# Stochastic homogenization on perforated domains I - Extension operators

Martin Heida

###### Abstract

In this first part of a series of 3 papers, we set up a framework to study the
existence of uniformly bounded extension and trace operators for
$W^{1,p}$-functions on randomly perforated domains, where the geometry is
assumed to be stationary ergodic. We drop the classical assumption of minimaly
smoothness and study stationary geometries which have no global John
regularity. For such geometries, uniform extension operators can be defined
only from $W^{1,p}$ to $W^{1,r}$ with the strict inequality $r<p$. In
particular, we estimate the $L^{r}$-norm of the extended gradient in terms of
the $L^{p}$-norm of the original gradient. Similar relations hold for the
symmetric gradients (for $\mathbb{R}^{d}$-valued functions) and for traces on
the boundary. As a byproduct we obtain some Poincaré and Korn inequalities of
the same spirit.

Such extension and trace operators are important for compactness in stochastic
homogenization. In contrast to former approaches and results, we use very weak
assumptions: local $(\delta,M)$-regularity to quantify statistically the local
Lipschitz regularity and isotropic cone mixing to quantify the density of the
geometry and the mesoscopic properties. These two properties are sufficient to
reduce the problem of extension operators to the connectivity of the geometry.

In contrast to former approaches we do not require a minimal distance between
the inclusions and we allow for globally unbounded Lipschitz constants and
percolating holes. We will illustrate our method by applying it to the Boolean
model based on a Poisson point process and to a Delaunay pipe process, for
which we can explicitly estimate the connectivity terms.

######  Acknowledgement.

I thank B. Jahnel for his helpful suggestions on literature. Furthermore, I
thank D.R.M. Renger and G. Friesecke for critical questions that pushed my
research further. Finally I thank the DFG for funding my reasearch via CRC
1114 Project C05.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Fundamental Notation and Geometric Objects
    2. 2.2 Simple Local Extensions and Traces
    3. 2.3 Local Nitsche-Extensions
    4. 2.4 Poincaré Inequalities
    5. 2.5 Korn Inequalities
    6. 2.6 Korn-Poincaré Inequalities
    7. 2.7 Voronoi Tessellations and Delaunay Triangulation
    8. 2.8 Local $\eta$-Regularity
    9. 2.9 Dynamical Systems
    10. 2.10 Random Measures and Palm Theory
    11. 2.11 Random Sets
    12. 2.12 Point Processes
    13. 2.13 Dynamical Systems on $\mathbb{Z}^{d}$
  3. 3 Quantifying Nonlocal Regularity Properties of the Geometry
    1. 3.1 Microscopic Regularity
    2. 3.2 Mesoscopic Regularity and Isotropic Cone Mixing
  4. 4 Extension and Trace Properties from $\left(\delta,M\right)$-Regularity
    1. 4.1 Preliminaries
    2. 4.2 Extensions preserving the Gradient norm via $\left(\delta,M\right)$-Regularity of $\partial\boldsymbol{\mathrm{P}}$
    3. 4.3 Extensions preserving the Symmetric Gradient norm via $\left(\delta,M\right)$-Regularity of $\partial\boldsymbol{\mathrm{P}}$
    4. 4.4 Support
    5. 4.5 Proof of Lemmas 4.7 and 4.9
    6. 4.6 Traces on $\left(\delta,M\right)$-Regular Sets, Proof of Theorem 1.7
  5. 5 The Issue of Connectedness
  6. 6 Sample Geometries
    1. 6.1 Delaunay Pipes for a Matern Process
    2. 6.2 Boolean Model for the Poisson Ball Process

##  1 Introduction

In 1979 Papanicolaou and Varadhan [22] and Kozlov [15] for the first time
independently introduced concepts for the averaging of random elliptic
operators. At that time, the periodic homogenization theory had already
advanced to some extend (as can be seen in the book [23] that had appeared one
year before) dealing also with non-uniformly elliptic operators [17] and
domains with periodic holes [3]. The most recent and most complete work for
extension operators on periodically perforated domains is [11].

In contrast, the homogenization on randomly perforated domains is still open
to a large extend. Recent results focus on minimally smooth domains [9, 24] or
on decreasing size of the perforations when the smallness parameter tends to
zero [8] (and references therein). The main issue in homogenization on
perforated domains compared to classical homogenization problems is
compactness. For elasticity, this is completely open.

The results presented below are meant for application in quenched convergence.
The estimates for the extension and trace operators which are derived strongly
depends on the realization of the geometry - thus on $\omega$. Nevertheless,
if the geometry is stationary, a corresponding estimate can be achieved for
almost every $\omega$.

#### The Problem

In order to illustrate the issues in stochastic homogenization on perforated
domains, we introduce the following example.

Let $\boldsymbol{\mathrm{P}}(\omega)\subset\mathbb{R}^{d}$ be a stationary
random open set and let $\varepsilon>0$ be the smallness parameter and let
$\tilde{\boldsymbol{\mathrm{P}}}{\left(\omega\right)}$ be an infinitely
connected component (i.e. an unbounded connected domain) of
$\boldsymbol{\mathrm{P}}{\left(\omega\right)}$. For a bounded open domain
$\boldsymbol{\mathrm{Q}}$, we consider
$\boldsymbol{\mathrm{Q}}_{\tilde{\boldsymbol{\mathrm{P}}}}^{\varepsilon}(\omega):=\boldsymbol{\mathrm{Q}}\cap\varepsilon\tilde{\boldsymbol{\mathrm{P}}}(\omega)$
and
$\Gamma^{\varepsilon}(\omega):=\boldsymbol{\mathrm{Q}}\cap\varepsilon\partial\tilde{\boldsymbol{\mathrm{P}}}(\omega)$
with outer normal $\nu_{\Gamma^{\varepsilon}(\omega)}$. For a sufficiently
regular and $\mathbb{R}^{d}$-valued function $u$ we denote
$\nabla^{s}u:=\frac{1}{2}\left(\nabla u+(\nabla u)^{\top}\right)$ the
symmetric part of $\nabla u$. A typical homogenization problem then is the
following::

$\displaystyle-\mathrm{div}\left(\left|\nabla^{s}u^{\varepsilon}\right|^{p-2}\nabla^{s}u^{\varepsilon}\right)$
$\displaystyle=g(u^{\varepsilon})$ $\displaystyle\mbox{on
}\boldsymbol{\mathrm{Q}}_{\tilde{\boldsymbol{\mathrm{P}}}}^{\varepsilon}(\omega)\,,$
$\displaystyle u$ $\displaystyle=0$ $\displaystyle\mbox{on
}\partial\boldsymbol{\mathrm{Q}}\cap\left(\varepsilon\boldsymbol{\mathrm{P}}\right)\,,$
(1.1) $\displaystyle\left|\nabla^{s}u^{\varepsilon}\right|^{p-2}\nabla
u^{\varepsilon}\cdot\nu_{\Gamma^{\varepsilon}(\omega)}$
$\displaystyle=f(u^{\varepsilon})$ $\displaystyle\mbox{on
}\Gamma^{\varepsilon}(\omega)\,.$

Note that for simplicity of illustration, the only randomness that we consider
in this problem is due to $\boldsymbol{\mathrm{P}}(\omega)$.

One way to prove homogenization of (1.1) is to prove $\Gamma$-convergence of

$\mathcal{E}_{\varepsilon,\omega}(u)=\int_{\boldsymbol{\mathrm{Q}}_{\tilde{\boldsymbol{\mathrm{P}}}}^{\varepsilon}(\omega)}\left(\frac{1}{p}\left|\nabla^{s}u\right|^{p}-G(u)\right)+\int_{\Gamma^{\varepsilon}(\omega)}F(u)\,,$

in a suitably chosen space where $G^{\prime}=g$ and $F^{\prime}=f$.
Conceptually, this implies convergence of the minimizers $u^{\varepsilon}$ to
a minimizer of a limit functional but if $G$ or $F$ are non-monotone, we need
compactness. However, the minimizers are elements of
$\boldsymbol{\mathrm{W}}^{1,p}(\boldsymbol{\mathrm{Q}}_{\tilde{\boldsymbol{\mathrm{P}}}}^{\varepsilon}):=W^{1,p}(\boldsymbol{\mathrm{Q}}_{\tilde{\boldsymbol{\mathrm{P}}}}^{\varepsilon};\mathbb{R}^{d})$
and since this space changes with $\varepsilon$, there is apriori no
compactness of $u^{\varepsilon}$, even though we have uniform apriori
estimates on the gradients.

The _canonical_ path to circumvent this issue in _periodic_ homogenization is
via uniformly bounded extension operators
$\mathcal{U}_{\varepsilon}:\,W^{1,p}(\boldsymbol{\mathrm{Q}}_{\tilde{\boldsymbol{\mathrm{P}}}}^{\varepsilon})\to
W^{1,p}(\boldsymbol{\mathrm{Q}})$ that share the property that for some $C>0$
independent from $\varepsilon$ it holds for all $u\in
W^{1,p}(\boldsymbol{\mathrm{Q}}_{\tilde{\boldsymbol{\mathrm{P}}}}^{\varepsilon})$
with $u|_{\mathbb{R}^{d}\backslash\boldsymbol{\mathrm{Q}}}\equiv 0$

$\left\|\nabla\mathcal{U}_{\varepsilon}u\right\|_{L^{p}(\boldsymbol{\mathrm{Q}})}\leq
C\left\|\nabla
u\right\|_{L^{p}(\boldsymbol{\mathrm{Q}}_{\tilde{\boldsymbol{\mathrm{P}}}}^{\varepsilon})}\,,\qquad\left\|\mathcal{U}_{\varepsilon}u\right\|_{L^{p}(\boldsymbol{\mathrm{Q}})}\leq
C\left\|u\right\|_{L^{p}(\boldsymbol{\mathrm{Q}}_{\tilde{\boldsymbol{\mathrm{P}}}}^{\varepsilon})}\,,$
(1.2)

see [11, 12], combined with uniformly bounded trace operators, see [7, 9].
Such operators have also been provided for elasticity problems [11, 21, 30,
31], i.e.

$\left\|\nabla^{s}\mathcal{U}_{\varepsilon}u\right\|_{L^{p}(\boldsymbol{\mathrm{Q}})}\leq
C\left\|\nabla^{s}u\right\|_{L^{p}(\boldsymbol{\mathrm{Q}}_{\tilde{\boldsymbol{\mathrm{P}}}}^{\varepsilon})}\,.$

The last estimate then allows to use Korn’s inequality combined with Sobolev’s
embedding theorem to find
$\mathcal{U}_{\varepsilon}u^{\varepsilon}\rightharpoonup u_{0}$ weakly in
$\boldsymbol{\mathrm{W}}^{1,p}(\boldsymbol{\mathrm{Q}})$.

##### What is the classical strategy?

The existing results on extension and trace operators for random domains are
focused on a.s. minimally smooth domains. A connected domain
$\boldsymbol{\mathrm{P}}\subset\mathbb{R}^{d}$ is minimally smooth [26] if
there exist $(\delta,M)$ such that for every
$x\in\partial\boldsymbol{\mathrm{P}}$ the set
$\partial\boldsymbol{\mathrm{P}}\cap\mathbb{B}_{\delta}{\left(x\right)}$ is
the graph of a Lipschitz continuous function with Lipschitz constant less than
$M$. It is further assumed that the complement
$\mathbb{R}^{d}\backslash\boldsymbol{\mathrm{P}}$ consists of uniformly
bounded sets. This concept leads to almost sure construction of uniformly
bounded extension operators
$W_{\mathrm{loc}}^{1,p}(\boldsymbol{\mathrm{P}})\to W^{1,p}(\mathbb{R}^{d})$
[9] in the sense that for every bounded $\boldsymbol{\mathrm{Q}}$ and every
$u\in W^{1,p}(\boldsymbol{\mathrm{Q}}\cap\boldsymbol{\mathrm{P}})$ with
$u|_{\mathbb{R}^{d}\backslash\boldsymbol{\mathrm{Q}}}\equiv 0$ holds

$\left\|\nabla\mathcal{U}u\right\|_{L^{p}(\boldsymbol{\mathrm{Q}})}\leq
C\left\|\nabla
u\right\|_{L^{p}(\boldsymbol{\mathrm{Q}}\cap\boldsymbol{\mathrm{P}})}\,,\qquad\left\|\mathcal{U}u\right\|_{L^{p}(\boldsymbol{\mathrm{Q}})}\leq
C\left\|u\right\|_{L^{p}(\boldsymbol{\mathrm{Q}}\cap\boldsymbol{\mathrm{P}})}\,,$
(1.3)

with $C$ independent from $\boldsymbol{\mathrm{Q}}$. Similarly, one obtains
for the trace $\mathcal{T}$ that [24]

$\left\|\mathcal{T}u\right\|_{L^{p}(\boldsymbol{\mathrm{Q}}\cap\partial\boldsymbol{\mathrm{P}})}\leq
C\left(\left\|u\right\|_{L^{p}(\boldsymbol{\mathrm{Q}}\cap\boldsymbol{\mathrm{P}})}+\left\|\nabla
u\right\|_{L^{p}(\boldsymbol{\mathrm{Q}}\cap\boldsymbol{\mathrm{P}})}\right)\,.$

Using a scaling argument to obtain e.g. (1.2), such extension and trace
operators are typically used in order to treat nonlinearities in
homogenization problems.

##### Why does this work?

The theory cited above is directly connected to the theory of Jones [13] and
Duran and Muschietti [5] on so-called John domains. These are precisely the
bounded domains $\boldsymbol{\mathrm{P}}$ that admit extension operators
$W^{1,p}(\boldsymbol{\mathrm{P}})\to W^{1,p}(\mathbb{R}^{d})$ satisfying

$\left\|\mathcal{U}u\right\|_{W^{1,p}(\mathbb{R}^{d})}\leq
C\left\|u\right\|_{W^{1,p}(\boldsymbol{\mathrm{Q}}\cap\boldsymbol{\mathrm{P}})}\,.$

######  Definition (John domains).

A bounded domain $\boldsymbol{\mathrm{P}}\subset\mathbb{R}^{d}$ is a John
domain (a.k.a $(\varepsilon,\delta)$-domain) if there exists
$\varepsilon,\delta>0$ such that for every $x,y\in\boldsymbol{\mathrm{P}}$
with $\left|x-y\right|<\delta$ there exists a rectifiable path
$\gamma:\,[0,1]\to\boldsymbol{\mathrm{P}}$ from $x$ to $y$ such that

$\displaystyle\mathrm{length}\gamma$
$\displaystyle\leq\frac{1}{\varepsilon}\left|x-y\right|\qquad\text{and}$
$\displaystyle\forall
t\in(0,1):\quad\inf_{z\in\mathbb{R}^{d}\backslash\boldsymbol{\mathrm{P}}}\left|\gamma(t)-z\right|$
$\displaystyle\geq\frac{\varepsilon\left|x-\gamma(t)\right|\left|\gamma(t)-y\right|}{\left|x-y\right|}\,.$

Because of the locality implied by $\delta$, it is possible to glue together
local extension operators on John domains such as done in [11] for periodic or
[9] for minimally smooth domains. In the stochastic case one benefits a lot
from the uniform boundedness of the components of
$\mathbb{R}^{d}\backslash\boldsymbol{\mathrm{P}}$, which allows to split the
extension problem into independent extension problems on uniformly John-
regular domains.

##### Why this is not enough for general random domains!

As one could guess from the emphasis that is put on the above explanations,
random geometries are merely minimally smooth. On an unbounded random domain
$\boldsymbol{\mathrm{P}}$, the constant $M$ can locally become very large in
points $x\in\partial\boldsymbol{\mathrm{P}}$, while simultaneously, $\delta$
can become very small in the very same $x$. In fact, they are not even
“uniformly John” as the following, yet deterministic example illustrates.

######  Example 1.1.

Considering

$\boldsymbol{\mathrm{P}}:=\left\\{\left(x_{1},x_{2}\right)\in\mathbb{R}^{2}:\;\exists
n\in\mathbb{N}:\,x_{1}-(2n+1)\in(-1,1],\,x_{2}<\max\left\\{1,\,n\left|x_{1}-(2n+1)\right|\right\\}\right\\}$

the Lipschitz constant on $(2n,2n+2)$ is $n$ and it is easy to figure out that
this non-uniformly Lipschitz domain violates the John condition due to the
cups. Hence, a uniform estimate of the form (1.3) cannot exist.

Therefor, an alternative concept to measure the large scale regularity of a
random geometry is needed. Since the classical results do not excluded the
existence of an estimate

$\frac{1}{\left|\boldsymbol{\mathrm{Q}}\right|}\int_{\boldsymbol{\mathrm{Q}}}\left|\nabla\mathcal{U}u\right|^{r}\leq
C\left(\frac{1}{\left|\boldsymbol{\mathrm{Q}}\right|}\int_{\boldsymbol{\mathrm{Q}}\cap\mathbb{B}_{\mathfrak{r}}{\left(\boldsymbol{\mathrm{P}}\right)}}\left|\nabla
u\right|^{p}\right)^{\frac{r}{p}}\,,\qquad\frac{1}{\left|\boldsymbol{\mathrm{Q}}\right|}\int_{\boldsymbol{\mathrm{Q}}}\left|\mathcal{U}u\right|^{r}\leq
C\left(\frac{1}{\left|\boldsymbol{\mathrm{Q}}\right|}\int_{\boldsymbol{\mathrm{Q}}\cap\mathbb{B}_{\mathfrak{r}}{\left(\boldsymbol{\mathrm{P}}\right)}}\left|u\right|^{p}\right)^{\frac{r}{p}}\,,$
(1.4)

or

$\frac{1}{\left|\boldsymbol{\mathrm{Q}}\right|}\int_{\boldsymbol{\mathrm{Q}}}\left|\nabla^{s}\mathcal{U}u\right|^{r}\leq
C\left(\frac{1}{\left|\boldsymbol{\mathrm{Q}}\right|}\int_{\boldsymbol{\mathrm{Q}}\cap\mathbb{B}_{\mathfrak{r}}{\left(\boldsymbol{\mathrm{P}}\right)}}\left|\nabla^{s}u\right|^{p}\right)^{\frac{r}{p}}\,,\qquad\frac{1}{\left|\boldsymbol{\mathrm{Q}}\right|}\int_{\boldsymbol{\mathrm{Q}}}\left|\mathcal{U}u\right|^{r}\leq
C\left(\frac{1}{\left|\boldsymbol{\mathrm{Q}}\right|}\int_{\boldsymbol{\mathrm{Q}}\cap\mathbb{B}_{\mathfrak{r}}{\left(\boldsymbol{\mathrm{P}}\right)}}\left|u\right|^{p}\right)^{\frac{r}{p}}\,,$
(1.5)

where $1\leq r<p$ and $C$ is independent from $\boldsymbol{\mathrm{Q}}$, such
inequalities will be our goal.

##### Our results in a nutshell

We will provide inequalities of the form (1.4)–(1.5) for a Voronoi-pipe model
and for a Boolean model. On the way, we will provide several concepts and
intermediate results that can be reused in further examples and general
considerations such as planed in part III of this series. Scaled versions
(replacing $\varepsilon=m^{-1}$ in Theorems 1.16 and 1.18) of (1.4)–(1.5) can
be formulated for functions

$u\in
W_{0,\partial\boldsymbol{\mathrm{Q}}}^{1,p}(\varepsilon\boldsymbol{\mathrm{P}}\cap\boldsymbol{\mathrm{Q}}):=\left\\{u\in
W^{1,p}(\boldsymbol{\mathrm{Q}}\cap\varepsilon\boldsymbol{\mathrm{P}}):\,u|_{(\varepsilon\boldsymbol{\mathrm{P}})\cap\partial\boldsymbol{\mathrm{Q}}}\equiv
0\right\\}\,,$

and will be of the form

$\frac{1}{\left|\boldsymbol{\mathrm{Q}}\right|}\int_{\mathbb{R}^{d}}\left|\nabla\mathcal{U}_{\varepsilon}u\right|^{r}\leq
C\left(\frac{1}{\left|\boldsymbol{\mathrm{Q}}\right|}\int_{\boldsymbol{\mathrm{Q}}\cap\varepsilon\boldsymbol{\mathrm{P}}}\left|\nabla
u\right|^{p}\right)^{\frac{r}{p}}\,,\qquad\frac{1}{\left|\boldsymbol{\mathrm{Q}}\right|}\int_{\mathbb{R}^{d}}\left|\mathcal{U}_{\varepsilon}u\right|^{r}\leq
C\left(\frac{1}{\left|\boldsymbol{\mathrm{Q}}\right|}\int_{\boldsymbol{\mathrm{Q}}\cap\varepsilon\boldsymbol{\mathrm{P}}}\left|u\right|^{p}\right)^{\frac{r}{p}}\,,$

resp.

$\frac{1}{\left|\boldsymbol{\mathrm{Q}}\right|}\int_{\mathbb{R}^{d}}\left|\nabla^{s}\mathcal{U}_{\varepsilon}u\right|^{r}\leq
C\left(\frac{1}{\left|\boldsymbol{\mathrm{Q}}\right|}\int_{\boldsymbol{\mathrm{Q}}\cap\varepsilon\boldsymbol{\mathrm{P}}}\left|\nabla^{s}u\right|^{p}\right)^{\frac{r}{p}}\,,\qquad\frac{1}{\left|\boldsymbol{\mathrm{Q}}\right|}\int_{\mathbb{R}^{d}}\left|\mathcal{U}_{\varepsilon}u\right|^{r}\leq
C\left(\frac{1}{\left|\boldsymbol{\mathrm{Q}}\right|}\int_{\boldsymbol{\mathrm{Q}}\cap\varepsilon\boldsymbol{\mathrm{P}}}\left|u\right|^{p}\right)^{\frac{r}{p}}\,,$

where the support of $\mathcal{U}_{\varepsilon}u$ lies within
$\mathbb{B}_{\varepsilon^{\beta}}{\left(\boldsymbol{\mathrm{Q}}\right)}$ for
$\varepsilon$ small enough and some arbitrarily chosen but fixed
$\beta\in(0,1)$.

#### Quantifying properties of random geometries

As a replacement for periodicity, we introduce the concept of mesoscopic
regularity of a stationary random open set:

######  Definition 1.2 (Mesoscopic regularity).

Let $\boldsymbol{\mathrm{P}}$ be a stationary ergodic random open set, let
$\tilde{f}$ be a positive, monotonically decreasing function $\tilde{f}$ with
$\tilde{f}(R)\to 0$ as $R\to\infty$ and let $\mathfrak{r}>0$ s.t.

$\mathbb{P}{\left(\exists
x\in\mathbb{B}_{R}{\left(0\right)}:\,\mathbb{B}_{4\sqrt{d}\mathfrak{r}}{\left(x\right)}\subset\mathbb{B}_{R}{\left(0\right)}\cap\boldsymbol{\mathrm{P}}\right)}\geq
1-\tilde{f}{\left(R\right)}\,.$ (1.6)

Then $\boldsymbol{\mathrm{P}}$ is called $(\mathfrak{r},\tilde{f})$-mesoscopic
regular. $\boldsymbol{\mathrm{P}}$ is called polynomially (exponentially)
regular if $1/\tilde{f}$ grows polynomially (exponentially).

As a consequence of Lemmas 3.14, 3.16 and 3.17 we obtain the following.

######  Corollary 1.3 (All stationary ergodic random open sets are mesoscopic
regular).

Let $\boldsymbol{\mathrm{P}}(\omega)$ be a stationary ergodic random open set.
Then there exists $\mathfrak{r}>0$ and a monotonically decreasing function
with $\tilde{f}(R)\to 0$ as $R\to\infty$ such that $\boldsymbol{\mathrm{P}}$
is $(\mathfrak{r},\tilde{f})$-mesoscopic regular. Furthermore, there exists a
jointly stationary random point process
$\mathbb{X}_{\mathfrak{r}}(\omega)=\left(x_{a}\right)_{a\in\mathbb{N}}$ and
for every $a\in\mathbb{N}$ it holds
$\mathbb{B}_{\frac{\mathfrak{r}}{2}}{\left(x_{a}\right)}\subset\boldsymbol{\mathrm{P}}$
and for all $a,b\in\mathbb{N}$, $a\neq b$, it holds
$\left|x_{a}-x_{b}\right|>2\mathfrak{r}$. Construct from
$\mathbb{X}_{\mathfrak{r}}$ a Voronoi tessellation of cells $G_{a}$ with
diameter $d_{a}=d(x_{a})$. Then for some constant $C>0$ and some monotone
decreasing $f:\,(0,\infty)\to\mathbb{R}$ and $C>0$ with $f(R)\leq
C\tilde{f}(C^{-1}R)$ it holds

$\mathbb{P}{\left(d(x_{a})>D\right)}<f{\left(D\right)}\,.$

$\mathfrak{r}$, $\mathbb{X}_{\mathfrak{r}}$ and $f$ from Corollary 1.3 will
play a central role in the analysis. We summarize some of these properties in
the following.

######  Assumption 1.4.

Let $\boldsymbol{\mathrm{P}}$ be a Lipschitz domain and assume there exists
$\mathbb{X}_{\mathfrak{r}}=\left(x_{a}\right)_{a\in\mathbb{N}}$ be a set of
points having mutual distance $\left|x_{a}-x_{b}\right|>2\mathfrak{r}$ if
$a\neq b$ and with
$\mathbb{B}_{\frac{\mathfrak{r}}{2}}{\left(x_{a}\right)}\subset\boldsymbol{\mathrm{P}}$
for every $a\in\mathbb{N}$ (e.g.
$\mathbb{X}_{\mathfrak{r}}(\boldsymbol{\mathrm{P}})$, see (2.51)).

The second important concept to quantify in a stochastic manner is that of
local Lipschitz regularity.

######  Definition 1.5 (Local $\left(\delta,M\right)$-Regularity).

Let $\boldsymbol{\mathrm{P}}\subset\mathbb{R}^{d}$ be an open set.
$\boldsymbol{\mathrm{P}}$ is called _$(\delta,M)$ -regular _in
$p_{0}\in\partial\boldsymbol{\mathrm{P}}$ if there exists an open set
$U\subset\mathbb{R}^{d-1}$ and a Lipschitz continuous function
$\phi:\,U\to\mathbb{R}$ with Lipschitz constant greater or equal to $M$ such
that
$\partial\boldsymbol{\mathrm{P}}\cap\mathbb{B}_{\delta}{\left(p_{0}\right)}$
is subset of the graph of the function
$\varphi:\,U\to\mathbb{R}^{d}\,,\;\tilde{x}\mapsto\left(\tilde{x},\phi{\left(\tilde{x}\right)}\right)$
in some suitable coordinate system.

Every Lipschitz domain $\boldsymbol{\mathrm{P}}$ is _locally
$(\delta,M)$-regular _in every $p_{0}\in\partial\boldsymbol{\mathrm{P}}$. In
what follows, we bound $\delta$ from above by $\mathfrak{r}$ only for
practical reasons in the proofs. The following quantities can be derived from
local $(\delta,M)$-regularity.

######  Definition 1.6.

For a Lipschitz domain $\boldsymbol{\mathrm{P}}\subset\mathbb{R}^{d}$ and for
every $p\in\partial\boldsymbol{\mathrm{P}}$ and $n\in\mathbb{N}\cup\\{0\\}$

$\displaystyle\Delta{\left(p\right)}$
$\displaystyle:=\sup_{\delta<\mathfrak{r}}\left\\{\exists
M>0:\,\boldsymbol{\mathrm{P}}\text{ is }\left(\delta,M\right)\text{-regular in
}p\right\\}\,,\quad\delta_{\Delta}{\left(p\right)}:=\frac{\Delta{\left(p\right)}}{2}\,,$
(1.7) $\displaystyle M_{r}(p)$
$\displaystyle:=\inf_{\eta>r}\inf\left\\{M:\,\boldsymbol{\mathrm{P}}\text{ is
}\left(\eta,M\right)\text{-regular in }p\right\\}\,,$ (1.8)
$\displaystyle\rho_{n}{\left(p\right)}$
$\displaystyle:=\sup_{r<\delta{\left(p\right)}}r\left(4M_{r}{\left(p\right)}^{2}+2\right)^{-\frac{n}{2}}\,,$
(1.9)

If no confusion occurs, we write $\delta=\delta_{\Delta}$. Furthermore, for
$c\in(0,1]$ let $\eta(p)=c\delta_{\Delta}(p)$ or $\eta(p)=c\rho_{n}(p)$,
$n\in\mathbb{N}$ and $r\in C^{0,1}(\partial\boldsymbol{\mathrm{P}})$ and
define

$\displaystyle\eta_{[r],\mathbb{R}^{d}}{\left(x\right)}$
$\displaystyle:=\inf\left\\{\eta{\left(\tilde{x}\right)}\,:\;\tilde{x}\in\partial\boldsymbol{\mathrm{P}}\,\text{s.t.
}x\in\mathbb{B}_{r{\left(\tilde{x}\right)}}{\left(\tilde{x}\right)}\right\\}\,,$
(1.10) $\displaystyle M_{[r,\eta],\mathbb{R}^{d}}(x)$
$\displaystyle:=\sup\left\\{M_{r(\tilde{x})}(\tilde{x})\,:\;\tilde{x}\in\partial\boldsymbol{\mathrm{P}}\,\text{s.t.
}x\in\overline{\mathbb{B}_{\eta{\left(\tilde{x}\right)}}{\left(\tilde{x}\right)}}\right\\}\,,$
(1.11)

where $\inf\emptyset=\sup\emptyset:=0$ for notational convenience. We also
write $M_{[\eta],\mathbb{R}^{d}}(x):=M_{[\eta,\eta],\mathbb{R}^{d}}(x)$ and
$\eta_{\mathbb{R}^{d}}{\left(x\right)}:=\eta_{[\eta],\mathbb{R}^{d}}{\left(x\right)}$.
Of course, we can also consider
$M_{[r],\partial\boldsymbol{\mathrm{P}}}:\,p\mapsto M_{r(p)}(p)$ as a function
on $\partial\boldsymbol{\mathrm{P}}$, and we will do this once in Lemma 3.8.

When it comes to application of the abstract results found below, it is
important to have in mind that $\eta$ and $M_{r}$ are quantities on
$\partial\boldsymbol{\mathrm{P}}$, while $\eta_{[r],\mathbb{R}^{d}}$ and
$M_{[r,\eta],\mathbb{R}^{d}}$ are quantities on $\mathbb{R}^{d}$. Hence, while
trivially

$\mathbb{P}(\eta_{[r],\mathbb{R}^{d}}\in(\eta_{1},\eta_{2}))=\lim_{n\to\infty}n^{-d}\left|\boldsymbol{\mathrm{Q}}\right|^{-1}\left|\left\\{x\in
n\boldsymbol{\mathrm{Q}}:\,\eta_{[r],\mathbb{R}^{d}}\in(\eta_{1},\eta_{2})\right\\}\right|$

(and similarly for $M_{[r,\eta],\mathbb{R}^{d}}$) for every convex bounded
open $\boldsymbol{\mathrm{Q}}$, we have in mind

$\mathbb{P}(\eta\in(\eta_{1},\eta_{2}))=\left(\lim_{n\to\infty}\mathcal{H}^{d-1}{\left(\partial\boldsymbol{\mathrm{P}}\cap
n\boldsymbol{\mathrm{Q}}\right)}\right)^{-1}\mathcal{H}^{d-1}{\left(\left\\{x\in(n\boldsymbol{\mathrm{Q}})\cap\partial\boldsymbol{\mathrm{P}}:\,\eta\in(\eta_{1},\eta_{2})\right\\}\right)}\,.$

We will prove measurability of $\eta_{[r],\mathbb{R}^{d}}$ and
$M_{[r,\eta],\mathbb{R}^{d}}$ in Lemma 3.11 and see how the weighted
expectations of $\eta_{[r],\mathbb{R}^{d}}$ and $M_{[r,\eta],\mathbb{R}^{d}}$
can be estimated by weighted expectations of $M$ and $\eta$ in Lemma 3.12.

#### Traces

The first important result is the boundedness of the trace operator.

######  Theorem 1.7.

Let $\boldsymbol{\mathrm{P}}\subset\mathbb{R}^{d}$ be a Lipschitz domain,
$\frac{1}{8}>\mathfrak{r}>0$ and let
$\boldsymbol{\mathrm{Q}}\subset\mathbb{R}^{d}$ be a bounded open set and let
$1\leq r<p_{0}<p$. Then the trace operator $\mathcal{T}$ satisfies for every
$u\in W_{\mathrm{loc}}^{1,p}{\left(\boldsymbol{\mathrm{P}}\right)}$

$\frac{1}{\left|\boldsymbol{\mathrm{Q}}\right|}\int_{\boldsymbol{\mathrm{Q}}\cap\partial\boldsymbol{\mathrm{P}}}\left|\mathcal{T}u\right|^{r}\leq
C\left(\frac{1}{\left|\boldsymbol{\mathrm{Q}}\right|}\int_{\mathbb{B}_{\frac{1}{4}}{\left(\boldsymbol{\mathrm{Q}}\right)}\cap\boldsymbol{\mathrm{P}}}\left|u\right|^{p}+\left|\nabla
u\right|^{p}\right)^{\frac{r}{p}}$

where for some constant $C_{0}$ depending only on $p_{0}$, $p$ and $r$ and $d$
and for $\tilde{\rho}=2^{-5}\rho_{1}$ one has

$\displaystyle C$
$\displaystyle=C_{0}\left(\frac{1}{\left|\boldsymbol{\mathrm{Q}}\right|}\int_{\mathbb{B}_{\frac{1}{4}\mathfrak{r}}{\left(\boldsymbol{\mathrm{Q}}\right)}\cap\partial\boldsymbol{\mathrm{P}}}\tilde{\rho}_{\mathbb{R}^{d}}^{-\frac{1}{p_{0}-r}}\right)^{\frac{p_{0}-r}{p_{0}}}\left(\frac{1}{\left|\boldsymbol{\mathrm{Q}}\right|}\int_{\mathbb{B}_{\frac{1}{4}\mathfrak{r}}{\left(\boldsymbol{\mathrm{Q}}\right)}\cap\boldsymbol{\mathrm{P}}}\left(1+\tilde{M}_{[\frac{1}{32}\delta],\mathbb{R}^{d}}\right)^{\left(\frac{1}{p_{0}}+1+\hat{d}\right)\frac{p}{p-p_{0}}}\right)^{\frac{p-p_{0}}{p_{0}p}}\,,$
(1.12) $\displaystyle C$
$\displaystyle=C_{0}\left(\frac{1}{\left|\boldsymbol{\mathrm{Q}}\right|}\int_{\mathbb{B}_{\frac{1}{4}\mathfrak{r}}{\left(\boldsymbol{\mathrm{Q}}\right)}\cap\partial\boldsymbol{\mathrm{P}}}\left(\tilde{\rho}_{\mathbb{R}^{d}}\left(1+\tilde{M}_{[\frac{1}{32}\delta],\mathbb{R}^{d}}\right)\right)^{-\frac{1}{p-r}}\right)^{\frac{p-r}{p}}\,.$
(1.13)

###### Proof.

This is proved in Section 4.6. ∎

#### Local Covering of $\partial\boldsymbol{\mathrm{P}}$

In view of Corollary 3.7, for every $n=1$ or $n=2$ there exist a complete
covering of $\partial\boldsymbol{\mathrm{P}}$ by balls
$\mathbb{B}_{\tilde{\rho}_{n}\left(p_{i}^{n}\right)}{\left(p_{i}^{n}\right)}$,
$\left(p_{i}^{n}\right)_{i\in\mathbb{N}}$, where
$\tilde{\rho}_{n}{\left(p\right)}:=2^{-5}\rho_{n}{\left(p\right)}$. We write
$\tilde{\rho}_{n,i}:=\tilde{\rho}_{n}(p_{i}^{n})$.

######  Definition 1.8 (Microscopic regularity and extension order).

The inner microscopic regularity $\alpha$ is

$\alpha:=\inf\left\\{\tilde{\alpha}\geq 0:\;\forall
p\in\partial\boldsymbol{\mathrm{P}}\exists
y\in\boldsymbol{\mathrm{P}}:\,\mathbb{B}_{\tilde{\rho}(p)/32(1+M_{\tilde{\rho}(p)}(p)^{\tilde{\alpha}})}{\left(p\right)}\subset\mathbb{B}_{\tilde{\rho}(p)/8}{\left(p\right)}\right\\}\,.$

In Lemma 3.1 we will see that indeed $\alpha\leq 1$.

######  Definition 1.9 (Extension order).

The geometry is _of extension order_ $n\in\mathbb{N}\cup\\{0\\}$ if there
exists $C>0$ such that for almost every $p\in\partial\boldsymbol{\mathrm{P}}$
there exists a local extension operator

$\displaystyle\mathcal{U}:\,W^{1,p}(\mathbb{B}_{\frac{1}{8}\delta(p)}{\left(p\right)}\cap\boldsymbol{\mathrm{P}})$
$\displaystyle\to
W^{1,p}(\mathbb{B}_{\frac{1}{8}\rho_{n}(p)}{\left(p\right)})\,,$
$\displaystyle\left\|\nabla\mathcal{U}u\right\|_{L^{p}(\mathbb{B}_{\frac{1}{8}\rho_{n}(p)}{\left(p\right)})}$
$\displaystyle\leq C\left(1+M_{\frac{1}{8}\delta(p)}(p)\right)\left\|\nabla
u\right\|_{L^{p}(\mathbb{B}_{\frac{1}{8}\delta(p)}{\left(p\right)})}\,.$
(1.14)

The geometry is _of symmetric extension order_ $n\in\mathbb{N}\cup\\{0\\}$ if
there exists $C>0$ such that for almost every
$p\in\partial\boldsymbol{\mathrm{P}}$ there exists a local extension operator

$\displaystyle\mathcal{U}:\,\boldsymbol{\mathrm{W}}^{1,p}(\mathbb{B}_{\frac{1}{8}\delta(p)}{\left(p\right)}\cap\boldsymbol{\mathrm{P}})$
$\displaystyle\to\boldsymbol{\mathrm{W}}^{1,p}(\mathbb{B}_{\frac{1}{8}\rho_{n}(p)}{\left(p\right)})\,,$
$\displaystyle\left\|\nabla^{s}\mathcal{U}u\right\|_{L^{p}(\mathbb{B}_{\frac{1}{8}\rho_{n}(p)}{\left(p\right)})}$
$\displaystyle\leq
C\left(1+M_{\frac{1}{8}\delta(p)}(p)\right)^{2}\left\|\nabla^{s}u\right\|_{L^{p}(\mathbb{B}_{\frac{1}{8}\delta(p)}{\left(p\right)})}\,.$
(1.15)

Corollary 3.6 shows that every locally Lipschitz geometry is of extension
order $n=1$ and every locally Lipschitz geometry is of symmetric extension
order $n=2$. However, better results for $n$ are possible, as we will see
below.

#### Global Tessellation of $\boldsymbol{\mathrm{P}}$

Let $\mathbb{X}=\left(x_{a}\right)_{a\in\mathbb{N}}$ be a jointly stationary
point process with $\boldsymbol{\mathrm{P}}$ such that
$\mathbb{B}_{\mathfrak{r}}{\left(\mathbb{X}\right)}\subset\boldsymbol{\mathrm{P}}$.
In this work, we will often assume that
$\left|x_{a}-x_{b}\right|>2\mathfrak{r}$ for all $a\neq b$ for simplicity in
Sections 5 and 6. The existence of such a process is always guarantied by
Lemmas 3.14 and 3.16. Its choice in a concrete example is, however, delicate.
Worth mentioning, for most of the theory developed until the end of Section 4
(Except for Lemmas 3.17 and 3.18 which are not used before Section 5), is
completely independent from this mutual minimal distance assumption.

From $\mathbb{X}$ we construct a Voronoi tessellation with cells
$\left(G_{a}\right)_{a\in\mathbb{N}}$ and we chose for each $x_{a}$ a radius
$\mathfrak{r}_{a}\leq\mathfrak{r}$ with
$\mathbb{B}_{\mathfrak{r}_{a}}{\left(x_{a}\right)}\subset
G_{a}\cap\boldsymbol{\mathrm{P}}$. Again, using Corollary 1.3, we assume that
$\mathfrak{r}_{a}=\mathfrak{r}$ is constant for simplicity.

#### Extensions I: Gradients

######  Notation 1.10.

Given $n\in\\{0,1\\}$ and $\alpha\in[0,1]$ we chose

$\mathfrak{r}_{n,\alpha,i}:=\tilde{\rho}_{n,i}/32(1+M_{\tilde{\rho}_{n,i}}(p_{n,i})^{\alpha})$
(1.16)

and some $y_{n,\alpha,i}$ such that

$B_{n,\alpha,i}:=\mathbb{B}_{\mathfrak{r}_{n,\alpha,i}}{\left(y_{n,\alpha,i}\right)}\subset\boldsymbol{\mathrm{P}}\cap\mathbb{B}_{\frac{1}{8}\tilde{\rho}_{n,i}}{\left(p_{n,i}\right)}\,.$
(1.17)

and for every $i$ and $a$, we define

$\tau_{n,\alpha,i}u:=\fint_{B_{n,\alpha,i}}u\,,\qquad\mathcal{M}_{a}u:=\fint_{\mathbb{B}_{\frac{\mathfrak{r}_{a}}{16}}{\left(x_{a}\right)}}u\,,$

local averages close to $\partial\boldsymbol{\mathrm{P}}$ and in $x_{a}$. We
say that $x_{a}\sim\sim x_{b}$ if
$G_{a}\cap\mathbb{B}_{\mathfrak{r}}{\left(G_{b}\right)}\neq\emptyset$ and we
say $x_{a}\in\mathbb{X}_{\mathfrak{r}}(\boldsymbol{\mathrm{Q}})$ if
$\mathbb{B}_{\mathfrak{r}}{\left(G_{a}\right)}\cap\boldsymbol{\mathrm{Q}}\neq\emptyset$.
Based on (4.14) we obtain the following extension result.

######  Theorem 1.11.

Let $\mathfrak{r}>0$ and let $\boldsymbol{\mathrm{P}}\subset\mathbb{R}^{d}$ be
a stationary ergodic random Lipschitz domain such that Assumption 1.4 holds
for $\mathbb{X}=\left(x_{a}\right)_{a\in\mathbb{N}}$ and
$\boldsymbol{\mathrm{P}}$ has microscopic regularity $\alpha$ with extension
order $n$. Let $\boldsymbol{\mathrm{Q}}\subset\mathbb{R}^{d}$ be a bounded
open set with
$\mathbb{B}_{\frac{1}{4}}{\left(0\right)}\subset\boldsymbol{\mathrm{Q}}$ and
let $1\leq r<p$. Furthermore, let

$\mathbb{E}{\left(\left(\left(1+M_{[\frac{3\delta}{8},\frac{\delta}{8}],\mathbb{R}^{d}}\right)^{nd}\left(1+M_{[\frac{1}{8}\delta],\mathbb{R}^{d}}\right)^{r}\left(1+M_{[\tilde{\rho}_{n}],\mathbb{R}^{d}}\right)^{\alpha(d-1)}\right)^{\frac{p}{p-r}}\right)}<\infty$

then there exist $C>0$ depending only on $d$, $r$ and $p$ such that for a.e.
$\omega$ there exists an extension operator
$\mathcal{U}_{\omega}:\,W_{\mathrm{loc}}^{1,p}(\boldsymbol{\mathrm{P}}(\omega))\to
W_{\mathrm{loc}}^{1,p}(\mathbb{R}^{d})$ and $C_{\omega}>0$ such that for every
$m\geq 1$ and every $u\in W^{1,p}(\boldsymbol{\mathrm{P}}(\omega))$ with
$u|_{\boldsymbol{\mathrm{P}}(\omega)\backslash m\boldsymbol{\mathrm{Q}}}\equiv
0$ it holds

$\displaystyle\frac{1}{\left|m\boldsymbol{\mathrm{Q}}\right|}\int_{m\boldsymbol{\mathrm{Q}}}\left|\nabla\left(\mathcal{U}_{\omega}u\right)\right|^{r}$
$\displaystyle\leq
C_{\omega}\left(\frac{1}{m^{d}}\int_{\boldsymbol{\mathrm{P}}\cap\mathbb{B}_{\mathfrak{r}}{\left(m\boldsymbol{\mathrm{Q}}\right)}}\left|\nabla
u\right|^{p}\right)^{\frac{r}{p}}$
$\displaystyle+C\frac{1}{m^{d}}\int_{\boldsymbol{\mathrm{P}}\cap\mathbb{B}_{\mathfrak{r}}{\left(m\boldsymbol{\mathrm{Q}}\right)}}\sum_{\begin{subarray}{c}i\neq
0\end{subarray}}\sum_{a}\tilde{\rho}_{\boldsymbol{\mathrm{P}}}^{-r}\chi_{\mathbb{B}_{\frac{\mathfrak{r}}{2}}{\left(G_{a}\right)}}\chi_{\mathbb{B}_{\tilde{\rho}_{n,i}}{\left(p_{n,i}\right)}}\left|\tau_{n,\alpha,i}u-\mathcal{M}_{a}u\right|^{r}$
$\displaystyle\quad+C\left|\frac{1}{m^{d}}\int_{\boldsymbol{\mathrm{P}}\cap
m\boldsymbol{\mathrm{Q}}}\sum_{a}\sum_{a\sim\sim
b}\chi_{\mathbb{B}_{\mathfrak{r}}{\left(G_{a}\right)}}\left|\mathcal{M}_{a}u-\mathcal{M}_{b}u\right|\right|^{r}\,,$
$\displaystyle\frac{1}{\left|m\boldsymbol{\mathrm{Q}}\right|}\int_{m\boldsymbol{\mathrm{Q}}}\left|\mathcal{U}_{\omega}u\right|^{r}\leq$
$\displaystyle
C_{\omega}\left(\frac{1}{m^{d}}\int_{\boldsymbol{\mathrm{P}}\cap\mathbb{B}_{\mathfrak{r}}{\left(m\boldsymbol{\mathrm{Q}}\right)}}\left|u\right|^{p}\right)^{\frac{r}{p}}\,.$

###### Proof.

This is a consequence of Lemma 4.7. ∎

In case one is interested in a weaker estimate on the extension operator, we
propose the following:

######  Theorem 1.12.

Under the assumptions of Theorem 1.11 let additionally

$\mathbb{E}{\left(\tilde{\rho}_{\boldsymbol{\mathrm{P}}}^{-\frac{rp}{p-r}}\right)}<\infty$

then there exists an extension operator
$\mathcal{U}_{\omega}:\,W_{\mathrm{loc}}^{1,p}(\boldsymbol{\mathrm{P}}(\omega))\to
W_{\mathrm{loc}}^{1,p}(\mathbb{R}^{d})$ such that for every $m\geq 1$ and
every $u\in W^{1,p}(\boldsymbol{\mathrm{P}}(\omega))$ with
$u|_{\boldsymbol{\mathrm{P}}(\omega)\backslash m\boldsymbol{\mathrm{Q}}}\equiv
0$ it holds

$\frac{1}{\left|m\boldsymbol{\mathrm{Q}}\right|}\int_{m\boldsymbol{\mathrm{Q}}}\left(\left|\nabla\left(\mathcal{U}_{\omega}u\right)\right|^{r}+\left|\mathcal{U}_{\omega}u\right|^{r}\right)\leq
C_{\omega}\left(\frac{1}{m^{d}}\int_{\boldsymbol{\mathrm{P}}\cap\mathbb{B}_{\mathfrak{r}}{\left(m\boldsymbol{\mathrm{Q}}\right)}}\left(\left|\nabla
u\right|^{p}+\left|u\right|^{p}\right)\right)^{\frac{r}{p}}\,.$

###### Proof.

This is a consequence of the proof of Lemma 4.7, replacing $M_{a}u$ in the
definition of $\mathcal{U}u$ by $0$. ∎

#### Percolation and Connectivity

The terms depending on $\left|\tau_{n,\alpha,i}u-\mathcal{M}_{a}u\right|$ or
$\left|\mathcal{M}_{a}u-\mathcal{M}_{a}u\right|$ appearing on the right hand
side in Theorem 1.11 need to be replaced by an integral over $\left|\nabla
u\right|^{p}$. Here, the pathwise topology of the geometry comes into play. By
this we mean that we have to integrate the gradient of $u$ over a path
connecting e.g. $p_{i}$ and $x_{a}$. Here, the mesoscopic properties of the
geometry will play a role. In particular, we need pathwise connectedness of
the random domain, a phenomenon which is known as percolation in the theory of
random sets. We will discuss two different examples to see that these terms
can indeed be handled in application, but shift a general discussion of
arbitrary geometries to a later publication.

#### Extensions II: Symmetric gradients

We now turn to the situation that $u$ is a $\mathbb{R}^{d}$-valued function
and that the given PDE system yields only estimates for
$\nabla^{s}u=\frac{1}{2}\left(\nabla u+(\nabla u)^{T}\right)$. We introduce
the following quantities:

######  Definition 1.13.

Given $n\in\\{0,1,2\\}$ and $\alpha\in[0,1]$ such that such that (1.17) holds
for $\mathfrak{r}_{i}=\mathfrak{r}_{n,\alpha,i}$ for every $i$ let for $i,a$

$\displaystyle\bar{\nabla}^{\bot}_{n,\alpha,i}u$
$\displaystyle:=\fint_{\mathbb{B}_{\mathfrak{r}_{n,\alpha,i}}{\left(y_{n,\alpha,i}\right)}}\left(\nabla
u-\nabla^{s}u\right)\,,$
$\displaystyle\left[\tau_{n,\alpha,i}^{\mathfrak{s}}u\right](x):=$
$\displaystyle\bar{\nabla}^{\bot}_{n,\alpha,i}u\,\left(x-y_{2,i}\right)+\fint_{\mathbb{B}_{\mathfrak{r}_{n,\alpha,i}}{\left(y_{n,\alpha,i}\right)}}u\,,$
$\displaystyle\bar{\nabla}^{\bot}_{a}u$
$\displaystyle:=\fint_{\mathbb{B}_{\frac{\mathfrak{r}_{a}}{16}}{\left(x_{a}\right)}}\left(\nabla
u-\nabla^{s}u\right)\,,$
$\displaystyle\left[\mathcal{M}_{a}^{\mathfrak{s}}u\right](x):=$
$\displaystyle\bar{\nabla}^{\bot}_{a}u\,\left(x-x_{a}\right)+\fint_{\mathbb{B}_{\frac{\mathfrak{r}_{a}}{16}}{\left(x_{a}\right)}}u\,.$

Using above introduced notation and $\boldsymbol{\mathrm{W}}$ do denote
$\mathbb{R}^{d}$-valued Sobolev spaces, we find the following.

######  Theorem 1.14.

Let $\mathfrak{r}>0$ and let $\boldsymbol{\mathrm{P}}\subset\mathbb{R}^{d}$ be
a stationary ergodic random Lipschitz domain such that Assumption 1.4 holds
for $\mathbb{X}=\left(x_{a}\right)_{a\in\mathbb{N}}$ and
$\boldsymbol{\mathrm{P}}$ has microscopic regularity $\alpha$ with symmetric
extension order $n\leq 2$. Let $\boldsymbol{\mathrm{Q}}\subset\mathbb{R}^{d}$
be a bounded open set with
$\mathbb{B}_{\frac{1}{4}}{\left(0\right)}\subset\boldsymbol{\mathrm{Q}}$ and
let $1\leq r<p_{0}<p$. Furthermore, let

$\mathbb{E}{\left(\left(\left(1+M_{[\frac{3\delta}{8},\frac{\delta}{8}],\mathbb{R}^{d}}\right)^{nd}\left(1+M_{[\frac{1}{8}\delta],\mathbb{R}^{d}}\right)^{2r}\left(1+M_{[\tilde{\rho}_{n}],\mathbb{R}^{d}}\right)^{\alpha(d-1)}\right)^{\frac{p}{p-r}}\right)}<\infty$

then hen there exist $C>0$ depending only on $d$, $r$, $s$ and $p$ such that
for a.e. $\omega$ there exists an extension operator
$\mathcal{U}_{\omega}:\,\boldsymbol{\mathrm{W}}_{\mathrm{loc}}^{1,p}(\boldsymbol{\mathrm{P}}(\omega))\to\boldsymbol{\mathrm{W}}_{\mathrm{loc}}^{1,p}(\mathbb{R}^{d})$
and $C_{\omega}>0$ such that for every $m\geq 1$ and every
$u\in\boldsymbol{\mathrm{W}}^{1,p}(\boldsymbol{\mathrm{P}}(\omega))$ with
$u|_{\boldsymbol{\mathrm{P}}(\omega)\backslash\boldsymbol{\mathrm{Q}}}\equiv
0$ it holds

$\displaystyle\frac{1}{\left|\boldsymbol{\mathrm{Q}}\right|}\int_{\boldsymbol{\mathrm{Q}}}\left|\nabla^{s}\left(\mathcal{U}_{\omega}u\right)\right|^{r}$
$\displaystyle\leq
C_{\omega}\left(\frac{1}{\left|\boldsymbol{\mathrm{Q}}\right|}\int_{\mathbb{B}_{\mathfrak{r}}{\left(\boldsymbol{\mathrm{Q}}\right)}\cap\boldsymbol{\mathrm{P}}}\left|\nabla^{s}u\right|^{p}\right)^{\frac{r}{p}}$
$\displaystyle\quad+C\frac{1}{\left|\boldsymbol{\mathrm{Q}}\right|}\int_{\boldsymbol{\mathrm{Q}}\backslash\boldsymbol{\mathrm{P}}}\sum_{a}\sum_{i\neq
0}\rho_{1,i}^{-r}\chi_{A_{1,i}}\chi_{\mathfrak{A}_{1,a}}\left|\tau_{n,\alpha,i}^{\mathfrak{s}}u-\mathcal{M}_{a}^{\mathfrak{s}}u\right|^{r}$
$\displaystyle\quad+\frac{1}{\left|\boldsymbol{\mathrm{Q}}\right|}\int_{\boldsymbol{\mathrm{Q}}}\left|\sum_{l=1}^{d}\sum_{a:\,\partial_{l}\Phi_{a}>0}\sum_{b:\,\partial_{l}\Phi_{b}<0}\frac{\partial_{l}\Phi_{a}\left|\partial_{l}\Phi_{b}\right|}{D_{l+}^{\Phi}}\left(\mathcal{M}_{a}^{\mathfrak{s}}u-\mathcal{M}_{b}^{\mathfrak{s}}u\right)\right|^{r}$
$\displaystyle\frac{1}{\left|\boldsymbol{\mathrm{Q}}\right|}\int_{\boldsymbol{\mathrm{Q}}}\left|\mathcal{U}_{\omega}u\right|^{r}$
$\displaystyle\leq
C_{\omega}\left(\frac{1}{\left|\boldsymbol{\mathrm{Q}}\right|}\int_{\mathbb{B}_{\mathfrak{r}}{\left(\boldsymbol{\mathrm{Q}}\right)}\cap\boldsymbol{\mathrm{P}}}\left|u\right|^{p}\right)^{\frac{r}{p}}\,,$

###### Proof.

This is a consequence of Lemma 4.9. ∎

######  Theorem 1.15.

Under the assumptions of Theorem 1.14 let additionally

$\mathbb{E}{\left(\tilde{\rho}_{\boldsymbol{\mathrm{P}}}^{-\frac{rp}{p-r}}\right)}<\infty$

then there exists an extension operator
$\mathcal{U}_{\omega}:\,W_{\mathrm{loc}}^{1,p}(\boldsymbol{\mathrm{P}}(\omega))\to
W_{\mathrm{loc}}^{1,p}(\mathbb{R}^{d})$ such that for every $m\geq 1$ and
every $u\in W^{1,p}(\boldsymbol{\mathrm{P}}(\omega))$ with
$u|_{\boldsymbol{\mathrm{P}}(\omega)\backslash m\boldsymbol{\mathrm{Q}}}\equiv
0$ it holds

$\frac{1}{\left|m\boldsymbol{\mathrm{Q}}\right|}\int_{m\boldsymbol{\mathrm{Q}}}\left(\left|\nabla\left(\mathcal{U}_{\omega}u\right)\right|^{r}+\left|\mathcal{U}_{\omega}u\right|^{r}\right)\leq
C_{\omega}\left(\frac{1}{m^{d}}\int_{\boldsymbol{\mathrm{P}}\cap\mathbb{B}_{\mathfrak{r}}{\left(m\boldsymbol{\mathrm{Q}}\right)}}\left(\left|\nabla
u\right|^{p}+\left|u\right|^{p}\right)\right)^{\frac{r}{p}}\,.$

###### Proof.

This is a consequence of the proof of Lemma 4.7, replacing
$M_{a}^{\mathfrak{s}}u$ in the definition of $\mathcal{U}u$ by $0$. ∎

### Discussion: Random Geometries and Applicability of the Method

In Section 6 we discuss two standard models from the theory of stochastic
geometries. The first one is a system of random pipes: Starting from a Poisson
point process and deleting all points with nearest neighbor closer than
$2\mathfrak{r}$ and introducing the Delaunay neighboring condition on the
points, every two neighbors are connect through a pipe of random thickness
$2\delta,$ where $\delta$ is distributed i.i.d among the pipes and we complete
the geometry by adding a ball of radius $\frac{\mathfrak{r}}{2}$ around each
point. Defining for bounded open domains
$\boldsymbol{\mathrm{Q}}\subset\mathbb{R}^{d}$ and $n\in\mathbb{N}$

$u\in
W_{0,\partial(n\boldsymbol{\mathrm{Q}})}^{1,p}(\boldsymbol{\mathrm{P}}\cap
n\boldsymbol{\mathrm{Q}}):=\left\\{u\in W^{1,p}(\boldsymbol{\mathrm{P}}\cap
n\boldsymbol{\mathrm{Q}}):\,u|_{\partial(n\boldsymbol{\mathrm{Q}})}\equiv
0\right\\}\,,$

and using $\boldsymbol{\mathrm{W}}$ instead of $W$ for $\mathbb{R}^{d}$-valued
functions, we find our first result:

######  Theorem 1.16.

In the pipe model of Section 6.1 let $\mathbb{P}(\delta(x,y)<\delta_{0})\leq
C_{\delta}\delta_{0}^{\beta}$ and let $1\leq r<s<p$ be such that
$\max\left\\{\frac{p\left(s+d\right)}{p-s},\frac{p(2d-s-1)}{p-s}\right\\}\leq\beta$
and $\frac{sr}{s-r}\leq\beta+d-1$. Then $\alpha=n=0$ both for extension and
symmetric extension order and there almost surely exists an extension operator
$\mathcal{U}:\,W_{\mathrm{loc}}^{1,p}(\boldsymbol{\mathrm{P}})\to
W_{\mathrm{loc}}^{1,p}(\mathbb{R}^{d})$ and constants $C,R>1$ such that for
all $m\in\mathbb{N}$ and every $u\in
W_{0,\partial(m\boldsymbol{\mathrm{Q}})}^{1,p}(\boldsymbol{\mathrm{P}}\cap
m\boldsymbol{\mathrm{Q}})$ it holds

$\frac{1}{\left|m\boldsymbol{\mathrm{Q}}\right|}\int_{\mathbb{R}^{d}}\left|\nabla\left(\mathcal{U}u\right)\right|^{r}\leq
C\left(\frac{1}{m^{d}}\int_{\boldsymbol{\mathrm{P}}\cap
m\boldsymbol{\mathrm{Q}}}\left|\nabla u\right|^{p}\right)^{\frac{r}{p}}\,.$

Furthermore there almost surely exists an extension operator
$\mathcal{U}:\,\boldsymbol{\mathrm{W}}_{\mathrm{loc}}^{1,p}(\boldsymbol{\mathrm{P}})\to\boldsymbol{\mathrm{W}}_{\mathrm{loc}}^{1,p}(\mathbb{R}^{d})$
and a constant $C>0$ such that for all $m\in\mathbb{N}$ and every
$u\in\boldsymbol{\mathrm{W}}_{0,\partial(m\boldsymbol{\mathrm{Q}})}^{1,p}(\boldsymbol{\mathrm{P}}\cap
m\boldsymbol{\mathrm{Q}})$

$\frac{1}{\left|m\boldsymbol{\mathrm{Q}}\right|}\int_{\mathbb{R}^{d}}\left|\nabla^{s}\left(\mathcal{U}u\right)\right|^{r}\leq
C\left(\frac{1}{m^{d}}\int_{\boldsymbol{\mathrm{P}}\cap
m\boldsymbol{\mathrm{Q}}}\left|\nabla^{s}u\right|^{p}\right)^{\frac{r}{p}}\,.$

In both cases for every $\beta\in(0,1)$ the following holds: for some
$m_{0}>1$ depending on $\omega$ and every $m>m_{0}$ the support of
$\mathcal{U}u$ lies within
$\mathbb{B}_{m^{1-\beta}}{\left(m\boldsymbol{\mathrm{Q}}\right)}$.

###### Proof.

The proof is given at the very end of Section 6.1. ∎

######  Corollary 1.17.

If $\mathbb{P}(\delta(x,y)<\delta_{0})\leq
C_{\delta}e^{-\gamma\delta_{0}^{-1}}$ then the last theorem holds for every
$1\leq r<p$.

In Section 6.2 we study the Boolean model based on a Poisson point process in
the percolation case. Introduced in Example 2.48 we will consider a Poisson
point process
$\mathbb{X}_{\mathrm{pois}}(\omega)=\left(x_{i}(\omega)\right)_{i\in\mathbb{N}}$
with intensity $\lambda$ (recall Example 2.48). To each point $x_{i}$ a random
ball $B_{i}=\mathbb{B}_{1}{\left(x_{i}\right)}$ is assigned and the family
$\mathbb{B}:=\left(B_{i}\right)_{i\in\mathbb{N}}$ is called the Poisson ball
process. We say that $x_{i}\sim x_{j}$ if $\left|x_{i}-x_{j}\right|<2$. In
case $\lambda>\lambda_{c}$ the union of these balls has a unique infinite
connected component (that means we have percolation) and we denote
$\mathbb{X}_{\mathrm{pois},\infty}$ the sellection of all points that
contribute to the infinite component and
$\boldsymbol{\mathrm{P}}_{\infty}\left(\omega\right):=\bigcup_{i\in\mathbb{X}_{\mathrm{pois},\infty}}B_{i}$
this infinite open set and seek for a corresponding uniform extension
operator. The connectedness of $\boldsymbol{\mathrm{P}}_{\infty}$ is hereby
essential. We use results from percolation theory that otherwise would not
hold.

Here we can show that the micro- and mesoscopic assumptions are fulfilled, at
least in case $\boldsymbol{\mathrm{P}}$ is given as the union of balls. If we
choose $\boldsymbol{\mathrm{P}}$ as the complement of the balls, the situation
becomes more involved. On one hand, Theorem 6.8 shows that $\alpha$ and $n$
change in an unfortunate way. Furthermore, the connectivity estimate remains
open. However, some of these problems might be overcome using a Matern
modification of the Poisson process. For the moment, we state the following.

######  Theorem 1.18.

In the boolean model of Section 6.2 it holds $\alpha=0$ in case
$\boldsymbol{\mathrm{P}}=\boldsymbol{\mathrm{P}}_{\infty}$ and both the
extension order and the symmetric extension order are $n=0$. If $d<p$ and

$\frac{pr}{p-r}<2,\quad r<d+2$

Then there almost surely exists an extension operator
$\mathcal{U}:\,W_{\mathrm{loc}}^{1,p}(\boldsymbol{\mathrm{P}})\to
W_{\mathrm{loc}}^{1,p}(\mathbb{R}^{d})$ and a constant $C>0$ such that for all
$m\in\mathbb{N}$ and every $u\in
W_{0,\partial\boldsymbol{\mathrm{Q}}}^{1,p}(\boldsymbol{\mathrm{P}}\cap
m\boldsymbol{\mathrm{Q}})$

$\frac{1}{\left|m\boldsymbol{\mathrm{Q}}\right|}\int_{m\boldsymbol{\mathrm{Q}}}\left|\nabla\left(\mathcal{U}u\right)\right|^{r}\leq
C\left(\frac{1}{m^{d}}\int_{\boldsymbol{\mathrm{P}}\cap
m\boldsymbol{\mathrm{Q}}}\left|\nabla u\right|^{p}\right)^{\frac{r}{p}}\,.$

If furthermore

$r<\frac{d+2}{2}$

then there almost surele exists an extension operator
$\mathcal{U}:\,\boldsymbol{\mathrm{W}}_{\mathrm{loc}}^{1,p}(\boldsymbol{\mathrm{P}})\to\boldsymbol{\mathrm{W}}_{\mathrm{loc}}^{1,p}(\mathbb{R}^{d})$
and a constant $C>0$ such that for all $m\in\mathbb{N}$ and every
$u\in\boldsymbol{\mathrm{W}}_{0,\partial\boldsymbol{\mathrm{Q}}}^{1,p}(\boldsymbol{\mathrm{P}}\cap
m\boldsymbol{\mathrm{Q}})$

$\frac{1}{\left|m\boldsymbol{\mathrm{Q}}\right|}\int_{m\boldsymbol{\mathrm{Q}}}\left|\nabla^{s}\left(\mathcal{U}u\right)\right|^{r}\leq
C\left(\frac{1}{m^{d}}\int_{\boldsymbol{\mathrm{P}}\cap
m\boldsymbol{\mathrm{Q}}}\left|\nabla^{s}u\right|^{p}\right)^{\frac{r}{p}}\,.$

In both cases for every $\beta\in(0,1)$ the following holds: for some
$m_{0}>1$ depending on $\omega$ and every $m>m_{0}$ the support of
$\mathcal{U}u$ lies within
$\mathbb{B}_{m^{1-\beta}}{\left(m\boldsymbol{\mathrm{Q}}\right)}$.

###### Proof.

The proof is given at the very end of Section 6.2. ∎

### Notes

#### Structure of the article

We close the introduction by providing an overview over the article and its
main contributions. In Section 2 we collect some basic concepts and
inequalities from the theory of Sobolev spaces, random geometries and discrete
and continuous ergodic theory. We furthermore establish local regularity
properties for what we call $\eta$-regular sets, as well as a related covering
theorem in Section 2.8. In Section 2.13 we will demonstrate that stationary
ergodic random open sets induce stationary processes on $\mathbb{Z}^{d}$, a
fact which is used later in the construction of the mesoscopic Voronoi
tessellation in Section 3.2.

In Section 3 we introduce the regularity concepts of this work. More
precisely, in Section 3.1 we introduce the concept of local
$\left(\delta,M\right)$-regularity and use the theory of Section 2.8 in order
to establish a local covering result for $\partial\boldsymbol{\mathrm{P}}$,
which will allow us to infer most of our extension and trace results. In
Section 3.2 we show how isotropic cone mixing geometries allow us to construct
a stationary Voronoi tessellation of $\mathbb{R}^{d}$ such that all related
quantities like “diameter” of the cells are stationary variables whose
expectation can be expressed in terms of the isotropic cone mixing function
$f$. Moreover we prove the important integration Lemma 3.18.

In Sections 4–5 we finally provide the aforementioned extension operators and
prove estimates for these extension operators and for the trace operator. In
Section 6 we study the sample geometries.

#### A Remark on Notation

This article uses concepts from partial differential equations, measure
theory, probability theory and random geometry. Additionally, we introduce
concepts which we believe have not been introduced before. This makes it
difficult to introduce readable self contained notation (the most important
aspect being symbols used with different meaning) and enforces the use of
various different mathematical fonts. Therefore, we provide an index of
notation at the end of this work. As a rough orientation, the reader may keep
the following in mind:

We use the standard notation $\mathbb{N}$, $\mathbb{Q}$, $\mathbb{R}$,
$\mathbb{Z}$ for natural ($>0$), rational, real and integer numbers.
$\mathbb{P}$ denotes a probability measure, $\mathbb{E}$ the expectation.
Furthermore, we use special notation for some geometrical objects, i.e.
$\mathbb{T}^{d}=[0,1)^{d}$ for the torus ($\mathbb{T}$ equipped with the
topology of the torus), $\mathbb{I}^{d}=(0,1)^{d}$ the open interval as a
subset of $\mathbb{R}^{d}$ (we often omit the index $d$), $\mathbb{B}$ a ball,
$\mathbb{C}$ a cone and $\mathbb{X}$ a set of points. In the context of finite
sets $A$, we write $\\#A$ for the number of elements.

Bold large symbols ($\boldsymbol{\mathrm{U}}$, $\boldsymbol{\mathrm{Q}}$,
$\boldsymbol{\mathrm{P}}$,$\dots$) refer to open subsets of $\mathbb{R}^{d}$
or to closed subsets with
$\partial\boldsymbol{\mathrm{P}}=\partial\mathring{\boldsymbol{\mathrm{P}}}$.
The Greek letter $\Gamma$ refers to a $d-1$ dimensional manifold (aside from
the notion of $\Gamma$-convergence).

Calligraphic symbols ($\mathcal{A}$, $\mathcal{U}$, $\dots$) usually refer to
operators and large Gothic symbols ($\mathfrak{B},\mathfrak{C},\dots$)
indicate topological spaces, except for $\mathfrak{A}$.

### Outlook

This work is the first part of a triology. In part II, we will see how to
apply the extension and trace operators introduced above.

In part III we will discuss general quantifyable properties of the geometry
that are eventually accessible also to computer algorithms that will allow to
predict homogenization behavior of random geometries.

##  2 Preliminaries

We first collect some notation and mathematical concepts which will be
frequently used throughout this paper. We first start with the standard
geometric objects, which will be labeled by bold letters.

###  2.1 Fundamental Notation and Geometric Objects

Throughout this work, we use
$\left(\boldsymbol{\mathrm{e}}_{i}\right)_{i=1,\dots d}$ for the Euclidean
basis of $\mathbb{R}^{d}$. By $C>0$ we denote any constant that depends on $p$
and $d$ but no further dependencies unless explicitly mentioned. Such
mentioning may expressed in some cases through the notation $C(a,b,\dots)$.
Furthermore, we use the following notation.

Unit cube  The torus $\mathbb{T}=[0,1)^{d}$ is quipped with the topology of
the metric $d(x,y)=\min_{z\in\mathbb{Z}^{d}}\left|x-y+z\right|$. In contrast,
the open interval $\mathbb{I}^{d}:=(0,1)^{d}$ is considered as a subset of
$\mathbb{R}^{d}$. We often omit the index $d$ if this does not provoke
confusion.

Balls  Given a metric space $\left(M,d\right)$ we denote
$\mathbb{B}_{r}{\left(x\right)}$ the open ball around $x\in M$ with radius
$r>0$. The surface of the unit ball in $\mathbb{R}^{d}$ is $\mathbb{S}^{d-1}$.
Furthermore, we denote for every $A\subset\mathbb{R}^{d}$ by
$\mathbb{B}_{r}{\left(A\right)}:=\bigcup_{x\in
A}\mathbb{B}_{r}{\left(x\right)}$.

Points  A sequence of points will be labeled by
$\mathbb{X}:=\left(x_{i}\right)_{i\in\mathbb{N}}$.

A cone  in $\mathbb{R}^{d}$ is usually labeled by $\mathbb{C}$. In particular,
we define for a vector $\nu$ of unit length, $0<\alpha<\frac{\pi}{2}$ and
$R>0$ the cone

$\mathbb{C}_{\nu,\alpha,R}(x):=\left\\{z\in\mathbb{B}_{R}{\left(x\right)}\,:\;z\cdot\nu>\left|z\right|\cos\alpha\right\\}\quad\text{and}\quad\mathbb{C}_{\nu,\alpha}(x):=\mathbb{C}_{\nu,\alpha,\infty}(x)\,.$

Inner and outer hull  We use balls of radius $r>0$ to define for a closed set
$\boldsymbol{\mathrm{P}}\subset\mathbb{R}^{d}$ the sets

$\displaystyle\boldsymbol{\mathrm{P}}_{r}$
$\displaystyle:=\overline{\mathbb{B}_{r}{\left(\boldsymbol{\mathrm{P}}\right)}}:=\left\\{x\in\mathbb{R}^{d}\,:\;\mathrm{dist}\left(x,\boldsymbol{\mathrm{P}}\right)\leq
r\right\\}\,,$ (2.1) $\displaystyle\boldsymbol{\mathrm{P}}_{-r}$
$\displaystyle:=\mathbb{R}^{d}\backslash\left[\mathbb{B}_{r}{\left(\mathbb{R}^{d}\setminus\boldsymbol{\mathrm{P}}\right)}\right]:=\left\\{x\in\mathbb{R}^{d}\,:\;\mathrm{dist}\left(x,\mathbb{R}^{d}\setminus\boldsymbol{\mathrm{P}}\right)\geq
r\right\\}\,.$

One can consider these sets as inner and outer hulls of
$\boldsymbol{\mathrm{P}}$. The last definition resembles a concept of
“negative distance” of $x\in\boldsymbol{\mathrm{P}}$ to
$\partial\boldsymbol{\mathrm{P}}$ and “positive distance” of
$x\not\in\boldsymbol{\mathrm{P}}$ to $\partial\boldsymbol{\mathrm{P}}$. For
$A\subset\mathbb{R}^{d}$ we denote $\mathrm{conv}(A)$ the closed convex hull
of $A$.

The natural geometric measures we use in this work are the Lebesgue measure on
$\mathbb{R}^{d}$, written $\left|A\right|$ for $A\subset\mathbb{R}^{d}$, and
the $k$-dimensional Hausdorff measure, denoted by $\mathcal{H}^{k}$ on
$k$-dimensional submanifolds of $\mathbb{R}^{d}$ (for $k\leq d$).

###  2.2 Simple Local Extensions and Traces

In the following, we formulate some extension and trace results. Although it
is well known how such results are proved and the proofs are standard, we
include them for completeness since we are particularly interested in the
dependence of the operator norm on the local Lipschitz regularity of the
boundary.

The following is well known:

######  Lemma 2.1.

For every $1\leq p\leq\infty$ there exists $C_{p}>0$ such that for every $R>0$
there exists an extension operator
$\mathcal{U}:\,W^{1,p}(\mathbb{B}_{R}{\left(0\right)})\to
W^{1,p}(\mathbb{B}_{2R}{\left(0\right)})$ such that

$\left\|\nabla\mathcal{U}u\right\|_{L^{p}(\mathbb{B}_{2R}{\left(0\right)})}\leq
C_{p}\left\|\nabla u\right\|_{L^{p}(\mathbb{B}_{R}{\left(0\right)})}\,.$

Let $\boldsymbol{\mathrm{P}}\subset\mathbb{R}^{d}$ be an open set and let
$p\in\partial\boldsymbol{\mathrm{P}}$ and $\delta>0$ be a constant such that
$\mathbb{B}_{\delta}{\left(p\right)}\cap\partial\boldsymbol{\mathrm{P}}$ is
graph of a Lipschitz function. We denote

$\displaystyle M(p,\delta)$
$\displaystyle:=\inf\left\\{M:\,\,\exists\phi:U\subset\mathbb{R}^{d-1}\to\mathbb{R}\text{
}\right.$ $\displaystyle\phantom{:=\inf}\left.\phi\,\text{Lipschitz, with
constant }M\text{ s.t.
}\mathbb{B}_{\delta}{\left(p\right)}\cap\partial\boldsymbol{\mathrm{P}}\text{
is graph of }\phi\right\\}\,.$ (2.2)

######  Remark 2.2.

For every $p$, the function $M(p,\cdot)$ is monotone increasing in $\delta$.

######  Lemma 2.3 (Uniform Extension for Balls).

Let $\boldsymbol{\mathrm{P}}\subset\mathbb{R}^{d}$ be an open set,
$0\in\partial\boldsymbol{\mathrm{P}}$ and assume there exists $\delta>0$,
$M>0$ and an open domain
$U\subset\mathbb{B}_{\delta}{\left(0\right)}\subset\mathbb{R}^{d-1}$ such that
$\partial\boldsymbol{\mathrm{P}}\cap\mathbb{B}_{\delta}{\left(0\right)}$ is
graph of a Lipschitz function
$\varphi:\,U\subset\mathbb{R}^{d-1}\to\mathbb{R}^{d}$ of the form
$\varphi(\tilde{x})=(\tilde{x},\phi(\tilde{x}))$ in
$\mathbb{B}_{\delta}{\left(0\right)}$ with Lipschitz constant $M$ and
$\varphi(0)=0$. Writing $x=(\tilde{x},x_{d})$ and defining
$\rho=\delta\sqrt{4M^{2}+2}^{-1}$ there exist an extension operator

$\left(\mathcal{U}u\right)(x)=\begin{cases}u(x)&\mbox{ if
}x_{d}<\phi(\tilde{x})\\\
u\left(\tilde{x},{-}x_{d}+2\phi(\tilde{x})\right)&\mbox{ if
}x_{d}>\phi(\tilde{x})\end{cases}\,,$ (2.3)

such that for

$\mathcal{A}\left(0,\boldsymbol{\mathrm{P}},\rho\right):=\left\\{\left(\tilde{x},{-}x_{d}+2\phi(\tilde{x})\right)\,:\;\left(\tilde{x},x_{d}\right)\in\mathbb{B}_{\rho}{\left(0\right)}\backslash\boldsymbol{\mathrm{P}}\right\\}\subset\mathbb{B}_{\delta}{\left(0\right)}\,,$
(2.4)

and for every $p\in[1,\infty]$ the operator

$\mathcal{U}:\,W^{1,p}{\left(\mathcal{A}\left(0,\boldsymbol{\mathrm{P}},\rho\right)\right)}\to
W^{1,p}(\mathbb{B}_{\rho}{\left(0\right)})\,,$

is continuous with

$\left\|\mathcal{U}u\right\|_{L^{p}(\mathbb{B}_{\rho}{\left(0\right)}\backslash\boldsymbol{\mathrm{P}})}\leq\left\|u\right\|_{L^{p}\left(\mathcal{A}\left(0,\boldsymbol{\mathrm{P}},\rho\right)\right)}\,,\qquad\left\|\nabla\mathcal{U}u\right\|_{L^{p}(\mathbb{B}_{\rho}{\left(0\right)}\backslash\boldsymbol{\mathrm{P}})}\leq
2M\left\|\nabla
u\right\|_{L^{p}\left(\mathcal{A}\left(0,\boldsymbol{\mathrm{P}},\rho\right)\right)}\,.$
(2.5)

######  Remark 2.4.

In case $\phi(\tilde{x})\geq 0$ we find
$\mathcal{A}\left(0,\boldsymbol{\mathrm{P}},\rho\right)\subset\mathbb{B}_{\rho}{\left(0\right)}$.

###### Proof of Lemma 2.3.

In case $\phi(\tilde{x})\equiv 0$ we consider the extension operator
$\mathcal{U}_{+}:\,W^{1,p}(\mathbb{R}^{d-1}\times({-}\infty,0))\to
W^{1,p}(\mathbb{R}^{d})$ having the form (compare also [6, chapter 5], [1])

$\left(\mathcal{U}_{+}u\right)(x)=\begin{cases}u(x)&\mbox{ if }x_{d}<0\\\
u\left(\tilde{x},{-}x_{d}\right)&\mbox{ if }x_{d}>0\end{cases}\,.$

The general case follows from transformation. ∎

######  Lemma 2.5.

Let $\boldsymbol{\mathrm{P}}\subset\mathbb{R}^{d}$ be an open set,
$0\in\partial\boldsymbol{\mathrm{P}}$ and assume there exists $\delta>0$,
$M>0$ and an open domain
$U\subset\mathbb{B}_{\delta}{\left(0\right)}\subset\mathbb{R}^{d-1}$ such that
$\partial\boldsymbol{\mathrm{P}}\cap\mathbb{B}_{\delta}{\left(0\right)}$ is
graph of a Lipschitz function
$\varphi:\,U\subset\mathbb{R}^{d-1}\to\mathbb{R}^{d}$ of the form
$\varphi(\tilde{x})=(\tilde{x},\phi(\tilde{x}))$ in
$\mathbb{B}_{\delta}{\left(0\right)}$ with Lipschitz constant $M$ and
$\varphi(0)=0$ and define $\rho=\delta\sqrt{4M^{2}+2}^{-1}$. Writing
$x=(\tilde{x},x_{d})$ we consider the trace operator
$\mathcal{T}:\,C^{1}\left(\boldsymbol{\mathrm{P}}\cap\mathbb{B}_{\delta}{\left(0\right)}\right)\to
C\left(\partial\boldsymbol{\mathrm{P}}\cap\mathbb{B}_{\rho}{\left(0\right)}\right)$.
For every $p\in[1,\infty]$ and every
$r<\frac{p\left(1-d\right)}{\left(p-d\right)}$ the operator $\mathcal{T}$ can
be continuously extended to

$\mathcal{T}:\,W^{1,p}\left(\boldsymbol{\mathrm{P}}\cap\mathbb{B}_{\delta}{\left(0\right)}\right)\to
L^{r}(\partial\boldsymbol{\mathrm{P}}\cap\mathbb{B}_{\rho}{\left(0\right)})\,,$

such that

$\left\|\mathcal{T}u\right\|_{L^{r}(\partial\boldsymbol{\mathrm{P}}\cap\mathbb{B}_{\rho}{\left(0\right)})}\leq
C_{r,p}\rho^{\frac{d\left(p-r\right)}{rp}-\frac{1}{r}}\sqrt{4M^{2}+2}^{\frac{1}{r}+1}\left\|u\right\|_{W^{1,p}\left(\boldsymbol{\mathrm{P}}\cap\mathbb{B}_{\delta}{\left(0\right)}\right)}\,.$
(2.6)

###### Proof.

We proceed similar to the proof of Lemma 2.3.

Step 1: Writing $B_{r}=\mathbb{B}_{r}{\left(0\right)}$ together with
$B_{r}^{-}=\left\\{x\in B_{r}:\,x_{d}<0\right\\}$ and
$\Sigma_{r}:=\left\\{x\in B_{r}:\,x_{d}=0\right\\}$ we recall the standard
estimate

$\left(\int_{\Sigma_{1}}\left|u\right|^{r}\right)^{\frac{1}{r}}\leq
C_{r,p}\left(\left(\int_{B_{1}^{-}}\left|\nabla
u\right|^{p}\right)^{\frac{1}{p}}+\left(\int_{B_{1}^{-}}\left|u\right|^{p}\right)^{\frac{1}{p}}\right)\,,$

which leads to

$\left(\int_{\Sigma_{\rho}}\left|u\right|^{r}\right)^{\frac{1}{r}}\leq
C_{r,p}\rho^{\frac{d\left(p-r\right)}{rp}-\frac{1}{r}}\left(\rho\left(\int_{B_{\rho}^{-}}\left|\nabla
u\right|^{p}\right)^{\frac{1}{p}}+\left(\int_{B_{\rho}^{-}}\left|u\right|^{p}\right)^{\frac{1}{p}}\right)\,.$

Step 2: Using the transformation rule and the fact that $1\leq\left|\det
D\varphi\right|\leq\sqrt{4M^{2}+2}$ we infer (2.6) similar to Step 2 in the
proof of Lemma 2.3.

$\displaystyle\left(\int_{\partial\boldsymbol{\mathrm{P}}\cap\mathbb{B}_{\rho}{\left(0\right)}}\left|u\right|^{r}\right)^{\frac{1}{r}}$
$\displaystyle\leq\sqrt{4M^{2}+2}^{\frac{1}{r}}\left(\int_{\Sigma_{\rho}}\left|u\circ\varphi\right|^{r}\right)^{\frac{1}{r}}$
$\displaystyle\leq
C_{r,p}\rho^{\frac{d\left(p-r\right)}{rp}-\frac{1}{r}}\sqrt{4M^{2}+2}^{\frac{1}{r}}\left(\rho\left(\int_{B_{\rho}^{-}}\left|\nabla\left(u\circ\varphi\right)\right|^{p}\right)^{\frac{1}{p}}+\left(\int_{B_{\rho}^{-}}\left|u\circ\varphi\right|^{p}\right)^{\frac{1}{p}}\right)$
$\displaystyle\leq
C_{r,p}\rho^{\frac{d\left(p-r\right)}{rp}-\frac{1}{r}}\sqrt{4M^{2}+2}^{\frac{1}{r}+1}\,\cdot$
$\displaystyle\qquad\cdot\left(\rho\left(\int_{B_{\rho}^{-}}\left|\left(\nabla
u\right)\circ\varphi\right|^{p}\det
D\varphi\right)^{\frac{1}{p}}+\left(\int_{B_{\rho}^{-}}\left|u\circ\varphi\right|^{p}\det
D\varphi\right)^{\frac{1}{p}}\right)$

and from this we conclude the Lemma with
$\varphi^{-1}(B_{\rho}^{-})\subset\mathbb{B}_{\delta}{\left(0\right)}$. ∎

###  2.3 Local Nitsche-Extensions

In this work, we will use bold letters for $\mathbb{R}^{d}$-valued function
spaces. In particular, we introduce for $1\leq p\leq\infty$

$\displaystyle\boldsymbol{\mathrm{L}}^{p}(\boldsymbol{\mathrm{Q}})$
$\displaystyle:=L^{p}(u;\mathbb{R}^{d})\,,$
$\displaystyle\boldsymbol{\mathrm{W}}^{1,p}(\boldsymbol{\mathrm{Q}})$
$\displaystyle:=\left\\{u\in\boldsymbol{\mathrm{L}}^{p}(\boldsymbol{\mathrm{Q}})\,:\;\nabla
u\in L^{p}{\left(\boldsymbol{\mathrm{Q}};\mathbb{R}^{d\times
d}\right)}\right\\}\,.$

From [5] we know that on general Lipschitz domains an estimate like the
following holds:

######  Lemma 2.6.

For every $1\leq p\leq\infty$ there exists a constant $C>0$ depending only on
the dimension $d\geq 2$ such that the following holds: For every radius $R>0$
there exists an extension operator
$\mathcal{U}_{R}:\,W^{1,p}{\left(\mathbb{B}_{R}{\left(0\right)}\right)}\to
W^{1,p}{\left(\mathbb{B}_{2R}{\left(0\right)}\right)}$ such that

$\left\|\nabla^{s}\left(\mathcal{U}_{R}u\right)\right\|_{W^{1,p}{\left(\mathbb{B}_{2R}{\left(0\right)}\right)}}\leq
C\left\|\nabla^{s}u\right\|_{W^{1,p}{\left(\mathbb{B}_{R}{\left(0\right)}\right)}}\,.$

Again, we will need a refined estimate on extensions on Lipschitz domains
which explicitly accounts for the local Lipschitz constant.

######  Lemma 2.7 (Uniform Nitsche-Extension for Balls).

For every $d\geq 2$ there exists a constant $C_{\mathcal{N}}$ depending only
on the dimension $d$ such that the following holds: Let
$\boldsymbol{\mathrm{P}}\subset\mathbb{R}^{d}$ be an open set,
$0\in\partial\boldsymbol{\mathrm{P}}$ and assume there exists $\delta>0$,
$M>0$ and an open domain
$U\subset\mathbb{B}_{\delta}{\left(0\right)}\subset\mathbb{R}^{d-1}$ such that
$\partial\boldsymbol{\mathrm{P}}\cap\mathbb{B}_{\delta}{\left(0\right)}$ is
graph of a Lipschitz function
$\varphi:\,U\subset\mathbb{R}^{d-1}\to\mathbb{R}^{d}$ of the form
$\varphi(\tilde{x})=(\tilde{x},\phi(\tilde{x}))$ in
$\mathbb{B}_{\delta}{\left(0\right)}$ with Lipschitz constant $M$ and
$\varphi(0)=0$. Writing $x=(\tilde{x},x_{d})$ and defining
$\rho=\delta\sqrt{4M^{2}+2}^{-1}$ and

$\mathcal{A}\left(0,\boldsymbol{\mathrm{P}},\rho\right):=\left\\{\left(\tilde{x},x_{d}\right)\in\boldsymbol{\mathrm{P}}\,:\;\left|\tilde{x}\right|<\rho,\,x_{d}\leq
C_{\mathcal{N}}(1+M^{2})\right\\}\,,$ (2.7)

and for every $p\in[1,\infty]$ there exists a continuous operator

$\mathcal{U}:\,W^{1,p}{\left(\mathcal{A}\left(0,\boldsymbol{\mathrm{P}},\rho\right)\right)}\to
W^{1,p}(\mathbb{B}_{\rho}{\left(0\right)})\,,$

such that for some constant $C$ independent from $\left(\delta,M\right)$ and
$\boldsymbol{\mathrm{P}}$ it holds

$\left\|\nabla^{s}\mathcal{U}u\right\|_{L^{p}(\mathbb{B}_{\rho}{\left(0\right)}\backslash\boldsymbol{\mathrm{P}})}\leq
C\left(1+M\right)^{2}\left\|\nabla^{s}u\right\|_{L^{p}\left(\mathcal{A}\left(0,\boldsymbol{\mathrm{P}},\rho\right)\right)}\,.$
(2.8)

######  Remark 2.8.

In case $\phi(\tilde{x})\geq 0$ the proof reveals
$\mathcal{A}\left(0,\boldsymbol{\mathrm{P}},\rho\right)\subset\mathbb{B}_{c\rho}{\left(0\right)}$
for some $c$ depending only on the dimension $d$.

In order to prove such a result we need the following lemma.

######  Lemma 2.9 ([26] Chapter 6 Section 1 Theorem 2).

There exist constants $c_{1},c_{2},c_{3}>0$ such that for every open set
$\boldsymbol{\mathrm{P}}\subset\mathbb{R}^{d}$ with local Lipschitz boundary
there exists a function
$d_{\boldsymbol{\mathrm{P}}}:\,\mathbb{R}^{d}\backslash\overline{\boldsymbol{\mathrm{P}}^{\complement}}\to\mathbb{R}$
with

$\displaystyle c_{1}d_{\boldsymbol{\mathrm{P}}}(x)$
$\displaystyle\leq\mathrm{dist}{\left(x,\boldsymbol{\mathrm{P}}\right)}$
$\displaystyle\leq c_{2}d_{\boldsymbol{\mathrm{P}}}(x)\,,$
$\displaystyle\forall i\in\left\\{1,\dots,d\right\\}:$
$\displaystyle\left|\partial_{i}d_{\boldsymbol{\mathrm{P}}}(x)\right|$
$\displaystyle\leq c_{3}\,,$ $\displaystyle\forall
i,k\in\left\\{1,\dots,d\right\\}:$
$\displaystyle\left|\partial_{i}\partial_{k}d_{\boldsymbol{\mathrm{P}}}(x)\right|$
$\displaystyle\leq c_{3}\left|d_{\boldsymbol{\mathrm{P}}}(x)\right|^{-1}\,.$

From the theory presented by Stein [26] we will not get an explicit form of
$C_{\mathcal{N}}$ but only an upper bound that grows exponentially with
dimension $d$.

###### Proof of Lemma 2.7.

We use an idea by Nitsche [20], which we transfer from $p=2$ to the general
case, thereby explicitly quantifying the influence of $M$. For simplicity we
write
$\boldsymbol{\mathrm{P}}_{\delta}:=\boldsymbol{\mathrm{P}}\cap\mathbb{B}_{\delta}{\left(0\right)}$
and
$\boldsymbol{\mathrm{P}}_{\delta}^{\complement}:=\mathbb{B}_{\delta}{\left(0\right)}\backslash\boldsymbol{\mathrm{P}}$
and assume that $x\in\boldsymbol{\mathrm{P}}_{\delta}$ iff
$x\in\mathbb{B}_{\delta}{\left(0\right)}$ and $x_{d}<\phi(\tilde{x})$.

As observed by Nitsche, it holds

$\forall x\in\boldsymbol{\mathrm{P}}_{\delta}^{\complement}:\quad
0<\left(1+M^{2}\right)^{-\frac{1}{2}}\left(x_{d}-\phi(\tilde{x})\right)\leq\mathrm{dist}(x,\partial\boldsymbol{\mathrm{P}})\leq
x_{d}-\phi(\tilde{x})\,,$

and together with Lemma 2.9, we can define
$d_{\boldsymbol{\mathrm{P}},M}(x):=2c_{2}\left(1+M^{2}\right)^{\frac{1}{2}}d_{\boldsymbol{\mathrm{P}}}(x)$
and find for $c>\max\left\\{\frac{2c_{2}}{c_{1}},4c_{2}c_{3}\right\\}$ that

$\displaystyle
2\left(x_{d}-\phi(\tilde{x})\right)\leq\mathrm{d}_{\boldsymbol{\mathrm{P}},M}(x)$
$\displaystyle\leq
c\left(1+M^{2}\right)^{\frac{1}{2}}\left(x_{d}-\phi(\tilde{x})\right)\,,$
$\displaystyle\forall i\in\left\\{1,\dots,d\right\\}:$
$\displaystyle\left|\partial_{i}d_{\boldsymbol{\mathrm{P}},M}(x)\right|$
$\displaystyle\leq c\left(1+M^{2}\right)^{\frac{1}{2}}\,,$
$\displaystyle\forall i,k\in\left\\{1,\dots,d\right\\}:$
$\displaystyle\left|\partial_{i}\partial_{k}d_{\boldsymbol{\mathrm{P}},M}(x)\right|$
$\displaystyle\leq
c\left(1+M^{2}\right)\left|d_{\boldsymbol{\mathrm{P}},M}(x)\right|^{-1}\,.$

If $\psi\in C([1,2])$ satisfies

$\int_{1}^{2}\psi(t)\,\mathrm{d}t=1\,,\qquad\int_{1}^{2}t\,\psi(t)\,\mathrm{d}t=0.$
(2.9)

Nitsche introduced $x_{\lambda}:=(\tilde{x},x_{d}-\lambda
d_{\boldsymbol{\mathrm{P}},M}(x))$ and proposed the following extension on
$x\in\boldsymbol{\mathrm{P}}_{\delta}^{\complement}$:

$u_{i}(x):=\int_{1}^{2}\psi(\lambda)\left(u_{i}(x_{\lambda})+\lambda
u_{d}(x_{\lambda})\,\partial_{i}d_{\boldsymbol{\mathrm{P}},M}(x)\right)\mathrm{d}\lambda\,.$

One can quickly verify that this maps
$C(\overline{\boldsymbol{\mathrm{P}}_{\delta}})$ onto
$C{\left(\overline{\mathbb{B}_{\rho}{\left(0\right)}}\right)}$. In what
follows, we write $\varepsilon[u](x):=\nabla^{s}u(x)$ and particularly
$\varepsilon_{ij}[u](x):=\frac{1}{2}\left(\partial_{i}u_{j}+\partial_{j}u_{i}\right)$
as well as $\varepsilon_{ij}^{\lambda}[u](x)=\varepsilon_{ij}[u](x_{\lambda})$
for $x\in\boldsymbol{\mathrm{P}}_{\delta}^{\complement}$. Then for
$x\in\boldsymbol{\mathrm{P}}_{\delta}^{\complement}\cap\mathbb{B}_{\rho}{\left(0\right)}$

$\displaystyle\varepsilon_{ij}[u](x)=\int_{1}^{2}\psi(\lambda)$
$\displaystyle\left(\varepsilon_{ij}^{\lambda}(x)+\lambda\partial_{i}d_{\boldsymbol{\mathrm{P}},M}(x)\,\varepsilon_{jd}^{\lambda}(x)+\lambda\partial_{j}d_{\boldsymbol{\mathrm{P}},M}(x)\,\varepsilon_{id}^{\lambda}(x)\right.$
(2.10)
$\displaystyle\;\left.+\lambda^{2}\partial_{i}d_{\boldsymbol{\mathrm{P}},M}(x)\,\partial_{j}d_{\boldsymbol{\mathrm{P}},M}(x)\,\varepsilon_{dd}^{\lambda}(x)+\lambda\partial_{i}\partial_{j}d_{\boldsymbol{\mathrm{P}},M}(x)\,u_{d}(x_{\lambda})\right)$
(2.11)

From the fundamental theorem of calculus we find

$u_{d}(x_{\lambda})=u_{d}(x_{1})+\delta(\tilde{x})\int_{1}^{\lambda}\partial_{d}u_{d}(x_{t})\,\mathrm{d}t\,,$

which leads by (2.9) to

$\int_{1}^{2}\psi(\lambda)\lambda\partial_{i}\partial_{j}d_{\boldsymbol{\mathrm{P}},M}(x)\,u_{d}(x_{\lambda})\,\mathrm{d}\lambda=\partial_{i}\partial_{j}d_{\boldsymbol{\mathrm{P}},M}(x)\,d_{\boldsymbol{\mathrm{P}},M}(x)\int_{1}^{2}\varepsilon_{dd}[u](x_{t})\,\mathrm{d}t\int_{\mu}^{2}\psi(\lambda)\lambda\,\mathrm{d}\lambda\,.$

We may now apply $\left|\,\cdot\,\right|^{p}$ on both sides of (2.10),
integrate over
$\boldsymbol{\mathrm{P}}_{\delta}^{\complement}\cap\mathbb{B}_{\rho}{\left(0\right)}$
and use the integral transformation theorem for each $\lambda$ to find

$\left\|\varepsilon[u]\right\|_{L^{p}{\left(\boldsymbol{\mathrm{P}}_{\delta}^{\complement}\cap\mathbb{B}_{\rho}{\left(0\right)}\right)}}\leq
C\left(1+M^{2}\right)\left\|\varepsilon[u]\right\|_{L^{p}{\left(\boldsymbol{\mathrm{P}}_{\delta}\right)}}\,.$

∎

###  2.4 Poincaré Inequalities

We denote for bounded open $A\subset\mathbb{R}^{d}$

$W_{(0),r}^{1,p}(A):=\left\\{u\in W^{1,p}(A)\,:\;\exists x:\,B_{r}(x)\subset
A\;\vee\;\fint_{B_{r}(x)}u=0\right\\}\,.$

Note that this is not a linear vector space.

######  Lemma 2.10.

For every $p\in[1,\infty)$ there exists $C_{p}>0$ such that the following
holds: Let $0<r<R$ and $x\in\mathbb{B}_{R}{\left(0\right)}$ such that
$\mathbb{B}_{r}{\left(x\right)}\subset\mathbb{B}_{R}{\left(0\right)}$ then for
every $u\in W^{1,p}(\mathbb{B}_{R}{\left(0\right)})$

$\left\|u\right\|_{L^{p}(\mathbb{B}_{R}{\left(0\right)})}^{p}\leq
C_{p}\left(R^{p}\frac{R^{d-1}}{r^{d-1}}\left\|\nabla
u\right\|_{L^{p}(\mathbb{B}_{R}{\left(0\right)})}^{p}+\frac{R^{d}}{r^{d}}\left\|u\right\|_{L^{p}(\mathbb{B}_{r}{\left(x\right)})}^{p}\right)\,,$
(2.12)

and for every $u\in W_{(0),r}^{1,p}((\mathbb{B}_{R}{\left(0\right)})$ it holds

$\left\|u\right\|_{L^{p}(\mathbb{B}_{R}{\left(0\right)})}^{p}\leq
C_{p}R^{p}\left(\frac{r}{R}\right)^{1-d}\left(1+\left(\frac{r}{R}\right)^{p-1}\right)\left\|\nabla
u\right\|_{L^{p}(\mathbb{B}_{R}{\left(0\right)})}^{p}\,.$ (2.13)

######  Remark.

In case $p\geq d$ we find that (2.13) holds iff $u(x)=0$ for some
$x\in\mathbb{B}_{1}{\left(0\right)}$.

###### Proof.

In a first step, we assume $x=0$ and $R=1$. The underlying idea of the proof
is to compare every $u(y)$,
$y\in\mathbb{B}_{1}{\left(0\right)}\backslash\mathbb{B}_{r}{\left(0\right)}$
with $u(rx)$. In particular, we obtain for
$y\in\mathbb{B}_{1}{\left(0\right)}\backslash\mathbb{B}_{r}{\left(0\right)}$
that

$u(y)=u(ry)+\int_{0}^{1}\nabla
u{\left(ry+t(1-r)y\right)}\cdot(1-r)y\,\mathrm{d}t$

and hence by Jensen’s inequality

$\left|u(y)\right|^{p}\leq C\left(\int_{0}^{1}\left|\nabla
u{\left(ry+t(1-r)y\right)}\right|^{p}(1-r)^{p}\left|y\right|^{p}\,\mathrm{d}t+\left|u(ry)\right|^{p}\right)\,.$

We integrate the last expression over
$\mathbb{B}_{1}{\left(0\right)}\backslash\mathbb{B}_{r}{\left(0\right)}$ and
find

$\displaystyle\int_{\mathbb{B}_{1}{\left(0\right)}\backslash\mathbb{B}_{r}{\left(0\right)}}\left|u(y)\right|^{p}\mathrm{d}y$
$\displaystyle\leq\int_{S^{d-1}}\int_{r}^{1}C\left(\int_{0}^{1}\left|\nabla
u{\left(rs\nu+t(1-r)s\nu\right)}\right|^{p}(1-r)^{p}s^{p}\,\mathrm{d}t\right)s^{d-1}\mathrm{d}s\mathrm{d}\nu$
$\displaystyle\quad+\int_{\mathbb{B}_{1}{\left(0\right)}\backslash\mathbb{B}_{r}{\left(0\right)}}\left|u(ry)\right|^{p}\mathrm{d}y$
$\displaystyle\leq\int_{S^{d-1}}\int_{r}^{1}C\left(\int_{rs}^{s}\left|\nabla
u{\left(t\nu\right)}\right|^{p}(1-r)^{p-1}s^{p-1}\,\mathrm{d}t\right)s^{d-1}\mathrm{d}s$
$\displaystyle\quad+\int_{\mathbb{B}_{1}{\left(0\right)}\backslash\mathbb{B}_{r}{\left(0\right)}}\left|u(ry)\right|^{p}\mathrm{d}y$
$\displaystyle\leq
C\int_{r}^{1}\mathrm{d}s\;s^{d-1}\frac{1}{\left(rs\right)^{d-1}}\int_{rs}^{s}\mathrm{d}t\;t^{d-1}\int_{S^{d-1}}\left|\nabla
u{\left(t\nu\right)}\right|^{p}(1-r)^{p-1}s^{p-1}$
$\displaystyle\quad+\int_{\mathbb{B}_{1}{\left(0\right)}\backslash\mathbb{B}_{r}{\left(0\right)}}\left|u(ry)\right|^{p}\mathrm{d}y$
$\displaystyle\leq C\frac{1}{r^{d-1}}\left\|\nabla
u\right\|_{L^{p}(\mathbb{B}_{1}{\left(0\right)})}^{p}+\frac{1}{r^{d}}\left\|u\right\|_{L^{p}(\mathbb{B}_{r}{\left(0\right)})}^{p}\,.$

For general $x\in\mathbb{B}_{1}{\left(0\right)}$, use the extension operator
$\mathcal{U}:\,W^{1,p}(\mathbb{B}_{1}{\left(0\right)})\to W^{1,p}(B_{4}(0))$
such that $\left\|\mathcal{U}u\right\|_{W^{1,p}(B_{4}(0))}\leq
C\left\|u\right\|_{W^{1,p}(\mathbb{B}_{1}{\left(0\right)})}$ and
$\left\|\nabla\mathcal{U}u\right\|_{W^{1,p}(B_{4}(0))}\leq C\left\|\nabla
u\right\|_{W^{1,p}(\mathbb{B}_{1}{\left(0\right)})}$. Since
$\mathbb{B}_{1}{\left(0\right)}\subset B_{2}(x)\subset B_{4}(0)$ we infer

$\left\|u\right\|_{L^{p}(\mathbb{B}_{1}{\left(0\right)})}^{p}\leq\left\|\mathcal{U}u\right\|_{L^{p}(B_{2}(x))}^{p}\leq
C\left(\frac{1}{r^{d-1}}\left\|\nabla\mathcal{U}u\right\|_{L^{p}(B_{2}(x))}^{p}+\frac{1}{r^{d}}\left\|\mathcal{U}u\right\|_{L^{p}(B_{r}(x))}^{p}\right)\,.$

and hence (2.12). Furthermore, since there holds
$\left\|u\right\|_{L^{p}(\mathbb{B}_{1}{\left(0\right)})}^{p}\leq
C\left\|\nabla u\right\|_{L^{p}(\mathbb{B}_{1}{\left(0\right)})}^{p}$ for
every $u\in W_{(0)}^{1,p}(\mathbb{B}_{1}{\left(0\right)})$, a scaling argument
shows $\left\|u\right\|_{L^{p}(\mathbb{B}_{r}{\left(0\right)})}^{p}\leq
Cr^{p}\left\|\nabla u\right\|_{L^{p}(\mathbb{B}_{r}{\left(0\right)})}^{p}$ for
every $u\in W_{(0),r}^{1,p}(\mathbb{B}_{1}{\left(0\right)})$ and hence (2.13).
For general $R>0$ use a scaling argument. ∎

A similar argument leads to the following, where we remark that the difference
in the appearing of $\frac{1}{r}$ is due to the fact, that integrating the
cylinder needs no surface element $r^{d-1}$.

######  Corollary 2.11.

For every $p\in[1,\infty)$ and $r>0$ there exists $C_{p}>0$ such that the
following holds: Let $r<L$,
$P_{L,r}:=\mathbb{B}_{r}^{d-1}{\left(0\right)}\times(0,L)$ and $x\in P_{L,r}$
such that $\mathbb{B}_{r}{\left(x\right)}\subset P_{L,r}$ then for every $u\in
W^{1,p}(P_{L,r})$

$\left\|u\right\|_{L^{p}(P_{L,r})}^{p}\leq C_{p}\left(L^{p}\left\|\nabla
u\right\|_{L^{p}(P_{L,r})}^{p}+\frac{L}{r}\left\|u\right\|_{L^{p}(\mathbb{B}_{r}{\left(x\right)})}^{p}\right)\,,$
(2.14)

and if additionally $\fint_{\mathbb{B}_{r}{\left(x\right)}}u=0$ then

$\left\|u\right\|_{L^{p}(P_{L,r})}^{p}\leq C_{p}\left(L^{p}\left\|\nabla
u\right\|_{L^{p}(P_{L,r})}^{p}+Lr^{p-1}\left\|\nabla
u\right\|_{L^{p}(\mathbb{B}_{r}{\left(x\right)})}^{p}\right)\,,$ (2.15)

Let $y\in P_{L,r}$ such that $\mathbb{B}_{r}{\left(y\right)}\subset P_{L,r}$
then for every $u\in W^{1,p}(P_{L,r})$

$\left|\fint_{\mathbb{B}_{r}{\left(y\right)}}u-\fint_{\mathbb{B}_{r}{\left(x\right)}}u\right|^{p}\leq
C_{p}\left(L^{p-1}r^{1-d}\left\|\nabla
u\right\|_{L^{p}(P_{L,r})}^{p}\right)\,.$ (2.16)

###  2.5 Korn Inequalities

We introduce on open sets $A\subset\mathbb{R}^{d}$ the Sobolev space

$\boldsymbol{\mathrm{W}}_{\nabla^{\bot}(0)}^{1,p}{\left(A\right)}:=\left\\{u\in\boldsymbol{\mathrm{W}}^{1,p}{\left(A\right)}:\;\forall
i,j:\,\int_{A}\partial_{i}u_{j}-\partial_{j}u_{i}=0\right\\}\,.$

To the authors best knowledge, the following is the most general Korn
inequality in literature.

######  Theorem 2.12 ([5] Theorem 2.7 and Corollary 2.8).

Let $1\leq p\leq\infty$ and $\varepsilon\in(0,1)$ and $\tilde{\delta}>0$. Then
there exists a constant $C_{p}>0$ depending only on $d$, $p$, $\varepsilon$
and $\tilde{\delta}$ such that for every bounded open set $A\in\mathbb{R}^{d}$
with $\delta>0$ such that $\delta/\mathrm{diam}A\geq\tilde{\delta}$ and with
the property

$\left.\begin{aligned} &\forall x,y\in
A,\,\left|x-y\right|<\delta:\quad\exists\gamma\in
C^{1}([0,1];A),\,\gamma(0)=x,\,\gamma(1)=y\text{ such that:}\\\ &\qquad
l(\gamma)\leq\frac{1}{\varepsilon}\left|x-y\right|\quad\text{and}\quad\forall
t\in(0,1):\,\mathrm{dist}{\left(\gamma(t),\partial
A\right)}\geq\frac{\varepsilon\left|x-\gamma(t)\right|\left|y-\gamma(t)\right|}{x-y}\end{aligned}\right\\}$
(2.17)

it holds

$\forall
u\in\boldsymbol{\mathrm{W}}_{\nabla^{\bot}(0)}^{1,p}{\left(A\right)}:\,\left\|\nabla
u\right\|_{L^{p}{\left(A\right)}}\leq
C_{p}\left\|\nabla^{s}u\right\|_{L^{p}{\left(A\right)}}\,.$ (2.18)

######  Remark 2.13.

In the original work the claimed dependence of $C_{p}$ was on $d$, $p$,
$\varepsilon$, $\delta$ and $A$ with the observation that (2.18) is invariant
under scaling of $A$. However, this scale invariance results in the dependence
on $d$, $p$, $\varepsilon$ and $\delta/\mathrm{diam}A$ since $\varepsilon$,
$p$ and $d$ are not sensitive to scaling of $A$.

######  Definition 2.14.

Domains $A\subset\mathbb{R}^{d}$ satisfying (2.17) for some
$\varepsilon\in(0,1)$ and $\delta>0$ are called $(\varepsilon,\delta)$-John
domains or simply John domains.

######  Corollary 2.15.

For every $1\leq p\leq\infty$ there exists $C_{p}$ depending only on $d$ and
$p$ such that for every bounded open convex set $A\subset\mathbb{R}^{d}$ the
estimate (2.18) holds.

We furthermore introduce the set

$\boldsymbol{\mathrm{W}}_{\nabla^{\bot}(0),r}^{1,p}(A):=\left\\{u\in\boldsymbol{\mathrm{W}}^{1,p}(A)\,:\;\exists
x:\,\mathbb{B}_{r}{\left(x\right)}\subset A\;\vee\;\forall
i,j:\,\int_{\mathbb{B}_{r}{\left(x\right)}}\partial_{i}u_{j}-\partial_{j}u_{i}=0\right\\}$

which is not a vector space.

######  Lemma 2.16 (Mixed Korn inequality).

Let $1\leq p\leq\infty$ and $\varepsilon,\delta\in(0,1)$. Then there exists a
constant $\tilde{C}_{p}>0$ depending only on $d$, $p$, $\varepsilon$ and
$\delta$ such that for every $(\varepsilon,\delta)$-John domain
$A\subset\mathbb{B}_{1}{\left(0\right)}$ and for every $r\in(0,1)$ and every
$x\in A$ with $\mathbb{B}_{r}{\left(x\right)}\subset A$ it holds

$\forall u\in\boldsymbol{\mathrm{W}}^{1,p}{\left(A\right)}:\,\left\|\nabla
u\right\|_{L^{p}{\left(A\right)}}\leq\tilde{C}_{p}\left(\frac{\left|A\right|}{r^{d}}\right)^{\frac{1}{p}}\left(\left\|\nabla^{s}u\right\|_{L^{p}{\left(A\right)}}+\left\|\nabla
u\right\|_{L^{p}{\left(\mathbb{B}_{r}{\left(x\right)}\right)}}\right)\,.$
(2.19)

Furthermore,

$\forall
u\in\boldsymbol{\mathrm{W}}_{\nabla^{\bot}(0),r}^{1,p}(A):\,\left\|\nabla
u\right\|_{L^{p}{\left(A\right)}}\leq\tilde{C}_{p}\left(\frac{\left|A\right|}{\left|\mathbb{S}^{d-1}\right|r^{d}}\right)^{\frac{1}{p}}\left(\left\|\nabla^{s}u\right\|_{L^{p}{\left(A\right)}}\right)\,.$
(2.20)

Unfortunately, we do not have a reference for a comparable Lemma in the
literature except for [25] in case $p=2$. The author strongly supposes a proof
must exist somewhere, however, we provide it for completeness.

###### Proof.

Let $C_{p}$ be the constant from Theorem 2.12 for domains with a diameter less
than $2$ and suppose (2.19) was wrong. Then there exists a sequence of
$(\varepsilon,\delta)$-John domains
$A_{n}\subset\mathbb{B}_{1}{\left(0\right)}$ with $x_{n}\in A_{n}$,
$r_{n}\in(0,1)$ with $\mathbb{B}_{r_{n}}{\left(x_{n}\right)}\subset A_{n}$ and
functions $u_{n}\in\boldsymbol{\mathrm{W}}^{1,p}{\left(A_{n}\right)}$ such
that

$1=\left\|\nabla u_{n}\right\|_{L^{p}{\left(A_{n}\right)}}\geq
C_{p}\left(\frac{\left|A_{n}\right|}{\left|\mathbb{S}^{d-1}\right|r_{n}^{d}}\right)^{\frac{1}{p}}n\left(\left\|\nabla^{s}u_{n}\right\|_{L^{p}{\left(A_{n}\right)}}+\left\|\nabla
u_{n}\right\|_{L^{p}{\left(\mathbb{B}_{r_{n}}{\left(x_{n}\right)}\right)}}\right)\,.$

We define $\overline{\nabla_{n}^{\bot}}(u_{n}):=\fint_{A_{n}}\left(\nabla
u_{n}-\nabla^{s}u_{n}\right)$ and
$u_{n,\bot}(x):=u_{n}(x)-\overline{\nabla_{n}^{\bot}}(u_{n})\,x$ with
$\nabla^{s}u_{n,\bot}=\nabla^{s}u_{n}$. Hence by (2.18)

$\displaystyle\left\|\nabla
u_{n}-\overline{\nabla_{n}^{\bot}}(u_{n})\right\|_{L^{p}{\left(A_{n}\right)}}$
$\displaystyle\leq
C_{p}\left\|\nabla^{s}u_{n,\bot}\right\|_{L^{p}{\left(A_{n}\right)}}=C_{p}\left\|\nabla^{s}u_{n}\right\|_{L^{p}{\left(A_{n}\right)}}\,.$

We directly infer with
$C_{n}:=\frac{\left|A_{n}\right|}{\left|\mathbb{S}^{d-1}\right|r_{n}^{d}}$

$C_{p}C_{n}^{\frac{1}{p}}\left(\left\|\nabla^{s}u_{n}\right\|_{L^{p}{\left(A_{n}\right)}}+\left\|\nabla
u_{n}\right\|_{L^{p}{\left(\mathbb{B}_{r_{n}}{\left(x_{n}\right)}\right)}}\right)\to
0\,,\qquad C_{n}^{\frac{1}{p}}\left\|\nabla
u_{n}-\overline{\nabla_{n}^{\bot}}(u_{n})\right\|_{L^{p}{\left(A_{n}\right)}}\to
0\,.$ (2.21)

Furthermore, we find

$\displaystyle 1=\left\|\nabla u_{n}\right\|_{L^{p}{\left(A_{n}\right)}}$
$\displaystyle\geq\left\|\overline{\nabla_{n}^{\bot}}(u_{n})\right\|_{L^{p}{\left(A_{n}\right)}}-\left\|\nabla
u_{n}-\overline{\nabla_{n}^{\bot}}(u_{n})\right\|_{L^{p}{\left(A_{n}\right)}}\,,$
$\displaystyle\left\|\overline{\nabla_{n}^{\bot}}(u_{n})\right\|_{L^{p}{\left(A_{n}\right)}}$
$\displaystyle\geq\left\|\nabla
u_{n}\right\|_{L^{p}{\left(A_{n}\right)}}-\left\|\nabla
u_{n}-\overline{\nabla_{n}^{\bot}}(u_{n})\right\|_{L^{p}{\left(A_{n}\right)}}\,,$

and hence
$\left\|\overline{\nabla_{n}^{\bot}}(u_{n})\right\|_{L^{p}{\left(A_{n}\right)}}\to
1$ due to (2.21). Since $\overline{\nabla_{n}^{\bot}}(u_{n})$ are constant, it
holds

$C_{n}\,\left\|\overline{\nabla_{n}^{\bot}}(u_{n})\right\|_{L^{p}{\left(\mathbb{B}_{r_{n}}{\left(x_{n}\right)}\right)}}^{p}=\left\|\overline{\nabla_{n}^{\bot}}(u_{n})\right\|_{L^{p}{\left(A_{n}\right)}}^{p}$

and we infer from a similar calculation

$\displaystyle C_{n}^{\frac{1}{p}}\left(\left\|\nabla
u_{n}\right\|_{L^{p}{\left(\mathbb{B}_{r_{n}}{\left(x_{n}\right)}\right)}}+\left\|\nabla
u_{n}-\overline{\nabla_{n}^{\bot}}(u_{n})\right\|_{L^{p}{\left(\mathbb{B}_{r_{n}}{\left(x_{n}\right)}\right)}}\right)$
$\displaystyle\geq
C_{n}^{\frac{1}{p}}\left\|\overline{\nabla_{n}^{\bot}}(u_{n})\right\|_{L^{p}{\left(\mathbb{B}_{r_{n}}{\left(x_{n}\right)}\right)}}$
$\displaystyle\geq\left\|\overline{\nabla_{n}^{\bot}}(u_{n})\right\|_{L^{p}{\left(A_{n}\right)}}\,.$

This implies
$\left\|\overline{\nabla_{n}^{\bot}}(u_{n})\right\|_{L^{p}{\left(A_{n}\right)}}\to
0$ by (2.21), a contradiction. Hence, (2.19) holds with $\tilde{C}_{p}=nC_{p}$
for some $n\in\mathbb{N}$.

Estimate (2.20) now follows from (2.19) and (2.18) and the definition of
$\boldsymbol{\mathrm{W}}_{\nabla^{\bot}(0),r}^{1,p}(\mathbb{B}_{R}{\left(0\right)})$.
∎

###  2.6 Korn-Poincaré Inequalities

Generalizing the above Korn inequality to a Korn-Poincaré inequality, we
define

$\displaystyle\boldsymbol{\mathrm{W}}_{(0),\nabla^{\bot}(0),r}^{1,p}{\left(\mathbb{B}_{R}{\left(0\right)}\right)}$
$\displaystyle:=\left\\{u\in\boldsymbol{\mathrm{W}}^{1,p}(\mathbb{B}_{r}{\left(0\right)})\,:\;\exists
x:\,B_{r}(x)\subset\mathbb{B}_{R}{\left(0\right)}\;\vee\right.\phantom{:=W^{1,p}\int_{\mathbb{B}_{r}{\left(x\right)}}\quad}$
$\displaystyle\phantom{:=\left\\{u\in
W^{1,p}(\mathbb{B}_{r}{\left(0\right)})\,:\right\\}}\left.\int_{\mathbb{B}_{r}{\left(x\right)}}u_{i}=0\;\vee\;\forall
i,j:\,\int_{\mathbb{B}_{r}{\left(x\right)}}\partial_{i}u_{j}-\partial_{j}u_{i}=0\right\\}\,.$

######  Lemma 2.17 (Mixed Korn-Poincaré inequality on balls).

For every $p\in[1,\infty)$ there exists $C_{p}>0$ such that for every $R>0$,
$r\in(0,R)$ and every $x\in\mathbb{B}_{R}{\left(0\right)}$ with
$\mathbb{B}_{r}{\left(x\right)}\subset\mathbb{B}_{R}{\left(0\right)}$ it holds

$\displaystyle\forall
u\in\boldsymbol{\mathrm{W}}_{(0),\nabla^{\bot}(0),r}^{1,p}{\left(\mathbb{B}_{R}{\left(0\right)}\right)}:$
$\displaystyle\left\|\nabla
u\right\|_{L^{p}{\left(\mathbb{B}_{R}{\left(0\right)}\right)}}^{p}$
$\displaystyle\leq
C_{p}\left(\frac{R}{r}\right)^{d}\left\|\nabla^{s}u\right\|_{L^{p}{\left(\mathbb{B}_{R}{\left(0\right)}\right)}}^{p}\,,$
(2.22)
$\displaystyle\left\|u\right\|_{L^{p}{\left(\mathbb{B}_{R}{\left(0\right)}\right)}}^{p}$
$\displaystyle\leq
C_{p}\left(\frac{R}{r}\right)^{2d-1}\left(1+\left(\frac{R}{r}\right)^{1-p}\right)R^{p}\left\|\nabla^{s}u\right\|_{L^{p}{\left(\mathbb{B}_{R}{\left(0\right)}\right)}}^{p}\,.$
(2.23)

###### Proof.

Apply Lemma 2.16 for $R=1$ and use a simple scaling argument to obtain

$\left\|\nabla
u\right\|_{L^{p}{\left(\mathbb{B}_{R}{\left(0\right)}\right)}}\leq
C_{p}\left(\frac{R}{r}\right)^{d}\left(\left\|\nabla^{s}u\right\|_{L^{p}{\left(\mathbb{B}_{r}{\left(0\right)}\right)}}\right)\,.$

Afterwards apply Lemma 2.10. ∎

######  Lemma 2.18 (Mixed Korn-Poincaré inequality on cylinders).

For every $p\in[1,\infty)$ and $r>0$ there exists $C_{p}>0$ such that the
following holds: Let $r<L$,
$P_{L,r}:=(0,L)\times\mathbb{B}_{r}^{d-1}{\left(0\right)}$ and $x\in P_{L,r}$
such that $\mathbb{B}_{r}{\left(x\right)}\subset P_{L,r}$ then for every
$u\in\boldsymbol{\mathrm{W}}^{1,p}(P_{L,r})$

$\left\|\nabla u\right\|_{L^{p}(P_{L,r})}^{p}\leq
C_{p}\left(\left(\frac{L}{r}\right)^{p}\left\|\nabla^{s}u\right\|_{L^{p}(P_{L,r})}^{p}+\frac{L}{r}\left\|\nabla
u\right\|_{L^{p}(\mathbb{B}_{r}{\left(x\right)})}^{p}\right)\,.$ (2.24)

Furthermore,

$\left\|u\right\|_{L^{p}(P_{L,r})}^{p}\leq
C_{p}\left(\frac{L^{2p}}{r^{p}}\left\|\nabla^{s}u\right\|_{L^{p}(P_{L,r})}^{p}+\frac{L^{p+1}}{r}\left\|\nabla
u\right\|_{L^{p}(\mathbb{B}_{r}{\left(x\right)})}^{p}+\frac{L}{r}\left\|u\right\|_{L^{p}(\mathbb{B}_{r}{\left(x\right)})}^{p}\right)\,,$
(2.25)

and if additionally
$u\in\boldsymbol{\mathrm{W}}_{(0),\nabla^{\bot}(0),r}^{1,p}(P_{L,r})$ then

$\left\|\nabla u\right\|_{L^{p}(P_{L,r})}^{p}\leq
C_{p}\frac{L^{p}}{r^{p}}\left\|\nabla^{s}u\right\|_{L^{p}(P_{L,r})}^{p}\,,\quad\left\|u\right\|_{L^{p}(P_{L,r})}^{p}\leq
C_{p}\frac{L^{2p}}{r^{p}}\left\|\nabla^{s}u\right\|_{L^{p}(P_{L,r})}^{p}\,,$
(2.26)

Defining
$\overline{\nabla_{a,\delta}^{\bot}}u:=\fint_{\mathbb{B}_{\delta}{\left(a\right)}}\left(\nabla
u-\nabla^{s}u\right)$ and

$\left[\mathcal{M}_{a}^{\mathfrak{s},\delta}u\right](x):=\overline{\nabla_{a,\delta}^{\bot}}u{\left(x-a\right)}+\fint_{\mathbb{B}_{\delta}{\left(a\right)}}u$
(2.27)

we find for $a,b$ with
$\mathbb{B}_{\delta}{\left(a\right)},\mathbb{B}_{\delta}{\left(b\right)}\subset
P_{L,r}$ for every $u\in\boldsymbol{\mathrm{W}}^{1,p}(P_{L,r})$ that

$\left|\left[\mathcal{M}_{a}^{\mathfrak{s},\delta}u\right](x)-\left[\mathcal{M}_{b}^{\mathfrak{s},\delta}u\right](x)\right|^{p}\leq
C\left|x-a\right|^{p}\frac{\left|a-b\right|^{2p}}{\delta^{p+d}}\left(\int_{\mathrm{conv}{\left(\mathbb{B}_{\delta}{\left(a\right)}\cup\mathbb{B}_{\delta}{\left(b\right)}\right)}}\left|\nabla^{s}u\right|^{p}\right)\,.$
(2.28)

Furthermore, for every $\delta<r$ we find

$\left|\left[\mathcal{M}_{a}^{\mathfrak{s},r}u\right](x)-\left[\mathcal{M}_{a}^{\mathfrak{s},\delta}u\right](x)\right|^{p}\leq
C\left(\left(\frac{\delta}{r}\right)^{-d}\left|x-a\right|^{p}+\left(\frac{\delta}{r}\right)^{1-d}(1+\left(\frac{\delta}{r}\right)^{p-d})\right)r^{p-d}\left\|\nabla^{s}u\right\|_{L^{p}{\left(\mathbb{B}_{r}{\left(a\right)}\right)}}^{p}\,.$
(2.29)

###### Proof.

_Step1:_ W.l.o.g we assume $L\in\mathbb{N}$,
$a=\frac{1}{2}\boldsymbol{\mathrm{e}}_{1}$,
$b=(L-\frac{1}{2})\boldsymbol{\mathrm{e}}_{1}$, $r=\frac{1}{2}$ and define

$\displaystyle\boldsymbol{\mathrm{P}}_{k}$
$\displaystyle:=\left(k\boldsymbol{\mathrm{e}}_{1}+[0,1)\times\mathbb{B}_{\frac{1}{2}}^{d-1}{\left(0\right)}\right)\,,\quad\boldsymbol{\mathrm{B}}_{k}:=k\boldsymbol{\mathrm{e}}_{1}+\mathbb{B}_{\frac{1}{2}}{\left(\frac{1}{2}\boldsymbol{\mathrm{e}}_{1}\right)}$
$\displaystyle\tau_{k}^{\mathfrak{s}}u(x)$
$\displaystyle:=\left[\mathcal{M}_{\left(k+\frac{1}{2}\right)\boldsymbol{\mathrm{e}}_{1}}^{\mathfrak{s},\frac{1}{2}}u\right](x)=\left[\fint_{\boldsymbol{\mathrm{B}}_{k}}\left(\nabla
u-\nabla^{s}u\right)\right]x+\fint_{\boldsymbol{\mathrm{B}}_{k}}u\,.$

Then we find by Lemma 2.16

$\displaystyle\left\|\nabla
u\right\|_{L^{p}(\boldsymbol{\mathrm{P}}_{K})}^{p}$ $\displaystyle\leq
C\left(\left\|\nabla\left(u-\tau_{K}^{\mathfrak{s}}u\right)\right\|_{L^{p}(\boldsymbol{\mathrm{P}}_{K})}^{p}+\left\|\nabla\tau_{K}^{\mathfrak{s}}u\right\|_{L^{p}(\boldsymbol{\mathrm{P}}_{K})}^{p}\right)$
$\displaystyle\leq
C\left(\left\|\nabla^{s}u\right\|_{L^{p}(\boldsymbol{\mathrm{P}}_{K})}^{p}+\left\|\nabla\tau_{K}^{\mathfrak{s}}u\right\|_{L^{p}(\boldsymbol{\mathrm{P}}_{K})}^{p}\right)\,.$

Since $\nabla\tau_{k}^{\mathfrak{s}}u$ is constant, we find

$\displaystyle\left\|\nabla\tau_{K}^{\mathfrak{s}}u\right\|_{L^{p}(\boldsymbol{\mathrm{P}}_{K})}^{p}$
$\displaystyle\leq
C\left\|\nabla\tau_{0}^{\mathfrak{s}}u\right\|_{L^{p}(\boldsymbol{\mathrm{P}}_{0})}^{p}+C\left(\sum_{k=0}^{K-1}\left\|\nabla\left(\tau_{k+1}^{\mathfrak{s}}u-\tau_{k}^{\mathfrak{s}}u\right)\right\|_{L^{1}{\left(\boldsymbol{\mathrm{P}}_{k+1}\right)}}\right)^{p}\,.$

Furthermore, we find

$\displaystyle\tau_{k}^{\mathfrak{s}}{\left(u-\tau_{k+1}^{\mathfrak{s}}u\right)}$
$\displaystyle=\fint_{\boldsymbol{\mathrm{B}}_{k}}\left(\nabla
u-\fint_{\boldsymbol{\mathrm{B}}_{k+1}}\left(\nabla
u-\nabla^{s}u\right)-\nabla^{s}u\right)x+\fint_{\boldsymbol{\mathrm{B}}_{k}}\left(u-\fint_{\boldsymbol{\mathrm{B}}_{k+1}}u\right)$
$\displaystyle=\tau_{k}^{\mathfrak{s}}u-\tau_{k+1}^{\mathfrak{s}}u=\tau_{k+1}^{\mathfrak{s}}{\left(u-\tau_{k}^{\mathfrak{s}}u\right)}\,.$

This implies by
$\nabla\tau_{k+1}^{\mathfrak{s}}\left(u-\tau_{k}^{\mathfrak{s}}u\right)=\fint_{\boldsymbol{\mathrm{B}}_{k+1}}\left(\nabla-\nabla^{s}\right)\left(u-\tau_{k}^{\mathfrak{s}}u\right)$
and Lemma 2.16 and Theorem 2.12

$\displaystyle\left\|\nabla\left(\tau_{k+1}^{\mathfrak{s}}u-\tau_{k}^{\mathfrak{s}}u\right)\right\|_{L^{p}{\left(\boldsymbol{\mathrm{P}}_{k+1}\right)}}^{p}$
$\displaystyle\leq
C\left\|\nabla\tau_{k+1}^{\mathfrak{s}}\left(u-\tau_{k}^{\mathfrak{s}}u\right)\right\|_{L^{p}{\left(\boldsymbol{\mathrm{B}}_{k+1}\right)}}^{p}$
$\displaystyle\leq
C\left\|\nabla\left(u-\tau_{k}^{\mathfrak{s}}u\right)\right\|_{L^{p}{\left(\boldsymbol{\mathrm{B}}_{k+1}\right)}}^{p}$
$\displaystyle\stackrel{{\scriptstyle\ref{lem:General-Korn}}}{{\leq}}\leq
C\left(\left\|\nabla^{s}\left(u-\tau_{k}^{\mathfrak{s}}u\right)\right\|_{L^{p}{\left(\boldsymbol{\mathrm{P}}_{k+1}\cup\boldsymbol{\mathrm{P}}_{k}\right)}}^{p}+\left\|\nabla\left(u-\tau_{k}^{\mathfrak{s}}u\right)\right\|_{L^{p}{\left(\boldsymbol{\mathrm{B}}_{k}\right)}}^{p}\right)$
$\displaystyle\stackrel{{\scriptstyle\ref{thm:Duran-
Korn}}}{{\leq}}C\left\|\nabla^{s}u\right\|_{L^{p}{\left(\boldsymbol{\mathrm{P}}_{k+1}\cup\boldsymbol{\mathrm{P}}_{k}\right)}}^{p}\,.$

Since the last inequality implies

$\left(\sum_{k=0}^{K-1}\left\|\nabla\left(\tau_{k+1}^{\mathfrak{s}}u-\tau_{k}^{\mathfrak{s}}u\right)\right\|_{L^{1}{\left(\boldsymbol{\mathrm{P}}_{k+1}\right)}}\right)^{p}\leq
K^{p-1}C\left\|\nabla^{s}u\right\|_{L^{p}{\left((0,K)\times\mathbb{B}_{1}^{d-1}{\left(0\right)}\right)}}^{p}$

and
$\left\|\nabla\tau_{0}^{\mathfrak{s}}u\right\|_{L^{p}(\boldsymbol{\mathrm{P}}_{0})}^{p}\leq
C\left(\left\|\nabla^{s}u\right\|_{L^{p}(\boldsymbol{\mathrm{P}}_{0})}^{p}+\left\|\nabla
u\right\|_{L^{p}(\boldsymbol{\mathrm{B}}_{0})}^{p}\right)$ by Lemma 2.16 we
find in total

$\left\|\nabla\tau_{K}^{\mathfrak{s}}u\right\|_{L^{p}(\boldsymbol{\mathrm{P}}_{K})}^{p}\leq
C\left\|\nabla
u\right\|_{L^{p}(\boldsymbol{\mathrm{B}}_{0})}^{p}+CK^{p-1}\left\|\nabla^{s}u\right\|_{L^{p}{\left((0,K)\times\mathbb{B}_{1}^{d-1}{\left(0\right)}\right)}}^{p}\,.$

Adding the last inequality from $K=0$ to $K=L$ implies (2.24) through scaling.
Applying Corollary 2.11 we infer that (2.25) and (2.26).

Step 2: We observe that Step 1 also holds for $P_{L,r}$ being replaced by
$\mathrm{conv}(\mathbb{B}_{\delta}{\left(a\right)}\cup\mathbb{B}_{\delta}{\left(b\right)})$.
Writing $u_{b}:=u-\mathcal{M}_{b}^{\mathfrak{s},\delta}u$ we find from the
above calculations

$\displaystyle\left|\mathcal{M}_{a}^{\mathfrak{s},\delta}u-\mathcal{M}_{b}^{\mathfrak{s},\delta}u\right|^{p}(x)$
$\displaystyle=\left|\mathcal{M}_{a}^{\mathfrak{s},\delta}\left(u-\mathcal{M}_{b}^{\mathfrak{s},\delta}u\right)\right|^{p}(x)$
$\displaystyle\leq
C\frac{1}{\delta^{d}}\left(\left|x-a\right|^{p}\int_{\mathbb{B}_{\delta}{\left(a\right)}}\left|\nabla
u_{b}-\nabla^{s}u_{b}\right|^{p}+\int_{\mathbb{B}_{\delta}{\left(a\right)}}\left|u_{b}\right|^{p}\right)\,.$

Using that
$u_{b}\in\boldsymbol{\mathrm{W}}_{(0),\nabla^{\bot}(0),r}^{1,p}(\mathrm{conv}(\mathbb{B}_{\delta}{\left(a\right)}\cup\mathbb{B}_{\delta}{\left(b\right)}))$,
we find (2.28) with help of (2.26) and Lemma 2.17.

Step 3: W.l.o.g. $a=0$. Writing
$\bar{u}(y):=u(y)-{\left(\overline{\nabla_{a,\delta}^{\bot}}u\right)}\,y$ with
$\fint_{\mathbb{B}_{r}{\left(0\right)}}u=\fint_{\mathbb{B}_{r}{\left(0\right)}}\overline{u}$
we infer (2.29) from Lemmas 2.16 and 2.10 via

$\displaystyle\left|\left[\mathcal{M}_{0}^{\mathfrak{s},1}u\right](x)-\left[\mathcal{M}_{0}^{\mathfrak{s},\delta}u\right](x)\right|^{p}$
$\displaystyle\leq C\left|\int_{\mathbb{B}_{1}{\left(0\right)}}\nabla
u-\nabla^{s}u-\overline{\nabla_{a,\delta}^{\bot}}u\right|^{p}\left|x\right|^{p}+\left|\fint_{\mathbb{B}_{1}{\left(0\right)}}\overline{u}-\fint_{\mathbb{B}_{\delta}{\left(0\right)}}\overline{u}\right|^{p}$
$\displaystyle\leq C\int_{\mathbb{B}_{1}{\left(0\right)}}\left(\left|\nabla
u-\overline{\nabla_{a,\delta}^{\bot}}u\right|^{p}+\left|\nabla^{s}u\right|^{p}\right)\left|x\right|^{p}+\fint_{\mathbb{B}_{1}{\left(0\right)}}\left|\overline{u}-\fint_{\mathbb{B}_{\delta}{\left(0\right)}}\overline{u}\right|^{p}$
$\displaystyle\leq
C\left(\delta^{-d}\left|x\right|^{p}\left\|\nabla^{s}u\right\|_{L^{p}{\left(\mathbb{B}_{1}{\left(0\right)}\right)}}^{p}+\delta^{1-d}\left(1+\delta^{p-d}\right)\left\|\nabla\overline{u}\right\|_{L^{p}{\left(\mathbb{B}_{1}{\left(0\right)}\right)}}^{p}\right)\,.$

∎

###  2.7 Voronoi Tessellations and Delaunay Triangulation

######  Definition 2.19 (Voronoi Tessellation).

Let $\mathbb{X}=\left(x_{i}\right)_{i\in\mathbb{N}}$ be a sequence of points
in $\mathbb{R}^{d}$ with $x_{i}\neq x_{k}$ if $i\neq k$. For each
$x\in\mathbb{X}$ let

$G{\left(x\right)}:=\left\\{y\in\mathbb{R}^{d}\,:\;\forall\tilde{x}\in\mathbb{X}\backslash\left\\{x\right\\}:\,\left|x-y\right|<\left|\tilde{x}-y\right|\right\\}\,.$

Then $\left(G{\left(x_{i}\right)}\right)_{i\in\mathbb{N}}$ is called the
_Voronoi tessellation_ of $\mathbb{R}^{d}$ w.r.t. $\mathbb{X}$. For each
$x\in\mathbb{X}$ we define
$d{\left(x\right)}:=\mathrm{diam}G{\left(x\right)}$.

We will need the following result on Voronoi tessellation of a minimal
diameter.

######  Lemma 2.20.

Let $\mathfrak{r}>0$ and let $\mathbb{X}=\left(x_{i}\right)_{i\in\mathbb{N}}$
be a sequence of points in $\mathbb{R}^{d}$ with
$\left|x_{i}-x_{k}\right|>2\mathfrak{r}$ if $i\neq k$. For $x\in\mathbb{X}$
let
$\mathcal{I}{\left(x\right)}:=\left\\{y\in\mathbb{X}\,:\;G{\left(y\right)}\cap\mathbb{B}_{\mathfrak{r}}{\left(G{\left(x\right)}\right)}\not=\emptyset\right\\}$.
Then $y\in\mathcal{I}(x)$ implies $\left|x-y\right|\leq 4d(x)$ and

$\\#\mathcal{I}{\left(x\right)}\leq\left(\frac{4d{\left(x\right)}}{\mathfrak{r}}\right)^{d}\,.$
(2.30)

###### Proof.

Let
$\mathbb{X}_{k}=\left\\{x_{j}\in\mathbb{X}\,:\;\mathcal{H}^{d-1}{\left(\partial
G_{k}\cap\partial G_{j}\right)}\geq 0\right\\}$ the neighbors of $x_{k}$ and
$d_{k}:=d{\left(x_{k}\right)}$. Then all $x_{j}\in\mathbb{X}$ satisfy
$\left|x_{k}-x_{j}\right|\leq 2d_{k}$. Moreover, every
$\tilde{x}\in\mathbb{X}$ with $\left|\text{$\tilde{x}$}-x_{k}\right|>4d_{k}$
has the property that $\mathrm{dist}{\left(\,\partial
G\left(\tilde{x}\right),\,x_{k}\,\right)}>2d_{k}>d_{k}+\mathfrak{r}$ and
$\tilde{x}\not\in\mathcal{I}_{k}$. Since every Voronoi cell contains a ball of
radius $\mathfrak{r}$, this implies that
$\\#\mathcal{I}_{k}\leq\left|\mathbb{B}_{4d_{k}}{\left(x_{k}\right)}\right|/\left|\mathbb{B}_{\mathfrak{r}}{\left(0\right)}\right|=\left(\frac{4d_{k}}{\mathfrak{r}}\right)^{d}$.
∎

######  Definition 2.21 (Delaunay Triangulation).

Let $\mathbb{X}=\left(x_{i}\right)_{i\in\mathbb{N}}$ be a sequence of points
in $\mathbb{R}^{d}$ with $x_{i}\neq x_{k}$ if $i\neq k$. The Delaunay
triangulation is the dual unoriented graph (see Def. LABEL:def:unoriented-
graph below) of the Voronoi tessellation, i.e. we say
$\mathbb{D}(\mathbb{X}):=\left\\{(x,y):\;\mathcal{H}^{d-1}{\left(\partial
G(x)\cap\partial G(y)\right)}\neq 0\right\\}$.

###  2.8 Local $\eta$-Regularity

Figure 1: An illustration of $\eta$-regularity. In Theorem 2.25 we will rely
on a “gray” region like in this picture.

######  Definition 2.22 ($\eta$\- regularity).

For a function $\eta:\,\partial\boldsymbol{\mathrm{P}}\to(0,r]$ we call
$\boldsymbol{\mathrm{P}}$ $\eta$-regular if

$\forall
p\in\partial\boldsymbol{\mathrm{P}},\,\varepsilon\in{\left(0,\frac{1}{2}\right)}\,,\,\tilde{p}\in\mathbb{B}_{\varepsilon\eta(p)}{\left(p\right)}\cap\partial\boldsymbol{\mathrm{P}}:\,\eta(\tilde{p})>(1-\varepsilon)\eta(p)\,.$
(2.31)

######  Remark 2.23.

This concept and its consequences from Lemma 2.24 and Theorem 2.25 will be
extensively used later to cover $\partial\boldsymbol{\mathrm{P}}$ by a
suitable family of open balls.

######  Lemma 2.24.

Let $\boldsymbol{\mathrm{P}}$ be a locally $\eta$-regular set for
$\eta:\partial\boldsymbol{\mathrm{P}}\to(0,\mathfrak{r})$. Then
$\eta:\,\boldsymbol{\mathrm{P}}\to\mathbb{R}$ is locally Lipschitz continuous
with Lipschitz constant $1$ and for every
$\varepsilon\in\left(0,\frac{1}{2}\right)$ and
$\tilde{p}\in\mathbb{B}_{\varepsilon\eta}{\left(p\right)}\cap\boldsymbol{\mathrm{P}}$
it holds

$\frac{1-\varepsilon}{1-2\varepsilon}\eta{\left(p\right)}>\eta{\left(\tilde{p}\right)}>\eta{\left(p\right)}-\left|p-\tilde{p}\right|>\left(1-\varepsilon\right)\eta{\left(p\right)}\,.$
(2.32)

Furthermore,

$\left|p-\tilde{p}\right|\leq\varepsilon\max\left\\{\eta{\left(p\right)},\eta{\left(\tilde{p}\right)}\right\\}\quad\Rightarrow\quad\left|p-\tilde{p}\right|\leq\frac{\varepsilon}{1-\varepsilon}\min\left\\{\eta{\left(p\right)},\eta{\left(\tilde{p}\right)}\right\\}$
(2.33)

###### Proof.

Let $p,\tilde{p}$ such that $\left|\tilde{p}-p\right|<\frac{1}{2}\eta(p)$ with
$\varepsilon_{p,\tilde{p}}:=\inf\left\\{\varepsilon:\,\left|\tilde{p}-p\right|<\varepsilon\eta(p)\right\\}$.
This means $\varepsilon\in[\varepsilon_{p,\tilde{p}},\frac{1}{2})$ iff
$\eta{\left(\tilde{p}\right)}\geq\left(1-\varepsilon\right)\eta(p)$ and we
find

$\displaystyle\eta{\left(\tilde{p}\right)}$
$\displaystyle\geq\eta{\left(p\right)}-\left|p-\tilde{p}\right|=\eta{\left(p\right)}-\varepsilon_{p,\tilde{p}}\eta{\left(p\right)}>\left(1-\varepsilon\right)\eta{\left(p\right)}$

which implies
$\left|\tilde{p}-p\right|<\frac{\varepsilon}{1-\varepsilon}\eta{\left(\tilde{p}\right)}$
and the local Lipschitz continuity by a symmetry argument in $p$, $\tilde{p}$.
This in turn leads to
$\eta{\left(p\right)}>\left(1-\frac{\varepsilon}{1-\varepsilon}\right)\eta{\left(\tilde{p}\right)}$
or

$\eta{\left(p\right)}=\frac{1-\varepsilon}{1-\varepsilon}\eta{\left(p\right)}<\frac{1}{1-\varepsilon}\left(\eta{\left(p\right)}-\left|p-\tilde{p}\right|\right)<\frac{1}{1-\varepsilon}\eta{\left(\tilde{p}\right)}\leq\frac{1}{1-2\varepsilon}\eta{\left(p\right)}\,,$

implying (2.32) and continuity of $\eta$.

In order to prove (2.33), w.l.o.g. let
$\eta{\left(\tilde{p}\right)}\leq\eta{\left(p\right)}$. Then

$\left|p-\tilde{p}\right|\leq\varepsilon\eta{\left(p\right)}\leq\frac{\varepsilon}{1-\varepsilon}\eta{\left(\tilde{p}\right)}\,.$

∎

######  Theorem 2.25.

Let $\Gamma\subset\mathbb{R}^{d}$ be a closed set and let
$\eta{\left(\cdot\right)}\in C{\left(\Gamma\right)}$ be bounded and satisfy
for every $\varepsilon\in\left(0,\frac{1}{2}\right)$ and for
$\left|p-\tilde{p}\right|<\varepsilon\eta{\left(p\right)}$

$\frac{1-\varepsilon}{1-2\varepsilon}\,\eta{\left(p\right)}>\eta{\left(\tilde{p}\right)}>\eta{\left(p\right)}-\left|p-\tilde{p}\right|>\left(1-\varepsilon\right)\eta{\left(p\right)}\,.$
(2.34)

and define $\tilde{\eta}{\left(p\right)}=2^{-K}\eta{\left(p\right)}$, $K\geq
2$. Then for every $C\in(0,1)$ there exists a locally finite covering of
$\Gamma$ with balls
$\mathbb{B}_{\tilde{\eta}{\left(p_{k}\right)}}{\left(p_{k}\right)}$ for a
countable number of points
${\left(p_{k}\right)}_{k\in\mathbb{N}}\subset\Gamma$ such that for every
$i\neq k$ with
$\mathbb{B}_{\tilde{\eta}{\left(p_{i}\right)}}{\left(p_{i}\right)}\cap\mathbb{B}_{\tilde{\eta}{\left(p_{k}\right)}}{\left(p_{k}\right)}\neq\emptyset$
it holds

$\displaystyle\frac{2^{K-1}-1}{2^{K-1}}\tilde{\eta}{\left(p_{i}\right)}\leq\tilde{\eta}{\left(p_{k}\right)}\leq\frac{2^{K-1}}{2^{K-1}-1}\tilde{\eta}{\left(p_{i}\right)}$
(2.35) and
$\displaystyle\frac{2^{K}-1}{2^{K-1}-1}\min{\left\\{\tilde{\eta}{\left(p_{i}\right)},\tilde{\eta}{\left(p_{k}\right)}\right\\}}\geq\left|p_{i}-p_{k}\right|\geq
C\max{\left\\{\tilde{\eta}{\left(p_{i}\right)},\tilde{\eta}{\left(p_{k}\right)}\right\\}}$

###### Proof.

We chose $\delta>0$, $n\in\mathbb{N}$ such that
$\left(1-\frac{1}{n}\right)\left(1-\delta\right)>C$. W.o.l.g. assume
$\tilde{\eta}<(1-\delta)$. Consider
$\tilde{Q}:=\left[0,\frac{1}{n}\right]^{d}$, let $q_{1,\dots,n^{d}}$ denote
the $n^{d}$ elements of $[0,1)^{d}\cap\frac{\mathbb{Q}^{d}}{n}$ and let
$\tilde{Q}_{z,i}=\tilde{Q}+z+q_{i}$, $z\in\mathbb{Z}^{d}$. We set
$B_{(0)}:=\emptyset$, $\Gamma_{1}=\Gamma$,
$\eta_{k}:=\left(1-\delta\right)^{k}$ and for $k\geq 1$ we construct the
covering using inductively defined open sets $B_{(k)}$ and closed set
$\Gamma_{k}$ as follows:

  1. 1.

Define $\Gamma_{k,1}=\Gamma_{k}$. For $i=1,\dots,n^{d}$ do the following:

    1. (a)

For every $z\in\mathbb{Z}^{d}$ do

$\displaystyle\text{if }\exists
p\in\left(\eta_{k}\tilde{Q}_{z,i}\right)\cap\Gamma_{k,i},\,\tilde{\eta}{\left(p\right)}\in(\eta_{k},\eta_{k-1}]$
then set $\displaystyle
b_{z,i}=\mathbb{B}_{\tilde{\eta}{\left(p\right)}}{\left(p\right)}\,,\;\mathbb{X}_{z,i}=\left\\{p\right\\}$
otherwise set $\displaystyle
b_{z,i}=\emptyset\,,\;\mathbb{X}_{z,i}=\emptyset\,.$

    2. (b)

Define $B_{(k),i}:=\bigcup_{z\in\mathbb{Z}^{d}}b_{z,i}$ and
$\Gamma_{k,i+1}=\Gamma_{k,i}\backslash B_{(k),i}$ and
$\mathbb{X}_{(k),i}:=\bigcup_{z\in\mathbb{Z}^{d}}\mathbb{X}_{z,i}$.
Observe: $p_{1},p_{2}\in\mathbb{X}_{(k),i}$ implies
$\left|p_{1}-p_{2}\right|>\left(1-\frac{1}{n}\right)\eta_{k}$ and
$p_{3}\in\mathbb{X}_{(k),j}$, $j<i$ implies
$p_{1}\not\in\mathbb{B}_{\eta_{k}}{\left(p_{3}\right)}$ and hence
$\left|p_{1}-p_{3}\right|>\eta_{k}$. Similar, $p_{3}\in\mathbb{X}_{l}$, $l<k$,
implies $\left|p_{1}-p_{3}\right|>\eta_{l}>\eta_{k}$.

  2. 2.

Define $\Gamma_{k+1}:=\Gamma_{k,n^{d}+1}$,
$\mathbb{X}_{k}:=\bigcup_{i}\mathbb{X}_{(k),i}$.

The above covering of $\Gamma$ is complete in the sense that every
$x\in\Gamma$ lies in one of the balls (by contradiction). We denote
$\mathbb{X}:=\bigcup_{k}\mathbb{X}_{k}=\left(p_{i}\right)_{i\in\mathbb{N}}$
the family of centers of the above constructed covering of $\Gamma$ and find
the following properties: Let $p_{1},p_{2}\in\mathbb{X}$ be such that
$\mathbb{B}_{\tilde{\eta}{\left(p_{1}\right)}}{\left(p_{1}\right)}\cap\mathbb{B}_{\tilde{\eta}{\left(p_{2}\right)}}{\left(p_{2}\right)}\neq\emptyset$.
W.l.o.g. let
$\tilde{\eta}{\left(p_{1}\right)}\geq\tilde{\eta}{\left(p_{2}\right)}$. Then
the following two properties are satisfied due to (2.34)

  1. 1.

It holds $\left|p_{1}-p_{2}\right|\leq
2\tilde{\eta}{\left(p_{1}\right)}\leq\frac{1}{2^{K-1}}\eta{\left(p_{1}\right)}$
and hence
$\mathbb{B}_{\tilde{\eta}{\left(p_{2}\right)}}{\left(p_{2}\right)}\subset\mathbb{B}_{2^{2-K}\eta{\left(p_{1}\right)}}{\left(p_{1}\right)}$
and
$\eta{\left(p_{2}\right)}\geq\frac{2^{K-1}-1}{2^{K-1}}\eta{\left(p_{1}\right)}$.
Furthermore
$\tilde{\eta}{\left(p_{1}\right)}\geq\tilde{\eta}{\left(p_{2}\right)}\geq\frac{2^{K-1}-1}{2^{K-1}}\tilde{\eta}{\left(p_{1}\right)}$.

  2. 2.

Let $k$ such that
$\tilde{\eta}{\left(p_{1}\right)}\in\left(\eta_{k},\eta_{k+1}\right]$. If also
$\tilde{\eta}{\left(p_{2}\right)}\in\left(\eta_{k},\eta_{k+1}\right]$ then the
observation in Step 1.(b) implies
$\left|p_{1}-p_{2}\right|\geq\left(1-\frac{1}{n}\right)\eta_{k}\geq\left(1-\frac{1}{n}\right)\left(1-\delta\right)\tilde{\eta}{\left(p_{1}\right)}$.
If $\tilde{\eta}{\left(p_{2}\right)}\not\in\left[\eta_{k},\eta_{k+1}\right)$
then $\tilde{\eta}{\left(p_{2}\right)}<\eta_{k}$ and hence
$p_{2}\not\in\mathbb{B}_{\tilde{\eta}{\left(p_{1}\right)}}{\left(p_{1}\right)}$,
implying $\left|p_{1}-p_{2}\right|>\tilde{\eta}{\left(p_{1}\right)}$.

Due to our choice of $n$ and $\delta$, this concludes the proof. ∎

###  2.9 Dynamical Systems

######  Assumption 2.26.

Throughout this work we assume that $(\Omega,\mathscr{F},\mathbb{P})$ is a
probability space with countably generated $\sigma$-algebra $\mathscr{F}$.

Due to the insight in [10], shortly sketched in the next two subsections,
after a measurable transformation the probability space $\Omega$ can be
assumed to be metric and separable, which always ensures Assumption 2.26.

######  Definition 2.27 (Dynamical system).

A dynamical system on $\Omega$ is a family $(\tau_{x})_{x\in\mathbb{R}^{d}}$
of measurable bijective mappings $\tau_{x}:\Omega\mapsto\Omega$ satisfying
(i)-(iii):

  1. (i)

$\tau_{x}\circ\tau_{y}=\tau_{x+y}$ , $\tau_{0}=id$ (Group property)

  2. (ii)

$\mathbb{P}(\tau_{-x}B)=\mathbb{P}(B)\quad\forall
x\in\mathbb{R}^{d},\,\,B\in\mathscr{F}$ (Measure preserving)

  3. (iii)

$A:\,\,\mathbb{R}^{d}\times\Omega\rightarrow\Omega\qquad(x,\omega)\mapsto\tau_{x}\omega$
is measurable (Measurability of evaluation)

A set $A\subset\Omega$ is almost invariant if
$\mathbb{P}\left(\left(A\cup\tau_{x}A\right)\backslash\left(A\cap\tau_{x}A\right)\right)=0$.
The family

$\mathscr{I}=\left\\{A\in\mathscr{F}\,:\;\forall
x\in\mathbb{R}^{d}\,\mathbb{P}\left(\left(A\cup\tau_{x}A\right)\backslash\left(A\cap\tau_{x}A\right)\right)=0\right\\}$
(2.36)

of almost invariant sets is $\sigma$-algebra and

$\mathbb{E}\left(f|\mathscr{I}\right)\text{denotes the expectation of
}f:\,\Omega\to\mathbb{R}\text{ w.r.t. }\mathscr{I}\,.$ (2.37)

A concept linked to dynamical systems is the concept of stationarity.

######  Definition 2.28 (Stationary).

Let $X$ be a measurable space and let $f:\Omega\times\mathbb{R}^{d}\to X$.
Then $f$ is called (weakly) stationary if $f(\omega,x)=f(\tau_{x}\omega,0)$
for (almost) every $x$.

######  Definition 2.29.

A family $\left(A_{n}\right)_{n\in\mathbb{N}}\subset\mathbb{R}^{d}$ is called
convex averaging sequence if

  1. (i)

each $A_{n}$ is convex

  2. (ii)

for every $n\in\mathbb{N}$ holds $A_{n}\subset A_{n+1}$

  3. (iii)

there exists a sequence $r_{n}$ with $r_{n}\to\infty$ as $n\to\infty$ such
that $B_{r_{n}}(0)\subseteq A_{n}$.

We sometimes may take the following stronger assumption.

######  Definition 2.30.

A convex averaging sequence $A_{n}$ is called regular if

$\left|A_{n}\right|^{-1}\\#\left\\{z\in\mathbb{Z}^{d}\,:\;\left(z+\mathbb{T}\right)\cap\partial
A_{n}\neq\emptyset\right\\}\to 0\,.$

The latter condition is evidently fulfilled for sequences of cones or balls.
Convex averaging sequences are important in the context of ergodic theorems.

######  Theorem 2.31 (Ergodic Theorem [4] Theorems 10.2.II and also [27]).

Let $\left(A_{n}\right)_{n\in\mathbb{N}}\subset\mathbb{R}^{d}$ be a convex
averaging sequence, let $(\tau_{x})_{x\in\mathbb{R}^{d}}$ be a dynamical
system on $\Omega$ with invariant $\sigma$-algebra $\mathscr{I}$ and let
$f:\,\Omega\to\mathbb{R}$ be measurable with
$\left|\mathbb{E}(f)\right|<\infty$. Then for almost all $\omega\in\Omega$

$\left|A_{n}\right|^{-1}\int_{A_{n}}f{\left(\tau_{x}\omega\right)}\,\mathrm{d}x\to\mathbb{E}{\left(f|\mathscr{I}\right)}\,.$
(2.38)

We observe that $\mathbb{E}\left(f|\mathscr{I}\right)$ is of particular
importance. For the calculations in this work, we will particularly focus on
the case of trivial $\mathscr{I}$. This is called ergodicity, as we will
explain in the following.

######  Definition 2.32 (Ergodicity and mixing).

A dynamical system $(\tau_{x})_{x\in\mathbb{R}^{d}}$ on a probability space
$(\Omega,\mathscr{F},\mathbb{P})$ is called _mixing_ if for every measurable
$A,B\subset\Omega$ it holds

$\lim_{\left\|x\right\|\to\infty}\mathbb{P}{\left(A\cap\tau_{x}B\right)}=\mathbb{P}(A)\,\mathbb{P}(B)\,.$
(2.39)

A dynamical system is called _ergodic_ if

$\lim_{n\to\infty}\frac{1}{\left(2n\right)^{d}}\int_{[-n,n]^{d}}\mathbb{P}{\left(A\cap\tau_{x}B\right)}\mathrm{d}x=\mathbb{P}(A)\,\mathbb{P}(B)\,.$
(2.40)

######  Remark 2.33.

a) Let $\Omega=\left\\{\omega_{0}=0\right\\}$ with the trivial
$\sigma$-algebra and $\tau_{x}\omega_{0}=\omega_{0}$. Then $\tau$ is evidently
mixing. However, the realizations are constant functions $f_{\omega}(x)=c$ on
$\mathbb{R}^{d}$ for some constant $c$.

b) A typical ergodic system is given by $\Omega=\mathbb{T}$ with the Lebesgue
$\sigma$-algebra and $\mathbb{P}=\mathcal{L}$ the Lebesgue measure. The
dynamical system is given by $\tau_{x}y:=\left(x+y\right)\\!\mod\mathbb{T}$.

c) It is known that $(\tau_{x})_{x\in\mathbb{R}^{d}}$ is ergodic if and only
if every almost invariant set $A\in\mathscr{I}$ has probability
$\mathbb{P}(A)\in\left\\{0,1\right\\}$ (see [4] Proposition 10.3.III) i.e.

$\left[\,\,\forall x\,\mathbb{P}{\left(\left(\tau_{x}A\cup
A\right)\backslash\left(\tau_{x}A\cap
A\right)\right)}=0\,\,\right]\;\Rightarrow\;\mathbb{P}{\left(A\right)}\in\left\\{0,1\right\\}\,.$
(2.41)

d) It is sufficient to show (2.39) or (2.40) for $A$ and $B$ in a ring that
generates the $\sigma$-algebra $\mathscr{F}$. We refer to [4], Section 10.2,
for the later results.

A further useful property of ergodic dynamical systems, which we will use
below, is the following:

######  Lemma 2.34 (Ergodic times mixing is ergodic).

Let $(\tilde{\Omega},\tilde{\mathscr{F}},\tilde{\mathbb{P}})$ and
$(\hat{\Omega},\hat{\mathscr{F}},\hat{\mathbb{P}})$ be probability spaces with
dynamical systems $(\tilde{\tau}_{x})_{x\in\mathbb{R}^{d}}$ and
$(\hat{\tau}_{x})_{x\in\mathbb{R}^{d}}$ respectively. Let
$\Omega:=\tilde{\Omega}\times\hat{\Omega}$ be the usual product measure space
with the notation $\omega=(\tilde{\omega},\hat{\omega})\in\Omega$ for
$\tilde{\omega}\in\tilde{\Omega}$ and $\hat{\omega}\in\hat{\Omega}$. If
$\tilde{\tau}$ is ergodic and $\hat{\tau}$ is mixing, then
$\tau_{x}(\tilde{\omega},\hat{\omega}):=(\tilde{\tau}_{x}\tilde{\omega},\hat{\tau}_{x}\hat{\omega})$
is ergodic.

###### Proof.

Relying on Remark 2.33.c) we verify (2.40) by proving it for sets
$A=\tilde{A}\times\hat{A}$ and $B=\tilde{B}\times\hat{B}$ which generate
$\mathscr{F}:=\tilde{\mathscr{F}}\otimes\hat{\mathscr{F}}$. We make use of
$A\cap
B=\left(\tilde{A}\cap\tilde{B}\right)\times\left(\hat{A}\cap\hat{B}\right)$
and observe that

$\displaystyle\mathbb{P}{\left(A\cap\tau_{x}B\right)}$
$\displaystyle=\mathbb{P}{\left(\left(\tilde{A}\cap\tilde{\tau}_{x}\tilde{B}\right)\times\left(\hat{A}\cap\hat{\tau}_{x}\hat{B}\right)\right)}=\hat{\mathbb{P}}{\left(\hat{A}\cap\hat{\tau}_{x}\hat{B}\right)}\,\tilde{\mathbb{P}}{\left(\tilde{A}\cap\tilde{\tau}_{x}\tilde{B}\right)}$
$\displaystyle=\hat{\mathbb{P}}{\left(\hat{A}\cap\hat{B}\right)}\,\tilde{\mathbb{P}}{\left(\tilde{A}\cap\tilde{\tau}_{x}\tilde{B}\right)}+\left[\hat{\mathbb{P}}{\left(\hat{A}\cap\hat{\tau}_{x}\hat{B}\right)}-\hat{\mathbb{P}}{\left(\hat{A}\cap\hat{B}\right)}\right]\,\tilde{\mathbb{P}}{\left(\tilde{A}\cap\tilde{\tau}_{x}\tilde{B}\right)}\,.$

Using ergodicity, we find that

$\displaystyle\lim_{n\to\infty}\frac{1}{\left(2n\right)^{d}}\int_{[-n,n]^{d}}\hat{\mathbb{P}}{\left(\hat{A}\cap\hat{B}\right)}\,\tilde{\mathbb{P}}{\left(\tilde{A}\cap\tilde{\tau}_{x}\tilde{B}\right)}\mathrm{d}x$
$\displaystyle=\hat{\mathbb{P}}{\left(\left(\hat{A}\cap\hat{B}\right)\right)}\,\tilde{\mathbb{P}}{\left(\tilde{A}\cap\tilde{B}\right)}$
$\displaystyle=\mathbb{P}{\left(A\cap B\right)}\,.$ (2.42)

Since $\hat{\tau}$ is mixing, we find for every $\varepsilon>0$ some $R>0$
such that $\left\|x\right\|>R$ implies
$\left|\hat{\mathbb{P}}{\left(\hat{A}\cap\hat{\tau}_{x}\hat{B}\right)}-\hat{\mathbb{P}}{\left(\hat{A}\cap\hat{B}\right)}\right|<\varepsilon$.
For $n>R$ we find

$\frac{1}{\left(2n\right)^{d}}\int_{[-n,n]^{d}}\left|\hat{\mathbb{P}}{\left(\hat{A}\cap\hat{\tau}_{x}\hat{B}\right)}-\hat{\mathbb{P}}{\left(\hat{A}\cap\hat{B}\right)}\right|\,\tilde{\mathbb{P}}{\left(\tilde{A}\cap\tilde{\tau}_{x}\tilde{B}\right)}\\\
\leq\frac{1}{\left(2n\right)^{d}}\int_{[-n,n]^{d}}\varepsilon+\frac{1}{\left(2n\right)^{d}}\int_{[-R,R]^{d}}2\to\varepsilon\quad\text{as
}n\to\infty\,.$ (2.43)

The last two limits (2.42) and (2.43) imply (2.40). ∎

######  Remark 2.35.

The above proof heavily relies on the mixing property of $\hat{\tau}$. Note
that for $\hat{\tau}$ being only ergodic, the statement is wrong, as can be
seen from the product of two periodic processes in
$\mathbb{T}\times\mathbb{T}$ (see Remark 2.33). Here, the invariant sets are
given by
$I_{A}:=\left\\{\left(\left(y+x\right)\\!\mod\mathbb{T}\,,\,x\right)\,:\;y\in
A\right\\}$ for arbitrary measurable $A\subset\mathbb{T}$.

###  2.10 Random Measures and Palm Theory

We recall some facts from random measure theory (see [4]) which will be needed
for homogenization. Let $\mathfrak{M}(\mathbb{R}^{d})$ denote the space of
locally bounded Borel measures on $\mathbb{R}^{d}$ (i.e. bounded on every
bounded Borel-measurable set) equipped with the Vague topology, which is
generated by the sets

$\left\\{\mu\,:\;\int f\,\mathrm{d}\mu\in A\right\\}\text{ for every open
}A\subset\mathbb{R}^{d}\text{ and }f\in C_{c}{\left(\mathbb{R}^{d}\right)}\,.$

This topology is metrizable, complete and countably generated. A random
measure is a measurable mapping

$\mu_{\bullet}:\;\Omega\to\mathfrak{M}(\mathbb{R}^{d})\,,\qquad\omega\mapsto\mu_{\omega}$

which is equivalent to both of the following conditions

  1. 1.

For every bounded Borel set $A\subset\mathbb{R}^{d}$ the map
$\omega\mapsto\mu_{\omega}(A)$ is measurable

  2. 2.

For every $f\in C_{c}(\mathbb{R}^{d})$ the map $\omega\mapsto\int
f\,\mathrm{d}\mu_{\omega}$ is measurable.

A random measure is stationary if the distribution of $\mu_{\omega}(A)$ is
invariant under translations of $A$ that is $\mu_{\omega}(A)$ and
$\mu_{\omega}(A+x)$ share the same distribution. From stationarity of
$\mu_{\omega}$ one concludes the existence ([10, 22] and references therein)
of a dynamical system $\left(\tau_{x}\right)_{x\in\mathbb{R}^{d}}$ on $\Omega$
such that $\mu_{\omega}\left(A+x\right)=\mu_{\tau_{x}\omega}\left(A\right)$.
By a deep theorem due to Mecke (see [19, 4]) the measure

$\mu_{\mathcal{P}}(A)=\int_{\Omega}\int_{\mathbb{R}^{d}}g(s)\,\chi_{A}(\tau_{s}\omega)\,\mathrm{d}\mu_{\omega}(s)\,\mathrm{d}\mathbb{P}(\omega)$

can be defined on $\Omega$ for every positive $g\in L^{1}(\mathbb{R}^{d})$
with compact support. $\mu_{\mathcal{P}}$ is independent from $g$ and in case
$\mu_{\omega}=\mathcal{L}$ we find $\mu_{\mathcal{P}}=\mathbb{P}$.
Furthermore, for every
$\mathcal{B}(\mathbb{R}^{d})\times\mathcal{B}(\Omega)$-measurable non negative
or $\mu_{\mathcal{P}}\times\mathcal{L}$\- integrable functions $f$ the
Campbell formula

$\int_{\Omega}\int_{\mathbb{R}^{d}}f(x,\tau_{x}\omega)\,\mathrm{d}\mu_{\omega}(x)\,\mathrm{d}\mathbb{P}(\omega)=\int_{\mathbb{R}^{d}}\int_{\Omega}f(x,\omega)\,\mathrm{d}\mu_{\mathcal{P}}(\omega)\,\mathrm{d}x$

holds. The measure $\mu_{\omega}$ has finite intensity if
$\mu_{\mathcal{P}}(\Omega)<+\infty$.

We denote by

$\mathbb{E}_{\mu_{\mathcal{P}}}{\left(f|\mathscr{I}\right)}:=\int_{\Omega}f\text{
the expectation of }f\text{ w.r.t. the }\sigma\text{-algebra
}\mathscr{I}\text{ and }\mu_{\mathcal{P}}\,.$ (2.44)

For random measures we find a more general version of Theorem 2.31.

######  Theorem 2.36 (Ergodic Theorem [4] 12.2.VIII).

Let $\left(\Omega,\mathscr{F},\mathbb{P}\right)$ be a probability space,
$\left(A_{n}\right)_{n\in\mathbb{N}}\subset\mathbb{R}^{d}$ be a convex
averaging sequence, let $(\tau_{x})_{x\in\mathbb{R}^{d}}$ be a dynamical
system on $\Omega$ with invariant $\sigma$-algebra $\mathscr{I}$ and let
$f:\,\Omega\to\mathbb{R}$ be measurable with
$\int_{\Omega}\left|f\right|\mathrm{d}\mu_{\mathcal{P}}<\infty$. Then for
$\mathbb{P}$-almost all $\omega\in\Omega$

$\left|A_{n}\right|^{-1}\int_{A_{n}}f{\left(\tau_{x}\omega\right)}\,\mathrm{d}\mu_{\omega}(x)\to\mathbb{E}_{\mu_{\mathcal{P}}}{\left(f|\mathscr{I}\right)}\,.$
(2.45)

Given a bounded open (and convex) set $\boldsymbol{\mathrm{Q}}\subset\Omega$,
it is not hard to see that the following generalization holds:

######  Theorem 2.37 (General Ergodic Theorem).

Let $\left(\Omega,\mathscr{F},\mathbb{P}\right)$ be a probability space,
$\boldsymbol{\mathrm{Q}}\subset\mathbb{R}^{d}$ be a bounded open set with
$0\in\boldsymbol{\mathrm{Q}}$, let $(\tau_{x})_{x\in\mathbb{R}^{d}}$ be a
dynamical system on $\Omega$ with invariant $\sigma$-algebra $\mathscr{I}$ and
let $f:\,\Omega\to\mathbb{R}$ be measurable with
$\int_{\Omega}\left|f\right|\mathrm{d}\mu_{\mathcal{P}}<\infty$. Then for
$\mathbb{P}$-almost all $\omega\in\Omega$ it holds

$\forall\varphi\in C_{0}(\boldsymbol{\mathrm{Q}}):\quad
n^{-d}\int_{n\boldsymbol{\mathrm{Q}}}\varphi(\frac{x}{n})f{\left(\tau_{x}\omega\right)}\,\mathrm{d}\mu_{\omega}(x)\to\mathbb{E}_{\mu_{\mathcal{P}}}{\left(f|\mathscr{I}\right)}\int_{\boldsymbol{\mathrm{Q}}}\varphi\,.$
(2.46)

###### Sketch of proof.

Chose a countable dense family of functions $\varphi\in
C_{0}(\boldsymbol{\mathrm{Q}})$ that spans $L^{1}(\boldsymbol{\mathrm{Q}})$
and that have support on a ball. Use a Cantor argument and Theorem 2.36 to
prove the statement for a countable dense family of
$C_{0}(\boldsymbol{\mathrm{Q}})$. From here, we conclude by density.

The last result can be used to prove the most general ergodic theorem which we
will use in this work: ∎

######  Theorem 2.38 (General Ergodic Theorem for the Lebesgue measure).

Let $\left(\Omega,\mathscr{F},\mathbb{P}\right)$ be a probability space,
$\boldsymbol{\mathrm{Q}}\subset\mathbb{R}^{d}$ be a bounded open set with
$0\in\boldsymbol{\mathrm{Q}}$, let $(\tau_{x})_{x\in\mathbb{R}^{d}}$ be a
dynamical system on $\Omega$ with invariant $\sigma$-algebra $\mathscr{I}$ and
let $f\in L^{p}(\Omega;\mu_{\mathcal{P}})$ and $\varphi\in
L^{q}(\boldsymbol{\mathrm{Q}})$, where $1<p,q<\infty$,
$\frac{1}{p}+\frac{1}{q}=1$. Then for $\mathbb{P}$-almost all
$\omega\in\Omega$ it holds

$n^{-d}\int_{n\boldsymbol{\mathrm{Q}}}\varphi(\frac{x}{n})f{\left(\tau_{x}\omega\right)}\,\mathrm{d}x\to\mathbb{E}{\left(f\right)}\int_{\boldsymbol{\mathrm{Q}}}\varphi\,.$

###### Proof.

Let $\varphi_{\delta}\in C(\overline{\boldsymbol{\mathrm{Q}}})$ with
$\left\|\varphi-\varphi_{\delta}\right\|_{L^{q}(\boldsymbol{\mathrm{Q}})}<\delta$.
Then

$\displaystyle\left|n^{-d}\int_{n\boldsymbol{\mathrm{Q}}}\varphi(\frac{x}{n})f{\left(\tau_{x}\omega\right)}\,\mathrm{d}x-\mathbb{E}{\left(f\right)}\int_{\boldsymbol{\mathrm{Q}}}\varphi\right|$
$\displaystyle\qquad\qquad\leq\left\|\varphi-\varphi_{\delta}\right\|_{L^{q}(\boldsymbol{\mathrm{Q}})}\left(n^{-d}\int_{n\boldsymbol{\mathrm{Q}}}\left|f{\left(\tau_{x}\omega\right)}\right|^{p}\,\mathrm{d}x\right)^{\frac{1}{p}}$
$\displaystyle\qquad\qquad\quad+\left|n^{-d}\int_{n\boldsymbol{\mathrm{Q}}}\varphi_{\delta}(x)f{\left(\tau_{x}\omega\right)}\,\mathrm{d}x-\mathbb{E}{\left(f\right)}\int_{\boldsymbol{\mathrm{Q}}}\varphi_{\delta}\right|+\mathbb{E}_{\mu_{\mathcal{P}}}{\left(f|\mathscr{I}\right)}\int_{\boldsymbol{\mathrm{Q}}}\left|\varphi-\varphi_{\delta}\right|\,,$

which implies the claim. ∎

###  2.11 Random Sets

The theory of random measures and the theory of random geometry are closely
related. In what follows, we recapitulate those results that are important in
the context of the theory developed below and shed some light on the
correlations between random sets and random measures.

Let $\mathfrak{F}(\mathbb{R}^{d})$ denote the set of all closed sets in
$\mathbb{R}^{d}$. We write

$\displaystyle\mathfrak{F}_{V}:=$
$\displaystyle\left\\{F\in\mathfrak{F}(\mathbb{R}^{d})\,:\;F\cap
V\not=\emptyset\right\\}$ $\displaystyle\mbox{if
}V\subset\mathbb{R}^{d}\quad\textnormal{is an open set}\,,$ (2.47)
$\displaystyle\mathfrak{F}^{K}:=$
$\displaystyle\left\\{F\in\mathfrak{F}(\mathbb{R}^{d})\,:\;F\cap
K=\emptyset\right\\}$ $\displaystyle\mbox{if
}K\subset\mathbb{R}^{d}\quad\textnormal{is a compact set}\,.$ (2.48)

The _Fell-topology_ $\mathscr{T}_{F}$ is created by all sets
$\mathfrak{F}_{V}$ and $\mathfrak{F}^{K}$ and the topological space
$(\mathfrak{F}(\mathbb{R}^{d}),\mathscr{T}_{F})$ is compact, Hausdorff and
separable[18].

######  Remark 2.39.

We find for closed sets $F_{n},F$ in $\mathbb{R}^{d}$ that $F_{n}\to F$ if and
only if [18]

  1. 1.

for every $x\in F$ there exists $x_{n}\in F_{n}$ such that
$x=\lim_{n\to\infty}x_{n}$ and

  2. 2.

if $F_{n_{k}}$ is a subsequence, then every convergent sequence $x_{n_{k}}$
with $x_{n_{k}}\in F_{n_{k}}$ satisfies $\lim_{k\to\infty}x_{n_{k}}\in F$.

If we restrict the Fell-topology to the compact sets
$\mathfrak{K}(\mathbb{R}^{d})$ it is equivalent with the Hausdorff topology
given by the Hausdorff distance

$\mathrm{d}{\left(A,B\right)}=\max\left\\{\sup_{y\in B}\inf_{x\in
A}\left|x-y\right|\,,\,\,\sup_{x\in A}\inf_{y\in
B}\left|x-y\right|\right\\}\,.$

######  Remark 2.40.

For $A\subset\mathbb{R}^{d}$ closed, the set

$\mathfrak{F}(A):=\left\\{F\in\mathfrak{F}(\mathbb{R}^{d})\,:\;F\subset
A\right\\}$

is a closed subspace of $\mathfrak{F}{\left(\mathbb{R}^{d}\right)}$. This
holds since

$\mathfrak{F}{\left(\mathbb{R}^{d}\right)}\backslash\mathfrak{F}{\left(A\right)}=\left\\{B\in\mathfrak{F}{\left(\mathbb{R}^{d}\right)}\,:\;B\cap\left(\mathbb{R}^{d}\backslash
A\right)\not=\emptyset\right\\}=\mathfrak{F}_{\mathbb{R}^{d}\backslash
A}\quad\text{is open.}$

.

######  Lemma 2.41 (Continuity of geometric operations).

The maps $\tau_{x}:\,A\mapsto A+x$ and
$b_{\delta}:\,A\mapsto\overline{\mathbb{B}_{\delta}{\left(A\right)}}$ are
continuous in $\mathfrak{F}\left(\mathbb{R}^{d}\right)$.

###### Proof.

We show that preimages of open sets are open. For open sets $V$ we find

$\displaystyle\tau_{x}^{-1}\left(\mathfrak{F}_{V}\right)$
$\displaystyle=\left\\{F\in\mathfrak{F}(\mathbb{R}^{d})\,:\;\tau_{x}F\cap
V\not=\emptyset\right\\}=\left\\{F\in\mathfrak{F}(\mathbb{R}^{d})\,:\;F\cap\tau_{-x}V\not=\emptyset\right\\}=\mathfrak{F}_{\tau_{-x}V}\,,$
$\displaystyle b_{\delta}^{-1}\left(\mathfrak{F}_{V}\right)$
$\displaystyle=\left\\{F\in\mathfrak{F}(\mathbb{R}^{d})\,:\;\overline{\mathbb{B}_{\delta}{\left(F\right)}}\cap
V\not=\emptyset\right\\}=\left\\{F\in\mathfrak{F}(\mathbb{R}^{d})\,:\;F\cap\mathbb{B}_{\delta}{\left(V\right)}\not=\emptyset\right\\}=\mathfrak{F}_{\left(b_{\delta}V\right)^{\circ}}\,.$

The calculations for
$\tau_{x}^{-1}{\left(\mathfrak{F}^{K}\right)}=\mathfrak{F}^{\tau_{-x}K}$ and
$b_{\delta}^{-1}{\left(\mathfrak{F}^{K}\right)}=\mathfrak{F}^{b_{\delta}K}$
are analogue. ∎

######  Remark 2.42.

The _Matheron- $\sigma$-field $\sigma_{\mathfrak{F}}$_ is the
Borel-$\sigma$-algebra of the Fell-topology and is fully characterized either
by the class $\mathfrak{F}_{V}$ of $\mathfrak{F}^{K}$.

######  Definition 2.43 (Random closed / open set according to Choquet (see
[18] for more details)).

  1. a)

Let $(\Omega,\sigma,\mathbb{P})$ be a probability space. Then a _Random Closed
Set (RACS)_ is a measurable mapping

$A:(\Omega,\sigma,\mathbb{P})\longrightarrow(\mathfrak{F},\sigma_{\mathfrak{F}})$

  2. b)

Let $\tau_{x}$ be a dynamical system on $\Omega$. A random closed set is
called stationary if its characteristic functions $\chi_{A(\omega)}$ are
stationary, i.e. they satisfy
$\chi_{A(\omega)}(x)=\chi_{A(\tau_{x}\omega)}(0)$ for almost every
$\omega\in\Omega$ for almost all $x\in\mathbb{R}^{d}$. Two random sets are
jointly stationary if they can be parameterized by the same probability space
such that they are both stationary.

  3. c)

A random closed set
$\Gamma:(\Omega,\sigma,P)\longrightarrow(\mathfrak{F},\sigma_{\mathfrak{F}})\quad\omega\mapsto\Gamma(\omega)$
is called a _Random closed $C^{k}$-Manifold_ if $\Gamma(\omega)$ is a piece-
wise $C^{k}$-manifold for P almost every $\omega$.

  4. d)

A measurable mapping

$A:(\Omega,\sigma,\mathbb{P})\longrightarrow(\mathfrak{F},\sigma_{\mathfrak{F}})$

is called _Random Open Set (RAOS)_ if $\omega\mapsto\mathbb{R}^{d}\backslash
A(\omega)$ is a RACS.

The importance of the concept of random geometries for stochastic
homogenization stems from the following Lemma by Zähle. It states that every
random closed set induces a random measure. Thus, every stationary RACS
induces a stationary random measure.

######  Lemma 2.44 ([32] Theorem 2.1.3 resp. Corollary 2.1.5).

Let $\mathfrak{F}_{m}\subset\mathfrak{F}$ be the space of closed m-dimensional
sub manifolds of $\mathbb{R}^{d}$ such that the corresponding Hausdorff
measure is locally finite. Then, the $\sigma$-algebra
$\sigma_{\mathfrak{F}}\cap\mathfrak{F}_{m}$ is the smallest such that

$M_{B}:\mathfrak{F}_{m}\rightarrow\mathbb{R}\quad
M\mapsto\mathcal{H}^{m}(M\cap B)$

is measurable for every measurable and bounded $B\subset\mathbb{R}^{d}$.

This means that

$M_{\mathbb{R}^{d}}:\mathfrak{F}_{m}\rightarrow\mathfrak{M}(\mathbb{R}^{d})\quad
M\mapsto\mathcal{H}^{m}(M\cap\cdot)$

is measurable with respect to the $\sigma$-algebra created by the Vague
topology on $\mathfrak{M}(\mathbb{R}^{d})$. Hence a random closed set always
induces a random measure. Based on Lemma 2.44 and on Palm-theory, the
following useful result was obtained in [10] (See Lemma 2.14 and Section 3.1
therein). We can thus assume w.l.o.g that $\Omega$ is a separable metric
space.

######  Theorem 2.45.

Let $(\Omega,\sigma,P)$ be a probability space with an ergodic dynamical
system $\tau$. Let
$A:(\Omega,\sigma,P)\longrightarrow(\mathfrak{F},\sigma_{\mathfrak{F}})$ be a
stationary random closed $m$-dimensional $C^{k}$-Manifold.

There exists a separable metric space
$\tilde{\Omega}\subset\mathfrak{M}{\left(\mathbb{R}^{d}\right)}$ with an
ergodic dynamical system $\tilde{\tau}$ and a mapping
$\tilde{A}:\,(\tilde{\Omega},\mathcal{B}_{\tilde{\Omega}},\mathbb{P})\to(\mathfrak{F},\sigma_{\mathfrak{F}})$
such that $A$ and $\tilde{A}$ have the same law and such that $\tilde{A}$
still is stationary. Furthermore, $(x,\omega)\mapsto\tau_{x}\omega$ is
continuous. We identify $\tilde{\Omega}=\Omega$, $\tilde{A}=A$ and
$\tilde{\tau}=\tau$.

Also the following result will be useful below.

######  Lemma 2.46.

Let $\mu$ be a Radon measure on $\mathbb{R}^{d}$ and let
$\boldsymbol{\mathrm{Q}}\subset\mathbb{R}^{d}$ be a bounded open set. Let
$\mathfrak{F}_{0}\subset\mathfrak{F}\left(\overline{\boldsymbol{\mathrm{Q}}}\right)$
be such that $\mathfrak{F}_{0}\to\mathbb{R}$, $A\mapsto\mu(A)$ is continuous.
Then

$m:\,\mathfrak{F}\times\mathfrak{F}_{0}\to\mathfrak{M}{\left(\mathbb{R}^{d}\right)}\,,\quad\left(P,B\right)\mapsto\begin{cases}A\mapsto\mu{\left(A\cap
B\right)}&B\subset P\\\ 0&\text{else}\end{cases}$

is measurable.

###### Proof.

For $f\in C_{c}(\mathbb{R}^{d})$ we introduce $m_{f}$ through

$m_{f}:\,\left(P,B\right)\mapsto\begin{cases}\int_{B}f\,\mathrm{d}\mu&B\subset
P\\\ 0&\text{else}\end{cases}$

and observe that $m$ is measurable if and only if for every $f\in
C_{c}{\left(\mathbb{R}^{d}\right)}$ the map $m_{f}$ is measurable (see Section
2.10). Hence, if we prove the latter property, the lemma is proved.