of Lemma 5.4.1, the set D\D^ε_b contains only the holes of H_g^ε, which are all disjoint balls, each strictly contained in the concentric cube Q^ε_i of size ε. We define w₂^ε ≡ 1 on D_b^ε, and explicitly construct w^ε₂ on D\D_b^ε as done in [CM97]: For each zi ∈ n^ε, with n^ε defined in the statement of Lemma 5.4.1, we write T_i^ε= B

ε

d d−2ρi

(εz_i) and B_i= B^ε

2(εz_i) and define w₂^ε= 1 − X

zi∈n^ε

w^ε,i₂ , (5.4.19)

with each w^ε,i₂ solving







−∆w^ε,i₂ = 0 in B_i\ T_i w₂^ε,i= 1 in T_i

w₂^ε,i= 0 in D \ Bi.

(5.4.20)

Since by Lemma 5.4.1 all the balls B_i are disjoint and contained in D\D_b^ε, definitions (5.4.19) and (5.4.20) yield that w^ε₂ satisfies (5.4.17) of Step 1. We thus constructed w₁^ε, w₂^ε satisfying (5.4.17) and (5.4.18) of Step 1. We conclude this step by remarking that definition (5.4.20) also implies that

w₂^ε,i= 1 − argminCap(T_i^ε, B_i^ε) ,

and that each w₂^ε,i may be written explicitly as











w^ε,i₂ (x) = ^|x−εzi|

−(d−2)−(^ε₂)^−(d−2) ε^−dρ^−(d−2)_i −(^ε

2)^−(d−2) in B_i\ T_i w^ε,i₂ = 1 in Ti

w^ε,i₂ = 0 in D \ B_i.

(5.4.21)

Step 3. Equipped with w₁^ε and w^ε₂ constructed above, we show that w^ε = w^ε₁∧ w₂^ε satisfies properties (H1)-(H3). As already discussed in Step 1, it suffices to prove that {w₂^ε}_ε>0 satisfies (H2) and (H3).

We begin with (H2): By (5.4.19), (5.4.21) and (5.4.8) of Lemma 5.4.1, a direct calculation leads to

k∇w^ε₂k²_L2(D). ε^d X

zi∈n^ε

ρ^d−2_i ≤ ε^d X

zi∈Z^d∩¹_εD

ρ^d−2_i . (5.4.22)

By Lemma 5.5.2 applied to the right hand side, we infer that, almost surely, lim sup

ε↓0⁺

k∇w^ε₂k²_L2(D)≤ C. (5.4.23)

In addition, since 1 − w^ε₂ = 0 in R^d\ S

zi∈n^εBi
, and the balls {B_i}_zi∈n^ε are essentially disjoint, by Poincar´e’s inequality we obtain also

k1 − w^ε₂k²_L2(D)≤ X

zi∈n^ε

k1 − w^ε₂k²_L2(Bi). ε² X

zi∈n^ε

k∇w^ε₂k²_L2(Bi).

This, together with (5.4.22) and (5.4.23), yields that almost surely w^ε₂ * 1 in H¹(D) when ε ↓ 0⁺. We thus established (H2).

To prove (H3) for w^ε₂, we first use (5.4.19) and (5.4.21) to decompose

−∆w₂^ε=

n^ε

X

i=1

(µ^ε,i− γ^ε,i), µ^ε,i= −∂_νw₂^ε,iδ_∂Bi, γ^ε,i= −∂_νw₂^ε,iδ_∂Ti,

with ν denoting the outer normal and δ_∂Bⁱ and δ_∂Tⁱ being the (d − 1)-dimensional Hausdorff measure restricted to ∂B_i and ∂T_i, respectively. We start by remarking (see (H5)’ of [CM97]) that, since in (H3) the functions vε are always assumed to be vanishing on each Ti, we only need to focus on the

convergence (H3) for the sequence of measures µ^ε:= −X

i∈n^ε

∂_νw^ε,i₂ δ_∂Bi := X

i∈n^ε

µ^ε,i.

More precisely, we claim that for every v_ * v in H₀¹(D) such that v_ε∈ H₀¹(D^ε), it holds (µ^ε, v_ε)_H−1,H₀¹(D)→ C₀

ˆ

D

v, (5.4.24)

where C₀ := (d − 2)σ_dh ρ^d−2i corresponds to the definition (5.2.6) for the case Φ = Z^d under consideration.

We begin by arguing that it suffices to prove (5.4.24) above for any truncated process (Z^d, R^M), with M ∈ N and R^M defined in (5.4.4). From now on, we use the lower index M to distinguish the

Case (a): Periodic centres 137

objects constructed with the truncated marks R^M and the ones coming from R. For instance, we denote by w^ε_2,M, µ^ε_M the analogues of w^ε and µ^ε introduced above. Note that, for R^M, the constant in (5.4.24) reads C0,M = (d − 2)σdh ρ^d−2_M i.

For any M ∈ N, since |C0− C_0,M| . h ρ^d−21ρ≥Mi, we bound by the triangular inequality, an integration by parts and Cauchy-Schwarz inequality



By an argument similar to the one in (5.4.22), we estimate lim sup establish (H3) by assumption (5.1.5).

To argue that almost surely and for every M ∈ N the convergence in (5.4.24) holds for µ^ε_M, we follow [CM97]. First, by the definition of w_i,M^ε , we compute

µ^ε_M = X

To show this, we fix M ∈ N and split the convergence (5.4.24) into the two following steps: If we define

To show (5.4.26), we consider the auxiliary problems

(−∆q_i,M^ε = 2^d(d − 2)dρ^d−2_i,M in B_i^ε

∂q^ε_i,M

∂ν = 2^d−1(d − 2)ρ^d−2_i,Mε on ∂B_i^ε, (5.4.28) which are in particular satisfied by the functions

q_i,M^ε (x) = 2^d−1(d − 2)ρ^d−2_i,M



|x − z_i|²−ε 2

2 . As q_i,M^ε = 0 on ∂B_i^ε, we may extend q_i,M^ε by zero outside of B_i^ε and estimate

k∇q^ε_i,Mk_L^∞_(Bi) = 2^d−1(d − 2)ρ^d−2_i,Mε . M^d−2ε.

Using Poincar´e’s inequality, and since the balls B_i^ε are disjoint, we infer that q^ε_M := X

zi∈Z^d∩¹_εD

q_i,M^ε → 0 in W^1,∞(R^d). (5.4.29)

We observe that by (5.4.28)

η^ε_M− ˜µ^ε_M = −∆q_M^ε + X

zi∈(Z^d∩¹_εD)\nε

2^d(d − 2)dρ^d−2_M,i1_Bi =: −∆q_M^ε + R^M_ε .

We have by (5.4.29) that −∆q_M^ε → 0 in W^−1,∞(R^d). On the other hand, for the term R^M_ε above, we have that by Lemma 5.5.3 and (5.4.8)

lim

ε↓0⁺

kR^εk_L1 . lim

ε↓0⁺ε^d X

zi∈(Z^d∩¹_εD)\nε

ρ^d−2_M,i = 0,

almost surely. Since R^ε is bounded in L^∞, we also have that R^ε * 0 in L^{∗ ∞}(D) and R^ε → 0 in W^−1,∞(D). This yields (5.4.26).

In order to show (5.4.27), we first remark that it suffices to argue that η_M^ε * C^∗ _0,M in L^∞(D).

Since the family of functions {η^ε_M}_ε>0 is uniformly bounded in L^∞(D), we identify the w^∗-limit by testing η^ε_M with any function ζ ∈ C₀¹(D): Indeed, by Lemma 5.5.4 applied to (Z^d, {ρ^d−2_i,M}_i∈Zd) in the domain B = D we infer almost surely that

(η_M^ε , ζ)_H−1,H₀¹(D)→ C_0,M ˆ

D

ζ.

This establishes (5.4.27) and thus concludes the proof for (H3) and for the whole lemma.

5.4.2 Proof of Lemma 5.3.1 in the general case

Let (Φ, R) be a marked point process satisfying the assumptions of Theorem 5.2.1. In contrast with the previous subsection, when the centres of the holes are distributed according to a general point process Φ, there is not a deterministic positive lower bound for the minimal distance between the points of Φ. This requires some technical changes in the arguments of Subsection 5.4.1 but still allows us to obtain a statement on the asymptotic geometry of H^ε similar to Lemma 5.4.1 and to prove Lemma 5.3.1.

Proof of Lemma 5.3.1 in the general case 139

Lemma 5.4.2. There exist an ε0 = ε0(d) and a family of random variables {rε}_ε>0 ⊂ R+ such that for P-almost every ω ∈ Ω

lim

ε↓0⁺rε(ω) = 0, (5.4.30)

and for any ε ≤ ε₀ there exist H_g^ε(ω), H_b^ε(ω), D_b^ε(ω) ⊂ R^d such that H^ε(ω) = H_g^ε(ω) ∪ H_b^ε(ω), H_b^ε(ω) ⊂ D^ε_b(ω),

dist H_g^ε(ω), D^ε_b(ω)
≥ εr_ε(ω)

2 , (5.4.31)

where

lim

ε↓0⁺Cap(H_b^ε(ω), D_b^ε(ω)) = 0. (5.4.32) Moreover, H_g^ε(ω) may be written as the following union of disjoint balls centred in n^ε(ω) ⊂ Φ(¹_εD):

H_g^ε(ω) := [

zj∈n^ε

Bε^d−2d ρj

(εz_j),

zi6=zmin_j∈n^εε|zi− z_j| ≥ 2r_εε, ε^d−2d ρj ≤ εrε(ω)

2 , lim

ε↓0⁺ε^d#(n^ε) = hN (Q)i|D|. (5.4.33) Furthermore, if for δ > 0 the process Φδ is defined as in (5.4.3), then

lim

ε↓0⁺ε^d# z_i∈ Φ^ε_2δ(D)(ω) : dist(zi, D_b^ε) ≤ δε
= 0. (5.4.34)

We remark that the lower bounds in (5.4.31) and (5.4.33) differ from the ones of Lemma 5.4.1 by the factor r_ε. This implies, by (5.4.30), that the minimal distance between the balls of the “good”

set H_g^ε is only o(ε) for ε ↓ 0⁺ and not of order ε as required in the construction of the functions {w_ε}_ε>0 carried out in the previous subsection. We overcome this technical issue by comparing wε

again with the oscillating test functions w^ε_M obtained by approximating H_g^ε by a simpler set Hg^ε,M. Here, Hg^ε,M is obtained not only by truncating the radii at size M as in the previous subsection, but also by considering in Hg^ε,M only the balls whose centres (in n^ε ⊂ Φ^ε(D)) satisfy (5.4.31) and (5.4.33) with M⁻¹ε instead of rεε. As in the previous subsection, we show that the sets H^ε,M are a good approximation of H^ε, in the sense that the associated functions w^ε_M and w^ε are close in H¹. This follows from the fact that the centres removed from H_g^ε, which are either too close to each other or to the “safety layer” D_b^ε, are few and may be taken care of by studying the properties of the thinnings Φ_δ of a process Φ defined in (5.4.3). In fact, the last limit (5.4.34) of the previous lemma states that the main error in considering the approximate holes Hg^ε,M is given by the points which are too close to each other.

Proof of Lemma 5.4.2. As in the previous subsection, we suppress the argument ω ∈ Ω for the random sets involved in the argument below. Let us fix an α ∈ (0,_d−2² ) and let us define

rε:= (ε^d−2d max

zj∈Φ^ε(D)ρj)^1d ∨ ε^α4. (5.4.35)

With this choice, we prove that the decomposition of H^ε required by the lemma holds true. We begin by showing that with this definition rε satisfies (5.4.30): Let

F^ε:= {z_j ∈ Φ^ε(D) | ε^d−2d ρ_j ≥ ε}.

If F^ε= ∅, then r_ε≤ ε^1d ∨ ε^α4. If otherwise, we estimate ε^d max

zj∈Φ^ε(D)ρ^d−2_j = ε^d max

zj∈F^ερ^d−2_j ≤ ε^d X

zj∈F^ε

ρ^d−2_j .

To get (5.4.30), it suffices to show that almost surely the right hand side above tends to zero in the limit ε ↓ 0⁺. By Lemma 5.5.3, this holds provided

ε^d#(F^ε) → 0 ε ↓ 0⁺. We show this by bounding

ε^d#(F^ε) . ε²ε^d X

zj∈Φ^ε(D)

ρ^d−2_j ,

and using Lemma 5.5.2. We thus established (5.4.30).

Equipped with (5.4.35), we now set ηε= rεε and begin by constructing the sets H_b^ε and D_b^ε. As in Lemma 5.4.1, we denote by I_b^ε the set of points in Φ^ε(D) which generate the sets H_b^ε and D_b^ε. We start by requiring that I_b^ε contains the points in Φ^ε(D) whose associated radii are “too big”, namely the set

J_b^ε =n

z_j ∈ Φ^ε(D) : ε^d−2d ρ_j ≥ η_ε 2

o

. (5.4.36)

Similarly to the periodic case, we set

H˜_b^ε:= [

zj∈J_b^ε

B

2ε

d d−2ρj

(εz_j). (5.4.37)

We now include in I_b^ε also the points in Φ^ε(D)\J_b^ε which, in spite of having radii below the threshold set in definition (5.4.36), are too close to each other. We indeed define

K_b^ε:= Φ^ε(D)\ Φ^ε_2rε(D) ∪ J_b^ε
. (5.4.38) Finally, we include into I_b^ε also the set of points which are not in J_b^ε∪ K_b^ε, but which might are close to ˜H_b^ε: We denote them by

I˜_b^ε:=n

z_j ∈ Φ^ε(D)\(J_b^ε∪ K_b^ε) : ˜H_b^ε∩ B_ηε(εz_j) 6= ∅o

. (5.4.39)

We thus set

I_b^ε= ˜I_b^ε∪ J_b^ε∪ K_b^ε, (5.4.40) H_b^ε:= [

zj∈I_b^ε

Bε^d−2d ρj

(εzj), H_g^ε:= H^ε\ H_b^ε, D_b^ε:= [

zj∈I_b^ε

B2ε^d−2d ρj

(εzj). (5.4.41)

Proof of Lemma 5.3.1 in the general case 141

It remains to show that, with H_b^ε, H_g^ε and D_b^ε defined as in (5.4.41), properties (5.4.31), (5.4.32), (5.4.33) and (5.4.34) are satisfied. We start with (5.4.32). As in the proof of Lemma 5.4.1, the

definition of D_b^ε allows us to estimate by the sub-additivity of the capacity Cap(H_b^ε, D^ε_b) . ε^d X

zj∈I_b^ε

ρ^d−2_j .

Thanks to Lemma 5.5.3, we conclude that, almost surely, when ε ↓ 0⁺, the right-hand side above vanishes provided that almost surely

lim

ε↓0⁺ε^d#(I_b^ε) = 0. (5.4.42)

We show this by using definition (5.4.40) and proving that each of the sets which constitute I_b^ε satisfies the limit above. We begin with J_b^ε: Definitions (5.4.35) and (5.4.36) yield

ε^d#(J_b^ε) . ε²r_ε^−(d−2)ε^d X

zj∈Φ^ε(D)

ρ^d−2_j ≤ ε^{2−α(d−2)}ε^d X

zj∈Φ^ε(D)

ρ^d−2_j .

By Lemma 5.5.2 and the assumption α < _d−2² , the right-hand side almost surely vanishes in the limit ε ↓ 0⁺. We thus established

lim

ε↓0⁺ε^d#(J_b^ε) = 0, (5.4.43)

i.e. limit (5.4.42) for J_b^ε. We now turn to K_b^ε: Let {δk}_k∈N be any sequence such that δk ↓ 0⁺. Since rε satisfies (5.4.30), we estimate for any δ_k

lim sup

ε↓0⁺

ε^d#(K_b^ε)^(5.4.38)= lim sup

ε↓0⁺

ε^d N^ε(D) − N_r^ε_ε(D)
^(5.4.3)

≤ lim sup

ε↓0⁺

ε^d N^ε(D) − N_δ^ε_k(D)
.

We now apply Lemma 5.5.2 to Φ and each Φ_δk to deduce that almost surely and for every δ_k lim sup

ε↓0⁺

ε^d#(K_b^ε) ≤ hN (Q) − Nδ_k(Q)i|D|, where Q is a unit cube. By sending δ_k↓ 0⁺, Lemma 5.5.2 yields

ε↓0lim⁺ε^d#(K_b^ε) = 0. (5.4.44)

To conclude the proof of (5.4.42), it remains to show that almost surely also

ε^d#( ˜I_b^ε) → 0 ε ↓ 0⁺. (5.4.45) By definitions (5.4.36), (5.4.38) and (5.4.39), for each z_i∈ Φ^ε(D)\(J_b^ε∪ K_b^ε), we have

min

zj∈Φ^ε(D)\{zi}ε|z_j− z_i| ≥ 2η_ε, ε^d−2d ρ_i < η_ε

2. (5.4.46)

On the one hand, by the first inequality above, the balls {B_ηε(εz_i)}_z

i∈ ˜I_b^ε are all disjoint and satisfy ε^d#( ˜I_b^ε) . ε^d X

zi∈ ˜I_b^ε

η^−d_ε |B_ηε(εzi)| = r_ε^−d X

zi∈ ˜I_b^ε

|B_ηε(εzi)|.

In addition, the inequalities (5.4.46), definitions (5.4.36), (5.4.38) and (5.4.37) also imply that

By definition (5.4.35), the inequality above reduces to ε^d#( ˜I_b^ε) . ε^dX

Thanks to (5.4.43), we apply Lemma 5.5.3 and deduce (5.4.45). This, together with (5.4.43) and (5.4.44), yields (5.4.42) and concludes the proof of (5.4.32).

To show (5.4.31), we recall the definitions of D_b^ε, H_b^ε and H_g^ε in (5.4.41) and set n^ε:= Φ^ε(D)\I_b^ε.

Finally, the properties (5.4.33) of the set H_g^εare a consequence of (5.4.46), definition (5.4.42) and (5.5.8) of Lemma 5.5.2.

To show (5.4.34), we resort to the definition of D^ε_b to estimate

z_i∈ Φ^ε_2δ(D)(ω) : dist(zi, D^ε_b) ≤ δε

This follows by an argument similar to the one for (5.4.45): Then, we may choose ε0= ε0(d) such that for all ε ≤ ε₀, property (5.4.30) yields η_ε= εr_ε≤ δε. By definition of J_b^ε in (5.4.36) and of E^ε

Proof of Lemma 5.3.1 in the general case 143

By Lemma 5.5.3 and (5.4.43), almost surely the right hand side tends to zero in the limit ε ↓ 0⁺. We conclude the argument for (5.4.34) by showing that the set C^ε is empty when ε is small: In fact, by construction, if zi∈ n_ε satisfies

dist

The proof of Lemma 5.4.2 is complete.

Proof of Lemma 5.3.1, general case. We split the proof of the lemma in the same steps as in the proof for the case of periodic centres (case (a)). Some of these steps may be proven exactly as in the previous subsection by relying on Lemma 5.4.2 instead of Lemma 5.4.1. We thus focus below only on the parts of the proof which differ from Subsection 5.4.1.

Step 1. Since by Lemma 5.4.2, the sets H_g^ε and D^ε_b are disjoint, the splitting w^ε= w₁^ε∧ w₂^ε with w^ε₁, w₂^ε solving (5.4.16), (5.4.17) and (5.4.18) remains unchanged from the case of periodic centres.

Step 2. Again by Lemma 5.4.2, we construct the sequence {w^ε₁}_ε>0 satisfying (5.4.16) and (5.4.18) as in Subsection 5.4.1. We thus only need to focus on the construction of the functions {w^ε₂}_ε>0, which we set equal to 1 on D^ε_b. For each zj ∈ n^ε, with n^ε being the set of centers of the particles in H_g^ε (see Lemma 5.4.2), we denote the random variables

d^ε_j := min

we consider the function w^ε,j₂ solving (5.4.20) in B^j\T^j and hence defined as

w^ε,j₂ (x) =

Note, that by definition of d^ε_j, (5.4.49), the functions ∇w^ε,j have disjoint support. Moreover, for

which immediately satisfies condition (5.4.17). Therefore, as discussed in Step 1, the function w^ε= w^ε₁∧ w^ε₂ satisfies (H1) and it suffices to prove (H2)-(H3) only for w₂^ε.

Step 3. We begin by showing that w₂^ε satisfies (H2): By the triangular inequality and definitions (5.4.50) and (5.4.52), we estimate

k∇w^ε₂k²₂ = X

By Lemma 5.5.2, the right-hand side above is almost surely bounded in the limit ε ↓ 0⁺. This, together with Poincar´e’s inequality for 1 − w₂^ε in D, yields that almost surely, up to a subsequence, we have w₂^ε* w in H¹(D) when ε ↓ 0⁺.

We claim that w ≡ 1. To this purpose, it is useful to consider the following “truncated” processes (n^ε_M, {ρ_j,M}_j∈n^ε) which we construct in the following way: For any M ∈ N, we set disjoint and have radii in [M⁻¹ε, ε] e may argue as in Subsection 5.4.1 and infer that almost surely, and for every M ∈ N, w^ε,M₂ * 1 in H¹(D), and therefore also strongly in L²(D). This implies by the

Proof of Lemma 5.3.1 in the general case 145

Condition (H2) holds for w^ε₂ provided that the limit on the right-hand side above vanishes. By Poincar´e’s inequality in D, it suffices to prove that

M ↑∞lim lim sup

ε↓0⁺

k∇(w^ε₂− w^ε,M₂ )k²₂ = 0. (5.4.54) To show this, we argue as follows: By construction

w₂^ε= w^ε,M₂ in Bi ⊂ D\D^ε_b, whenever ρi ≤ M and d_i≥ M⁻¹ε, w₂^ε,M ≡ 1 in B_i, whenever d_i ≤ M⁻¹ε.

This implies that the L²-norm on the left-hand side above reduces to ∇ w^ε− w^ε,M₂

Similarly to (5.4.53), we use the explicit formulation for w₂^ε, w₂^ε,M to control the first term on the right-hand side above by we obtain that almost surely

M ↑+∞lim lim sup Hence, from (5.4.57) we obtain

lim sup

On the one hand, by (5.4.34) and Lemma 5.5.3, the second term on the right-hand side vanishes. On the other hand, Lemma 5.5.2 for both Φ and Φ^2M−1 imply that

lim sup

ε↓0+

X

zi∈n^ε

k∇w₂^ε,ik²₂1_di≤M⁻¹_ε≤ hN (Q) − N_2M⁻¹(Q)ihρ^d−2i,

where Q is a unit cube. Finally, the right-hand side in the above estimate converges to zero in the limit M → ∞ again by Lemma 5.5.2. If we now wrap up the previous estimate with (5.4.56) and

(5.4.55), we conclude (5.4.54). We thus established (H1) and (H2) for the sequence w₂^ε(and thus also fo w^ε).

It remains to prove (H3). We consider again the truncated sequences {w^ε,M₂ }_ε>0 above and start by arguing that it is enough to show that, for every M ∈ N fixed, condition (H3) is satisfied by w2^ε,M, namely inequality yield that to prove (H3) it suffices to show (5.4.58).

The proof of (5.4.58) follows the same lines of Step 3. in Subsection 5.4.1 (in particular, (5.4.24) for w₂^ε,M), so we just point out the differences: By arguing as in that case, it suffices to prove that

η^ε_M := X

zi∈n^ε_M

d(d − 2)ρ^d−2_i,M ε^d

d^d_i1_Bi * C^∗ _0,M in L^∞(D) (5.4.59) The factor ε^dd^−d_i in the above expression is due to the fact that the balls B_i have now radii d_i instead of ε. Hence, by including this factor, we have

kε^dd^−d_i 1Bik_L1 = k1_Bε(εzi)k_L1 = ε^d as in the periodic case.

Since

then as in Step 3 of Subsection 5.4.1, we use Lemma 5.5.2 for Φ_2M⁻¹ and Lemma 5.5.4 applied to (Φ_2M⁻¹, {ρ^d−2_i,M}) to infer that, almost surely and for all ζ ∈ C₀¹(D),

We conclude (5.4.59) by arguing that, almost surely and for all ζ ∈ C₀¹(D), we have 

Auxiliary results 147

Indeed,

    ˆ

D

(η^ε_M − ˜η_M^ε )ζ   

. M^d−2 X

zi∈Φ^ε

2M −1(D)\n^ε_M

ˆ

Bε(εzi)

|ζ|

. M^d−2kζk_∞ε^d#n

z_i∈ Φ^ε_2M−1(D) : d_i ≤ ε M

o

.

By definition of d_i and (5.4.34), the right-hand side tends to zero almost surely in the limit ε → 0.

This concludes the proof of (5.4.59) and thus establishes (H3) for the sequence {w₂^ε}_ε>0. The proof of Lemma 5.3.1 is complete.

In document Sedimentation of particle suspensions in Stokes flows (Page 137-153)