Convergence rates for the homogenization of the Poisson problem in randomly perforated domains

1.   Introduction

In this paper we obtain convergence rates for the homogenization of the Poisson problem in a bounded domain of $\mathbb{R}^d$, $d \geqslant 3$, that is perforated by many small random holes $H^ \varepsilon$. We impose Dirichlet boundary conditions on the boundary of the set and of the holes $H^ \varepsilon$. In other words, given $f \in H^{-1}(D)$ and $D \subseteq \mathbb{R}^d$ bounded and regular, we define the perforated set $D^ \varepsilon: = D\backslash H^ \varepsilon$ and study the boundary value problem

We assume that, for $\varepsilon > 0$, the holes $H^ \varepsilon$ are a union of spherical holes having random centres and radii. Let $(\Phi, \mathcal{R})$ be marked point process where $\Phi$ is either the lattice $\mathbb{Z}^d$ or a Poisson point process of intensity $\lambda > 0$. We assume that the associated marks $\mathcal{R} = \{ \rho_z \}_{z \in \Phi}$ are independent and identically distributed random variables that satisfy the moment condition

Here and below, $\mathbb{E}_\rho [ \, \cdot \, ]$ denotes the expectation under $\rho$. The set $H^ \varepsilon$ is thus defined by

As shown in [13], if $\beta = 0$ in (1.2), then for $\mathbb{P}$-almost every realization of the random set $H^ \varepsilon$, the solutions to (1.1) converge weakly in $H^1_0(D)$ to the homogenized problem

The constant $C_0 > 0$ is the limit of the density of harmonic capacity generated by the set $H^ \varepsilon$: If $S^{d}$ denotes the $d$-dimensional unit sphere and $\mathcal{H}^{d}$ is the $d$-dimensional Hausdorff measure, then

In this paper, we strengthen the condition (1.2) on the integrability of the marks $\mathcal{R}$ from $\beta = 0$ to $\beta > 0$ and study the convergence rates of $u_ \varepsilon$ to the homogenized solution $u$.

By the Strong Law of Large Numbers, assumption (1.2) with $\beta = 0$ is minimal in order to ensure that for $\mathbb{P}$-almost every realization of $H^ \varepsilon$, its density of capacity admits a finite limit. However, it does not prevent the balls in $H^ \varepsilon$ from having radii that are much bigger than $\varepsilon^\frac{d}{d-2}$. This gives rise to clustering phenomena with overwhelming probability. In particular the expected number of balls of $H^ \varepsilon$ that intersect, namely such that their radius $\varepsilon^\frac{d}{d-2} \rho_z$ is bigger than the typical distance $\varepsilon$ between the centres, is of order $\varepsilon^{-d+2}$ (over an expected total of $\varepsilon^{-d}$ balls). The analogue holds also under assumption (1.2) for values $\beta < \frac{(d-2)^2}{2}$, with the expected number of overlapping balls being of order $\varepsilon^{-d +2 +\frac{2}{d-2}\beta}$.

The presence of balls that overlap is the main challenge in the proof of the qualitative homogenization statement obtained in [13] and is one of the challenges of the current paper. It requires a careful treatment of the set $H^ \varepsilon$ to ensure that the presence of long chains of overlapping balls does not destroy the homogenization process. For a more detailed discussion on this issue we refer to the introductory section in [13] and to Subsection 2.2 of the present paper.

The main results contained in this paper provide an annealed (i.e. averaged in probability) estimate for the $H^1$-norm of the homogenization error $u_ \varepsilon- W_ \varepsilon u$. The function $W_ \varepsilon$ is a suitable corrector function that is related to the so-called oscillating test function [6,23]. We assume that $\Phi$ is the lattice $\mathbb{Z}^d$ or that it is a Poisson point process in dimension $d = 3$. For a comment on the case $d >3$, we refer to Remark 2.2. If $\mathbb{E} \bigl[ \, \cdot \, \bigr]$ denotes the expectation under the probability measure associated to the process $(\Phi, \mathcal{R})$, we show that¹

¹ In the case of $\Phi$ being a Poisson point process, there is a factor $\log \varepsilon$ on the right-hand side. We refer to Theorem 2.1 for the precise statement.

We stress that in the case of periodic holes, namely when $\Phi = \mathbb{Z}^d$ and the radii $\rho_z = r$, $r > 0$ are constant and deterministic, the optimal rate on the right-hand side of (1.6) is $\varepsilon$ [18]. In the current setting, the unboundedness and randomness of the radii seems to yield slower convergence rates. We refer to Subsection 2.2 for a discussion on the exponents obtained in (1.6) and how they relate to the techniques used in the present paper to treat the case of unbounded and random radii.

The main quantity that governs the decay of the homogenization error $u_ \varepsilon - W_ \varepsilon u$ is the convergence of the capacity density of $H^ \varepsilon$ to the constant term $C_0$ that appears in the homogenized equation (1.4). In the periodic case mentioned in the previous paragraph, the term $C_0 = c_d r^{d-2}$ (c.f. (1.5)) is close to the density of capacity of $H^ \varepsilon$ already at scale $\varepsilon$. Heuristically, indeed, if $A \subseteq D$ we have

and this chain of identities is true as long as $|A|$ is at least of order $\varepsilon$. On the other hand, in the random setting, this identity is expected to hold at scales that are larger than $\varepsilon$ due to the fluctuations of the process $(\Phi, \mathcal{R})$. We also remark that the threshold $d-2$ in the parameter $\beta$ obtained in (1.6) is related to the $L^2$-nature of the norm considered for the homogenization error. Roughly speaking, the norm considered in (1.6) requires a control on the expectation of the square of the capacity generated by the balls in $H^ \varepsilon$. This quantity depends on the $2(d-2)$-moments of the random variable $\rho$.

Starting with [6] and [21], there is a large amount of literature devoted to the homogenization of (1.1), both for deterministic and random holes $H^ \varepsilon$ [3,20]; similar problems have also been studied in the case of the fractional laplacian $(-\Delta)^s$, [2] or for nonlinear elliptic operators [4,24]. All the models considered in the deterministic case contain assumptions that ensure that, for $\varepsilon$ small enough, the holes in $H^ \varepsilon$ do not to overlap. In the random models mentioned above, a non-overlapping condition is as well imposed, at least for $\mathbb{P}$-almost every realization and $\varepsilon>0$ small enough. For a complete and more detailed description of these works, we refer to the introduction of [13].

For what concerns quantitative rates of convergence for (1.1) to (1.4), the first result in the periodic case is contained in [18]. When the holes are randomly distributed, the first quantitative result has been obtained in [9]. In this paper, the authors study the analogue of (1.1) for the operator $-\Delta + \lambda$ in an unbounded domain of $\mathbb{R}^3$, that is perforated by $m$ spherical holes of identical radius $\sim m^{-1}$. The centres of the holes are independent and distributed according to a compactly supported and continuous potential $V$. If $u_m$ denotes the analogue of $u_ \varepsilon$, when the massive term $\lambda$ is big enough compared to the size of $V$, the authors provide rates of convergence for the $L^2$-norm of the difference $u_m -u$ in the limit $m \to +\infty$. Furthermore, they prove the Gaussianity of the fluctuations of $u_m$ around the homogenized solution $u$ in the CLT-scaling. In [16], this result has been obtained in the same setting of [9] without any constraint on the massive term $\lambda > 0$.

The quantitative estimates developed in this paper are also used in [10] to obtain homogenization results for solutions to (1.1) and the analogous Stokes system in the regimes leading to Darcy's law. In [10], the radii in (1.3) are rescaled by a factor $\varepsilon^{\alpha}$, $1 < \alpha< \frac{d}{d-2}$ and the random variables $\{ \rho_z \}_{z\in \Phi}$ satisfy a suitable moment condition. Like (1.2), also this condition does not prevent the holes from overlapping and give rise to clusters.

We conclude this introduction mentioning that the analogue of (1.1) for a Stokes (and Navier-Stokes) system with no-slip boundary conditions on the holes $H^ \varepsilon$ has been considered in [1,22] in the periodic case and then extended to more general configurations of holes (see, e.g., [5,8,15]). In the case of the Stokes operator, the limit equation contains an additional zero-th order term similar to $C_0$ in (1.4). Under the same assumptions of this paper, the analogue of the homogenization result contained in [13] has been proven for a Stokes system in [11,12]. We believe that techniques similar to the one of this paper may be used to prove the same result of (1.6) also in the case of a Stokes system. However, in this case, we expect that the role played by the pressure would yield a less explicit definition of the corrector function $W_ \varepsilon$.

2.   Setting and main result

Let $d \geqslant 3$ and $D \subseteq \mathbb{R}^d$ be a bounded and smooth domain that is star-shaped with respect to the origin. For $\varepsilon > 0$, we define the punctured set $D^ \varepsilon = D \backslash H^ \varepsilon$, with $H^ \varepsilon$ as in (1.3). We assume that the union of balls $H^ \varepsilon$ is generated by a marked point process $(\Phi, \mathcal{R})$ on $\mathbb{R}^d \times \mathbb{R}_+$: We generate the centres of the balls in $H^ \varepsilon$ via a point process $\Phi$. To each point $z\in \Phi$, we associate a mark $\rho_z \geqslant 0$ that determines the radius of the ball. We refer to [7,Chapter 9,Definitions 9.1.I-9.1.IV] for an extensive and rigorous definition of marked point processes and their associated measures on $\mathbb{R}^d \times \mathbb{R}_+$.

We denote by $(\Omega; \mathcal{F}, \mathbb{P})$ the probability space associated to $(\Phi, \mathcal{R})$, so that the random sets in (1.3) and the random field solving (1.1) may be written as $H^ \varepsilon = H^ \varepsilon(\omega)$, $D^ \varepsilon = D^ \varepsilon(\omega)$ and $u_ \varepsilon(\omega; \cdot)$, respectively. The set of realizations $\Omega$ may be seen as the set of atomic measures $\sum_{n \in \mathbb{N}} \delta_{(z_n, \rho_n)}$ in $\mathbb{R}^d \times \mathbb{R}_+$ or, equivalently, as the set of (unordered) collections $\{ (z_n , \rho_n) \}_{ n\in \mathbb{N}} \subseteq \mathbb{R}^d \times \mathbb{R}_+$.

Throughout this paper we assume that $(\Phi, \mathcal{R})$ satisfies the following conditions:

(i) $\Phi$ is either the lattice $\mathbb{Z}^d$ or $\Phi = \mathop{Poi}(\lambda)$, i.e. a Poisson point process of intensity $\lambda>0$;

(ii) The marks $\{ \rho_z\}_{z \in \Phi}$ are independent and identically distributed: The marginal of the marks with respect to the process $\Phi$, has n-correlation function, $n \in \mathbb{N}$, that may be written as the product

(iii) The marks $\mathcal{R}$ have finite $(d-2+\beta)$-moment, namely the density function $g$ in (ii) satisfies

We stress that conditions (i)-(ii) yield that $(\Phi, \mathcal{R})$ is stationary. In the case $\Phi = \mathop{Poi}(\lambda)$, the process $(\Phi, \mathcal{R})$ is stationary with respect to the action of the group of translations $\{\tau_x \}_{x\in \mathbb{R}^d}$. This means that the probability measure $\mathbb{P}$ is invariant under the action of the transformation $\tau_x: \Omega \to \Omega$, $\omega = \{ ( z_i; \rho_{z_i}) \}_{i \in \mathbb{N}} \mapsto \tau_x \omega : = \{ ( z_i + x; \rho_{z_i}) \}_{i \in \mathbb{N}}$. In the case $\Phi = \mathbb{Z}^d$ the same holds under the action of the group $\{\tau_z \}_{z\in \mathbb{Z}^d}$.

Notation. When no ambiguity occurs, we skip the argument $\omega \in \Omega$ in the notation for $H^ \varepsilon(\omega), D^ \varepsilon(\omega)$, $u_ \varepsilon(\omega ; \cdot)$ and in all the other random objects. We denote by $\mathbb{E}\bigl[ \, \cdot \, \bigr]$ and $\mathbb{E}_{\Phi}\bigl[ \, \cdot \, \bigr]$ the expectations under the total probability measure $\mathbb{P}$ and the probability measure $\mathbb{P}_{\Phi}$ associated to the point process $\Phi$, respectively. For $\varepsilon > 0$ and a set $A \subseteq \mathbb{R}^d$, we define

and the random variables

For any $\mu \in H^{-1}(D)$, we write $\langle \, \cdot \, ; \, \cdot \, \rangle$ for the duality product with $H^1_0(D)$; we use the notation for the averaged sum $\#(I)^{-1}\sum_{i\in I}$ and $\lesssim$ and $\gtrsim$ instead of $\leqslant C$ and $\geqslant C$ with the constant $C$ depending on the dimension $d$, the domain $D$ and, in the case of $\Phi = \mathop{Poi}(\lambda)$, the intensity rate $\lambda$.

For two sets $A \subseteq B \subseteq \mathbb{R}^d$, we denote by $\mathop{Cap}( A ; B)$ the relative harmonic capacity of the set $A$ in $B$ (c.f., for instance, [13][(4.17)]).

2.1.   Main result

Before stating the main results, we need to define a suitable corrector function $W_ \varepsilon$ that appears in the homogenization error $u_ \varepsilon - W_ \varepsilon u$. For $x \in \mathbb{R}^d$ we set

Note that, if $\Phi = \mathbb{Z}^d$, then $R_{ \varepsilon, x} = \frac \varepsilon 4$ for every $x \in \Phi$. For $\delta > 0$, we denote by $\Phi_{\delta}^ \varepsilon(D) \subseteq \Phi_ \varepsilon(D)$ the set

where here and through out the paper $a \vee b = \max\bigl\{ a ; b \bigr\}$.

For each $z \in \Phi^ \varepsilon_\delta(D)$, let $w_{ \varepsilon, z} \in H^1(B_{\frac \varepsilon 4}( \varepsilon z))$ be the solution to

We thus define

We stress that (2.5) ensures that definitions (2.6) and (2.7) are well-posed since the set $\{ B_{R_{ \varepsilon, z}}( \varepsilon z)\}_{z \in \Phi_\delta^ \varepsilon(D)}$ is made of disjoint balls and, for every $z \in \Phi^ \varepsilon_\delta(D)$, it holds $B_{ \varepsilon^\frac{d}{d-2} \rho_z}( \varepsilon z) \subseteq B_{R_{ \varepsilon, z}}( \varepsilon z)$. Note that in the above definition the function $W_ \varepsilon \in H^1(D)$ depends on the choice of the parameter $\delta$ used to select the subset $\Phi_\delta^ \varepsilon(D)$. The optimal parameter $\delta$ will be fixed in Theorem 2.1. We finally stress that, in the periodic case $\Phi = \mathbb{Z}^d$ and $\rho_z \equiv r$, for any $\delta > 0$ and $\varepsilon$ small enough, the function $W_ \varepsilon$ coincides with the oscillating test function constructed in [6,18].

Theorem 2.1. Let $(\Phi, \mathcal{R})$ satisfy conditions (i)-(iii). For $\varepsilon>0$ and $p > d$, let $f \in W^{1, p}(D)$ with $\|f\|_{W^{1, p}(D)} = 1$ and $u_ \varepsilon$ and $u$ be as in $(1.1)$ and $(1.4)$, respectively. We consider the random field $W_ \varepsilon$ in $(2.7)$ with

(a) If $\Phi = \mathbb{Z}^d$, there exists a constant $C = C(d, D, p)> 0$ such that

(b) If $\Phi = \mathop{Poi}(\lambda)$ with $\lambda >0$ and $d = 3$, there exists a constant $C = C(\lambda, D, p)>0$ such that

Remark 2.2. As shown throughout Section 4, the argument of Theorem 2.1 (b) applies also to higher dimensions $d \geqslant 4$. In this case, the homogenization error decays as $\varepsilon^{\alpha}$, for an exponent $\alpha = \alpha(d, \beta)$. The exponent $\alpha$, however, is generally smaller than the one in the periodic case and it is the same only for $\beta$ small enough ($\beta \leqslant \frac{2}{d-1}(d-2)$).

We expect that the techniques used to prove Theorem 2.1 do extend to other examples of stationary random distributions of centres. For instance, to point processes that satisfy a finite-range of dependence assumption and for which the expected number of elements in a finite set $A$ scales like its (Lebesgue-)measure. We also believe that Theorem 2.1 may be adapted to sets of holes $H_m$, $m \in \mathbb{N}$ that are a collection of $m$ balls having centres that are independently distributed according to a fixed density $\Psi \in C^\infty_c(D)$. In this case, the balls have random radii rescaled by $m^{-\frac{1}{d-2}}$ and the homogenization limit is obtained for $m \to \infty$. For this choice of holes, the deterministic control on the cardinality of the set of centres simplifies some of the estimates.

Remark 2.3. As it becomes apparent in the proof of Theorem 2.1, the choice of $W_ \varepsilon$ is not unique. The same result holds, for instance, if $W_ \varepsilon$ is replaced with the oscillating test function $w_ \varepsilon$ constructed in [13,Section 3] and in Subsection 3.2 of the present paper. The function $W_ \varepsilon$, however, has a simpler and more explicit construction that may be implemented numerically with more efficiency. It is, indeed, an oscillating test function basically restricted to the balls of $H^ \varepsilon$ that do not overlap and have radius smaller than $\varepsilon^{1+\delta}$. We emphasize that the condition on the minimal distance being at least $\varepsilon^2$ is purely technical and may be avoided by either assuming that the radii $\{\rho_z\}_{z\in \Phi}$ are (uniformly) bounded from below, or that their inverse $\rho_z^{-1}$ satisfies a suitable moment condition.

2.2.   Ideas of the proofs

The proof of Theorem 2.1 is inspired to the proof of the same result in the case of periodic holes shown in [18]. The latter, in turn, upgrades the result of [6] from the qualitative statement $u_ \varepsilon \rightharpoonup u$ in $H^1_0(D)$ to an estimate on the convergence of the homogenization error. Both arguments rely on the construction of suitable oscillating test functions $\{ w_ \varepsilon \}_{ \varepsilon> 0} \subseteq H^1(D)$. In the qualitative statement of [6], these functions allow to pass to the limit $\varepsilon \downarrow 0$ in the weak formulation of (1.1) and infer the homogenized equation (1.4).

The functions $\{ w_ \varepsilon \}_{ \varepsilon >0}$ may be constructed as $W_ \varepsilon$ in (2.7), where the set $\Phi^ \varepsilon_\delta$ coincides with the whole set $\Phi = \mathbb{Z}^d$ and the functions $\{ w_{ \varepsilon, z}\}_{z\in \Phi^ \varepsilon(D)}$ introduced in (2.6) satisfy $w_{ \varepsilon, z}(\cdot) = w_{ \varepsilon, 0}( \cdot + z)$. The additional term $C_0 = c_d r^{d-2}$ that appears in the homogenized equation (1.4) is the limit of the measures $-\Delta w_ \varepsilon$ when tested against the function $\rho u_ \varepsilon \in H^1_0(D^ \varepsilon)$, $\rho \in C^\infty_0(D)$. It is not hard to see from (2.7) that, for functions that vanish on the holes $H^ \varepsilon$, the action of $-\Delta w_ \varepsilon$ reduces the periodic measure

that is concentrated on the spheres $\{ \partial B_{\frac \varepsilon 4}( \varepsilon z) \}_{z \in \mathbb{Z}^d}$. Here, $\partial_n$ is the outer normal derivative on $\partial B_{\frac \varepsilon 4}( \varepsilon z)$.

In [18], the corrector $W_ \varepsilon$ is chosen as the oscillating test function $w_ \varepsilon$ itself. As a first step, it is shown that the decay of $\| u_ \varepsilon - W_ \varepsilon u \|_{H^1_0(D)}$ boils down to controlling the convergence of the density of capacity of $H^ \varepsilon$ to its limit $C_0$ (c.f. (1.5)). The latter is expressed in terms of the decay of the norm $\|\mu_ \varepsilon- C_0\|_{H^{-1}(D)}$, with $\mu_ \varepsilon$ as in (2.8). As a second step, the authors appeal to a result of [19] to estimate the decay of $\parallel {\mu _\varepsilon } - {C_0}_D{\parallel _{{H^{ - 1}}(D)}}$ in terms of the size $\varepsilon$ of the periodic cell $C_ \varepsilon: = [-\frac \varepsilon 2; \frac \varepsilon 2]$ of $\mu_ \varepsilon$. The crucial feature is that, up to a correction of order $\varepsilon^2$, the measure $\mu_ \varepsilon -C_0$ has zero average in $C_ \varepsilon$. In other words, we have

In this paper we adapt to the random setting the previous two-step argument. The first main difference is strictly related to the randomness of the radii in $H^ \varepsilon$ and needs to be addressed also in the case of bounded radii (i.e. if $\beta = +\infty$ in (2.1)) and periodic centres. In this case, the measure $\mu_ \varepsilon$ is defined as in (2.8) but, on each sphere $\partial B_{\frac \varepsilon 4}( \varepsilon z)$, the term $\partial_n w_ \varepsilon$ depends on the random mark $\rho_z$. Therefore, contrarily to the periodic case, (2.9) may not hold in each cube $\varepsilon z + C_ \varepsilon$. Nevertheless, by the Law of Large Numbers, we may expect that the average of $\mu_ \varepsilon - C_0$ is close to zero over cubes of size $k \varepsilon$, $k >>1$, as the left-hand side in (2.9) turns into an averaged sum of $k^d$ random variables. This motivates the introduction of a partition of the set $D$ into cubes of mesoscopic size $k \varepsilon$ (c.f. Section 3.1) that plays the role of the cells $\varepsilon z + C_ \varepsilon$ of the periodic case. This allows us to adapt the result by [19] and obtain

Here, the last term accounts for the difference between the average of $\mu_ \varepsilon$ in each cube of size $(k \varepsilon)$, $k \in \mathbb{N}$ and the value $C_0$. This inequality, implies an estimate of the form:

The optimal choice of $k$ yields the exponent $\frac{d}{d+2}$ of Theorem 2.1. If $\rho_z \equiv r$ for all $z \in \mathbb{Z}^d$, then the second term vanishes and the above estimate with $k = 1$ gives the optimal rate of [18].

In the case of centres distributed according to a Poisson point process, the argument for Theorem 2.1 follows the same ideas sketched above. Although the centres of the holes in $H^ \varepsilon$ have random positions, their typical distance is still of size $\varepsilon$. This feature gives rise to the additional logarithmic factor in the rate of Theorem 2.1. The main technical challenge is related to the construction of the mesoscopic partition of $D$ that allows to obtain the analogue of (2.10). In contrast with the case $\Phi = \mathbb{Z}^d$, indeed, there are ($\mathbb{P}$-many) realizations of $H^ \varepsilon$ where the support of the measure $\mu_ \varepsilon$ defined in (2.8) intersects the boundary of the covering. In other words, the spheres $\{ \partial B_{\frac \varepsilon 4}( \varepsilon z) \}_{z\in \Phi^ \varepsilon_\delta(D)}$ might fall across two cubes of size $\varepsilon k$ that cover $D$. This, in particular, implies that to the covering does not correspond to a well-defined partition of the spheres where the measure $\mu_ \varepsilon$ is supported. We tackle this issue by constructing a suitable random covering. We refer to Subsection 3.1 for the precise construction.

A second challenge that arises in the proof of Theorem 2.1 is related to the presence of overlapping holes in the case $\beta < +\infty$ in (2.1). The strategy to deal with this issue is very similar to the one used in [13]: We partition, indeed, the set of holes as $H^ \varepsilon = H^ \varepsilon_b \cup H^ \varepsilon_g$, where the subset $H^ \varepsilon_b$ contains all the holes that overlap (c.f. Lemma 3.1). As shown in [13], the contribution of $H^ \varepsilon_b$ to the density of capacity is negligible in the limit $\varepsilon \downarrow 0$. As a consequence, we may modify the estimates of [18], to prove that $\parallel u_ \varepsilon- W_ \varepsilon u \parallel_{H^1_0(D)}$ is controlled by the norm $\| \mu_ \varepsilon- C_0{\bf{1}}_D \|_{H^{-1}(D)}$, where the measure $\mu_ \varepsilon$ is now only related to the union of disjoint balls $H^ \varepsilon_g$.

3.   Proof of Theorem 2.1, $(a)$

3.1.   Partition of the holes $H^ \varepsilon$ and mesoscopic covering of $D$

This subsection contains some technical tools that will be crucial to prove the main result: The first one is an adaptation of [13] and provides a suitable way of dividing the holes $H^ \varepsilon$ between the ones that may overlap due to the unboundedness of the marks $\{ \rho_z\}_{z \in \Phi}$ and the ones that, instead, are disjoint and have radii $\varepsilon^\frac{d}{d-2} \rho_z$ much smaller than the distance $\varepsilon$ between the centres.

Lemma 3.1. Let $\delta \in (0, \frac{2}{d-2}]$ be fixed. There exists an $\varepsilon_0 = \varepsilon_0(\delta, d)$ such that for every $\varepsilon \leqslant \varepsilon_0$ and $\omega \in \Omega$ we may find a partition of the realization of the holes

with the following properties:

● There exists a subset of centres $n^ \varepsilon(D) \subseteq \Phi^ \varepsilon(D)$ such that

● There exists a set $D^ \varepsilon_b \subseteq \{ x \in \mathbb{R}^3 \, \colon \, \mathop{dist}(x, D) \leqslant 2\}$ satisfying

and

Proof of Lemma 3.1. The construction for the sets $H^ \varepsilon_g, H^ \varepsilon_b$ and $D^ \varepsilon_b$ is the one implemented in the proof of [13,Lemma 2.2]. We fix $\delta \in (0, \frac{2}{d-2}]$ throughout the proof.

We denote by $I_ \varepsilon^b \subseteq \Phi_ \varepsilon(D)$ the set that generates the holes $H_b^ \varepsilon$. We construct it in the following way: We first set $J^ \varepsilon_b: = \Phi^ \varepsilon(D) \backslash \Phi^ \varepsilon_\delta(D)$ with $\Phi^ \varepsilon_\delta$ as in (2.5). Since $\Phi = \mathbb{Z}^d$, this turns into

Given the holes $\tilde H^ \varepsilon_b : = \bigcup_{z \in J^ \varepsilon_b} B_{2 ( \varepsilon^\frac{d}{d-2} \rho_z \wedge 1)}( \varepsilon z)$, we include in $I_b^ \varepsilon$ also the set of points in $\Phi_ \varepsilon(D) \backslash J^ \varepsilon_b$ that are "too close'' to the set $\tilde H_b^ \varepsilon$, i.e.

We define

It remains to show that the sets defined above satisfy properties (3.1)-(3.3). Property (3.1) is an immediate consequence of definition (3.4). The first inclusion in (3.2) follows by the definition of $H^ \varepsilon_b$ and $D^ \varepsilon_b$ in (3.6); for the inequality in (3.2) we instead appeal to the subadditivity of the capacity to bound

Moreover, by the monotonicity property ${\rm{Cap}}(A; C) \leqslant {\rm{Cap}}(A; B)$ for every $A \subseteq B \subseteq C$, this turns into

i.e. the estimate in (3.2).

To conclude the proof of this lemma, it remains to argue (3.3): By construction (see (3.6)), it holds that

On the one hand, by (3.5) and the definition of $n^ \varepsilon(D)$ in (3.6), for each $z \in n^ \varepsilon(D)$ we have that

On the other hand, by (3.4) and (3.5), if $w \in \tilde I^ \varepsilon_b$, then $4 \varepsilon^\frac{d}{d-2} \rho_w \leqslant \varepsilon^{1+\delta}$ so that

whenever $\varepsilon$ is such that $\varepsilon^\delta < \frac 1 4$. Hence, also

Combining this with (3.8) and (3.7), we infer (3.3). The proof of Lemma 3.1 is complete.

We now construct a suitable covering of $D$ that, as explained in Subsection 2.2, plays a fundamental role in the proof of Theorem 2.1. We recall that, by our assumption, the set $D$ is any smooth domain that is star-shaped with respect to the origin.

For $z \in \mathbb{R}^d$, we define the "microscopic cubes''

while for $k \in \mathbb{N}$ and any $z \in \mathbb{Z}^d$ we set

Let $N_k \subseteq \mathbb{Z}^d$ be such that the collection $\{ Q_{k, z} \}_{z \in N_k}$ is an essentially disjoint covering of $D$. Since $D$ is bounded, we may assume that

Let

Since $D$ is smooth and has compact boundary, it is easy to see that there exist $C_1 = C_1(D)$ such that, whenever $k \varepsilon \leqslant C_1$ it holds

Finally, for each $z \in N_k$ we denote by $N_{k, z} \subseteq \Phi$ the set of points of $\Phi_\delta^ \varepsilon(D)$ that, when rescaled, are contained into the cube $Q_{k, z}$, i.e. such that

Note that, since in this section we assumed that $\Phi = \mathbb{Z}^d$, it follows that for every $z \in N_k$, we have $\bigcup_{w \in N_{k, z}} Q_{ \varepsilon, w} \subseteq Q_{k, z}$ where $Q_{ \varepsilon, w}$ are defined as in (3.9). Morevoer, for every $z \in \mathring{N}_{k, z}$, the collection $\{Q_{ \varepsilon, w}\}_{w \in N_{k, z}}$ provides a refinement of $Q_{k, z}$.

3.2.   Quenched estimates for the homogenization error

All the results contained in this subsection are quenched, in the sense that they hold for any fixed realization of the holes $H^ \varepsilon$. The main result of this section is Lemma 3.2 that allows to control the norm of the homogenization error $u_ \varepsilon- W_ \varepsilon u$ in terms of suitable averaged sums of the random marks $\{\rho_z \}_{z \in \Phi}$.

Before giving the statement of Lemma 3.2, we recall the construction of the oscillating test function $w_ \varepsilon \in H^1(D)$ implemented in [13]. As mentioned in the introduction and in Subsection 2.2, the main feature of this function is to vanish on the holes $H^ \varepsilon$ and "approximate'' the density of the capacity of $H^ \varepsilon$. We note that the unboundedness of the marks $\{ \rho_z\}_{z\in \Phi}$ implies that the set $\Phi^ \varepsilon_\delta(D) \subsetneq \Phi^ \varepsilon(D)$ and that the function $W_ \varepsilon$ in (2.7) does not vanish in all the holes contained in $H^ \varepsilon$.

Let $H^ \varepsilon_g, H^ \varepsilon_b$ and $D^ \varepsilon_b$ be as in Lemma 3.1. For every $z \in \Phi^ \varepsilon(D)$, let²

² We assume that the minimizer exists. If this is not the case, it suffices to take $v_ \varepsilon$ in the minimizing class such that $\int_{D_b^ \varepsilon} |\nabla v_ \varepsilon |^2 \leqslant 2 {\rm{Cap}}( H^ \varepsilon_b; D^ \varepsilon_b)$.

We pick as oscillating test function

where $w_ \varepsilon^g$ and $w_ \varepsilon^b$ are defined as follows:

and

For each $z \in n_ \varepsilon(D)$, the function $w_{z, \varepsilon}$ is as in (2.6). We remark that each $w_{ \varepsilon, z}$ admits the explicit formulation

For $k \in \mathbb{N}$, let $\{ Q_{k, z}\}_{z\in N_k}$ be the covering of $D$ constructed at the end of Subsection 3.1. For every $z \in N_k$, we define the random variables

Lemma 3.2. Let $\delta \in (0, \frac{2}{d-2}]$ be fixed. Then for every $\varepsilon >0$ and $k \in \mathbb{N}$ with $k \varepsilon \leqslant 1$ the following inequality holds: If $u_ \varepsilon, u$ are as in Theorem 2.1 and $W_ \varepsilon$ as in $(2.7)$, then

Lemma 3.2 relies on the next lemma, that is an adaptation of [18][Theorem 3.2] and shows that controlling the error $u_ \varepsilon- W_ \varepsilon u$ considered in Theorem 2.1 boils down to controlling the convergence to $C_0$ of the density of capacity generated by $H^ \varepsilon$.

Lemma 3.3. Let $\delta \in (0; \frac{2}{d-2}]$ be fixed; let $u_ \varepsilon, u, w_ \varepsilon$ and $W_ \varepsilon$ be as in Lemma 3.2. Let $w_ \varepsilon$ be as in $(3.15)$. Then

with

Proof of Lemma 3.2. The statement follows from Lemma 3.3, provided that we show that

and that for every $\varepsilon >0$ and $k \in \mathbb{N}$ such that $k \varepsilon \leqslant 1$

We first argue (3.21): By definition (3.16) for $w_b^ \varepsilon$ and Lemma 3.1, we have that

Since by Lemma 3.1 the sets $\bigcup_{z \in n^ \varepsilon(D)} B_{\frac \varepsilon 4}( \varepsilon z)$ and $D_b^ \varepsilon$ are disjoint, we appeal to (3.15) to estimate

The function $w_ \varepsilon^g-1$ vanishes on $\bigcup_{z \in n_ \varepsilon(D)} \partial B_{\frac \varepsilon 4}( \varepsilon z )$: Since the balls $\{ B_{\frac{ \varepsilon}{4}}( \varepsilon z)\}_{z \in n^ \varepsilon(D)}$ are all disjoint, Poincaré's inequality in each ball $B_{\frac \varepsilon 4}( \varepsilon z)$ yields

Using definitions (3.17), (2.6) and property (3.1) of Lemma 3.1, we may rewrite

and, inserting this into (3.24), also

To conclude the proof of (3.21) for $w_ \varepsilon -1$, it thus remains to estimate the last term on the right-hand side. By construction (c.f. (3.16)), it holds $w_ \varepsilon^b -1 = 0$ on $\partial D_ \varepsilon^b$; appealing to Lemma 3.1, we also have that $D^ \varepsilon_b \subseteq \{ x \in \mathbb{R}^3 \, \colon \, \mathop{dist}(x, D) \leqslant 2 \}$. We thus apply Poincaré's inequality in this set and conclude that

To establish (3.21) for $w_ \varepsilon -1$, it only remains to combine this last inequality with (3.25).

We now argue (3.21) for $\nabla(w_ \varepsilon - W_ \varepsilon)$: By definition (2.5) and (3.1) of Lemma 3.1, it holds

Thanks to definition (3.15) for $w_ \varepsilon$ and the fact that, by Lemma 3.1 the support of $\nabla w_g^ \varepsilon$ and $\nabla w_b^ \varepsilon$ is disjoint, we use the triangle inequality to infer that

Comparing definition (3.17) for $w_g^ \varepsilon$ with definition (2.7) for $W_ \varepsilon$ and using inclusion (3.26), we observe that

Since the balls $\{ B_{\frac \varepsilon 4}( \varepsilon z) \}_{z \in \Phi^ \varepsilon_\delta(D)}$ are disjoint, the previous identity and the triangle inequality imply that

Inserting this bound into (3.27) yields (3.21) also for the norm of $\nabla(w_ \varepsilon- W_ \varepsilon)$.

We now turn to (3.22) and claim that we may apply Lemma 5.1 with $M = \mu_ \varepsilon$, $\mathcal{Z} = \{ \varepsilon w \}_{w \in \Phi^ \varepsilon_\delta(D)}$, $\mathcal{X} = \{ \varepsilon^\frac{d}{d-2} \rho_w\}_{w \in \Phi^ \varepsilon_\delta(D)}$ and $r_w\equiv \frac \varepsilon 4$ for every $w \in \Phi^ \varepsilon_\delta(D)$. We use as covering $\{K_j\}_{j \in J}$ the sets $\{ Q_{k, z}\}_{z \in N_k}$. Conditions (5.1) and (5.3) are satisfied thanks to (2.5) and by construction (see Subsection 3.1), respectively. Appealing to Lemma 5.1, we therefore have that

By the triangle inequality and the previous estimate, we thus bound

so that, to prove (3.22), it only remains to control the last term on the right-hand side above. We do this by observing that, since $\{ Q_{k, z}\}_{z \in N_k}$ is a disjoint covering of $D$, for each $\phi \in H^1_0(D)$ we have

By the triangle inequality, also

We claim that

This is an easy consequence of the properties of the covering $\{ Q_{k, z} \}_{z \in N_k}$ of $D$, (3.11), together with Cauchy-Schwarz's inequality and Poincaré's inequality for $\phi$ in $D$.

We now turn to the second term in (3.29). We note that, by definition (3.12), the set

Since $\phi \in H^1_0(D)$ and $D$ is a smooth and bounded set, we may appeal to Poincaré's inequality in the previous set on the right-hand side to bound

Appealing once again to Cauchy-Schwarz's inequality and using the above estimate, we control

Hence, provided $k \varepsilon \lesssim 1$, we may appeal to (3.13) and infer that

Combining this with (3.30) and (3.29) allows us to infer that for every $\phi \in H^1_0(D)$

or, equivalently, that

This, together with (3.28), establishes (3.22). The proof of Lemma 3.2 is complete.

Proof of Lemma 3.3. The argument for this lemma is very similar to the one of [18,Theorem 3.1]. Since $f \in W^{1, p}$, $p > d$ and since $D$ is smooth, by standard elliptic regularity we infer that the solution $u$ of (1.4) satisfies $u \in W^{2, \infty}(D)$. By computing the (distributional) Laplacian of $u_ \varepsilon - w_ \varepsilon u$ we obtain that in $D^ \varepsilon$

We now smuggle the term $(-\Delta W_ \varepsilon) u \in H^{-1}(D)$ in the right-hand side so that the previous identity turns into

We stress that, since $u \in W^{2, +\infty}(D)\cap H^1_0(D)$, $u_ \varepsilon \in H^1_0(D^ \varepsilon)$, $w_ \varepsilon \in H^1(D)$, the above equation holds in the sense that for every $\phi \in H^1_0(D_ \varepsilon)$

Since the balls $\{ B_{\frac \varepsilon 4}( \varepsilon z)\}_{z \in \Phi_\delta^ \varepsilon(D)}$ are all mutually disjoint, by definition (2.7) and equations (2.6) we have that

Since $\phi \in H^1_0(D^ \varepsilon)$ and therefore it vanishes on the spheres $\{\partial B_{ \varepsilon^\frac{d}{d-2}\rho_z}( \varepsilon z)\}_{z \in \Phi_\delta^ \varepsilon(D)}$, the above identity implies that

Inserting this last identity in (3.32), we infer that

We now choose $\phi = u_ \varepsilon- w_ \varepsilon u$ and apply Hölder's and Poincaré's inequalities to bound

To obtain the claim of Lemma 3.3 it remains to use that, by the triangle inequality and Hölder's inequality, we have

and that, by definitions (2.7) and (3.15), the difference $W_ \varepsilon- w_ \varepsilon$ is compactly supported in $\{ x \in \mathbb{R}^d \, \colon \, \mathop{dist}(x; \partial D) \leqslant 4 \}$ (see also Lemma 3.1).

3.3.   Annealed estimates (Proof of Theorem 2.1, $(a)$)

In this subsection we rely on the quenched estimate of Lemma 3.2 to prove the statement of Theorem 2.1 in the case of periodic centres. The first ingredient is the following annealed bound:

Lemma 3.4. Let $(\Phi , \mathcal{R})$ satisfy the assumptions of Theorem 2.1, $(a)$. For every $\delta \in (0, \frac{2}{d-2}]$, let $n^ \varepsilon(D) \subseteq \Phi^ \varepsilon(D)$ be the random subset constructed in Lemma 3.1. Then

Proof of Theorem 2.1, $(a)$. By the assumptions on $D$, we may assume that for $\varepsilon > 0$ and $k \in \mathbb{N}$ such that $\varepsilon k \lesssim 1$, the cube $Q_{k, 0} \subseteq D$. We restrict to the values of $k\in \mathbb{N}$ satisfying the previous bound.

Combining Lemma 3.2 and Lemma 3.4, we bound for every $\varepsilon> 0$ and $k\in \mathbb{N}$ as above

Since the sets $N_k, \mathring{N}_k$ are deterministic and $\{ S_{k, z}\}_{z \in \mathring{N}_k}$ are identically distributed, we infer that

We observe that, by (2.5), (2.1) and the inequality $N^ \varepsilon(D) \lesssim \varepsilon^{-d}$ (c.f. (2.3)), we have that

Since $k \geqslant 1$, the previous two displays thus imply

We now claim that for every $k \in \mathbb{N}$ such that $\varepsilon k \leqslant C_1$ with $C_1$ as in (3.13), then

where $Y_{ \varepsilon, 0}$ is defined as in (3.19). We begin by showing how to conclude the proof of the theorem provided (3.34) holds.

Let us first assume that (2.1) holds with $\beta \geqslant d-2$; in this case, we have that

and therefore that

Estimate of Theorem 2.1 for $\beta \geqslant d-2$ follows from this inequality if we minimize the right-hand side above in $k$, i.e. if we choose $k = \lfloor \varepsilon^{-\frac{2}{d+2}} \rfloor$, and set $\delta$ as in Theorem 2.1.

Let us now assume that $\beta < d-2$ in (2.1): In this case, we bound

so that (3.34) turns into

Also in this case, we infer the estimate of Theorem 2.1 by minimizing the right-hand side in $k$ and $\delta$, i.e. choosing $k = \lfloor \varepsilon^{-\frac{2}{d+2}} \rfloor$ and $\delta$ as in Theorem 2.1.

To complete the proof of the theorem it only remains to argue (3.34) from (3.33). We first tackle the second term on the right-hand side of (3.33) and show that

This may be done after noticing that the left-hand side may be written, up to an error, as the sum of $(2k +1)^d$ centred and independent random variables: Definitions (2.5) and (3.14) for $\Phi^ \varepsilon_{\delta}(D)$ and $N_{k, z}$ imply that

Since $0 \in \mathring{N}_k$, this, (3.19) and the triangle inequality allows us to bound

Thanks to Chebyshev's inequality and assumption (2.1) we have

and thus we may rewrite (3.37) as

Since

the independence of the random variables $\{\rho_z \}_{z \in \Phi}$, and the fact that $N^ \varepsilon(Q_{k, 0}) = (2k+1)^{d}$ (c.f. (2.3)), allows us to obtain (3.35), by means of standard CLT arguments.

We now turn to the remaining term in (3.33) and argue that

By the triangle inequality and assumption (2.1), the left-hand side is bounded by

To establish (3.39) from this it suffices to remark that, by (3.19) and (3.36), we have

so that this, and the fact that the random variables $\{ \rho_z \}_{z \in \Phi^ \varepsilon(D)}$ are identically distributed, yields

Inserting this into (3.40) implies (3.39). To establish (3.34) it remains to combine (3.39), (3.35) and (3.33). The proof of Theorem 2.1, $(a)$ is complete.

Proof of Lemma 3.4. We resort to the construction of the set $n^ \varepsilon(D)$ implemented in Lemma 3.1: By (3.6), (3.4) and (3.5) in the proof of Lemma 3.1 we decompose

and prove the statement of Lemma 3.4 for each one of the two sums. We begin with the first one: Using (3.4) we write

Taking the expectation and using that $\{ \rho_z\}_{\Phi^ \varepsilon(D)}$ are identically distributed and that $N^ \varepsilon(D) \lesssim \varepsilon^{-d}$, we immediately bound

i.e. the claim of Lemma 3.4 for the first sum in (3.41).

We now turn to the second sum in (3.41): By definition (3.5), if $z \in \tilde I_b^ \varepsilon$, then $\rho_z \leqslant \varepsilon^{-\frac{2}{d-2} +\delta}$ and there exists an element $w \in J_b^ \varepsilon$ such that $\varepsilon |z -w| \lesssim \varepsilon + ( \varepsilon^\frac{d}{d-2}\rho_w \wedge 1)$. This allows us to bound

We now take the expectation and use that $\Phi = \mathbb{Z}^d$ and that $\{\rho_z \}_{z\in \Phi}$ are independent and identically distributed: This implies that

Since for every $w \in J^ \varepsilon_b$, the set

we obtain that

This, together with identities (3.41) and (3.42), establishes Lemma 3.4.

4.   Proof of Theorem 2.1, $(b)$

In this section we adapt the argument of the previous section to Theorem 2.1 in case $(b)$. As mentioned in Subsection 2.2, the main challenge is related to the construction of a mesoscopic covering $\{ K_{k, z} \}_{z \in N_k}$ that plays the same role of $\{ Q_{k, z} \}_{z \in N_k}$ of Subsection 3.1. In the present case, the random positions of the centres imply that (with positive probability) there are configurations in which some of the spheres $\{ \partial B_{\frac \varepsilon 4}( \varepsilon z)\}_{z\in \Phi}$ intersect the boundary of $\{ Q_{k, z} \}_{z \in N_k}$. This prevents us from appealing to Lemma 5.1 as condition (5.3) is violated.

All the results contained in this section besides hold for any dimension $d \geqslant 3$. However, in the proof of Theorem 2.1, $(b)$ we obtain the same decay rate of case $(a)$ only in $d = 3$. In higher dimensions we obtain a slower (but still algebraic) rate. In order to best appreciate this dimensional constraint, in the whole section we work in a general dimension $d \geqslant 3$.

Throughout this section we set $\delta$ as in Theorem 2.1 and define the parameters

4.1.   Partition of the holes and mesoscopic covering of $D$

This subsection contains an adaptation to the case of random centres of Lemma 3.1 and of the sets $\{ Q_{k, z}\}_{z \in N_k}$.

Lemma 4.1. Let $\delta$ be as in Theorem 2.1. We recall the definition $(2.4)$ of $R_{ \varepsilon, z}$. For $\omega \in \Omega$, we consider a realization of the marked point process $(\Phi; \mathcal{R})$ and of the associated set of holes $H^ \varepsilon$. Then, there exists a partition

with the following properties:

● There exists a subset of centres $n^ \varepsilon(D) \subseteq \Phi^ \varepsilon(D)$ such that

and such that $2 \sqrt d \varepsilon^\frac{d}{d-2} \rho_z \leqslant R_{ \varepsilon, z}$, for every $z \in n^ \varepsilon(D)$.

● There exists a set $D^ \varepsilon_b(\omega) \subseteq \{ x \in \mathbb{R}^d \, \colon \, \mathop{dist}(x, D) \leqslant 2\}$ satisfying

and for which $B_{R_{ \varepsilon, z}}( \varepsilon z) \cap D^ \varepsilon_b = \emptyset$, for every $z \in n^ \varepsilon(D)$.

Proof of Lemma 4.1. The construction for the sets $H^ \varepsilon_g, H^ \varepsilon_b$ and $D^ \varepsilon_b$ is analogous to the one implemented in the proof of Lemma 3.1, with the only difference that in this case, we set

and in the definition (3.5) of $\tilde I_b^ \varepsilon$ the ball $B_{\frac \varepsilon 4}( \varepsilon z)$ is replaced by $B_{R_{ \varepsilon, z}}( \varepsilon z)$.

For $k$ as in (4.1), let $\{ Q_{k, z} \}_{z\in N_k}$ be as in Subsection 3.1. For every $z \in N_k$ we define the sets $N_{k, z}$ as in (3.14). We stress that, in this case, (3.14) is ill-defined for the realizations of $\Phi$ such that there are points in $\Phi^ \varepsilon_\delta(D)$ that fall on the boundary of the cubes $\{Q_{k, z}\}_{z \in N_k}$. This issue may be easily solved by fixing a deterministic rule to assign these points to a particular cube that shares the boundary considered. We stress that all the arguments of this section do not depend on this rule since the set of the boundaries of the covering $\{Q_{k, z}\}_{z \in N_k}$ has zero (Lebesgue)-measure.

For $z\in N_k$ and $w \in N_{k, z}$, we define the modification of the minimal distance $R_{ \varepsilon, w}$:

We aim at obtaining a (random) collection of disjoint sets $\{ K_{k, z}\}_{z\in N_k}$ having size $\simeq \varepsilon (2k +1)$ and such that for every $z \in N_k$ and $w \in \Phi^ \varepsilon_\delta(D)$

We modify $\{Q_{k, z}\}_{z \in N_k}$ as follows: For $\kappa$ as in (4.1), any $z \in N_k$ and $w \in N_{k, z}$, we consider the cubes

Note that, by definition (2.4), all the cubes above are disjoint. For every $z\in N_k$, we thus set (see Figure (1))

Since the cubes $\{ Q_{ \varepsilon, z} \}_{z \in \Phi_\delta^ \varepsilon(D)}$ are disjoint we have that

We emphasize that the previous properties hold for every realization of the point process $\Phi$. The introduction of the modified random variable $\tilde R_{ \varepsilon, z}$ is needed to ensure that the second property in (4.4) holds with $\varepsilon^\kappa$ instead of $1$. This yields that the difference between the volume of the set $K_{k, z}$ and the cube $Q_{k, z}$ is of order $\varepsilon^{d +\kappa} k^{d-1}$ instead of $\varepsilon^d k^{d-1}$. This condition plays a crucial role in the proof of the theorem (see (4.14)) and is the main term that forces the dimensional constraint $d = 3$ in the rates of convergence.

4.2.   Quenched estimates for the homogenization error

In this section we adapt Lemma 3.2 to the current setting. As in the case of Lemma 3.2, the next result relies on a variation of Lemma 3.3 that allows us to replace in the definition (3.20) of $\mu_ \varepsilon$ the radii $\frac{ \varepsilon}{4}$ with $\tilde R_{ \varepsilon, z}$ defined in (4.2).

We define the oscillating test function $w_ \varepsilon \in H^1(D)$ as done in Subsection 3.2, this time using the sets $H^ \varepsilon_b, H^ \varepsilon_g$ and $D^ \varepsilon_b$ of Lemma 4.1 with $\delta$ as in Theorem 2.1, and $R_{ \varepsilon, z}$ instead of $\frac \varepsilon 4$ in (3.17). We also define the analogues of (3.19), this time associated to the covering $\{ K_{k, z} \}_{z \in N_k}$ constructed in the previous subsection: For every $z \in N_k$ we indeed set

Lemma 4.2. Let $W_ \varepsilon$ be as in $(2.7)$ and let $u_ \varepsilon, u$ as in Theorem 2.1. Then, for every $\varepsilon >0$ and $k \in \mathbb{N}$ such that $\varepsilon k \lesssim 1$ we have that

Lemma 4.3. Let $u_ \varepsilon, u$ and $W_ \varepsilon$ be as in Lemma 3.2 and let $w_ \varepsilon$ be as defined above. Then

where $\tilde W_ \varepsilon$ is defined as in $(2.7)$ with $R_{ \varepsilon, z}$ substituted by $\tilde R_{ \varepsilon, z}$. Furthermore, in this case

with $\tilde w_{ \varepsilon, z}$ as in $(2.6)$ with $\tilde R_{ \varepsilon, z}$ instead of $R_{ \varepsilon, z}$.

Proof of Lemma 4.2. Analogously to the proof of Lemma 3.2, we appeal to Lemma 4.3 and reduce to showing that

and

Inequality (4.6) may be argued exactly as done for (3.21) in the proof of Lemma 3.2, this time appealing to Lemma 4.1 instead of Lemma 3.1.

We thus turn to (4.7). We begin by remarking that $\tilde W_{ \varepsilon}$ is well-defined: Indeed, by definition (2.4) and (4.5), we have that $\tilde R_{ \varepsilon, z} \leqslant R_{ \varepsilon, z} \leqslant \frac \varepsilon 4$ for every $z \in \Phi^ \varepsilon_\delta(D)$. Furthermore, since $\kappa < \delta$ (c.f. (4.1) and Theorem 2.1), it follows from (2.5) that $2 \varepsilon^\frac{d}{d-2} \rho_z \leqslant \tilde R_{ \varepsilon, z}$, for every $z\in \Phi^ \varepsilon_\delta(D)$. Therefore, comparing the two definitions of $W_ \varepsilon$ and $\tilde W_ \varepsilon$, we use (4.2) to bound:

Since, if $R_{ \varepsilon, w} \neq \tilde R_{ \varepsilon, w}$, then $\varepsilon^{1+\kappa} \leqslant \tilde R_{ \varepsilon, w} \leqslant R_{ \varepsilon, w}$, we have that

Appealing to (2.7), (2.6) and the adaptation of (3.18) for both $\tilde W_ \varepsilon$ and $W_ \varepsilon$, the previous integrals may be bounded by

Since $w \in \Phi^ \varepsilon_\delta(D)$, we have that $\rho_w \leqslant \varepsilon^{-\frac{2}{d-2}+\delta}$ so that

Inserting this into (4.9) and appealing to (4.1) for $\kappa$ yields (4.7).

We finally tackle (4.8): As done for (3.22) of Lemma 3.2, we aim at applying Lemma 5.1. We thus pick $\mathcal{Z} = \Phi^ \varepsilon_\delta(D)$ and $\mathcal{X} = \{ \varepsilon^\frac{d}{d-2} \rho_z \}_{z \in \mathcal{Z}}, \mathcal{R}: = \{ \tilde R_{ \varepsilon, z}\}_{z \in \mathcal{Z}}$. As shown above before the argument for (4.7), condition (5.1) is satisfied. Moreover, thanks to (4.3), the collection $\{ K_{k, z} \}_{z \in N_k}$ satisfies (5.3). Hence, by Lemma 5.1, we have that

where, thanks to (4.5), we have that $m_k \stackrel{}{ = } c_d \sum_{z \in N_k} S_{k, z} {\bf{1}}_{K_{k, z}}$. By the triangle inequality it thus only remains to control the norm $\| m_k - C_0 \|_{H^{-1}(D)}$. Using (4.4), this may be done exactly as in the proof of Lemma 3.2. The proof of Lemma 4.2 is complete.

Proof of Lemma 4.3. This lemma may be argued as done for Lemma 3.3. The only difference is that, in (3.31), we smuggle in $-\Delta \tilde W_ \varepsilon$ instead of $-\Delta W_ \varepsilon$ and apply the triangle inequality to bound $\| \nabla (\tilde W_ \varepsilon- w_ \varepsilon )\|_{L^2} \leqslant \| \nabla (W_ \varepsilon- w_ \varepsilon )\|_{L^2} + \| \nabla (\tilde W_ \varepsilon- W_ \varepsilon )\|_{L^2}$.

4.3.   Annealed estimates (Proof of Theorem 2.1, $(b)$)

As in case $(a)$, the next lemma provides annealed bounds for some of the quantities appearing in the right-hand side of Lemma 4.2.

Lemma 4.4. Let $n^ \varepsilon(D) \subseteq \Phi^ \varepsilon(D)$ the (random) subset constructed in Lemma 4.1. Then

Proof of Theorem 2.1, $(b)$. We recall that $k$ satisfies (4.1). Combining Lemma 4.2 and Lemma 4.4, we bound

As done in the proof of Theorem 2.1 (a), this also turns into

We now claim that, thanks to (4.1), the previous estimate reduces to

If the previous estimate holds, by the choice of $\delta$ and (4.1), we infer that

which establishes Theorem 2.1, $(b)$ if $d = 3$.

To conclude the proof, we only need to obtain (4.11) from (4.10). The sum over $z \in N_k \backslash \mathring{N}_k$ may be treated as in (3.39). We thus obtain inequality (4.11) provided that

and

We argue (4.13): Recalling the definition of the covering $\{ Q_{k, z} \}_{z \in N_k}$, we decompose

and rewrite

Since the process $(\Phi , \mathcal{R})$ is stationary, we bound

Let us partition the cube $Q_{k, 0}$ into $(2k+1)^{d}$ cubes of size $\varepsilon$ and let $Q$ be as in (3.10); the definitions of $\Phi^ \varepsilon_\delta(D)$ and $\tilde R_{ \varepsilon, z}$ (c.f. (2.5), (4.5)) and the stationarity of $(\Phi, \mathcal{R})$ imply that

We now apply Lemma 5.2 with $G((x, \rho); \omega) = \bigl(\frac{ \varepsilon}{R_{ \varepsilon, x}}\bigr)^d {\bf{1}}_{R_{ \varepsilon, x} \geqslant \varepsilon^2} \rho^{2(d-2)} {\bf{1}}_{\rho< \varepsilon^{-\frac{2}{d-2} +\delta}}$ to infer that

We rewrite

We claim that

This inequality, indeed, is obtained by decomposing the indicator function above into

and using that, by definition (2.4), it holds

where $B_r(0)$ is the ball of radius $r>0$ centred at the origin. Hence,

Inserting this into (4.17) we obtain that for $\varepsilon \lesssim 1$

Thus, inequality (4.15) turns into

This yields (4.13) provided that

Let $Q_{r}$ be the cube of size $r>0$ centred at the origin. Using (2.5) we bound

Since we may decompose the set $Q_k \backslash Q_{k-1}$ into $\lesssim k^{d-1}$ unitary cubes, we use again the stationarity of $(\Phi; \mathcal{R})$ and infer that

where $Q_1$ is any unitary cube that is contained in $Q_k \backslash Q_{k-1}$³. We now decompose $Q_1 = \sum_{n = 1}^{ \lceil -\kappa\log \varepsilon \rceil}A_n$ with

³ In this last step one should distinguish between unitary cubes according to the number of faces that they share with $\partial Q_k$. However, the argument shown below is immediately adapted to any of the previous cubes.

and use (4.2) to rewrite

We now appeal again to Lemma 5.2 as for (4.16) to reduce to

Arguing as for (4.18) and using the stationarity of $\Phi$ we infer that

To establish (4.19) it only remains to combine the previous inequality with (4.20) and use (4.1). The proof of (4.13) is therefore complete.

Inequality (4.12) may be obtained in a similar way as to that of (4.13): Since we may decompose the set $\bigcup_{z \in N_k} Q_{k, z}\backslash Q_{k-1, z}$ into $n \lesssim ( \varepsilon k)^{-d} k^{d-1}$ disjoint cubes $\{ Q_{ \varepsilon, i} \}_{i = 1}^n$ of size $\varepsilon$, we use definition (2.5) and the stationarity of $(\Phi, \mathcal{R})$ to bound

so that, again by Lemma 5.2, we obtain

We establish (4.12) after observing that, thanks to (4.1), it holds $k^{-1} \varepsilon^{\frac{2d}{d+2}} \leqslant ( \varepsilon k)^2$.

We now tackle (4.14): As in the proof of case $(a)$, we may assume that for $\varepsilon \lesssim 1$ the set $Q_{0, k} \subseteq D$. By construction (see definition (4.3)), the (random) set $K_{ \varepsilon, 0}$ satisfies

where $Q_k$ is, as above, the cube of size $2k+1$ centred at the origin. Hence, decomposing $Q_k = \bigcup_{i = 1}^{(2k+1)^d}Q_i$ into unitary cubes, definitions (4.5) and (2.5) allow us to rewrite

with

We rewrite

so that the triangle inequality, assumption (2.1) and the quenched bounds in (4.4) yield

Appealing to definitions (4.21), (4.5) and (2.4), we observe that $Z_i$ and $Z_j$ are independent whenever $i, j$ are such that $Q_{j}$ and $Q_{i}$ are not adjacent. We emphasize that they are not identically distributed as in definition (4.21) the random variables $Y_{ \varepsilon, z}$ contain the modified radii $\tilde R_{ \varepsilon, z}$ (see (4.5) and (4.2)). Hence, by taking the expectation in the previous inequality, we estimate

To establish (4.14) from (4.22) it suffices to bound

We emphasize that in the first bound the right-hand side may be bounded by $\lesssim \mathbb{E}\bigl[ \rho^{2(d-2)} {\bf{1}}_{\rho < \varepsilon^{-\frac{2}{d-2}+\delta}}\bigr]$.

Inequality (4.23) follows from Cauchy-Schwarz's inequality, the triangle inequality and definitions (4.21) and (4.5). We thus turn to (4.24) and fix $i = 1, \cdots, (2k+1)^d$. Since by definition (4.5) it holds

we use this and (4.21) to rewrite

Observing that $\mathbb{E} \bigl[ \sum_{z\in \Phi(Q_i)} \rho_z^{d-2} \bigr] = \lambda \mathbb{E} \bigl[ \rho^{d-2} \bigr]$, and writing

we infer that

We now appeal to Lemma 5.2 with $G((x, \rho); \omega) = \rho {\bf{1}}_{\rho \leqslant \varepsilon^{-\frac{2}{d-2}+\delta}} {\bf{1}}_{R_{ \varepsilon, x} \leqslant \varepsilon^\frac{d}{d-2} \rho \vee \varepsilon^2}$ to infer that

By the properties of the Poisson point process and definition (2.4) this yields

Hence

To establish (4.24) it remains to bound the last term above by $\varepsilon^{(\frac{2}{d-2}-\delta)\beta}$: By stationarity we have

We now observe that if $Q_i \in Q_{k-1}$, then by (4.2) we have that $\tilde R_{ \varepsilon, z} = R_{ \varepsilon, z}$: In this case, the expectation on right-hand side above may be bounded similarly to (4.18) so that

If, otherwise, $Q_i \in Q_{k}\backslash Q_{k-1}$, then (4.2) and a decomposition of $Q_i$ similar to the one performed in (4.20) implies that

Using that, if $d > 3$, we have $\kappa < \frac{2}{d-3}$ (c.f. (4.1)) and dealing with the remaining expectation as done in (4.26), yields that

Combining (4.27) and (4.26) with (4.25) implies (4.24) and, in turn, (4.14). The proof of Theorem 2.1 is complete.

Proof of Lemma 4.4. The proof of this lemma follows the same lines of the argument for Lemma 3.4. We resort to the construction made in Lemma 4.1 (c.f. (3.6)) to decompose

The expectation of the first sum may be bounded by by $\varepsilon^{(\frac{2}{d-2}-\delta)\beta} + \varepsilon^d$ by arguing in a way analogue to the one for (3.42) in Lemma 3.4. In this case, besides (2.1), we also appeal to assumption $(ii)$ and to the properties of the Poisson point process. Hence, it only remains to estimate the last sum in (4.28). As done for the same sum in (3.41), we use the definiton of $n^ \varepsilon(D)$ and the stationarity of $(\Phi, \mathcal{R})$ to rewrite

By Lemma 5.2 applied to

we infer that

Since the marks $\{\rho_z\}$ are identically distributed and independent, we use (2.1) to bound

Since $\mathbb{E}_\Phi\bigl[ \# \{ z \in \Phi^ \varepsilon(D) \backslash\{ 0 \} \, \colon \, \varepsilon |z| \leqslant \varepsilon + \varepsilon^\frac{d}{d-2} \rho \wedge 1 \} \bigr] \lesssim (\rho^{d-2} + 1)$, the term corresponding to the first sum on the right-hand side above is easily bounded by

We now turn to the second term in (4.29): Since this term reduces to the values $\rho < \varepsilon^{-\frac{2}{d-2} +\delta}$, we have that

Using Hölder's inequality with exponents {$\frac{d}{d-1}$ and $d$ in the inner expectation, definition (2.4) and the fact that $\Phi$ is a Poisson point process, this implies that

We control the last term by

Since $d \geqslant 3$ and $\delta \leqslant \frac{2}{d-2}$, the last term on the right-hand side above is bounded by $\varepsilon^2$. Combining this with (4.31), (4.29) and (4.30) yields $\mathbb{E} \bigl[ \varepsilon^d \sum_{z \in \tilde I^ \varepsilon_b} \rho_z^{d-2} \bigr] \lesssim \varepsilon^{(\frac{2}{d-2} - \delta)\beta}+ \varepsilon^2$. This concludes the proof of Lemma 4.4.

5.   Auxiliary results

Let $\mathcal{Z}: = \{ z_i \}_{i \in I} \subseteq D$ be a collection of points and let $\mathcal{X}: = \{ X_i \}_{i\in I}, \mathcal{R}: = \{ r_i \}_{i\in I} \subseteq \mathbb{R}_+$. We assume that

We define the measure

where each $v_i \in H^1(B_{r_i}(z_i))$ solves (2.6) with $B_{ \varepsilon \rho_z}( \varepsilon z)$ and $B_{R_{ \varepsilon, z}}( \varepsilon z)$ replaced by $B_{X_i}(z_i)$ and $B_{r_i}(z_i)$, respectively.

The next lemma is a generalization of the result by [19] used in [18] to show the analogue of Theorem 2.1 in the case of periodic holes $H_ \varepsilon$.

Lemma 5.1. Let $\mathcal{Z}$, $\mathcal{X}$ and $\mathcal{R}$ be as above and let $M$ be as in $(5.2)$. Then, there exists a constant $C = C(d)< +\infty$ such that for every Lipschitz and (essentially) disjoint covering $\{ K_j\}_{j \in J}$ of $D$ such that

we have that

with

Here, the constant $c_d$ is as in (1.5).

The next result is a consequence of the assumptions (i)-(iii) on the marked point process $(\Phi, \mathcal{R})$. Since it is used extensively in the proof of Theorem 2.1, in the sake of a self-contained presentation, we give below the statement and its brief proof. Let $(\Omega, \mathcal{F}, \mathbb{P})$ the underlying probability space for $(\Phi, \mathcal{R})$. Let $G: \mathbb{R}^d \times \mathbb{R}_+ \times \Omega \to \mathbb{R}$ be a stationary random field. This means that for almost every $(x, \rho) \in \mathbb{R}^d \times \mathbb{R}_+$, the expectation $\mathbb{E}\bigl[ G( (x, \rho); \omega)\bigr] = \mathbb{E}\bigl[ G((0, \rho); \omega )\bigr]$.

Lemma 5.2. Let $A \subseteq \mathbb{R}^d$ be bounded and such that $0\in A$. Let $(\Phi; \mathcal{R})$ satisfy (i)-(iii) with $\Phi = \mathop{Poi}(\lambda)$. For every $\varepsilon > 0$ and $x \in \mathbb{R}^d$, let $R_{ \varepsilon, z}$ be as in $(2.4)$. Then for every stationary $G: \mathbb{R}^d \times \mathbb{R}_+ \times \Omega \to \mathbb{R}$ it holds

Proof of Lemma 5.1. With no loss of generality, we give the proof for $d = 3$. We start by remarking that, thanks to (5.3), we may rewrite the measure $M$ in (5.2) as

By the definition of the capacitary functions $\{ v_i \}_{i\in I}$ (see also (3.18)), $m$ in (5.4) satisfies

For every $j\in J$, we thus define $q_j \in H^1(K_j)$ as the (weak) solution to

We stress that $q_j$ exists since $K_{j}$ is a Lipschitz domain and, thanks to (5.3) and (5.5)-(5.6), the compatibility condition $\langle M_j ; 1 \rangle - \int_{K_j} m_j = 0$ is satisfied.

By (5.7) and (5.5)-(5.6), we thus have that

The statement of Lemma 5.1 holds, provided that we show that for each $j \in J$

We argue (5.8) as follows: testing (5.7) with $q_{j}$ itself and using that $\int_{K_j} q_j = 0$, we obtain

By Cauchy-Schwarz's inequality and by the definition of $v_i$ (see also (3.18)), this implies that

By the trace estimate for functions $u \in H^1(B_r)$, $r>0$

inequality (5.9) turns into

Since by (5.3) we have $r_i \leqslant \mathop{diam}(K_j)$, we infer that

This, Cauchy-Schwarz's inequality, and (5.1) further yield

Since by (5.7) the function $q_j$ has zero mean, we may apply Poincaré-Wirtinger's inequality to conclude that

This establishes (5.8) and, in turn, concludes the proof of Lemma 5.1.

Proof of Lemma 5.2. Without loss of generality we assume that $|A| = 1$. By the assumption (i)-(ii) on $(\Phi, \mathcal{R})$ and by symmetry we have that

Appealing to Fubini's theorem and relabelling the elements $\{ (x_i, \rho_i)\}_{i = 1}^n$, this implies

i.e. $\mathbb{E}\bigl[ \sum_{z \in \Phi(A) } G( (z, \rho_z), \omega \backslash \{(z, \rho_z)\} ) \bigr] = \lambda \int_{A} \mathbb{E}_\rho \biggl[ \mathbb{E} \bigl[ G((x, \rho) , \omega) \bigr] \biggr] \, {\mathrm{d}} x$. Since $G$ is stationary, the above equality immediately implies Lemma 5.2.

Acknowledgments

The author thanks the anonymous referees for the helpful comments and suggestions on the first version of the paper.