# On bounds for the remainder term of counting functions of the Neumann
Laplacian on domains with fractal boundary

S<EMAIL_ADDRESS>School of Mathematics, University of
Birmingham, Edgbaston, Birmingham, B15 2TT, UK  L.
<EMAIL_ADDRESS>The second author was supported by EPSRC DTP
and the University of Birmingham. School of Mathematics, University of
Birmingham, Edgbaston, Birmingham, B15 2TT, UK

###### Abstract

We provide a new constructive method for obtaining explicit remainder
estimates of eigenvalue counting functions of Neumann Laplacians on domains
with fractal boundary. This is done by establishing estimates for first non-
trivial eigenvalues through Rayleigh quotients. A main focus lies on domains
whose boundary can locally be represented as a limit set of an IFS, with the
classic Koch snowflake and certain Rohde snowflakes being prototypical
examples, to which the new method is applied. Central to our approach is the
construction of a novel foliation of the domain near its boundary.

##  1 Introduction

#### Historic remarks:

In the celebrated 1966 article [19], Kac asked which geometric information can
be inferred from the Laplace spectrum of a domain coining the famous question:
_Can one hear the shape of a drum?_ This inverse problem, i. e. computing the
“drum” given its spectrum, was shown to be ultimately unsolvable in 1992 by
Gordon, Webb and Wolpert [13] for domains with piecewise smooth boundary and
subsequently in 2000 by Chen and Sleemann [8] for domains with fractal
boundary. The focus of the present article lies on the complementary forward
problem, namely to decipher the influence of a domain’s geometry on its
Laplace spectrum. This forward problem has attracted an immense amount of
attention throughout the years (see for example [3, 17] for a summary). Since
the Laplace eigenvalue equation can be interpreted physically as the wave
equation of a function that is periodic in time, an eigenvalue $\lambda$ can
be understood as acoustic frequency of a wave mode. Heuristically, higher
harmonics correspond to higher order eigenfunctions displaying similar
geometric aspects of their domain. As a consequence, it is particularly
interesting how the geometry of a domain dictates the asymptotic behaviour of
eigenvalues of the Laplace operator. The idea goes back to 1911 when Weyl
established a remarkable law for the eigenvalue counting function
$N_{D}(\Omega,t)$ (i. e. the number of eigenvalues $\leq t$ for the Dirichlet
Laplace problem counted with multiplicity) for open and bounded
$\Omega\subset\mathbb{R}^{2}$ in [47], which had a tremendous impact in
mathematics:

$\displaystyle
N_{D}(\Omega,t)=\frac{\operatorname{vol}_{2}(\Omega)}{4\pi}t+o(t)\,,\qquad\text{
as }t\to\infty.$ (1.1)

In higher dimensions, the classic Weyl law for an open
$\Omega\subset\mathbb{R}^{n}$

$\displaystyle
N_{D}(\Omega,t)=C_{W}^{(n)}\operatorname{vol}_{n}(\Omega)t^{n/2}+o\left(t^{n/2}\right)\,,\qquad\text{
as }t\to\infty$ (1.2)

with $C_{W}^{(n)}:=2^{-n}\pi^{-n/2}/\Gamma(1+n/2)$ was shown to hold under the
minimal hypothesis that $\operatorname{vol}_{n}(\Omega)<\infty$ in [36, 37].
Based on his result (1.1), Weyl conjectured that for sufficiently regular
domains $\Omega\subset\mathbb{R}^{n}$, the asymptotic behaviour is given by

$\displaystyle
N_{D}(\Omega,t)=C_{W}^{(n)}\operatorname{vol}_{n}(\Omega)t^{n/2}-\frac{1}{4}C_{W}^{(n-1)}\operatorname{vol}_{n-1}(\partial\Omega)t^{(n-1)/2}+o\left(t^{(n-1)/2}\right)\text{
as }t\to\infty.$ (1.3)

A similar conjecture has been formulated for the Neumann counting function; it
differs from the Dirichlet-case only by the sign of the second term. These
conjectures can be motivated by Weyl’s proof of approximating a planar domain
$\Omega$ with unions of squares, which suggests that correction terms to the
leading order are dictated by the deviation of $\Omega$ from the union of
squares and thus by the geometry of the boundary $\partial\Omega$. Supposing
that $\partial\Omega$ is smooth and additionally assuming that Billiard
trajectories are almost never periodic, the Weyl conjecture was proved by
Ivrii in 1980 (see [16, 38, 39] and the references therein) for several
boundary conditions including Dirichlet and Neumann type.

Regularity assumptions in the Weyl conjecture are essential. Indeed, if the
boundary is irregular such a result cannot be expected and is wrong in
general. In particular, the remainder term in (1.3) is meaningless when the
boundary shows fractal structure with Hausdorff-dimension
$\dim_{H}\partial\Omega>n-1$ as then
$\operatorname{vol}_{n-1}\partial\Omega=\infty$. Notice that under Ivrii’s
conditions, the exponent in the second term coincides with half the
topological dimension $\dim(\partial\Omega)/2=(\dim(\Omega)-1)/2$ of the
boundary $\partial\Omega$ and that the associated coefficient is linked to the
domain’s surface area. It is therefore natural to expect analogue substitutes
of these quantities in case of non-integer dimensions.

Motivated by physical observations, in a first step in this direction, Berry
conjectured that the remainder term would be linked to the Hausdorff dimension
and the Hausdorff measure of the boundary in [4, 5] (see also [23]). However,
this quickly turned out to be false as a number of counterexamples were found
in [7, 25] leading to the formulation of the modified Weyl-Berry conjecture in
[22]. The modified Weyl-Berry conjecture links the power law of the remainder
term in the eigenvalue counting function to the Minkowski dimension and its
coefficient to the Minkowski content (see Sec. 2.2). In case of Dirichlet
boundary conditions, Lapidus and Pomerance verified the modified Weyl-Berry
conjecture under the conditions that $n=1$ and that $\partial\Omega$ is
Minkowski measurable with Minkowski dimension $\delta\in(0,1)$ and Minkowski
content $\mathcal{M}_{\delta}(\partial\Omega)\in(0,\infty)$ by proving
$N_{D}(\Omega,t)=C_{W}^{(1)}\operatorname{vol}_{1}(\Omega)t^{1/2}-c_{\delta}\mathcal{M}_{\delta}(\partial\Omega)t^{\delta/2}+o(t^{\delta/2})$
as $t\to\infty$. Moreover, they showed that the analogue is incorrect in
higher dimensions in general [25, 26]. Here, $c_{\delta}$ denotes a constant
only depending on $\delta$. In arbitrary dimensions $n$, Lapidus showed an
asymptotic law
$N_{N}(\Omega,t)=C_{W}^{(n)}\operatorname{vol}_{n}(\Omega)t^{n/2}+\mathcal{O}\left(t^{\delta/2}\right)$
for domains $\Omega$ with an extension property which ensures that the
essential spectrum is empty, and under the conditions that $\delta\in(n-1,n]$
and that the upper Minkowski content of $\partial\Omega$ exists in [22]. In
contrast, the existence of the upper inner Minkowski content was shown to be
sufficient for estimates of the Dirichlet counting function in the same
article. Of particular relevance to the present article is the work [30],
where an exact rather than an asymptotic bound on
$N_{N}(\Omega,t)-C_{W}^{(n)}\operatorname{vol}_{n}(\Omega)t^{n/2}$ was
obtained for domains whose boundary locally is a graph of a $BV$-function.

#### Overview of main results:

Motivated by the fact that among counting functions for mixed boundary
conditions the Neumann case provides an upper bound, the present article
focuses on remainder estimates of Neumann eigenvalue counting functions
$N_{N}(\Omega,t)$. To obtain these, we introduce the concept of well-covered
domains (see Def. 3.4). For such domains $\Omega$, a main result, stated in
the below theorem provides an explicit upper bound of
$N_{N}(\Omega,t)-C_{W}^{(n)}\operatorname{vol}_{n}(\Omega)t^{n/2}$ for all
$t\geq t_{0}$ for a determined $t_{0}$.

######  Theorem (Thm. 3.10).

Let $\Omega\subset\mathbb{R}^{n}$ be a domain whose boundary $\partial\Omega$
has upper inner Minkowski dimension $\delta>n-1$ (see Sec. 2.2). Suppose that
$\Omega$ is well-covered (see Def. 3.4), meaning that for all sufficiently
small $\varepsilon>0$ the set $\Omega_{-\varepsilon}:=\\{x\in\Omega\ :\
\operatorname{dist}(x,\partial\Omega)<\varepsilon\\}$ admits a cover by
domains with uniformly comparable diameter that satisfies the following. (i)
The cardinality of the cover scales like $\varepsilon^{-\delta}$ and (ii) each
covering domain admits a well-behaved foliation (see Def. 3.1). Then there is
an explicitly determined $M_{\Omega}$ and $t_{0}\in\mathbb{R}$ such that

$\displaystyle
N_{N}(\Omega,t)-C_{W}^{(n)}\operatorname{vol}_{n}(\Omega)t^{n/2}\leq
M_{\Omega}t^{\delta/2}$ (1.4)

for all $t\geq t_{0}$.

Examples of domains with fractal boundary to which the above theorem applies
are Koch and certain Rohde snowflakes (see Sec. 3.4). Rhode snowflakes are
particularly important domains as, up to bi-Lipschitz transformation, every
planar quasidisk (i. e. a domain that is bounded by the image of a circle
under a quasi-conformal map) can be realised as such a snowflake (see [35]).
In this context, it is worth noting that bi-Lipschitz maps preserve upper
inner Minkowski dimension, the property of being well-covered and hence all
prerequisites of Thm. 3.10. Another significant subclass of domains $\Omega$
for which Thm. 3.10 holds is the class of domains whose boundary locally is a
graph of a $BV$-function (see Sec. 3.2). For this subclass an estimate of the
form (1.4) is presented in [30]. Indeed, the work of [30] gave the motivation
for developing the novel concept of well-covered domains in order to achieve
the estimate (1.4) for snowflake domains. For showing that a snowflake domain
is well-covered we develop a new way of explicitly constructing a foliation of
the domains covering the inner $\varepsilon$-neighbourhood
$\Omega_{-\varepsilon}$, see Sec. 3.1, addressing (ii) in the above theorem.
Note that the existence of the upper Minkowski content is not necessary in
Thm. 3.10, but that the assumptions of Thm. 3.10 imply the existence of the
upper inner Minkowski content.

Under the assumptions of Thm. 3.10, we can also treat the case $\delta=n-1$,
in which we obtain
$N_{N}(\Omega,t)-C_{W}^{(n)}\operatorname{vol}_{n}(\Omega)t^{n/2}=\mathcal{O}\left(t^{(n-1)/2}\right)$
as $t\to\infty$. This addresses a question in [23], where the weaker estimate
$N_{N}(\Omega,t)-C_{W}^{(n)}\operatorname{vol}_{n}(\Omega)t^{n/2}=\mathcal{O}\left(t^{(n-1)/2}\log(t)\right)$
is given, see Rem. 3.11(i).

###### 1.1. Remark.

Having the estimate (1.4) for all $t\geq t_{0}$ with explicitly determined
$M_{\Omega}$ and $t_{0}$ rather than asymptotically is key for obtaining
Dirichlet Weyl laws for self-similar sprays $U$, whose generator is a finite
union of pairwise disjoint well-covered domains. (Loosely, a self-similar
spray is a countable union of scaled copies of its generator.) Indeed, in the
companion paper [20] by the authors, Thm. 3.10, in conjunction with an
estimate by van den Berg and Lianantonakis from [43], is used as a pivotal
tool to achieve the asymptotics

$N_{D}(U,t)=C_{W}^{(2)}\operatorname{vol}_{2}(U)t+\sum_{j\in
J}A_{j}(t)t^{j}+\mathcal{O}(t^{\frac{\log 4}{2\log 3}})$ (1.5)

for a family of self-similar sprays $U$ whose generator is a finite union of
pairwise disjoint Koch snowflakes. Here, the $A_{j}$ are bounded and $J$ is of
finite cardinality. Cor. 2.3, which allows to combine Thm. 3.10 with the
estimate from [43], hinges on common conditions for the existence of estimates
of Neumann and Dirichlet counting functions. Depending on the structure of
$U$, the asymptotic (1.5) can exhibit several asymptotic terms of faster
growth than the error term $O(t^{\frac{\log 4}{2\log 3}})$. Explicit examples
for $U$ for which (1.5) gives two, three and four summands of faster growth
than $t^{\frac{\log 4}{2\log 3}}$ are presented in [20], and hints given for
which sprays Thm. 3.10 leads to even more asymptotic terms of faster growth
than the error term. Notably, the exponents in (1.5), i. e. the elements of
$J$, are shown to be in one-to-one correspondence with the exponents in the
volume expansion of
$\varepsilon\mapsto\operatorname{vol}_{2}\left(U_{-\varepsilon}\right)$,
providing valuable insights as to how the fine geometric structure of a self-
similar spray influences the distribution of the Dirichlet Laplace
eigenvalues.

The rough outline for obtaining the explicit bound on $N_{N}(\Omega,t)$ for
well-covered domains $\Omega$ in Thm. 3.10 is to partition $\Omega$ into
$\Omega_{-\varepsilon}$ and $\Omega\setminus\Omega_{-\varepsilon}$. We use a
Whitney cover of $\Omega$ and restrict it to a cover of
$\Omega\backslash\Omega_{-\varepsilon}$ by Whitney cubes
$\\{Q\,:\,Q\in\mathcal{W}_{\varepsilon}\\}$. For this cover we show that
$\operatorname{vol}_{n-1}\left(\partial\overline{\bigcup_{Q\in\mathcal{W}_{\varepsilon}}Q}\right)=\mathcal{O}\left(\varepsilon^{(n-1)-\delta}\right)$
as $\varepsilon\to 0$ (see Prop. 2.5). This is used for an asymptotic law of
$N_{N}\left(\overline{\bigcup_{Q\in\mathcal{W}_{\varepsilon}}Q},t\right)$. For
handling $\Omega_{-\varepsilon}$ we use that $\Omega$ is well-covered,
denoting the covering domains of $\Omega_{-\varepsilon}$ by
$D_{i}^{\varepsilon}$. The well-behaved foliation of $D_{i}^{\varepsilon}$ is
used to show a Poincaré-Wirtinger-like inequality for all non-constant $u\in
H^{1}(D_{i}^{\varepsilon})$ by separating $D_{i}^{\varepsilon}=E\cup
D_{i}^{\varepsilon}\backslash E$ where $E$ has a known first non-trivial
eigenvalue $\lambda_{2}^{N}(E)$. The inverse of the first non-trivial
eigenvalue of a domain $D_{i}^{\varepsilon}$ with respect to Neumann boundary
conditions, denoted by $\lambda_{2}^{N}(D_{i}^{\varepsilon})$, is the best
Poincaré constant for non-constant functions $u\in
H^{1}(D_{i}^{\varepsilon})$. We use a version of the variational technique
introduced in [30] and thus avoid the explicit need for existence of bounded
extensions $W^{1,2}(\Omega)\hookrightarrow W^{1,2}(\mathbb{R}^{n})$ of the
Sobolev spaces involved.

######  Corollary (cf. Cor. 3.6).

Let $\Omega$ be well-covered and let $\\{D^{\varepsilon}_{i}\\}_{i\in
I_{\varepsilon}}$ be a cover of $\Omega_{-\varepsilon}$ consisting of well-
foliated domains as in Def. 3.4. Then by Lem. 3.2, there exists
$C_{1}(\Omega_{-\varepsilon})$ with
$\inf_{\varepsilon>0}C_{1}(\Omega_{-\varepsilon})>0$ such that

$\displaystyle\lambda_{2}^{N}(D^{\varepsilon}_{i})\geq
C_{1}(\Omega_{-\varepsilon})\varepsilon^{-2}.$

Notice that the uniform comparable geometry of the covering domains of
$\Omega_{-\varepsilon}$ as used in Thm. 3.10 above is precisely the condition
to ensure a scaling behaviour
$\lambda_{2}^{N}(D_{i}^{\varepsilon})\gtrsim\operatorname{diam}(D_{i}^{\varepsilon})^{-2}$.
In particular for a family of such domains $\\{D_{i}^{\varepsilon}\\}_{i\in
I_{\varepsilon}}$, $\sum_{i\in
I_{\varepsilon}}N_{N}(D_{i}^{\varepsilon},t)=\\#\\{I_{\varepsilon}\\}$
whenever $\varepsilon$ is small enough. Choosing $\varepsilon$ depending on
$t$, these estimates allow for sufficient control of
$N_{N}(\Omega,t)\leq\sum_{\\{i\in
I_{\varepsilon}\\}}N_{N}(D_{i}^{\varepsilon},t)+\sum_{Q\in\mathcal{W}_{\varepsilon}}N_{N}(Q,t)$
to prove Thm. 3.10.

#### Outline:

The present article is structured as follows: We first introduce the relevant
notions of Laplace eigenvalue counting functions, Minkowski content and
Whitney covers in Sec. 2. In Sec. 3.1 we show how one can construct families
of paths ending at the boundary of a domain $D$ and show how this structure on
$D$ gives rise to a lower bound for the first non-trivial Neumann eigenvalue
$\lambda_{2}^{N}(D)$ of the form
$\lambda_{2}^{N}(D)\gtrsim\operatorname{diam}(D)^{-2}$ (see Lem. 3.2). In Sec.
3.3, we use this bound to obtain explicit and asymptotic estimates for the
remainder term of counting functions in Thm. 3.10, providing an estimate for
an $M_{\Omega}$ such that (1.4) holds for all sufficiently large $t$. Sec. 3.2
is devoted to proving that sets of bounded variation are well-covered domains.
Finally, in Sec. 3.4 we apply Thm. 3.10 to the classic Koch snowflake and
certain Rohde snowflakes, providing explicit constructions and results for
these cases. In the final Sec. 4, important constants, that are used
throughout the article, are collected.

##  2 Preliminaries

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain, i. e. a bounded open
connected subset of $\mathbb{R}^{n}$ and denote its boundary by
$\partial\Omega$. By $H^{1}(\Omega)=W^{1,2}(\Omega)$ we denote the usual
Sobolev space, i. e. the set of all $u\in L^{2}(\Omega)$ with a weak
derivative $\nabla u\in L^{2}(\Omega)$. We equip $H^{1}(\Omega)$ with the
usual inner product $(u,v)_{H^{1}(\Omega)}:=(u,v)_{L^{2}(\Omega)}+(\nabla
u,\nabla v)_{L^{2}(\Omega)}$ so that $H^{1}(\Omega)$ becomes a Hilbert space.
Further, $H^{1}_{0}(\Omega)\subset H^{1}(\Omega)$ shall denote the closure of
the set of compactly supported $C^{\infty}(\Omega)$-functions in
$H^{1}(\Omega)$. On $H^{1}(\Omega)$, resp. $H^{1}_{0}(\Omega)$, we consider
the Laplacian $\Delta:=\sum_{i=1}^{n}\partial_{i}^{2}$ and focus on the
eigenvalue equation of $-\Delta$ subject to Neumann (2.1) or Dirichlet (2.2)
boundary conditions.

$\displaystyle\begin{cases}-\Delta u=\lambda u&\text{ in }\Omega\\\
\frac{\partial u}{\partial\mathbf{n}}=0&\text{ on }\partial\Omega\end{cases}$
(2.1)

$\displaystyle\begin{cases}-\Delta u=\lambda u&\text{ in }\Omega\\\ u=0&\text{
on }\partial\Omega\end{cases}$ (2.2)

where $\mathbf{n}$ denotes the exterior normal to $\partial\Omega$. The
variational formulation of the problem (2.1) is stated as follows: Find $u\in
H^{1}(\Omega)$ such that $(\nabla u,\nabla
v)_{L^{2}{(\Omega)}}=\lambda(u,v)_{L^{2}{(\Omega)}}$ for all $v\in
H^{1}(\Omega)$. Note that the space in which this problem is studied dictates
the boundary condition and that the variational formulation of (2.2) is: Find
$u\in H^{1}_{0}(\Omega)$ such that $(\nabla u,\nabla
v)_{L^{2}{(\Omega)}}=\lambda(u,v)_{L^{2}{(\Omega)}}$ for all $v\in
H^{1}_{0}(\Omega)$. Replacing $H^{1}(\Omega)$ resp. $H^{1}_{0}(\Omega)$ with
any other closed space $V$ satisfying $H^{1}_{0}(\Omega)\subset V\subset
H^{1}(\Omega)$ gives rise to variational problems with more general boundary
conditions.

The corresponding non-negative spectrum will be denoted by $\sigma(-\Delta)$
and the essential spectrum by $\sigma_{\text{ess}}(-\Delta)$. Supposing that
the non-essential spectrum of the eigenvalue problem (2.1) contains at least
two isolated points
$0=\lambda_{1}^{N}(\Omega)<\lambda_{2}^{N}(\Omega)<\inf\sigma_{\text{ess}}(-\Delta)$,
it is well known that

$\displaystyle\lambda_{2}^{N}=\inf_{\begin{subarray}{c}v\in
H^{1}(\Omega)\backslash\\{0\\}:\\\
\int_{\Omega}vdx=0\end{subarray}}\frac{\|\nabla
v\|^{2}_{L^{2}(\Omega)}}{\|v\|^{2}_{L^{2}(\Omega)}}.$ (2.3)

Writing $1^{\perp}$ for the set of non-constant functions, we then have:

###### 2.1. Corollary (cf. [30]).

Let $\lambda_{2}^{N}(\Omega)$ be the first non-trivial Neumann eigenvalue of
$\Omega$. Then

$\displaystyle\lambda_{2}^{N}(\Omega)\geq
L^{-1}\Leftrightarrow\left(\|u\|^{2}_{L^{2}(\Omega)}\leq L\|\nabla
u\|^{2}_{L^{2}(\Omega)}\,\,\forall u\in H^{1}(\Omega)\cap 1^{\perp}\right).$

This classic variational result will be crucial in proving lower bounds for
the first non-trivial eigenvalue under Neumann conditions (see Lemma 3.2). The
analogous statements to (2.3) and Cor. 2.1 hold true for other boundary
conditions by replacing $H^{1}(\Omega)$ with any closed space $V$ satisfying
$H^{1}_{0}(\Omega)\subset V\subset H^{1}(\Omega)$ in both statements, and by
replacing $1^{\perp}$ with ’orthogonal to the eigenfunction to $\lambda_{1}$’
(see for example [10, 22]).

In order to define a counting function
$N(\Omega,t):=\\#\\{\lambda\in\sigma(-\Delta):\lambda\leq t\\}$, it is
necessary that $\sigma(-\Delta)$ is discrete with the only accumulation point
being at $\infty$. While this is always satisfied in case of Dirichlet
boundary conditions, there are several occasions where this may fail to be
true under Neumann boundary conditions as the essential spectrum can be non-
empty in this case. Indeed, it was shown in [15] that any closed subset of
$\mathbb{R}_{\geq 0}$ can be realised as the essential spectrum of a Laplacian
on a domain $\subset\mathbb{R}^{2}$ subject to Neumann boundary conditions. On
the other side, several criteria have been found which ensure that the
essential spectrum is empty, see for example [29]. In [33] Rellich showed that
the Neumann spectrum is discrete whenever the inclusion
$\iota:H^{2}(\Omega)\hookrightarrow L^{2}(\Omega)$ is compact.333The original
German publication uses the concept of completely continuous operators
(„vollstetige Operatoren“). Whenever the domain is a Hilbert space, this
notion coincides with the usual notion of a compact operator. Maz’ya showed in
[27] that $\iota$ is compact iff $\lim_{M\to
0}\inf_{t\in(0,M)}t^{-1}\inf_{\begin{subarray}{c}\operatorname{vol}_{n}(F_{1})\geq
t\\\ \operatorname{vol}_{n}(F_{2})\leq M\end{subarray}}c(F_{1},F_{2})=0$.
Here, $c(F_{1},F_{2}):=\inf_{f\in V(F_{1},F_{2})}\int_{\Omega}|\nabla
f|^{2}dx$ is called _relative capacity_ where $V(F_{1},F_{2}):=\\{f\in
C^{\infty}(\Omega):f|_{F_{1}}=1,f|_{F_{2}}=0\\}$ for any two disjoint
$F_{1},F_{2}\subset\Omega$ that are closed in $\Omega$ giving possible
descriptions of such domains.

Any domain $\Omega$ that admits a bounded linear extension operator
$E:H^{1}(\Omega)\to H^{1}(\mathbb{R}^{n})$ (called an _extension domain_) has
a discrete Neumann spectrum, as the Neumann counting function exists in this
case (see e. g. [22]). In the planar case, a simply connected domain $\Omega$
is an extension domain iff $\partial\Omega$ is a _quasicircle_ , as was shown
in [45, 18]. Recall that a quasicircle is the image of the planar unit circle
under a quasiconformal map. It can be shown (see [1] or [28] and the
references therein) that a Jordan curve $\gamma\subset\mathbb{R}^{2}$ is
quasicircular if there is some $c\in\mathbb{R}$ such that for any two points
$a,b\in\gamma$ one has $\operatorname{diam}\Gamma\leq c|a-b|$, where
$\Gamma\subset\gamma\backslash\\{a,b\\}$ is the component with smaller
diameter. Moreover, Rohde has shown in [35] that any quasicircle is a
snowflake up to a bi-Lipschitz map, in particular showing that any possible
snowflake is also a quasicircle, see Rem. 3.12. Higher dimensional domains
that are bounded by quasispheres or the $(\varepsilon,\delta)$-domains
introduced by Jones in [18] are also extension domains but this set is no
longer exhaustive [10]; see also [14] for a complete description of such
extension domains and [2, 34] for further characterisation of such maps.

###  2.1 Dirichlet-Neumann bracketing

For a domain $\Omega\subset\mathbb{R}^{n}$ a _volume cover_
$\\{\Omega_{i}\\}_{i\in I}$ of $\Omega$ consists of at most countably many
open sets $\Omega_{i}\subset\Omega$ with
$\operatorname{vol}_{n}(\Omega)=\operatorname{vol}_{n}\left(\bigcup_{i\in
I}\Omega_{i}\right)$. We write $N_{N}(\Omega,t)$ (resp. $N_{D}(\Omega,t)$) for
the number of eigenvalues (with multiplicity) of $-\Delta$ on $\Omega$ subject
to Neumann (resp. Dirichlet) conditions on $\partial\Omega$. In addition to
the classic Dirichlet-Neumann bracketing linking the counting functions of
Dirichlet and Neumann eigenvalues, we mention a version that allows for non-
disjoint covers as long as the elements of the cover do not intersect too
often. More precisely one has the following result which also follows from the
Min-Max-Principle. For any volume cover $\\{\Omega_{i}\\}_{i\in I}$ of
$\Omega$, let $\mu:=\sup_{x\in\Omega}\\#\\{i\in I:x\in\Omega_{i}\\}$ denote
its _multiplicity_.

###### 2.2. Proposition (Dirichlet-Neumann bracketing with multiplicity,
[30]).

Let $\\{\Omega_{i}\\}_{i\in I}$ be a volume cover of $\Omega$. If the
$\Omega_{i}$ are pairwise disjoint, then

$\displaystyle\sum_{i\in I}N_{D}(\Omega_{i},t)\leq N_{D}(\Omega,t)\leq
N_{N}(\Omega,t)\leq\sum_{i\in I}N_{N}(\Omega_{i},t)$

If the volume cover has finite multiplicity $\mu$, then

$\displaystyle N_{N}(\Omega,t)\leq\sum_{i\in I}N_{N}(\Omega_{i},\mu t).$

###### 2.3. Corollary.

Let $\\{\Omega_{n}\\}$ be a finite volume cover of a domain $\Omega$. Define
$Q(\Omega^{\prime},t):=N_{N}(\Omega^{\prime},t)-N_{D}(\Omega^{\prime},t)$ for
any bounded open set $\Omega^{\prime}$ and let
$R_{D}(t):=N_{D}(\Omega,t)-\sum_{n\in\mathbb{N}}N_{D}(\Omega_{n},t)$. Then

$\displaystyle R_{D}(t)\leq-Q(\Omega,t)+\sum_{n\in\mathbb{N}}Q(\Omega_{n},t).$

###### Proof.

Prop. 2.2 shows that $Q(\Omega^{\prime},t)\geq 0$ and
$N_{N}(\Omega,t)-\sum_{n}N_{N}(\Omega_{n},t)\leq 0$. Moreover,

$\displaystyle R_{D}(t)$
$\displaystyle=N_{D}(\Omega,t)-\sum_{n}N_{D}(\Omega_{n},t)=N_{N}(\Omega,t)-\sum_{n}N_{D}(\Omega_{n},t)-Q(\Omega,t)$
$\displaystyle\leq\sum_{n}N_{N}(\Omega_{n},t)-\sum_{n}N_{D}(\Omega_{n},t)-Q(\Omega,t)=\sum_{n}Q(\Omega_{n},t)-Q(\Omega,t).\qed$

A simple estimate for $Q$ was found in [11] based on estimates on the Neumann
counting function.

###  2.2 Minkowski content and dimension

Let $X$ be any non-empty open bounded subset of $\mathbb{R}^{n}$ with boundary
$\partial X$. We write

$\displaystyle
X_{\varepsilon}:=\\{x\in\mathbb{R}^{n}:\operatorname{dist}(x,X)<\varepsilon\\}\,\text{
and }X_{-\varepsilon}:=\\{x\in X:\operatorname{dist}(x,\partial
X)<\varepsilon\\}=(\partial X)_{\varepsilon}\cap X.$

Then for any $s>0$ one defines the corresponding _upper inner $s$-Minkowski
content_ as

$\displaystyle\overline{\mathcal{M}}_{s}(\partial
X,X):=\limsup_{\varepsilon\searrow
0}\underbrace{\operatorname{vol}_{n}(X_{-\varepsilon})\varepsilon^{-(n-s)}}_{=:\overline{\mathcal{M}}_{s}^{\varepsilon}(\partial
X,X)}$

whenever this exists. We call $\partial X$ _upper inner Minkowski measurable_
whenever $\overline{\mathcal{M}}_{s}(\partial X,X)\in(0,\infty)$ for some $s$.
We observe that (similar to the case of Hausdorff dimensions), one has for
$s,t>0$

$\displaystyle\overline{\mathcal{M}}_{t}^{\varepsilon}(\partial
X,X)=\varepsilon^{t-s}\overline{\mathcal{M}}_{s}^{\varepsilon}(\partial X,X).$

It follows that $s\mapsto\overline{\mathcal{M}}_{s}(\partial X,X)$ has a
critical „jump“ from $+\infty$ to $0$ at exactly one $s\in\mathbb{R}_{\geq
0}$. This is defined as the _upper inner Minkowski dimension_ :

$\displaystyle\overline{\dim_{M}}(\partial
X,X):=\inf\left\\{s:\overline{\mathcal{M}}_{s}(\partial X,X)=0\right\\}.$

Whenever $\partial X$ is upper inner Minkowski measurable, the upper inner
$s$-Minkowski content for $s=\overline{\dim_{M}}(\partial X,X)$ is called the
_upper inner Minkowski content of $\partial X$_ and is denoted by
$\overline{\mathcal{M}}(\partial X,X)$. The classic upper (i. e. non-inner)
Minkowski content and dimension are obtained by replacing $X_{-\varepsilon}$
with $X_{\varepsilon}$ in this definition.

###  2.3 Regularisation of
$\boldsymbol{\Omega\backslash\Omega_{-\varepsilon}}$ by Whitney covers

Whitney covers are a common construction used to prove estimates on counting
functions by approximating a domain by a discrete union of cubes. Let
$\Omega\subset\mathbb{R}^{n}$ be an arbitrary bounded domain. A Whitney cover
of $\Omega$ is a volume cover of $\Omega$ by cubes of different sizes such
that a cube containing $x\in\Omega$ has a diameter that is uniformly
comparable to $\operatorname{dist}(x,\partial\Omega)$. The construction of
such a cover is well known and sometimes attributed to Whitney, who used it to
study extensions of functions in [48]. However, the basic idea behind it had
in fact already been published by Courant and Hilbert in [9, pp. 355-356] in
1924 when Whitney was only 17. For the sake of completeness we include the
construction from [40, Chap. VI§1], as the notation introduced here will be
used in later results: Consider the nested family of lattices
$\left\\{2^{-k}\mathbb{Z}^{n}\right\\}_{k\in\mathbb{Z}}$. To each point
$p=(p_{1},\ldots,p_{n})$ in the lattice $2^{-k}\mathbb{Z}^{n}$ we take the
open cubes $\prod_{i=1}^{n}\left(p_{i},p_{i}+2^{-k}\right)$ with diagonal of
length $2^{-k}\sqrt{n}$ in the positive quadrant of $p$. The set of these
cubes is denoted by $\operatorname{Cube}(2^{-k}\mathbb{Z}^{n})$. We then slice
up $\Omega$ into sectors and cover them individually:

$\displaystyle\mathcal{W}_{k}$
$\displaystyle:=\left\\{Q\in\operatorname{Cube}(2^{-k}\mathbb{Z}^{n}):Q\cap\left(\overline{\Omega_{-2^{-k+2}\sqrt{n}}}\backslash\Omega_{-2^{-k+1}\sqrt{n}}\right)\neq\emptyset\right\\}$
(2.4) $\displaystyle\mathcal{W}^{\prime}$
$\displaystyle:=\bigcup_{k\in\mathbb{Z}}\mathcal{W}_{k}.$

Since
$\Omega=\bigcup_{k\in\mathbb{Z}}\left(\overline{\Omega_{-2^{-k+2}\sqrt{n}}}\backslash\Omega_{-2^{-k+1}\sqrt{n}}\right)$,
it follows that
$\operatorname{vol}_{n}\Omega=\operatorname{vol}_{n}\left(\bigcup_{Q\in\mathcal{W}^{\prime}}Q\right)$.
For the following, let $Q\in\mathcal{W}^{\prime}$ and let $k\in\mathbb{Z}$ be
so that $Q\in\mathcal{W}_{k}$. Then by definition,
$\operatorname{diam}(Q)=2^{-k}\sqrt{n}$ and there exists $x\in
Q\cap\left(\overline{\Omega_{-2^{-k+2}\sqrt{n}}}\backslash\Omega_{-2^{-k+1}\sqrt{n}}\right)$.
For this $x$ we have
$2^{-k+1}\sqrt{n}\leq\operatorname{dist}(x,\partial\Omega)\leq
2^{-k+2}\sqrt{n}$. Furthermore, $\operatorname{dist}(Q,\partial\Omega)\leq
2^{-k+2}\sqrt{n}=4\operatorname{diam}(Q)$. Moreover, we have
$\operatorname{diam}(Q)=2^{-k+1}\sqrt{n}-2^{-k}\sqrt{n}\leq\operatorname{dist}(x,\partial\Omega)-\operatorname{diam}(Q)\leq\operatorname{dist}(Q,\partial\Omega)$.
However, $\mathcal{W}^{\prime}$ might contain overlapping cubes. To rule out
such cubes, observe that any two intersecting cubes are nested. Suppose two
cubes $Q,Q^{\prime}$ intersect with $Q\subset Q^{\prime}$. Then by the above

$\displaystyle\operatorname{diam}(Q^{\prime})\leq\operatorname{dist}(Q^{\prime},\partial\Omega)\leq\operatorname{dist}(Q,\partial\Omega)\leq
4\operatorname{diam}(Q).$

This shows that to any nested sequence of intersecting cubes in
$\mathcal{W}^{\prime}$ there is a unique maximal cube containing all others.
We define $\mathcal{W}\subset\mathcal{W}^{\prime}$ as the set of all those
maximal cubes, which then are pairwise disjoint and $\mathcal{W}$ inherits the
other properties from $\mathcal{W}^{\prime}$. Such a volume cover will be
called a _Whitney cover_. This proves:

###### 2.4. Lemma (Existence of Whitney covers, [40]).

Let $\Omega\subset\mathbb{R}^{n}$ be a domain. Then there is a volume cover of
$\Omega$ consisting of pairwise disjoint cubes $\\{Q\\}_{Q\in\mathcal{W}}$ and

$\displaystyle\operatorname{diam}(Q)\leq\operatorname{dist}(Q,\partial\Omega)\leq
4\operatorname{diam}(Q)\qquad\forall Q\in\mathcal{W}.$ (2.5)

Based on this, we have an immediate estimate on the number of cubes of a
certain size in a Whitney cover.

###### 2.5. Proposition (Cardinality of slices of Whitney covers).

Let $\mathcal{W}$ be a Whitney cover of a domain
$\Omega\subset\mathbb{R}^{n}$, as constructed above. Suppose that there exists
$\delta\in[n-1,n)$ such that
$\overline{\mathcal{M}}_{\delta}(\partial\Omega,\Omega)\in(0,\infty)$. Then
there is $\mathfrak{M}_{\Omega}\in\mathbb{R}$ such that
$\\#\mathcal{W}_{k}\leq\mathfrak{M}_{\Omega}2^{k\delta}$.

Similar results concerning the Hausdorff measure were obtained for example by
Käenmäki, Lehrbäck and Vuorinen in [21].

###### Proof.

Let $d_{k}:=2^{-k}\sqrt{n}$. By construction of $\mathcal{W}_{k}$ as in (2.4),

$\displaystyle\\#\mathcal{W}_{k}$
$\displaystyle\leq\frac{\operatorname{vol}_{n}\left(\Omega_{-\left(4d_{k}+d_{k}\right)}\right)-\operatorname{vol}_{n}\left(\Omega_{-\left(2d_{k}-d_{k}\right)}\right)}{2^{-kn}}\leq\frac{\operatorname{vol}_{n}(\Omega_{-5\cdot
2^{-k}\sqrt{n}})}{2^{-kn}}.$

Defining
$\varepsilon^{\prime}(\varepsilon):=\varepsilon^{\delta-n}\operatorname{vol}_{n}(\Omega_{-\varepsilon})/\overline{\mathcal{M}}_{\delta}(\partial\Omega,\Omega)-1$,
we have

$\\#\mathcal{W}_{k}\leq\overline{\mathcal{M}}_{\delta}(\partial\Omega,\Omega)\left(1+\varepsilon^{\prime}\left(5\sqrt{n}2^{-k}\right)\right)\left(5\sqrt{n}\right)^{n-\delta}2^{k\delta}.$

Then for
$\mathfrak{M}_{\Omega}:=\sup_{k}\overline{\mathcal{M}}_{\delta}(\partial\Omega,\Omega)\left(1+\varepsilon^{\prime}\left(5\sqrt{n}2^{-k}\right)\right)\left(5\sqrt{n}\right)^{n-\delta}$
and the assertion follows from $\limsup_{\varepsilon\to
0}\varepsilon^{\prime}(\varepsilon)=0$. ∎

###### 2.6. Proposition.

Let $\Omega\subset\mathbb{R}^{n}$ be bounded with
$\delta:=\overline{\dim_{M}}(\partial\Omega,\Omega)<\infty$ and upper inner
Minkowski content
$\overline{\mathcal{M}}_{\delta}(\partial\Omega,\Omega)\in(0,\infty)$. Let
$\mathcal{W}$ be a Whitney cover of $\Omega$ and for any $\varepsilon>0$, let

$\displaystyle\mathcal{W}_{\varepsilon}:=\left\\{Q\in\mathcal{W}:Q\cap\left(\Omega\backslash\Omega_{-\varepsilon}\right)\neq\emptyset\right\\},$

in other words, $\mathcal{W}_{\varepsilon}$ is the smallest collection of
Whitney cubes in $\mathcal{W}$ that covers
$\\{x\in\Omega:\operatorname{dist}(x,\partial\Omega)\geq\varepsilon\\}$. Then
there is $A_{\Omega}\in\mathbb{R}$ such that

$\displaystyle\operatorname{vol}_{n-1}\partial\left(\overline{\bigcup_{Q\in\mathcal{W}_{\varepsilon}}Q}\right)\leq
A_{\Omega}\cdot\varepsilon^{(n-1)-\delta}.$

###### Proof.

Since the closure of the outermost cubes in $\mathcal{W}_{\varepsilon}$
intersect $\partial\Omega_{-\varepsilon}$, we have
$\operatorname{vol}_{n-1}\partial\left(\overline{\bigcup_{Q\in\mathcal{W}_{\varepsilon}}Q}\right)\leq\sum_{Q\in\mathcal{W}_{\varepsilon}:\overline{Q}\cap\partial\Omega_{-\varepsilon}\neq\emptyset}\operatorname{vol}_{n-1}\partial
Q$. Suppose that $\overline{Q}$ has non-empty intersection with
$\partial\Omega_{-\varepsilon}$. Then there exists $x\in\overline{Q}$ such
that $\operatorname{dist}(x,\partial\Omega)=\varepsilon$. Now with (2.5) and
$\operatorname{dist}(Q,\partial\Omega)+\operatorname{diam}(Q)\geq\sup_{x\in
Q}\operatorname{dist}(x,\partial\Omega)\geq\varepsilon\geq\operatorname{dist}(Q,\partial\Omega)$,
we observe that

$\displaystyle
5\operatorname{diam}(Q)\geq\varepsilon\geq\operatorname{diam}(Q)\geq\frac{\varepsilon}{5}.$

With $k_{\min}:=\min\\{k\in\mathbb{Z}:2^{-k}\sqrt{n}\leq\varepsilon\\}$, the
diameter of such a cube is therefore restricted to one of at most three
possible sizes,
$\operatorname{diam}(Q)\in\\{2^{-k_{\min}}\sqrt{n},2^{-(k_{\min}+1)}\sqrt{n},2^{-(k_{\min}+2)}\sqrt{n}\\}.$
Hence, an upper bound of the circumference of
$\overline{\bigcup_{Q\in\mathcal{W}_{\varepsilon}}Q}$ is

$\displaystyle\operatorname{vol}_{n-1}\partial\left(\overline{\bigcup_{Q\in\mathcal{W}_{\varepsilon}}Q}\right)$
$\displaystyle\leq\sum_{k=k_{\min}}^{k_{\min}+2}\sum_{Q\in\mathcal{W}_{k}}\operatorname{vol}_{n-1}\partial
Q=2n\sum_{k=k_{\min}}^{k_{\min}+2}\\#\mathcal{W}_{k}\cdot 2^{-k(n-1)}$
$\displaystyle\leq
2n\mathfrak{M}_{\Omega}\sum_{k=0}^{2}2^{-k((n-1)-\delta)}\cdot
2^{-k_{\min}((n-1)-\delta)}\leq A_{\Omega}\varepsilon^{(n-1)-\delta},$

where
$A_{\Omega}:=2n(2\sqrt{n})^{\delta-(n-1)}\mathfrak{M}_{\Omega}\sum_{k=0}^{2}2^{-k((n-1)-\delta)}$
since one has $2\cdot 2^{-k_{\min}}\sqrt{n}\geq\varepsilon$ and
$n-1\leq\delta$. ∎

In particular in the regular case (i. e. when $\delta=n-1$), Prop. 2.6 shows
that the circumference of the closure of the union of all large enough cubes
of a Whitney cover is bounded.

###### 2.7. Proposition.

In the setting of Prop. 2.6 almost all Billard trajectories in
$\overline{\bigcup_{Q\in\mathcal{W}_{\varepsilon}}Q}$ are non-periodic.

###### Proof.

We may assume that $\mathcal{W}_{\varepsilon}$ consists only of cubes that
have at least one face shared with another cube in
$\mathcal{W}_{\varepsilon}$, as the statement follows from known results for
isolated cubes. Then there is $k_{\max}\in\mathbb{Z}$ such that
$\mathcal{W}_{\varepsilon}\subset\bigcup_{k\leq k_{\max}}\mathcal{W}_{k}$ and
all cubes in $\mathcal{W}_{\varepsilon}$ have vertices in the
$2^{-k_{\max}}\mathbb{Z}^{n}$-lattice. Let
$x\in\partial\overline{\bigcup_{Q\in\mathcal{W}_{\varepsilon}}Q}$ and $v$ be
any direction of a Billard. Then, up to lattice symmetry, the set of
reflection points of such a trajectory in
$\overline{\bigcup_{Q\in\mathcal{W}_{\varepsilon}}Q}$ is contained in
$\\{x+tv:t\in\mathbb{R}\backslash\\{0\\}\\}$. Up to lattice symmetry, this set
contains the origin $x$ iff $v$ is rational up to normalisation. ∎

##  3 Estimates of the first non-trivial eigenvalue and of counting functions

From [30] by Netrusov and Safarov one can conclude that for bounded variation
domains $\Omega$ one has
$N_{N}(\Omega,t)=C_{W}^{(n)}t^{n/2}+\mathcal{O}(t^{\delta/2})$, where
$\delta:=\overline{\dim_{M}}(\partial\Omega,\Omega)$ is the upper inner
Minkowski dimension. The aim of this section is to obtain an analogue
asymptotic result on $N_{N}(\Omega,t)$ for more general domains, with focus on
domains with self-similar boundary, thus generalising the results and
partially resolving a question raised in [30, Sec. 5.4.2] in Thm. 3.8 and Rem.
3.9. Key for obtaining such an asymptotic result for $N_{N}(\Omega,t)$ is the
construction of a foliation in $\Omega_{-\varepsilon}$.

###  3.1 Foliations

The rough procedure for obtaining asymptotic results on $N_{N}(\Omega,t)$ is
to partition $\Omega$ into $\Omega_{-\varepsilon}$ and
$\Omega\setminus\Omega_{-\varepsilon}$. For
$\Omega\setminus\Omega_{-\varepsilon}$ we will use Whitney covers, and for
$\Omega_{-\varepsilon}$ we will use paths that connect points which are close
to the boundary $\partial\Omega$ to points located in a more controlled region
away from the boundary (close to $\partial\Omega_{-\varepsilon}$). If a family
of such paths partitions a domain, we will call it a _foliation_ of the
domain. If the corresponding volume form is bounded in a certain sense (see
Def. 3.1), such a family of paths allows an estimate of the second Neumann
eigenvalue of the domain. Following an idea by Brolin in [6], we develop a
method to construct foliations for domains whose boundary locally admits IFS
structure. In the below, we will restrict ourselves to the two-dimensional
setting and impose some assumptions for ease of exposition. Note that
generalisations to higher dimensions as well as more general settings of IFS
can be formulated (see for example Rem. 3.3).

To state our assumptions, let us recall some notions. By an _iterated function
system (IFS)_ we understand a family $\Phi=\\{\phi_{i}\\}_{i\in\Sigma}$ of
contracting maps $\phi_{i}\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$ with finite
index set $\Sigma$. The _limit set_ associated to the IFS $\Phi$ is the unique
non-empty compact set satisfying $M=\bigcup_{i\in\Sigma}\phi_{i}(M)=:\Phi(M)$.
The limit set is called _self-similar_ if all the $\phi_{i}$ are
_similarities_ , i. e. $|\phi_{i}(x)-\phi_{i}(y)|=r_{i}|x-y|$ for any
$x,y\in\mathbb{R}^{2}$ and some $r_{i}\in(0,1)$. We say that $\Phi$ satisfies
the _open set condition (OSC)_ if there exists an open and bounded
$O\subset\mathbb{R}^{2}$ such that $\phi_{i}(O)\subset O$ and
$\phi_{i}(O)\cap\phi_{j}(O)=\emptyset$ for $i\neq j\in\Sigma$.

The domains $\Omega\subset\mathbb{R}^{2}$ that we consider here shall satisfy
the following. $\partial\Omega$ has a (local) self-similar IFS structure, i.
e. $\partial\Omega$ is a finite union of self-similar sets. Suppose that
$M\subset\partial\Omega$ is the limit set of an IFS
$\Phi=\\{\phi_{i}\\}_{i\in\Sigma}$ with $\Sigma:=\\{1,\cdots,m\\}$ satisfying
the open set condition with all $\phi_{i}$ being similarities. There exists a
closed interval $I_{0}$ embedded in some affine plane $\mathbb{R}^{2-1}$, such
that $\Phi^{k}I_{0}$ converges to $M$ in the Hausdorff metric as $k\to\infty$.
By applying translation, rotation and scaling we can assume without loss of
generality that $I_{0}=[0,1]\times\\{0\\}\subset\mathbb{R}^{2}$. Suppose that
$\Phi(I_{0})\subset\mathbb{R}\times[0,\infty)$ and that
$\phi_{i}(I_{0})\cap\phi_{j}(I_{0})\neq\emptyset$ if and only if
$j\in\\{i-1,i+1\\}$. In other words the maps $\phi_{i}$ are numbered in such a
way that only neighbouring $\phi_{i}(I_{0})$ intersect and we assume that the
intersection is a singleton. We introduce a foliation (called _seed
foliation_) of the domain $G_{1}$ bounded by $I_{1}:=\Phi(I_{0})$ and $I_{0}$
for which any fibre runs from $I_{0}$ to $\Phi(I_{0})$. This gives rise to a
map $b_{1}\colon I_{0}\to\Phi(I_{0})$ that maps $x\in I_{0}$ to the
corresponding point in $\Phi(I_{0})$ that lies in the same fibre as $x$, see
Fig. 11(a) for an example.

$\phi_{1}(I_{0})$$\phi_{2}(I_{0})$$\phi_{3}(I_{0})$$\phi_{4}(I_{0})$$E$$I_{0}$
Figure 1(a) Example of a seed foliation in $G_{1}$. It connects $I_{0}$ (a
unit interval) and $\Phi(I_{0})$ (the first iteration of the Koch curve,
dashed) given by its classic four maps $\\{\phi_{i}\\}_{i=1,\ldots,4}$ in a
pre-described manner. Additionally a box $E$ has been added underneath $I_{0}$
as needed in Lem. 3.2 and the fibres have been extended (in dotted lines)
through $E$.

Figure 1(b) Example of a seed foliation of the conformally distorted Koch
curve described in Rem. 3.3 with $I_{0}=[2,17]$ under the conformal map
$f\colon\mathbb{R}^{2}\setminus B_{1}(0)\to\mathbb{R}^{2}\setminus B_{1}(0)$,
$(x,y)\mapsto(x^{2}-y^{2},2xy)$.

Next, we show how the IFS leads to a foliation of
$\bigcup_{k\in\mathbb{N}}\overline{\Phi^{k}(G_{1})}$. For $k\geq 1$, let
$G_{k}$ be the domain bounded by $I_{0}$ and $\Phi^{k}(I_{0})$ and let $H_{k}$
be the closure of the domain bounded by $\Phi^{k-1}(I_{0})$ and
$\Phi^{k}(I_{0})$, i. e. $H_{k}=\overline{G_{k}\backslash G_{k-1}}$. Then
$\Phi$ maps $H_{k}$ into $H_{k+1}$ and this mapping is surjective. Thus, any
point $x\in\bigcup_{k}G_{k}$ lies in some $H_{k}$ and there is a fibre in
$H_{1}$ whose image under some iteration of maps of the IFS runs through $x$.
In order to describe the foliation, it is sufficient to describe the
construction of such fibres through points in $\Phi^{k}(I_{0})$. For such a
point $x$ there is a finite word $w=(w_{1}w_{2}\cdots w_{k})\in\Sigma^{k}$
with
$x\in\phi_{w}(I_{0}):=\phi_{w_{k}}\phi_{w_{k-1}}\cdots\phi_{w_{1}}(I_{0})$.444Note
that this convention is useful here as it allows efficient use of the left-
shift operator $\sigma$ instead of a right-shift. This gives rise to a map
$b_{k}^{-1}:\Phi^{k}(I_{0})\to\Phi^{k-1}(I_{0})$ given by $\phi_{w}(I_{0})\ni
x\mapsto\phi_{\sigma w}b_{1}^{-1}\left(\phi_{\sigma
w}\right)^{-1}(x)\in\phi_{\sigma w}(I_{0})$ that maps each point
$x\in\Phi^{k}(I_{0})$ to the point at which the fibre through $x$ intersects
$\Phi^{k-1}(I_{0})$. The concatenation $b_{k}b_{k-1}\cdots b_{1}$ then
constructs a fibre $\gamma_{q}$ from a unique $q\in I_{0}$ to $x$ showing that
$\bigcup_{k}G_{k}$ is foliated.

Note that each $\gamma_{q}$ is of finite length
$\operatorname{len}\gamma_{q}$, which can be seen as follows. Let $L$ denote
the supremum over the lengths of all fibers that connect a point in $I_{0}$ to
a point in $\Phi(I_{0})$. The above construction ensures that the length of
any fibre connecting a point in $I_{0}$ to a point in $\Phi^{k}(I_{0})$ is
bounded from above by $L\cdot\sum_{\ell=0}^{k-1}r^{\ell}\leq L/(1-r)$, where
$r<1$ denotes the supremum over the contraction ratios of
$\phi_{1},\ldots,\phi_{m}$. As the upper bound is independent of $k$, it gives
a uniform upper bound. This observation implies that we can parameterise
fibres $\gamma_{q}$ by $\gamma_{q}(t)$ for
$t\in[0,\operatorname{len}\gamma_{q}]$. We let
$\varphi:(q,t)\mapsto\gamma_{q}(t)$ denote the associated change of
coordinates, let $q\in\text{int}(I_{0})$ be fixed and $t>0$ be such that
$x:=\gamma_{q}(t)\in\text{int}(H_{s+1})$ for some $s$. Further, we let $v_{s}$
denote the intersection point of $\Phi^{s}(I_{0})$ with the fibre through $x$
and set $\widetilde{b_{1}}:=\varphi|_{\varphi^{-1}(G_{1})}$. Analogously to
the construction of $b_{k}$ we define $\widetilde{b_{k}}$ mapping
$(q_{{k-1}},t)$ with $q_{k-1}\in\Phi^{k-1}(I_{0})$ and $t$ small enough to the
point in $H_{k}$ whose fibre runs through $q_{{k-1}}$ and has a length $t$
from $q_{{k-1}}$. Then $\varphi(q,t)=\widetilde{b_{s+1}}(b_{s}\cdots
b_{1}q,\tilde{t})$ where $\widetilde{t}$ is the length of the fibre through
$q$ from $b_{s}\cdots b_{1}q$ to $\varphi(q,t)$. Therefore $t-\widetilde{t}$
is constant for small enough variations of $t$. Supposing that
$w\in\Sigma^{s}$ is such that
$v_{s}\in\phi_{\omega}(I_{0})\subset\Phi^{s}(I_{0})$ the density function
$\beta:=\lvert\det D\varphi\rvert$ of the change of coordinates given by
$\varphi$ satisfies

$\displaystyle\beta(q,t)$
$\displaystyle=\left|\det\left(\prod_{\ell=1}^{s}Db_{\ell}|_{b_{\ell-1}\cdots
b_{1}(q)}\right)\right|\cdot\left|\det D\widetilde{b_{s+1}}|_{(b_{s}\cdots
b_{1}q,\tilde{t})}\right|$
$\displaystyle=\left|\det\left(\prod_{\ell=1}^{s}Db_{1}|_{(\phi_{w|_{\ell-1}})^{-1}b_{\ell-1}\cdots
b_{1}(q)}\right)\right|\cdot\underbrace{\left|\det
D\widetilde{b_{1}}|_{(\phi_{w})^{-1}b_{s}\cdots
b_{1}q,\tilde{t}}\right|}_{\in\text{image}_{x\in G_{1}}\beta(x)},$

since all $\phi_{i}$ are similarities. This quantity is readily computed for
triangles, which is the only relevant case here, see Fig. 2. The collection of
fibres thus constructed gives a partition of the domain into rectified curves
that are differentiable almost everywhere. We add a bounded domain $E$ with
$I_{0}\subset\partial E$ (e. g. a rectangle) to which we can extend each fibre
for some $t<0$. Thus, every fibre will run through $E$ before passing $I_{0}$.
We adjust the parametrisation so that any such extended path $\gamma$ is
parameterised by some interval $(0,\operatorname{len}\gamma)$.

$a$$b$$c$ Figure 2: Basic case of a ’natural’ foliation in a triangle. In such
a situation an elementary calculation reveals that $|\det
Db_{1}(q)|=\frac{b}{a}$.

###### 3.1. Definition.

A domain $D$ is called _well-foliated_ if there exists a collection
$\\{\gamma\\}_{\gamma\in\Gamma}$ of paths in $D$ as above with
$0<\inf_{\gamma\in\Gamma}\operatorname{len}(\gamma)\leq\sup_{\gamma\in\Gamma}\operatorname{len}(\gamma)<L$
such that the density $\beta$ is bounded in the sense that

$\displaystyle\operatorname{ess\,inf}_{x\in D}\beta(x)>0\ \text{and}$ (3.1)
$\displaystyle\sup_{\gamma\in\Gamma}\int_{0}^{\operatorname{len}(\gamma)}\beta(\gamma,t)dt<\infty.$
(3.2)

In this setting, let $r\in(0,\inf_{\gamma\in\Gamma}\operatorname{len}\gamma)$
be fixed and define $E:=\\{\gamma(t):\gamma\in\Gamma,0<t<r\\}$. For $S\subset
D$, we write $\beta_{\inf}(S):=\operatorname{ess\,inf}_{x\in S}\beta(x)$ and
$\mathcal{I}_{\beta}(S):=\sup_{\gamma\in\Gamma}\int_{0}^{\operatorname{len}\gamma}1_{S}(\gamma(t))\beta(\gamma(t))dt$.

Note that we can identify a path $\gamma\in\Gamma$ with its intersection point
$\gamma(0)=q\in I_{0}$.

###### 3.2. Lemma.

Let $D\subset\mathbb{R}^{n}$ be a domain that is well-foliated by
$\\{\gamma\\}_{\gamma\in\Gamma}$ as in Def. 3.1. Then for all $\alpha>0$

$\displaystyle\lambda_{2}^{N}(D)$
$\displaystyle\geq\left(\lambda_{2}^{N}(E)^{-1}\left(1+\frac{1+\alpha}{r}\frac{\mathcal{I}_{\beta}{(D\backslash
E)}}{\beta_{\inf}{(E)}}\right)+\left(1+\alpha^{-1}\right)L\frac{\mathcal{I}_{\beta}(D\backslash
E)}{\beta_{\inf}(D)}\right)^{-1}.$

###### Proof.

Let $u\in H^{1}(D)\cap 1^{\perp}$ and set
$\overline{u_{E}}:=\frac{1}{\operatorname{vol}E}\int_{E}udx$. Then
$\|u\|^{2}_{L^{2}(D)}\leq\|u-\overline{u_{E}}\|^{2}_{L^{2}(D)}$. In
particular, by Cor. 2.1, it suffices to show the following Poincaré-Wirtinger
inequality

$\displaystyle\|u-\overline{u_{E}}\|_{L^{2}(D)}^{2}\leq C\|\nabla
u\|_{L^{2}(D)}^{2},$ (3.3)

with $C^{-1}$ given by the right hand side of the claim. To this end, we
subdivide the expression

$\displaystyle\|u-\overline{u_{E}}\|_{L^{2}(D)}^{2}=\underbrace{\int_{E}|u(y)-\overline{u_{E}}|^{2}dy}_{:=I_{1}}+\underbrace{\int_{D\backslash
E}|u(x)-\overline{u_{E}}|^{2}dx}_{:=I_{2}},$

and estimate $I_{1}$ and $I_{2}$ separately. By Cor. 2.1, one has

$\displaystyle I_{1}\leq\lambda^{N}_{2}(E)^{-1}\|\nabla
u\|_{L^{2}(E)}^{2}\leq\lambda^{N}_{2}(E)^{-1}\|\nabla u\|_{L^{2}(D)}^{2}.$
(3.4)

Recall that for any $a,b,c\in\mathbb{C}$ and $\alpha>0$, there is the
following inequality:
$|a-b|^{2}\leq(1+\alpha)|a-c|^{2}+(1+\alpha^{-1})|b-c|^{2}$. We now consider

$\displaystyle I_{2}$ $\displaystyle=\int_{D\backslash
E}|u(x)-\overline{u_{E}}|^{2}dx=\int_{\Gamma}\int_{r}^{\operatorname{len}\gamma}|u(\gamma(t))-\overline{u_{E}}|^{2}\beta(\gamma,t)dtd\gamma,$
$\displaystyle=\frac{1}{r}\int_{\Gamma}\int_{r}^{\operatorname{len}\gamma}\int_{0}^{r}|u(\gamma(t))-\overline{u_{E}}|^{2}\beta(\gamma,t)dt^{\prime}dtd\gamma$
$\displaystyle\leq\underbrace{\frac{1+\alpha}{r}\int_{\Gamma}\int_{r}^{\operatorname{len}\gamma}\int_{0}^{r}|u(\gamma(t^{\prime}))-\overline{u_{E}}|^{2}\beta(\gamma,t)dt^{\prime}dtd\gamma}_{=:I_{2}^{\prime}}$
$\displaystyle\qquad\qquad+\underbrace{\frac{1+\alpha^{-1}}{r}\int_{\Gamma}\int_{r}^{\operatorname{len}\gamma}\int_{0}^{r}|u(\gamma(t))-u(\gamma(t^{\prime}))|^{2}\beta(\gamma,t)dt^{\prime}dtd\gamma}_{=:I_{2}^{\prime\prime}},$

which holds for any $\alpha>0$. By assumption,
$\int_{r}^{\operatorname{len}\gamma}\beta(\gamma,t)dt\leq\mathcal{I}_{\beta}(D\backslash
E)$ for all $\gamma\in\Gamma$. Hence

$\displaystyle I_{2}^{\prime}$
$\displaystyle=\frac{1+\alpha}{r}\int_{\Gamma}\int_{r}^{\operatorname{len}\gamma}\int_{0}^{r}|u(\gamma(t^{\prime}))-\overline{u_{E}}|^{2}\beta(\gamma,t)dt^{\prime}dtd\gamma$
$\displaystyle\leq\frac{1+\alpha}{r}\mathcal{I}_{\beta}{(D\backslash
E)}\int_{\Gamma}\int_{0}^{r}|u(\gamma(t^{\prime}))-\overline{u_{E}}|^{2}dt^{\prime}d\gamma\leq\frac{1+\alpha}{r}\frac{\mathcal{I}_{\beta}{(D\backslash
E)}}{\beta_{\inf}{(E)}}\int_{\Gamma}\int_{0}^{r}|u(\gamma(t^{\prime}))-\overline{u_{E}}|^{2}\beta(\gamma,t^{\prime})dt^{\prime}d\gamma$
$\displaystyle=\frac{1+\alpha}{r}\frac{\mathcal{I}_{\beta}{(D\backslash
E)}}{\beta_{\inf}{(E)}}\|u-\overline{u_{E}}\|^{2}_{L^{2}(E)}\leq\lambda_{2}^{N}(E)^{-1}\frac{1+\alpha}{r}\frac{\mathcal{I}_{\beta}{(D\backslash
E)}}{\beta_{\inf}{(E)}}\|\nabla u\|^{2}_{L^{2}(D)},$

and

$\displaystyle I_{2}^{\prime\prime}$
$\displaystyle=\frac{1+\alpha^{-1}}{r}\int_{\Gamma}\int_{r}^{\operatorname{len}\gamma}\int_{0}^{r}|u(\gamma(t))-u(\gamma(t^{\prime}))|^{2}\beta(\gamma,t)dt^{\prime}dtd\gamma$
$\displaystyle=\frac{1+\alpha^{-1}}{r}\int_{\Gamma}\int_{r}^{\operatorname{len}\gamma}\int_{0}^{r}\left|\int_{t^{\prime}}^{t}\partial_{s}u(\gamma(s))|_{s=\sigma}d\sigma\right|^{2}\beta(\gamma,t)dt^{\prime}dtd\gamma$
$\displaystyle\stackrel{{\scriptstyle(\dagger)}}{{\leq}}\frac{1+\alpha^{-1}}{r}\int_{\Gamma}\int_{r}^{\operatorname{len}\gamma}\int_{0}^{r}(t-t^{\prime})\int_{t^{\prime}}^{t}\left|\partial_{s}u(\gamma(s))|_{s=\sigma}\right|^{2}d\sigma\beta(\gamma,t)dt^{\prime}dtd\gamma$
$\displaystyle\leq\frac{1+\alpha^{-1}}{r}\int_{\Gamma}\int_{r}^{\operatorname{len}\gamma}\int_{0}^{r}(t-t^{\prime})\int_{t^{\prime}}^{t}\left|\nabla
u(x)|_{x=\gamma(\sigma)}\right|^{2}d\sigma\beta(\gamma,t)dt^{\prime}dtd\gamma$
$\displaystyle\leq\frac{1+\alpha^{-1}}{r}\int_{\Gamma}\int_{r}^{\operatorname{len}\gamma}\int_{0}^{r}L\int_{0}^{\operatorname{len}\gamma}\left|\nabla
u(x)|_{x=\gamma(\sigma)}\right|^{2}d\sigma\beta(\gamma,t)dt^{\prime}dtd\gamma\,,\qquad\text{
as }t-t^{\prime}\leq L$
$\displaystyle\leq(1+\alpha^{-1})L\mathcal{I}_{\beta}(D\backslash
E)\int_{\Gamma}\int_{0}^{\operatorname{len}\gamma}\left|\nabla
u(x)|_{x=\gamma(\sigma)}\right|^{2}d\sigma d\gamma$
$\displaystyle\leq(1+\alpha^{-1})L\frac{\mathcal{I}_{\beta}(D\backslash
E)}{\beta_{\inf}{(D)}}\int_{\Gamma}\int_{0}^{\operatorname{len}\gamma}\left|\nabla
u(x)|_{x=\gamma(\sigma)}\right|^{2}\beta(\gamma,\sigma)d\sigma
d\gamma=(1+\alpha^{-1})L\frac{\mathcal{I}_{\beta}(D\backslash
E)}{\beta_{\inf}{(D)}}\|\nabla u\|^{2}_{L^{2}(D)}.$

where we used an application of Jensen’s Theorem in $(\dagger)$.555Using
Jensen and $|\nabla\gamma_{q}(s)|^{2}=1$, $|s|\leq 1$ and
$|\partial_{s}u|=|\langle s,\nabla u\rangle|\leq|\nabla u|$,
$\displaystyle|u(x)-u(y)|^{2}$
$\displaystyle=\left|\int_{t_{0}}^{t}\partial_{s}u(\gamma_{q}(s))ds\right|^{2}\leq(t-t_{0})\int_{t_{0}}^{t}\left|\partial_{s}u(\gamma_{q}(s))\right|^{2}ds$
$\displaystyle\leq(t-t_{0})\int_{t_{0}}^{t}\left|\nabla
u\right|^{2}ds\leq(t-t_{0})\int_{0}^{\operatorname{len}\gamma_{q}}\left|\nabla
u|_{\gamma_{q}(s)}\right|^{2}ds.$ ∎

###### 3.3. Remark.

  1. (i).

Lem. 3.2 generalises results of [30]. A comparison with the results of [30]
will be presented in Sec. 3.2.

  2. (ii).

Note that Lem. 3.2 and all results below hold for the higher dimensional
setting, where $\Omega\subset\mathbb{R}^{n}$ with arbitrary $n\in\mathbb{N}$.
Merely the construction of a foliation is described only for the case $n=2$,
for ease of exposition of the construction and formulation of suitable
conditions.

  3. (iii).

The above construction is presented for IFS consisting of similarities for
ease of exposition. However, the construction extends to more general IFS. For
example, a seed foliation can be constructed for the image $f(V)$ of the Koch
curve $V$ under a conformal map $f$ as shown in Fig. 11(b). Notice that $f(V)$
is a self-conformal set and that the corresponding conformal IFS can be used
to transport the seed foliation to obtain a foliation of the region bounded
between the $x$-axis and the conformal Koch curve. In Fig. 11(b), we applied
$f$ to the seed foliation of $V$ shown in Fig. 11(a), but note that other
constructions for seed foliations are possible in more general settings.

###### 3.4. Definition.

A domain $\Omega\subset\mathbb{R}^{n}$ is called _well-covered_ if there
exists $\mu\in\mathbb{N}$ and $\varepsilon_{0}>0$ such that for any
$\varepsilon\in(0,\varepsilon_{0}]$ there is a cover of
$\Omega_{-\varepsilon}$ by well-foliated domains
$\\{D_{i}^{\varepsilon}\subset\Omega\\}_{i\in I_{\varepsilon}}$ satisfying
$\operatorname{vol}_{n}(\Omega_{-\varepsilon})=\operatorname{vol}_{n}(\Omega_{-\varepsilon}\cap(\bigcup_{i}D_{i}^{\varepsilon}))$
with corresponding quantities $r,L,\mathcal{I}_{\beta}$ (as in Def. 3.1) and
uniformly bounded multiplicity $\leq\mu$ such that

  1. (i).

There are $\delta\in\mathbb{R}$ and $C(\Omega)\in\mathbb{R}$ such that
$\\#I_{\varepsilon}\leq C(\Omega)\varepsilon^{-\delta}$ for all
$\varepsilon\in(0,\varepsilon_{0}]$.

  2. (ii).

There exist constants
$c_{r}^{\pm},c_{L}^{\pm},c_{\mathcal{I}}^{\pm},c_{\operatorname{diam}}^{\pm},c_{\operatorname{vol}}^{\pm}>0$
such that for all $\varepsilon\in(0,\varepsilon_{0}]$ and for each
$D_{i}^{\varepsilon}$ one has

$\tfrac{r}{\varepsilon}\in(c_{r}^{-},c_{r}^{+}),\
\tfrac{L}{\varepsilon}\in(c_{L}^{-},c_{L}^{+}),\
\tfrac{\mathcal{I}_{\beta}}{\varepsilon}\in(c_{\mathcal{I}}^{-},c_{\mathcal{I}}^{+}),\
\tfrac{\operatorname{diam}D_{i}^{\varepsilon}}{\varepsilon}\in(c_{\operatorname{diam}}^{-},c_{\operatorname{diam}}^{+}),\
\tfrac{\operatorname{vol}_{n}(D_{i}^{\varepsilon})}{\varepsilon^{n}}\in(c_{\operatorname{vol}}^{-},c_{\operatorname{vol}}^{+}).$

  3. (iii).

$\inf_{\varepsilon,i}\beta_{\inf}(D_{i}^{\varepsilon})>0$.

  4. (iv).

For any $D_{i}^{\varepsilon}$ the corresponding $E$ must have first non-
trivial Neumann eigenvalue $\lambda_{2}^{N}(E)\geq c_{E}r^{-2}$ for a fixed
$c_{E}$ independent of $\varepsilon$.

To simplify notation, we sometimes use $\ll$, $\gg$ and $\asymp$ with the
following meaning: For two real-valued functions $f,g$ with common domain of
definition $D$ we write $f\ll g$ if there exists a constant $c\in\mathbb{R}$
such that $f(x)\leq cg(x)$ holds for all $x\in D$. Moreover, we write $f\asymp
g$ if $f\ll g$ and $f\gg g$.

###### 3.5. Remark.

  1. (i).

Condition (i) in Def. 3.4 is closely related to upper inner Minkowski
measurabilty. To see this, notice that Def. 3.4(i) implies

$\displaystyle\operatorname{vol}_{n}(\Omega_{-\varepsilon})\leq\sum_{i\in
I_{\varepsilon}}\operatorname{vol}_{n}D_{i}^{\varepsilon}\leq
c_{\operatorname{vol}}^{+}\varepsilon^{n}\\#I_{\varepsilon}\leq
c_{\operatorname{vol}}^{+}C(\Omega)\varepsilon^{n-\delta}.$

If $\delta$ coincides with the upper inner Minkowski dimension of
$\partial\Omega$, the above equation implies that
$\overline{\mathcal{M}}_{\delta}(\partial\Omega,\Omega)<\infty$. Moreover, if
$\Omega$ is upper inner Minkowski measurable, then by definition of the inner
upper Minkowski content, there is an error $\varepsilon^{\prime}(\varepsilon)$
with $\limsup_{\varepsilon}\varepsilon^{\prime}(\varepsilon)=0$ (cf. proof of
Prop. 2.5) such that

$\displaystyle\\#I_{\varepsilon}$
$\displaystyle\leq\frac{1}{\varepsilon^{n}c_{\operatorname{vol}}^{-}}\sum_{i\in
I_{\varepsilon}}\operatorname{vol}_{n}D_{i}^{\varepsilon}\leq\frac{\mu}{c_{\operatorname{vol}}^{-}}\operatorname{vol}_{n}\Omega_{-\varepsilon{(c_{\operatorname{diam}}^{+}+1)}}\varepsilon^{-n}$
$\displaystyle\leq\frac{\mu}{c_{\operatorname{vol}}^{-}}\overline{\mathcal{M}}_{\delta}(\partial\Omega,\Omega)\varepsilon^{-\delta}(c_{\operatorname{diam}}^{+}+1)^{n-\delta}\left(1+\varepsilon^{\prime}(\varepsilon(c_{\operatorname{diam}}^{+}+1))\right),$

Notice that through $\varepsilon^{\prime}$, yet another geometric quantity of
$\partial\Omega$, namely the speed of convergence of
$\operatorname{vol}_{n}(\Omega_{-\varepsilon})\varepsilon^{\delta-n}$ enters
the final estimate.

  2. (ii).

The reverse condition in Def. 3.4.(iv) is automatically satisfied for all
domains to which (2.3) applies. More precisely, based on a result by Szegő in
[42], Weinberger showed in [46] a Faber-Krahn-type isoperimetric inequality:

$\displaystyle\lambda_{2}^{N}(\Omega)\leq\lambda_{2}^{N}(\Omega^{\ast})=\frac{p_{n}^{2}}{\left(\frac{1}{2}\operatorname{diam}\Omega^{\ast}\right)^{2}}=p_{n}^{2}\left(\frac{\omega_{n}}{\operatorname{vol}_{n}\Omega}\right)^{\frac{2}{n}},$

where $\Omega^{\ast}$ is the ball with
$\operatorname{vol}_{n}(\Omega^{\ast})=\operatorname{vol}_{n}(\Omega)$,
$\omega_{n}$ is the volume of the $n$-dimensional unit ball and $p_{n}$ is the
first positive zero of
$\left(x^{1-\frac{n}{2}}J_{\frac{n}{2}}(x)\right)^{\prime}$, i. e. the
smallest positive solution of $J_{\frac{n}{2}}(x)=xJ_{1+\frac{n}{2}}(x)$ where
$J_{m}(x)$ is the $m$-th spherical Bessel function. Therefore one may replace
the condition in Def. 3.4.(iv) with
$\lambda_{2}^{N}(E)\asymp\varepsilon^{-2}$.

  3. (iii).

Bi-Lipschitz maps preserve the property of being well-covered: Let
$f:\Omega\to\Omega^{\prime}$ be a bi-Lipschitz map and let $\Omega$ be well-
covered. Then $\Omega^{\prime}$ is also well-covered. Indeed, if a family of
covers $\\{D_{i}^{\varepsilon}\\}_{i\in I_{\varepsilon}}$ satisfies the
conditions of Def. 3.4, then $\\{f(D_{i}^{\varepsilon})\\}_{i\in
I_{\varepsilon}}$ can be used to cover $(f\Omega)_{-\varepsilon^{\prime}}$
with $\varepsilon\asymp\varepsilon^{\prime}$.

###### 3.6. Corollary
(“$\lambda_{2}^{N}(D)\asymp\operatorname{diam}(D)^{-2}$”).

Let $\Omega$ be well-covered and let $\\{D^{\varepsilon}_{i}\\}_{i\in
I_{\varepsilon}}$ be a cover of $\Omega_{-\varepsilon}$ consisting of well-
foliated domains as in Def. 3.4. Then by Lem. 3.2,

$\displaystyle\lambda_{2}^{N}(D^{\varepsilon}_{i})\geq
C_{1}(\Omega_{-\varepsilon})\varepsilon^{-2}$

###  3.2 Sets of bounded variation as well-foliated domains

In this section we show that the class of well-covered domains contains the
class of domains of bounded variation. For the class of domains of bounded
variation bounds on Laplace eigenvalue counting functions have been found in
[30].

###### 3.7. Definition (Def. 1.1-1.2 in [30]).

Let $\Omega^{\prime}\subset\mathbb{R}^{n-1}$ be a domain and let $f\in
C^{0}(\overline{\Omega^{\prime}},\mathbb{R})$. We define the _oscillation of
$f$ on $\Omega^{\prime}$_ as

$\displaystyle\operatorname{Osc}(f,\Omega^{\prime}):=\frac{1}{2}\left(\sup_{x\in\Omega^{\prime}}f(x)-\inf_{x\in\Omega^{\prime}}f(x)\right).$

Let $Q_{n}$ be an $n$-dimensional cube with arbitrary size whose edges are
parallel to the axes.

  1. (i).

Let $f:Q_{n}\to\mathbb{R}$ be bounded. Then for any $\delta>0$, we define
$\mathcal{V}_{\delta}(f,Q_{n})$ as the maximal number of disjoint open cubes
$Q_{n,i}\subset Q_{n}$ whose edges are parallel to coordinate axes with
$\operatorname{Osc}(f,Q_{n,i})\geq\delta$ for all $i$. If
$\operatorname{Osc}(f,Q_{n})<\delta$, we set
$\mathcal{V}_{\delta}(f,Q_{n}):=1$.

  2. (ii).

Let $\tau:\mathbb{R}_{>0}\to\mathbb{R}_{>0}$ be non-decreasing. The space
spanned by the continuous functions $f:\overline{Q_{n}}\to\mathbb{R}$ defined
on some cube $Q_{n}$ such that $\mathcal{V}_{1/t}(f,Q_{n})\leq\tau(t)$ for all
$t$, is called the set of functions with $(\tau,\infty)$-bounded variation and
denoted by $\operatorname{BV}_{\tau,\infty}(Q_{n})$.

For a domain $\Omega\subset\mathbb{R}^{n}$, we write
$\Omega\in\operatorname{BV}_{\tau,\infty}$ iff for any point
$p\in\partial\Omega$ there is an open neighbourhood
$U_{p}\subset\mathbb{R}^{n}$ such that (up to an $SO(n)$), $U_{p}\cap\Omega$
is the set of points below the graph of some $f\in BV_{\tau,\infty}(Q_{n-1})$
and a constant function $b<\inf f$.

###### 3.8. Theorem.

Let $\tau(t)\asymp t^{\delta}$ for some $\delta\geq 1$. Then the set of well-
covered domains contains $\operatorname{BV}_{\tau,\infty}$ as a proper subset.

###### Proof.

Let $\Omega\in\operatorname{BV}_{\tau,\infty}$ be a domain in $\mathbb{R}^{n}$
and let $p\in\partial\Omega$ with corresponding neighbourhood $U_{p}$. Then
$U_{p}\cap\Omega$ is bounded by a graph $\Gamma_{f}$ of a function
$f:\overline{Q_{n-1}}\to\mathbb{R}$ from above and a constant $b<\inf f$ from
below. Hence there is a foliation $\\{\gamma\in\Gamma\\}$ by straight lines:
For any $x\in Q_{n-1}$, let
$\gamma_{x}:t\mapsto(x_{1},x_{2},\cdots,x_{n-1},b)+t(0,\cdots,0,1)$ be so that
$\operatorname{len}\gamma_{x}=f(x)-b>0$. Such foliations satisfy conditions
(3.1)-(3.2) trivially, as the corresponding density $\beta$ is $1$ everywhere.
Then for some $r\in(b,\inf f)$, we set $E:=\\{x\in U_{p}:\exists\gamma\text{
s.\,t.\ }x=\gamma(r)\\}$ showing that $U_{p}$ is well-foliated for all
$p\in\partial\Omega$.

Since $\partial\Omega$ is compact (being closed and bounded), there is a
finite set of open neighbourhoods $U_{p}$ corresponding to some
$f\in\operatorname{BV}_{\tau,\infty}$ that cover $\partial\Omega$. By [30,
Thm. 3.5, Cor. 3.8], for any $\varepsilon>0$ small enough, each of these can
be covered by a family of well-foliated domains of size $\asymp\varepsilon$
with $r=\varepsilon$ and of cadinality
$\ll\mathcal{V}_{\varepsilon/2}(f,Q_{n-1})+\varepsilon^{-n}\operatorname{vol}_{n}(\Omega_{-\varepsilon})+\varepsilon^{-1}\int_{1}^{1/\varepsilon}t^{-2}\tau(t)dt\ll\tau(1/\varepsilon)=\varepsilon^{-\delta}$.

Since the Koch snowflake $K$ is not locally a graph,
$K\notin\operatorname{BV}_{\tau,\infty}$ while $K$ is well-covered as shown in
Sec. 3.4. ∎

###### 3.9. Remark.

Suppose there is a bi-Lipschitz function $f:\Omega\to\Omega^{\prime}$ with
Lipschitz constant $L$, such that
$\Omega^{\prime}\in\operatorname{BV}_{\tau,\infty}$ with $\tau(t)\sim
t^{-\delta}$ for some $\delta>1$. Then $\Omega$ is well-covered. Such domains
are studied in [30] but in many practical cases, it appears to be easier to
show a domain is well-covered instead of showing it is
$\operatorname{BV}_{\tau,\infty}$ up to a bi-Lipschitz transformation.
Moreover, Thm. 3.8 shows that Thm. 3.10 covers a larger class than
$\operatorname{BV}_{\tau,\infty}$ addressing the open problem raised in [30,
Sec. 5.4.2].

###  3.3 Bounds on eigenvalue counting functions

###### 3.10. Theorem.

Let $\Omega\subset\mathbb{R}^{n}$ be a domain. Suppose $\Omega$ is well-
covered and that the upper inner Minkowski dimension $\delta$ of $\Omega$
satisfies $\delta\in[n-1,n)$. Then there exists $M_{\Omega}$ given in Sec. 4
such that

$\displaystyle N_{N}(\Omega,t)\leq
C_{W}^{(n)}\operatorname{vol}_{n}(\Omega)t^{n/2}+M_{\Omega}t^{\delta/2},$

for all sufficiently large $t$.

###### Proof.

Let $\\{Q\\}_{Q\in\mathcal{W}}$ denote a Whitney cover of $\Omega$ (see Sec.
2.3). For $\varepsilon>0$ we restrict $\\{Q\\}_{Q\in\mathcal{W}}$ to the
smallest subset $\mathcal{W}_{\varepsilon}\subset\mathcal{W}$ of cubes volume
covering $\Omega\backslash\Omega_{-\varepsilon}$, i. e. those cubes that are
sufficiently far away from $\partial\Omega$ and hence are sufficiently large.
Suppose that $\varepsilon$ is small enough so that $\Omega_{-\varepsilon}$ is
well-covered by well-foliated domains $\\{D^{\varepsilon}_{i}\\}_{i\in
I_{\varepsilon}}$ with $L_{i}^{\varepsilon}\sim\varepsilon$. Let
$J_{\varepsilon}\subset I_{\varepsilon}$ be such that
$D_{i}^{\varepsilon}\cap\Omega\backslash\bigcup_{\mathcal{W}_{\varepsilon}}Q\neq\emptyset$
for $i\in J_{\varepsilon}$ and set
$U_{\varepsilon}:=\\{D^{\varepsilon}_{i}\\}_{i\in J_{\varepsilon}}$. Then
$\Omega=\bigcup_{i\in
J_{\varepsilon}}D^{\varepsilon}_{i}\cup\bigcup_{\mathcal{W}_{\varepsilon}}Q$
and this volume cover can be split into two disjoint volume subcovers. Let
$T_{\varepsilon}:=\\{Q\in\mathcal{W}_{\varepsilon}:Q\cap\bigcup_{i\in
J_{\varepsilon}}D_{i}^{\varepsilon}\neq\emptyset\\}$ and
$U_{\varepsilon}^{\mathcal{W}}:=U_{\varepsilon}\cup T_{\varepsilon}$ be the
set of all well-foliated domains $D_{i}^{\varepsilon}$ together with all cubes
in $\mathcal{W}_{\varepsilon}$ that interset some $D_{i}^{\varepsilon}$. Next,
we restrict the Whitney cover further to the minimal set of cubes necessary to
volume cover $\Omega$ given the volume cover $\\{U\\}_{U\in
U_{\varepsilon}^{\mathcal{W}}}$ by defining
$\widetilde{\mathcal{W}}_{\varepsilon}:=\\{Q\in\mathcal{W}_{\varepsilon}:Q\cap
U=\emptyset\,\,\forall U\in U_{\varepsilon}^{\mathcal{W}}\\}$. These volume
covers are disjoint and have multiplicity
$\mu(U_{\varepsilon}^{\mathcal{W}})\leq\mu+1$ and
$\mu(\widetilde{\mathcal{W}}_{\varepsilon})=1$ so that
$\operatorname{vol}_{n}\Omega=\operatorname{vol}_{n}\left[\left(\bigcup_{U_{\varepsilon}^{\mathcal{W}}}U\right)\sqcup\left(\operatorname{int}\bigcup_{\widetilde{\mathcal{W}}_{\varepsilon}}\overline{Q}\right)\right]$.
Therefore

$\displaystyle
N_{N}(\Omega,t)\leq\underbrace{N_{N}\bigg{(}\bigcup_{U_{\varepsilon}^{\mathcal{W}}}U,t\bigg{)}}_{:=S^{\varepsilon}_{1}(t)}+\underbrace{N_{N}\bigg{(}\operatorname{int}\bigcup_{\widetilde{\mathcal{W}}_{\varepsilon}}\overline{Q},t\bigg{)}}_{:=S^{\varepsilon}_{2}(t)}$

and we estimate both terms separately.

  1. $S_{1}^{\varepsilon}(t)$.

By construction one can estimate the first non-trivial eigenvalue of any
Whitney cube $Q\in U_{\varepsilon}^{\mathcal{W}}$ as,

$\displaystyle\lambda_{2}^{N}(Q)=\left(\frac{\sqrt{n}\pi}{\operatorname{diam}Q}\right)^{2}\geq\left(\frac{\sqrt{n}\pi}{\operatorname{dist}(Q,\partial\Omega)}\right)^{2}\geq\left(\frac{\sqrt{n}\pi}{c_{\operatorname{diam}}^{+}\varepsilon}\right)^{2}.$

For large enough $\lambda$ we can choose $\varepsilon$ s. t.
$(\mu+1)\lambda=C_{2}\varepsilon^{-2}$. We define the auxiliary constant
$1\leq
s_{\sup}:=\sup_{\varepsilon}\frac{1}{\varepsilon}\inf\left(\nu:\bigcup_{i\in
I_{\varepsilon}}D_{i}^{\varepsilon}\subset\Omega_{-\nu}\right)\leq
c_{\operatorname{diam}}^{+}$. Any $Q\in T_{\varepsilon}$ satisfies
$\operatorname{dist}(Q,\partial\Omega)\geq\varepsilon-\operatorname{diam}(Q)$
(i. e. $\operatorname{diam}Q\geq\frac{\varepsilon}{5}$) and
$\operatorname{diam}Q\leq\operatorname{dist}(Q,\partial\Omega)\leq
s_{\sup}\varepsilon$. Then, by Prop. 2.5,

$\displaystyle S_{1}^{\varepsilon}(\lambda)$ $\displaystyle\leq\sum_{U\in
U_{\varepsilon}^{\mathcal{W}}}N_{N}\left(U,(\mu+1)\lambda\right)\leq\\#I_{\varepsilon}+\sum_{k_{-}\leq
k\leq
k_{+}}\\#\mathcal{W}_{k}\leq\\#I_{\varepsilon}+2^{k_{-}\delta}\sum_{k=0}^{\lfloor
2+\log_{2}(5s_{\sup})\rfloor}{\mathfrak{M}_{\Omega}}2^{k\delta}$
$\displaystyle\leq
C(\Omega)\varepsilon^{-\delta}+{\mathfrak{M}_{\Omega}}\left(\frac{\sqrt{n}}{s_{\sup}}\right)^{\delta}\frac{(8\cdot
5s_{\sup})^{\delta}-1}{2^{\delta}-1}\varepsilon^{-\delta}\leq
C_{3}(\Omega)\varepsilon^{-\delta}=\frac{C_{3}(\Omega)(\mu+1)^{\delta/2}}{C_{2}(\Omega)^{\delta/2}}\lambda^{\delta/2},$

where
$k_{-}:=\left\lfloor\log_{2}\frac{\sqrt{n}}{s_{\sup}\varepsilon}\right\rfloor$
and $k_{+}:=\left\lceil\log_{2}\frac{5\sqrt{n}}{\varepsilon}\right\rceil$ and
hence $k_{+}-k_{-}\leq 2+\log_{2}(5s_{\sup})$.

  2. $S_{2}^{\varepsilon}(t)$.

Notice that
$\operatorname{int}\bigcup_{Q\in\widetilde{\mathcal{W}}_{\varepsilon}}\overline{Q}$
is a polygon whose volume is bounded by $\operatorname{vol}_{n}\Omega$. By
Prop. 2.6, Prop. 2.7 and a result on two-term asymptotics for polygons (see
Sec. 4.5 and Thm. 7.4.11 in [17] or [44]),

$\displaystyle\limsup_{t\to\infty}\frac{S_{2}^{\varepsilon}(t)}{C_{W}^{(n)}\operatorname{vol}_{n}(\Omega)t^{n/2}+\frac{C_{W}^{(n-1)}}{4}A_{\Omega}\varepsilon^{n-1-\delta}t^{(n-1)/2}}\leq
1.$

The estimates for $S_{1}^{\varepsilon}(t)$ and $S_{2}^{\varepsilon}(t)$
together show that for large enough $t$

$N_{N}(\Omega,t)\leq
C_{W}^{(n)}\operatorname{vol}_{n}(\Omega)t^{n/2}+M_{\Omega}t^{\delta/2}.\qed$

###### 3.11. Remark.

  1. (i).

The minimal value of $t$ for which the estimate of Thm. 3.10 holds true
depends entirely on the maximal value of $\varepsilon$ in Def. 3.4.(i). More
precisely, if $\Omega$ is well-covered for all $\varepsilon<\varepsilon_{0}$,
an upper bound for $N_{N}(\Omega,t)$ for all
$t>t_{0}:=\frac{C_{2}\varepsilon_{0}^{-2}}{\mu+1}$ is presented below. This
variant of the above estimate involves estimating
$S_{2}^{\varepsilon}(t)\leq\sum_{Q\in\widetilde{\mathcal{W}}_{\varepsilon}}N_{N}(Q,t)$.
More precisely, setting
$k^{\prime}_{-}:=\lfloor-\log_{2}\frac{\operatorname{diam}\Omega}{\sqrt{n}}\rfloor$,
one has

$\displaystyle S_{2}^{\varepsilon}(t)$
$\displaystyle\leq\sum_{Q\in\widetilde{\mathcal{W}}_{\varepsilon}}N_{N}(Q,t)\leq\sum_{k=k^{\prime}_{-}}^{k_{+}}\\#\mathcal{W}_{k}N_{N}(2^{-k}I^{n},t)$

While more accurate estimates are possible, this is sufficient to provide an
upper bound in terms of counting functions of cubes. Based on a simple lattice
counting argument and a result by Pólya in [32], the Neumann counting function
of a unit cube $I^{n}=(0,1)^{n}\subset\mathbb{R}^{n}$ has an upper bound of
$N_{N}(I^{n},t)\leq N_{D}(I^{n},(\sqrt{t}+2\pi\sqrt{n})^{2})\leq
C_{W}^{(n)}(\sqrt{t}+2\pi\sqrt{n})^{n}$.666Proof:
$N_{N}(I^{n},t)=\\#\\{k=(k_{1},\cdots,k_{n})\in\mathbb{N}_{0}^{n}:|k|_{2}^{2}\leq
t/\pi^{2}\\}$ and
$N_{D}(I^{n},t)=\\#\\{k=(k_{1},\cdots,k_{n})\in\mathbb{N}^{n}:|k|_{2}^{2}\leq
t/\pi^{2}\\}$. Defining
$\widetilde{N_{N}}(r):=\\#\\{k\in\mathbb{N}_{0}^{n}:|k|_{2}^{2}\leq r^{2}\\}$
one observes $N_{N}(I^{n},t)=\widetilde{N_{N}}(\sqrt{t}/\pi)$ and analogously
for $N_{D}$. Notice that
$\widetilde{N_{N}}(r)\leq\widetilde{N_{D}}(r+2\sqrt{n})$ since one increases
the radius sufficiently. Thus
$N_{N}(I^{n},t)=\widetilde{N_{N}}(\sqrt{t}/\pi)\leq\widetilde{N_{D}}(\sqrt{t}/\pi+2\sqrt{n})=N_{D}(I^{n},(\pi(\sqrt{t}/\pi+2\sqrt{n})^{2})$.
Finally $N_{D}(I^{n},t)\leq C_{W}^{(n)}t^{n/2}$ by Pólya’s result. Applying
this to the cubes in $\mathcal{W}_{\varepsilon}$ yields an explicit upper
bound of $S_{2}^{\varepsilon}(t)$ that is not only asymptotic but holds for
all $t>t_{0}$.

$\displaystyle S_{2}^{\varepsilon}(t)$ $\displaystyle\leq
C_{W}^{(n)}\sum_{k=k_{-}^{\prime}}^{k_{+}}\\#\mathcal{W}_{k}\left(2^{-k}\sqrt{t}+2\pi\sqrt{n}\right)^{n}$
$\displaystyle\leq
C_{W}^{(n)}\left(\operatorname{vol}(\Omega)t^{n/2}+\mathfrak{M}_{\Omega}\sum_{k=k_{-}^{\prime}}^{k_{+}}2^{k\delta}\left[\left(2^{-k}\sqrt{t}+2\pi\sqrt{n}\right)^{n}-\left(2^{-k}\sqrt{t}\right)^{n}\right]\right).$

With $S_{2}^{\varepsilon}(t)\leq S_{2}^{\varepsilon}(t_{0})$ if $t\leq t_{0}$
this extends to a global estimate of $N_{N}(\Omega,t)$ and the coefficient of
such an error term estimate will be denoted by $M_{K}^{\text{abs}}$. Since the
estimate of $S_{1}^{\varepsilon}$ is also correct for all $t$, this gives an
absolute estimate of the remainder term of the Neumann counting function. In
the regular case of $\delta=n-1$, this estimate then only provides a weaker
estimate of $t^{(n-1)/2}\log t$ instead of the boundary term $t^{(n-1)/2}$.
Similarly such an asymptotic term is also obtained if one uses the regular
asymptotic law of the counting functions $N_{N}(2^{-k}I^{n},t)$ as shown for
example by Lapidus in [22] whenever the upper Minkowski content agrees the
upper inner Minkowski content.

  2. (ii).

Let $\Omega\subset\mathbb{R}^{n}$ be any open set with finite volume such that
$\operatorname{vol}_{n}(\Omega_{-\varepsilon})\leq\widetilde{C}\varepsilon^{n-\delta}$
for some $\delta\in(n-1,n)$ and all sufficiently small $\varepsilon>0$. Then,
using Whitney covers and an estimate of the form of Prop. 2.5, one can find
lower bounds for $N_{D}(\Omega,t)$ as was shown by van den Berg and
Lianantonakis in [43]. Indeed, for any such domain $\Omega$ one has

$\displaystyle
N_{D}(\Omega,t)\geq\begin{cases}C_{W}^{(n)}\operatorname{vol}_{n}(\Omega)t^{n/2}-\frac{5\widetilde{C}}{(n-\delta)(\delta+1-n)}t^{\delta/2}&\colon\delta>n-1,\
t>0\\\
C_{W}^{(n)}\operatorname{vol}_{n}(\Omega)t^{n/2}-3\widetilde{C}t^{(n-1)/2}\left(\log\left((2\operatorname{vol}_{n}\Omega)^{2/n}t\right)\right)&\colon\delta=n-1,\
t>\frac{4}{\sqrt[n]{\operatorname{vol}_{n}(\Omega)^{2}}}.\end{cases}$

Together with Thm. 3.10, this gives explicit upper and lower bounds for the
remainder term of counting functions (for Neumann, Dirichlet and mixed
boundary conditions) of well-covered domains whenever $\delta>n-1$: Since
$\operatorname{vol}_{n}(\Omega_{-\varepsilon})\leq\operatorname{vol}_{n}\left(\bigcup_{i\in
I_{\varepsilon}}D_{i}^{\varepsilon}\right)\leq
c_{\operatorname{vol}}^{+}C(\Omega)\varepsilon^{n-\delta}$, we may set
$\widetilde{C}:=c_{\operatorname{vol}}^{+}C(\Omega)$ to obtain an estimate for
all $t>t_{0}$ for $|N_{N}(\Omega,t)-C_{W}^{(n)}\operatorname{vol}_{n}\Omega
t^{n/2}|\leq\max\\{M_{K}^{\text{abs}},\frac{5\widetilde{C}}{((n-\delta)(\delta+1-n))}\\}t^{\delta/2}$
with $M_{K}^{\text{abs}}$ taken from (i) above. This constant will be denoted
by $\widetilde{M_{K}}$. It plays a central role in the companion paper [20].
In [20] these estimates are used to deduce high order asymptotic terms for
$N_{D}(U,t)$ for self-similar sprays $U$ whose generators are finite unions of
pairwise disjoint Koch snowflakes.

  3. (iii).

By Thm. 3.10 there is an $M_{\Omega}\in\mathcal{O}(1)$ such that
$N_{N}(\Omega,t)=C_{W}^{(n)}\operatorname{vol}\Omega
t^{n/2}+M_{\Omega}(t)t^{\delta/2}$. Based on the fact that for any rescaling
$\alpha>0$ one has $N_{N}(\alpha\Omega,t)=N_{N}(\Omega,\alpha^{2}t)$, it is
clear that $M_{\alpha\Omega}(t)=\alpha^{\delta}M_{\Omega}(\alpha^{2}t)$.

  4. (iv).

Combining Thm. 1.10 of [30] with Thm. 3.8 it follows that for any $n$ there is
a well-covered domain $\Omega\subset\mathbb{R}^{n}$ and a
$t_{\Omega}\in\mathbb{R}_{+}$ for which
$N_{N}(\Omega,t)-C_{W}^{(n)}\operatorname{vol}_{n}\Omega t^{n/2}\geq
t_{\Omega}^{-1}t^{\delta/2}$ for all $t>t_{\Omega}$. In this sense, the result
of Thm. 3.10 is order-sharp.

###  3.4 Application to snowflakes

For snowflake domains and more generally any domain whose boundary has a self-
similar structure, the values of $\beta(\gamma,t)$ as introduced in Def. 3.1
can often be easily understood: They increase in discrete steps according to a
power-law as $t$ approaches $\operatorname{len}\gamma$ (cf. Fig. 6). For the
Koch snowflake for example, one might say, that $\beta$ doubles in value each
time a fibre passes through $\Phi^{k}(I_{0})$ for a $k\in\mathbb{N}$.

The remainder of this section is devoted to finding an explicit estimate for
the remainder term of the eigenvalue counting function of the Neumann
Laplacian for a family of snowflakes constructed below.

#### Construction of $p$-snowflakes of Rohde type.

We summarise the construction of a class of snowflakes introduced by Rohde in
[35]. Let $p\in\left(\frac{1}{4},\frac{\sqrt{3}-1}{2}\right)$. For this $p$,
we consider the IFS of four contractions, each of contraction ratio $p$, whose
action on the unit interval is depicted in Fig. 3. The limit set of this IFS
will be called $p$-Koch curve. Note that the classic Koch curve arises when
$p=1/3$.

$0$$1$$p$$p$$p$$p$$1-2p$ Figure 3: Illustration of the action of the IFS
dependent on $p$ on the unit interval, which is used in [35, 12]. The
resulting curve depicted here shall be denoted by $A_{p}$.

Replacing each of the four sides of a unit square with an outwards pointing
copy of the $p$-Koch curve produces a snowflake-like domain $R(p)$. Each thus
constructed $R(p)$ belongs to the class of homogeneous snowflake domains
considered by Rohde in [35]. We will discuss the more general construction of
Rohde and its significance in Rem. 3.12. We will also consider snowflakes
$K(p)$, where one replaces each side of an equilateral triangle with an
outwards pointing copy of the $p$-Koch curve and note that $K:=K(1/3)$ is the
classic Koch snowflake.

#### Bounds on the Neumann eigenvalue counting function for $p$-snowflakes of
Rohde type.

Fix $p\in\left(\frac{1}{4},\frac{\sqrt{3}-1}{2}\right)$ and let
$\delta:=\overline{\dim_{M}}\partial R(p)=\overline{\dim_{M}}\partial
K(p)=-\log_{p}4$ denote the upper inner Minkowski dimension of the boundary of
$R(p)$ and $K(p)$. In order to apply Thm. 3.10, we need to show that $K(p)$
and $R(p)$ are well-covered. To this end, let $\varepsilon\in
J_{k}^{(p)}:=\left(p^{k+1}\frac{1-2p}{\sqrt{4p-1}}\right.,\left.p^{k}\frac{1-2p}{\sqrt{4p-1}}\right]$.
Based on a modification of the geometric approximation of
$K(1/3)_{-\varepsilon}$ in [24], we construct a cover of $K(p)_{-\varepsilon}$
and $R(p)_{-\varepsilon}$ by well-foliated domains as in the proof of Thm.
3.10. This cover has multiplicity $\mu=2$. Such covers for the classic Koch
snowflake and $R(p)$ are depicted in Fig. 4 and Fig. 5. The cover consists of
three types of objects:

  1. (A)

Fringed rectangles (fR) $(1-2p)p^{k-1}\times\varepsilon$ with fractal top of
height $p^{k-1}\frac{\sqrt{4p-1}}{2}$.

  2. (B)

Short rectangles (sR) $p^{k}\times\varepsilon$ with a fractal top of height
$p^{k}\frac{\sqrt{4p-1}}{2}$.

  3. (C)

Long rectangles (lR)
$\left(p^{k}+\frac{\varepsilon\sqrt{4p-1}}{2p}\right)\times\varepsilon$ and a
fractal top of height $p^{k}\frac{\sqrt{4p-1}}{2}$.

Figure 4: Instance of the cover of an inner $\varepsilon$-neighbourhood
$K_{-\varepsilon}$ of the classic Koch snowflake $K$ and $\varepsilon\in
J_{2}^{(1/3)}$. There are three types of covering domains used here: fringed
Rectangles (fR, coloured gray), short rectangles (sR, horizontal lines) and
long rectangles (lR, vertical lines). An example of the underlying domain
$E_{\text{fR}}$ is marked with a grid – the other underlying domains are
analogous. Both end segments of this curve are parts of additional fR’s, as
the snowflake is build out of three equilateral Koch curves.  Figure 5: Cover
of an instance of $R(p)$ by domains as in (A)-(C). The only difference to the
Koch snowflake (see Fig. 4) is located at the four edges of the unit square
where one now uses two copies of sR instead of one fR. To better illustrate
the overlap, the leftmost and rightmost copies of sR are marked by vertical
lines instead of horizontal lines.
$\beta=1$$\beta=\frac{2p}{1-2p}$$\beta=\left(\frac{2p}{1-2p}\right)^{2}$$\beta=\left(\frac{2p}{1-2p}\right)^{3}$$E$$Q$$D$$r=\varepsilon$$(1-2p)p^{k-1}$
Figure 6: Systematic construction of the foliation of sets near the boundary
of the $p$-snowflake of Rohde type in the case of a “fringed rectangle (fR)”
(see Sec. 3.4(A)). The well-foliated domain is the dotted rectangle (the set
$E$) and the section of the snowflake above it. The gray areas within the
snowflake indicate areas of constant $\beta(\gamma,t)$. One notices that in
the particular case of $p=1/3$, a connection between the ternary expansion of
$\gamma(0)\in\mathbb{R}$ and the shape of the curve (and in particular its
number of turns) is evident.

We will now show that the fringed rectangles are well-foliated. Analoguously
it can be shown that the short rectangles and the long rectangles are well-
foliated, too. The foliation $\Gamma$, as constructed in Sec. 3.1, is shown in
Fig. 6. For every $\gamma=\gamma_{q}\in\Gamma$ with $q\in Q$ the integral
$\int_{0}^{\operatorname{len}\gamma}\beta(\gamma,t)dt$ is bounded by the sum
of integrals over $\beta(\gamma,t)$ along the longest fibre within each
iteration of triangles. More precisely, we have a partition
$0=t_{-1}<t_{0}<t_{1}\cdots$ of $[0,\operatorname{len}\gamma]$ such that
$\gamma_{q}(t_{\ell})\in\Phi^{\ell}(I_{0})$, yielding

$\displaystyle\begin{aligned}
\int_{0}^{\operatorname{len}\gamma}\beta(\gamma,t)dt=\sum_{\ell=0}^{\infty}\int_{t_{\ell-1}}^{t_{\ell}}\beta(\gamma,t)dt&<r+\sum_{\ell=1}^{\infty}\frac{\sqrt{4p-1}}{2}\cdot
p^{k+\ell-2}\left(\frac{2p}{1-2p}\right)^{\ell-1}\\\
&=r+p^{k-1}\frac{(1-2p)\sqrt{4p-1}}{2(1-2p-2p^{2})}<\infty.\end{aligned}$
(3.5)

With $r=\varepsilon$, this implies that
$\mathcal{I}_{\beta}<\varepsilon\left(1+\frac{4p-1}{2p^{2}(1-2p-2p^{2})}\right)$.
Moreover, in all of the cases of covering domains (fR, sR and lR), the
foliation is constructed as shown in Fig. 6 so that $\inf\beta=1$, showing
that fR, sR and lR are well-foliated. It also provides an intuition of how to
construct a suitable family of foliations in similar cases.

To show that a snowflake as constructed above is a well-covered domain, we now
need to check parts (i)-(iv) of Def. 3.4.

To (i): By the iteration dynamics, $R(p)_{-\varepsilon}$ is covered by
$\frac{4}{3}(4^{k}-1)$ instances of fR and $\frac{4}{3}(4^{k}+2)$ copies of sR
or lR. In total this amounts to a cover of cardinality
$\\#I_{\varepsilon}=\frac{4}{3}(2\cdot 4^{k}+1)\leq
C(R(p))\varepsilon^{-\delta}$ with
$C(R(p))=3\left(\frac{1-2p}{\sqrt{4p-1}}\right)^{\delta}$. Analogously one can
chose $C(K(p))=\frac{3}{4}C(R(p))$. For the classic Koch snowflake
$K:=K(1/3)$, the cardinality of the cover is $2\cdot 4^{k}-2\leq
C(K)\varepsilon^{-\delta}$ with $C(K)=1$.

To (ii): By construction $r=\varepsilon$, so that $c_{r}^{\pm}=1$. The length
$L$ of a fibre satisfies $\varepsilon=r\leq
L<r+\sum_{\ell=0}^{\infty}p^{\ell+k-1}\frac{\sqrt{4p-1}}{2}\leq\left(1+\frac{4p-1}{p^{2}(2-6p+4p^{2})}\right)\varepsilon$
implying $c_{L}^{-}\geq 1$ and $c_{L}^{+}\leq
1+\frac{4p-1}{p^{2}(2-6p+4p^{2})}$. From our previous estimate on
$\mathcal{I}_{\beta}$ we infer $c_{\mathcal{I}}^{-}\geq 1$ and
$c_{\mathcal{I}}^{+}\leq 1+\frac{4p-1}{2p^{2}(1-2p-2p^{2})}$. The diameters of
the covering domains (A)-(C) above are found to be bounded from above by
$\sqrt{[(1-2p)p^{k-1}]^{2}+\left(\varepsilon+p^{k-1}\frac{\sqrt{4p-1}}{2}\right)^{2}}$
so that
$c_{\operatorname{diam}}^{+}\leq\sqrt{\frac{4p-1}{p^{4}}+\left(1-\frac{4p-1}{2p^{2}(1-2p)}\right)^{2}}$.
Moreover, $c_{\operatorname{diam}}^{-}\geq 1$.

To (iii) and (iv): By minimising
$\lambda_{2}^{N}(E_{\text{lR}})\varepsilon^{2}$ as a function of
$\varepsilon\in J_{k}^{(p)}$ we obtain
$c_{E}(R(p))=\min\left(1,\frac{4(1-2p)^{2}p^{2}}{(3-2p)^{2}(4p-1)}\right)\pi^{2}$.
Using the above constants we compute $C_{1}(R(p)_{-\varepsilon})$ as given in
Cor.3.6 so that
$\lambda_{2}^{N}(D_{i}^{\varepsilon})\geq\varepsilon^{-2}C_{1}(R(p)_{-\varepsilon})$.
See Fig. 77(a) for a plot of $C_{1}(R(p)_{-\varepsilon})$.

Figure 7(a) Values of $C_{1}(R(p)_{-\varepsilon})$ for values for
$p\in(\frac{1}{4},\frac{\sqrt{3}-1}{2})$ as in Cor.3.6. The values approach
$0$ as $p\nearrow\frac{\sqrt{3}-1}{2}$.

Figure 7(b) Values of $M_{K(p)}$ for values for
$p\in(\frac{1}{4},\frac{\sqrt{3}-1}{2})$ as given by Thm. 3.10.

In particular in the case of the classic Koch snowflake one finds
$\lambda_{2}^{N}(D)\geq 0,0031\varepsilon^{-2}$ for any element of a cover of
$K_{-\varepsilon}$ as given by (A)-(C) above.

Based on the above considerations, one now obtains expressions for $M_{R(p)}$
as well as coefficients following Rem. 3.11.(i). As expected, the coefficients
$M_{R(p)}$ diverge as $p\nearrow\frac{\sqrt{3}-1}{2}$ and remain regular as
$p\searrow 1/4$, see Fig. 77(b).

In the case of the classic Koch snowflake $K$, a slighty better estimate is
possible based on an estimate of the inner tube area by Lapidus and Pearse in
[24]:

$\displaystyle\operatorname{vol}_{2}(K_{-\varepsilon})\leq
3\left[\varepsilon^{2-\delta}4^{-\\{x\\}}\left(\frac{3\sqrt{3}}{40}9^{\\{x\\}}+\frac{\sqrt{3}}{2}3^{\\{x\\}}+\frac{1}{6}\left(\frac{\pi}{3}-\sqrt{3}\right)\right)-\frac{\varepsilon^{2}}{3}\left(\frac{\pi}{3}+2\sqrt{3}\right)\right],$

where $\delta=\log_{3}4=\overline{\dim_{M}}(\partial K,K)$ and $\\{x\\}$ is
the fractional part of $x:=-\log_{3}(\varepsilon\sqrt{3})$. Therefore,

$\displaystyle\overline{\mathcal{M}}_{\delta}(\partial
K,K)\leq\frac{1}{480}\left(723\sqrt{3}+20\pi\right).$

Moreover, the monotonicity of the above upper bound shows that
$\varepsilon^{\prime}\leq 0$ in our estimate and hence $\mathfrak{M}_{K}\leq
11.61$. Based on this, $C_{3}(K)\approx 1354$ and $S_{1}^{\varepsilon}(t)\leq
104282t^{\delta/2}$. Computing $M_{K}$ we finally obtain an asymptotic upper
bound for the remainder term of the Neumann counting function of the Koch
snowflake of

$N_{N}(K,t)-C_{W}^{(n)}\operatorname{vol}_{2}(K)t\leq 104325{.}5t^{\delta/2}.$

Alternatively, an absolute upper bound can be found following Rem. 3.11.(i):

$\displaystyle N_{N}(K,t)-C_{W}^{(2)}\operatorname{vol}(K)t\leq
C_{W}^{(2)}\left(3.537\cdot 10^{6}t^{\delta/2}-353-911\sqrt{t}\right).$

Since $\varepsilon<1/9$ is more than sufficient to satisfy Def. 3.4, this
holds true for all $t\geq 0.1$. Some limited computational results on the
Neumann (or Dirichlet) spectrum of the Koch Snowflake can be found in [31,
41]. A few estimations in the above application were deliberately chosen sub-
optimally for the benefit of the presentation so that some improvements in
these constants are expected to be possible.

###### 3.12. Remark.

Fix $p\in\left(\frac{1}{4},\frac{1}{2}\right)$. By replacing each of the four
sides of a unit square $S_{0}$ with either an outwards pointing copy of the
curve $A_{p}$ depicted in Fig. 3 or with $A_{1/4}$ one arrives at an object
$S_{1}$ which is made up of $4\cdot 4$ intervals of lengths $p$ or
$\frac{1}{4}$. Applying either substitution individually to each of the
resulting intervals, and iterating the procedure leads to an object $S_{n}$
which is made up of $4\cdot 4^{n}$ intervals. This construction is presented
in [35] and we will refer to the limiting objects as _general snowflake-
domains of Rohde type_. For such snowflake-domains, estimates for Poincaré
constants were obtained in [12]. The snowflake-domains $R(p)$ that we consider
in the present section arise as the special cases where the curve $A_{p}$ is
chosen in every replacement step.

Rohde showed in [35] that the general snowflake-domains of Rohde type form an
exhaustive family of quasicircles up to bi-Lipschitz maps. Besides a more
involved computation of the upper inner Minkowski dimension and upper inner
Minkowski content of the boundary, the construction of a foliation and the
arguments applied in the present section carry over to the general snowflake-
domains of Rohde type for $p\in\left(\frac{1}{4},\frac{\sqrt{3}-1}{2}\right)$.
The restriction $p<\frac{\sqrt{3}-1}{2}$ ensures that
$\int_{0}^{\operatorname{len}\gamma}\beta(\gamma,t)dt<\infty$ in (3.5). Since
the above argument for Lem. 3.2 and hence Thm. 3.10 is not affected by bi-
Lipschitz transformations (Rem. 3.9), knowledge of the upper inner Minkowski
content would lead to estimates for the Neumann counting function for a large
family of quasicircular domains.

##  4 Constants

The following constants are used within this document. $\mathfrak{M}_{\Omega}$
and $A_{\Omega}$ are defined in Prop. 2.5 and Prop. 2.6, respectively.

$\displaystyle C_{1}(\Omega_{-\varepsilon})$
$\displaystyle:=\left[c_{E}^{-1}(c_{r}^{+})^{2}\left(1+\frac{c_{r}^{+}+\sqrt{c_{E}c_{L}^{+}c_{r}^{-}}}{c_{r}^{+}\inf_{i\in
I_{\varepsilon}}\beta_{\inf}}\frac{c_{\mathcal{I}}^{+}}{c_{r}^{-}}\right)+\frac{1+\frac{c_{r}^{+}}{\sqrt{c_{E}c_{L}^{+}c_{r}^{-}}}}{\inf_{i\in
I_{\varepsilon}}\beta_{\inf}}c_{L}^{+}c_{\mathcal{I}}^{+}\right]^{-1}$
$\displaystyle C_{2}(\Omega)$
$\displaystyle:=\min\left\\{\inf_{\varepsilon>0}C_{1}(\Omega_{-\varepsilon}),\frac{1}{2}\left(\sqrt{n}\pi/c_{\operatorname{diam}}^{+}\right)^{2}\right\\}$
$\displaystyle C_{3}(\Omega)$
$\displaystyle:=C(\Omega)+{\mathfrak{M}_{\Omega}}\frac{(40\sqrt{n})^{\delta}}{2^{\delta}-1}$
$\displaystyle M_{\Omega}$
$\displaystyle:=C_{3}(\Omega)\left(\frac{\mu+1}{C_{2}(\Omega)}\right)^{\delta/2}+\frac{C_{W}^{(n-1)}}{4}A_{\Omega}\left(\frac{\mu+1}{C_{2}(\Omega)}\right)^{\frac{\delta-(n-1)}{2}}.$

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