# Pleijel nodal domain theorem in non-smooth setting

Nicolò De Ponti , Sara Farinelli , Ivan Yuri Violo Dipartimento di Matematica
e Applicazioni, Università di Milano–Bicocca,
email: nicolo.deponti(at)unimib.itLagrange Mathematics and Computation
Research Center,
email: sfarinel(at)sissa.itCentro di Ricerca Matematica Ennio De Giorgi,
Scuola Normale Superiore di Pisa,
email: ivan.violo(at)sns.it

###### Abstract

We prove the Pleijel theorem in non-collapsed RCD spaces, providing an
asymptotic upper bound on the number of nodal domains of Laplacian
eigenfunctions. As a consequence, we obtain that the Courant nodal domain
theorem holds except at most for a finite number of eigenvalues. More in
general, we show that the same result is valid for Neumann (resp. Dirichlet)
eigenfunctions on uniform domains (resp. bounded open sets). This is new even
in the Euclidean space, where the Pleijel theorem in the Neumann case was open
under low boundary-regularity.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Sobolev spaces and Neumann eigenfunctions in uniform domains
  4. 4 From local to global isoperimetric inequality
  5. 5 Almost Euclidean Faber-Krahn inequality for small volumes
  6. 6 Proof of main theorem

##  1 Introduction

Given a continuous eigenfunction $u$ of a linear operator $L$, there is a lot
of interest in studying the properties of its nodal domains, the latter being
defined as the connected components of the set $\\{u\neq 0\\}$. In the usual
setting $L$ is an operator of differential nature, with discrete spectrum
$\lambda_{1}\leq\lambda_{2}\leq\ldots\leq\lambda_{k}\leq\ldots,$ and a
classical problem is to bound the number of nodal domains of $u_{k}$ in terms
of $k$. Here $u_{k}$ is an eigenfunction of eigenvalue $\lambda_{k}$.

There are two main results known in this direction. The first one, due to
Courant [35] (see also [36]), provides a pointwise bound: _for every_ $k$ the
number of nodal domains of $u_{k}$ is less than or equal to $k$. The second
one is due to Pleijel [78] and provides an asymptotic upper bound, which
implies that _for sufficiently large_ $k$ the number of nodal domains of
$u_{k}$ is strictly less than $k$.

The theorems of Courant and Pleijel have been deeply investigated in a
different number of situations, see [10, 14, 16, 17, 25, 30, 37, 40, 45, 56,
66, 69, 79, 86] for a non-exhaustive list. The main goal of the present paper
is to obtain an asymptotic upper bound on the number of nodal domains for
Dirichlet and Neumann Laplacian eigenfunctions in the setting of possibly non-
smooth metric measure spaces. In particular, we focus on the class of
$\mathrm{RCD}(K,N)$ spaces consisting of metric measure spaces satisfying a
synthetic notion of Ricci curvature bounded from below by $K$ and dimension
bounded from above by $N$ (see [3] and Section 2.5 for more details).
Nevertheless, our analysis is of interest already in the Euclidean case since
we prove the Pleijel theorem for Lipschitz and even more rough domains (see
Corollary 1.3 below and the subsequent discussion). Indeed, the validity of a
Pleijel result in the Neumann case with boundary regularity below $C^{1,1}$
was an open question in the field (see the comments after Remark 1.2 in [56]).

About the Courant nodal domain theorem, let us just briefly mention that its
validity is open for $\mathrm{RCD}$ spaces. This is mainly due to the fact
that the _weak unique continuation property_ for the Laplacian in this setting
is currently not known. We refer to [44, 43] for more on this problem, where
also the failure of the _strong_ unique continuation property in the
$\mathrm{RCD}$ setting is shown. We remark that a worse, but still pointwise,
upper bound on the number of nodal domains can be easily deduced from the
variational characterization of the eigenvalues (see [66]).

Before stating our main result, let us first introduce the setting and some
notations referring to Section 2 for the precise definitions. Our
investigation deals with eigenfunctions of the Dirichlet or Neumann Laplacian
$\Delta_{\mathcal{D}}$, $\Delta_{\mathcal{N}}$ in a bounded domain
$\Omega\subset{\rm X}$ in a $\mathrm{RCD}(K,N)$ space $({\rm X},{\sf
d},\mathcal{H}^{N})$, where $\mathcal{H}^{N}$ denotes the $N$-dimensional
Hausdorff measure in $({\rm X},{\sf d})$. As usual in this kind of problems,
some additional assumptions are required to deal with the Neumann case and we
will demand that $\Omega$ is a uniform domain (see Definition 3.1). We will
clarify below why we need to restrict our attention to RCD spaces endowed with
the Hausdorff measure, called non-collapsed in the literature, instead of
considering the full RCD class. Here we limit ourselves to mention that these
assumptions are sufficient for the Dirichlet and Neumann Laplacian in $\Omega$
to have discrete spectrum and for the eigenfunctions to be continuous. We list
the Dirichlet and Neumann eigenvalues respectively by

$\begin{split}&0\leq\lambda_{1}^{\mathcal{D}}(\Omega)\leq\lambda_{2}^{\mathcal{D}}(\Omega)\leq\dots\leq\lambda_{k}^{\mathcal{D}}(\Omega)\leq\dots\rightarrow+\infty,\\\
&0=\lambda_{1}^{\mathcal{N}}(\Omega)\leq\lambda_{2}^{\mathcal{N}}(\Omega)\leq\ldots\leq\lambda_{k}^{\mathcal{N}}(\Omega)\leq\ldots\rightarrow+\infty,\end{split}$

counted with multiplicity. Thanks to the continuity of a Laplacian
eigenfunction $u$ in our setting, it makes sense to define its _nodal domains_
, which are the connected components of $\Omega\setminus\\{u=0\\}$. For any
$k\in\mathbb{N}$ we can now define

$M_{\Omega}^{\mathcal{D}}(k):=\sup\left\\{\\#\text{ of nodal domains of }u:u\
\textrm{Dirichlet eigenfunction of eigenvalue }\
\lambda_{k}^{\mathcal{D}}(\Omega)\right\\}$

and analogously $M_{\Omega}^{\mathcal{N}}$ in the Neumann case (see Def. 6.3
for more detailed definitions of
$M_{\Omega}^{\mathcal{D}},M_{\Omega}^{\mathcal{N}}$). We finally denote by
$j_{\alpha}$ the first positive zero of the Bessel function of index
$\alpha>0$ and by $\omega_{N}$ the volume of the unit ball in the
$N$-dimensional Euclidean space.

######  Theorem 1.1 (Pleijel theorem in RCD setting - Neumann and Dirichlet
cases).

Let $({\rm X},{\sf d},\mathcal{H}^{N})$ be an $\mathrm{RCD}(K,N)$ space, with
$K\in\mathbb{R}$ and $N\geq 2$, and let $\Omega\subset{\rm X}$ be an open and
bounded set. Then

$\varlimsup_{k\to+\infty}\frac{M_{\Omega}^{\mathcal{D}}(k)}{k}\leq\frac{(2\pi)^{N}}{\omega^{2}_{N}j^{N}_{\frac{(N-2)}{N}}}<1\,.$
(1.1)

If moreover $\Omega\subset{\rm X}$ is a uniform domain, then

$\varlimsup_{k\to+\infty}\frac{M_{\Omega}^{\mathcal{N}}(k)}{k}\leq\frac{(2\pi)^{N}}{\omega^{2}_{N}j^{N}_{\frac{(N-2)}{N}}}<1\,.$
(1.2)

In particular, for every $k\in\mathbb{N}$ large enough every Dirichlet (resp.
Neumann) eigenfunction of eigenvalue $\lambda_{k}^{\mathcal{D}}(\Omega)$
(resp. $\lambda_{k}^{\mathcal{N}}(\Omega)$) in any $\Omega$ bounded open set
(resp. uniform domain) has less than $k$ nodal domains.

There has been recently a growing interest in the study of eigenvalues and
eigenfunctions of the Laplacian and their zero set in the setting of RCD
spaces (see [7, 6, 61, 62, 42, 18, 28, 44, 92, 8, 43]). However, to the best
of our knowledge, Theorem 1.1 is the first non-trivial result related to nodal
domains.

The class of non-collapsed $\mathrm{RCD}(K,N)$ space includes non-collapsed
Ricci limit spaces [32, 33] and finite dimensional Alexandrov spaces [77, 91],
and our result is new also for these classes of spaces where the Courant’s
nodal domain theorem is not known. Additionally, thanks to the recent [83], we
know that every $\mathrm{RCD}(K,N)$ space contains a rich class of non-trivial
uniform domains, hence it is possible to find many sets that satisfy the
assumptions of our result also in the Neumann case. We recall that, in the
somewhat easier Dirichlet case, Theorem 1.1 goes back to the work of Pleijel
[78] in the Euclidean plane and to Bérard and Meyer [17] for smooth Riemannian
manifolds.

When ${\rm X}$ is bounded it is allowed to take $\Omega={\rm X}$ in the
Theorem 1.1. In this case Neumann eigenfunctions coincide with the usual
Laplacian eigenfunctions on ${\rm X}$ and we have the following.

######  Corollary 1.2.

Let $({\rm X},{\sf d},\mathcal{H}^{N})$ be a compact $\mathrm{RCD}(K,N)$
space, with $K\in\mathbb{R}$ and $N\geq 2$. Denote by
$\\{\lambda_{k}\\}_{k\in\mathbb{N}}$ the eigenvalues of the Laplacian in ${\rm
X}$ and by $M(k)$ the maximal number of nodal domains of any Laplacian
eigenfunction of eigenvalue $\lambda_{k}.$ Then

$\varlimsup_{k\to+\infty}\frac{M(k)}{k}\leq\frac{(2\pi)^{N}}{\omega^{2}_{N}j^{N}_{\frac{(N-2)}{N}}}<1\,.$
(1.3)

The result in Theorem 1.1 in the case of Neumann eigenfunctions is interesting
already when taking $({\rm X},{\sf d},\mathcal{H}^{N})$ to be the
$N$-dimensional Euclidean space. We extract this version below in a self-
contained statement, for the convenience of the reader.

######  Corollary 1.3.

Let $\Omega\subset\mathbb{R}^{N}$, $N\geq 2,$ be a uniform domain. For every
$k\in\mathbb{N}$ denote by $\lambda_{k}^{\mathcal{N}}(\Omega)$ the Neumann
Laplacian eigenvalues in $\Omega$ and by $M_{\Omega}^{\mathcal{N}}(k)$ the
maximal number of nodal domains of a Neumann eigenfunction of eigenvalue
$\lambda_{k}^{\mathcal{N}}(\Omega).$ Then

$\varlimsup_{k\to+\infty}\frac{M_{\Omega}^{\mathcal{N}}(k)}{k}\leq\frac{(2\pi)^{N}}{\omega^{2}_{N}j^{N}_{\frac{(N-2)}{N}}}<1.$
(1.4)

Recall that the class of uniform domains in the Euclidean space includes
bounded _Lipschitz domains_ , but also more irregular domains such as _quasi
disks_ and in particular the interior of a _Koch Snowflake_ (see Section 3.1
for more details and references). A Pleijel theorem for Neumann eigenfunctions
of Euclidean domains was firstly proved by Polterovich [79], who considered
planar domains with piecewise real analytic boundary. The general
$N$-dimensional case was obtained in [69] for domains $\Omega$ with $C^{1,1}$
boundary, where the regularity assumption is required in order to apply to
eigenfunctions a reflection procedure across the boundary of $\Omega$. The
same limitation on the regularity of the boundary appears in [56] (in the
context of more general Robin problems), where it is explicitly stated the
problem of the validity of Pleijel theorem under a weaker regularity of the
boundary. Very recently, the techniques introduced in [69] were employed and
refined in [15] in the planar case, where the authors were able to treat
$2$-dimensional domains with smooth boundaries except for a finite number of
vertices.

Our work introduces a different strategy and avoids any reflection argument,
allowing us to handle more general domains without imposing any restriction on
the dimension. To explain the basic idea of our method we recall that a key
step in the original proof of the Pleijel theorem is to exploit the fact that
an eigenfunction $u$ in $\Omega$, when restricted to one of its nodal domains
$U\subset\Omega$, satisfies a zero-Dirichlet boundary condition in $U$ itself,
thus allowing to apply the Faber-Krahn inequality and get a lower bound for
the volume of $U$. While this is true for a Dirichlet-eigenfunction and for
all its nodal domains, it is in general false for a Neumann eigenfunction and
a nodal domain that touches the boundary. The reflection procedure in [69] is
needed precisely to handle this issue, but requires smoothness of the
boundary. Instead our observation is that, by the very definition of nodal
domain, an eigenfunction $u$ (even in the Neumann case) has indeed zero-
Dirichlet boundary conditions in $U$ but relative to the ambient domain
$\Omega$, i.e. ignoring the portion of $\partial U$ which is contained in
$\partial\Omega$. The key point is then to view $\Omega$ as a metric space in
its own right and prove that it is regular enough to satisfy a version of the
Faber-Krahn inequality, which then allows to carry out the rest of the
argument. This is where the uniform condition will enter into play ensuring
the required analytical properties of $\Omega$.

Even if we use mostly techniques coming from the metric setting, we also
develop some purely-Euclidean technical tools that we believe could be useful
to show other Pleijel-type results in $\mathbb{R}^{N}$ under low boundary-
regularity. In particular, we prove a Faber-Krahn-tpye inequality and a
Green’s formula for eigenfunctions of uniform domains (see Corollary 5.2 and
Corollary 6.2 respectively). Both the results were previously available only
assuming $C^{1,1}$-boundary.

We now comment further on the assumptions and the proof of Theorem 1.1.

The uniformity hypothesis on the domain guarantees the discreteness of the
spectrum of the Neumann Laplacian, a fact even needed to state the theorem.
Additionally we will make a crucial use of analytical properties of uniform
domains in metric measure spaces, such as Sobolev extension properties, the
validity of a Poincaré inequality and a Sobolev embedding (see Section 3 for
more details).

The non-collapsed assumption is more technical in nature, and we leave for
future investigations the general case of possibly collapsed
$\mathrm{RCD}(K,N)$ spaces. Let us notice that in collapsed $\mathrm{RCD}$
spaces the spectrum of the Laplacian can produce a singular and in some sense
unexpected behaviour in the asymptotic regime (see the recent [38]), and thus
this generalization seems non-trivial as we are going to further clarify in
the next lines commenting the proof.

The main scheme of the proof of Theorem 1.1 is similar to the one usually
employed in the smooth setting, e.g. in [69, 86, 78, 56, 45, 17]. The two
primary ingredients are the Weyl law and an almost-Euclidean Faber-Krahn
inequality for small volumes.

The Weyl law has already been investigated in the setting of
$\mathrm{RCD}(K,N)$ spaces (see [8] and [92]). In the non-collapsed case it
takes the usual formulation

$\lim_{\lambda\to+\infty}\frac{N(\lambda)}{\lambda^{N/2}}=\frac{\omega_{N}}{(2\pi)^{N}}\mathcal{H}^{N}(\Omega)\,,$

where
$N(\lambda):=\sharp\\{k\in\mathbb{N}:\lambda^{{\mathcal{D}}}_{k}(\Omega)\leq\lambda\\}$
is the eigenvalues counting function and
$\\{\lambda^{{\mathcal{D}}}_{k}(\Omega)\\}_{k\in\mathbb{N}}$ are the Dirichlet
eigenvalues of the domain $\Omega$ (see Definition 2.5). We stress that the
Weyl law in the Dirichlet case is sufficient for our purposes, even if in our
main statement we consider both Dirichlet and Neumann eigenfunctions. This
thanks to the elementary inequality
$\lambda_{k}^{\mathcal{N}}(\Omega)\leq\lambda_{k}^{\mathcal{D}}(\Omega)$
between Neumann and Dirichlet eigenvalues (see Lemma 2.8). We remark that
suitable forms of the Weyl law on the whole space have been studied under
slightly more general assumptions than non-collapsing, but the situation is
more intricate and there exist compact $\mathrm{RCD}(K,N)$ spaces for which
$N(\lambda)$ is not asymptotic to $\lambda^{\beta}$ for any $\beta\geq 0$. We
refer to [8, 38] for the details.

Concerning the almost-Euclidean Faber-Krahn inequality, it roughly states that
the first Dirichlet eigenvalue of an open set $U\subset{\rm X}$, of
sufficiently small volume, is bounded below by the first Dirichlet eigenvalue
of the Euclidean ball having the same volume and up to a small error. This
will be obtained starting from an almost-Euclidean isoperimetric inequality
for small volumes (similar to the one obtained in [17] in the smooth setting)
and rearrangement methods. In contrast with the proof in the smooth case, our
situation requires to deal with a set $C$ of possibly “bad” points, and to
work with sets $U$ that stay sufficiently far from $C$. We refer to Theorem
5.3 and Theorem 3.2 for the precise statements, and we suggest to compare them
with [17, Lemme 16,15]. For both these results the non-collapsed assumption
also plays a key role to ensure a more regular infinitesimal behaviour of the
ambient space.

## Acknowledgements

The authors thank Asma Hassannezhad for helpful discussions. The third author
was supported by the Academy of Finland project Incidences on Fractals, Grant
No.321896.

##  2 Preliminaries

###  2.1 Calculus in metric measure spaces

The triple $({\rm X},{\sf d},\mathfrak{m})$ will denote a metric measure
space, where $({\rm X},{\sf d})$ is a complete and separable metric space and
$\mathfrak{m}$ is a non-negative Borel measure, finite on bounded sets. We
will also always assume $\mathop{\rm supp}\nolimits(\mathfrak{m})=X$. For
every set $A\subset{\rm X}$ we will denote by $\overline{A}$ its topological
closure, by $A^{c}\coloneqq{\rm X}\setminus A$ its complement and by $\partial
A$ its topological boundary. We denote by $B_{r}(x):=\\{y\in{\rm X}:{\sf
d}(x,y)<r\\}$ the ball of radius $r$ and center $x$. The same set is also
denoted by $B^{{\rm X}}_{r}(x)$ whenever we want to emphasize the role of the
space ${\rm X}$. By ${\sf d}(A,B):=\inf\\{{\sf d}(x,y)\ :\ x\in A,y\in B\\}$
we denote the distance between two sets $A,B\subset{\rm X}$, so that ${\sf
d}(A,\emptyset)=+\infty$. The open $\varepsilon$-enlargement of a set
$A\subset{\rm X}$ is denoted by $A^{\epsilon}:=\\{x\in{\rm X}:{\sf
d}(A,x)<\varepsilon\\}$. Given a set $C\subset{\rm X}$, we denote by ${\sf
d}\lower 3.0pt\hbox{$|_{C}$}:={\sf d}\lower 3.0pt\hbox{$|_{C\times C}$}$ the
restriction of the distance to the set $C$. We will say that $({\rm X},{\sf
d})$ is proper if closed and bounded subsets of ${\rm X}$ are compact.

Given a metric space $({\rm X},{\sf d})$ and a rectifiable curve
$\gamma:[a,b]\to{\rm X}$, we denote by $l(\gamma)$ its length (see e.g. [57,
Chapter 5.1]). We say that $\gamma$ joins $x\in{\rm X}$ and $y\in{\rm X}$ if
$\gamma(a)=x$ and $\gamma(b)=y.$

######  Definition 2.1 (Nodal domain).

Let $({\rm X},{\sf d})$ be a metric space, $A\subset{\rm X}$ be any subset and
$f:A\to\mathbb{R}$ be a continuous function. The _nodal domains_ of $f$ (in
$A$) are the connected components of $A\setminus\\{x\in A\ :\ f(x)=0\\}.$

In the next result we recall some elementary properties of nodal sets.

######  Lemma 2.2.

Let $({\rm X},{\sf d})$ be a metric space, $A\subset{\rm X}$ be any subset and
$f:A\to\mathbb{R}$ be a continuous function. Let $U\subset A$ be a nodal
domain of $f.$ Then either $f>0$ or $f<0$ in $U.$ Moreover if $A$ is open and
$({\rm X},{\sf d})$ is locally connected then $U$ is also open.

###### Proof.

The set $U$ is connected by definition, hence $f(U)\subset\mathbb{R}$ is also
connected and does not contain zero. It follows that $f(U)\subset(0,\infty)$
or $f(U)\subset(-\infty,0).$ For a proof that $({\rm X},{\sf d})$ locally
connected implies that $U$ is open whenever $A$ is open see e.g. [75, Theorem
25.3]. ∎

For every open set $\Omega\subset{\rm X}$ we denote by $\mathsf{LIP}(\Omega)$,
$\mathsf{LIP}_{\mathsf{loc}}(\Omega)$ and $\mathsf{LIP}_{c}(\Omega)$
respectively the space of Lipschitz functions, locally Lipschitz functions and
Lipschitz functions with compact support in $\Omega$. We also denote by
$\mathsf{LIP}_{bs}(\Omega)$ the subset of $\mathsf{LIP}({\rm X})$ of functions
having support bounded and contained in $\Omega.$ The slope ${\rm lip}(f)(x)$
of a locally Lipschitz function $f\in\mathsf{LIP}_{\mathsf{loc}}(\Omega)$ at a
point $x\in\Omega$ is defined as

${\rm lip}(f)(x):=\varlimsup_{y\to x}\frac{|f(y)-f(x)|}{{\sf d}(y,x)},$

taken to be 0 when $x$ is isolated. The slope satisfies the following Leibniz
rule: ${\rm lip}(fg)\leq f{\rm lip}(g)+g{\rm lip}(f)$, for every
$f,g\in\mathsf{LIP}_{\mathsf{loc}}(\Omega).$

Given $p\in[1,\infty]$, we use the notation $L^{p}({\rm X},\mathfrak{m})$
(resp. $L^{p}_{\mathsf{loc}}({\rm X},\mathfrak{m})$) for the space of Lebesgue
$p$-integrable (resp. $p$-locally integrable) real functions on $X$ endowed
with the Borel $\sigma$-algebra. For brevity, the same function space is also
denoted by $L^{p}(\mathfrak{m})$. When $\Omega\subset{\rm X}$ is an open set,
we set $L^{p}(\Omega):=L^{p}(\Omega,\mathfrak{m}\lower
3.0pt\hbox{$|_{\Omega}$})$ where $\mathfrak{m}\lower 3.0pt\hbox{$|_{\Omega}$}$
is the restriction of the measure $\mathfrak{m}$ to $\Omega$. For a function
$u\in L^{p}(\Omega)$ we define its essential support $\mathop{\rm
supp}\nolimits(u)$ as the smallest closed set $C$ such that $u=0$
$\mathfrak{m}$-a.e. in $\Omega\setminus C.$

The Cheeger energy ${\sf Ch}\colon L^{2}(\mathfrak{m})\to[0,\infty]$ is
defined as the convex and lower semicontinuous functional

${\sf Ch}(f):=\inf\Big{\\{}\varliminf_{n\to\infty}\int_{{\rm X}}{\rm
lip}^{2}(f_{n})\,{\mathrm{d}}\mathfrak{m}\colon(f_{n})\subset
L^{2}(\mathfrak{m})\cap\mathsf{LIP}_{\mathsf{loc}}({\rm
X}),\lim_{n\to\infty}\|f-f_{n}\|_{L^{2}(\mathfrak{m})}=0\Big{\\}}.$

The Sobolev space $\mathit{W}^{1,2}({\rm X},{\sf d},\mathfrak{m})$ (or
$\mathit{W}^{1,2}({\rm X})$ for short) is then defined as
$\mathit{W}^{1,2}({\rm X},{\sf d},\mathfrak{m}):=\\{{\sf Ch}<\infty\\}$
equipped with the norm $\|f\|^{2}_{\mathit{W}^{1,2}({\rm
X})}:=\|f\|_{L^{2}(\mathfrak{m})}^{2}+{\sf Ch}(f)$, which makes it a Banach
space. This approach to the definition of Sobolev space was introduced in [5],
where it is also shown to be equivalent to the previous definitions given in
[31, 85]. For every $f\in\mathit{W}^{1,2}({\rm X})$ there exists a notion of
modulus of the gradient called _minimal weak upper gradient_ , minimal w.u.g.
for short, denoted by $|Df|\in L^{2}(\mathfrak{m})$ and satisfying

${\sf Ch}(f)=\int_{{\rm X}}|Df|^{2}{\mathrm{d}}\mathfrak{m}\,.$

For every $f\in\mathsf{LIP}_{\mathsf{loc}}({\rm X})$ we have $|Df|\leq{\rm
lip}(f)$ $\mathfrak{m}$-a.e.. Moreover, the following calculus rules are
satisfied (see e.g. [51]): for every $f,g\in\mathit{W}^{1,2}({\rm X})$ it
holds

$\begin{split}&\text{\emph{locality:} $|Df|=|Dg|$ $\mathfrak{m}$-a.e.\ in
$\\{f=g\\}$,}\\\ &\text{\emph{chain rule:} for every
$\varphi\in\mathsf{LIP}(\mathbb{R})$ with $\varphi(0)=0$,
$\varphi(f)\in\mathit{W}^{1,2}({\rm X})$ and
$|D\varphi(f)|=|\varphi^{\prime}(f)||Df|,$}\\\ &\text{\emph{Leibniz rule:} for
every $\eta\in\mathsf{LIP}\cap L^{\infty}({\rm X})$, $\eta
f\in\mathit{W}^{1,2}({\rm X})$ and $|D(\eta
f)|\leq|\eta||Df|+|D\eta||f|.$}\end{split}$ (2.1)

Given $\Omega\subset{\rm X}$ open we also define the following local Sobolev
spaces

$\begin{split}&\mathit{W}^{1,2}_{0}(\Omega)\coloneqq\overline{\mathsf{LIP}_{bs}(\Omega)}^{\mathit{W}^{1,2}({\rm
X})},\\\ &\mathit{W}^{1,2}(\Omega)\coloneqq\\{f\in L^{2}(\Omega)\ :\
f\eta\in\mathit{W}^{1,2}({\rm
X}),\,\forall\eta\in\mathsf{LIP}_{bs}(\Omega),\,|Df|\in
L^{2}(\Omega)\\},\end{split}$

where in the definition of $\mathit{W}^{1,2}(\Omega)$ the minimal w.u.g.
$|Df|\in L^{2}(\Omega)$ is defined by

$|Df|\coloneqq|D(f\eta_{n})|,\quad\text{$\mathfrak{m}$-a.e.\ in
$\\{\eta_{n}=1\\}$},$ (2.2)

with $\eta_{n}\in\mathsf{LIP}_{bs}(\Omega)$ is any sequence satisfying
$\\{\eta_{n}=1\\}\uparrow\Omega$ (there is no dependence on the chosen
sequence, by the locality property of the minimal weak upper gradient). We
endow $\mathit{W}^{1,2}(\Omega)$ with the norm given by

$\|f\|_{\mathit{W}^{1,2}(\Omega)}^{2}\coloneqq\|f\|_{L^{2}(\Omega)}^{2}+\||Df|\|_{L^{2}(\Omega)}^{2},$

which makes it a Banach space. Observe that by the Leibniz rule we have that
for every $f\in\mathit{W}^{1,2}({\rm X})$ it holds that $f\lower
3.0pt\hbox{$|_{\Omega}$}\in\mathit{W}^{1,2}(\Omega)$ and also $|Df|\lower
3.0pt\hbox{$|_{\Omega}$}=|Df\lower 3.0pt\hbox{$|_{\Omega}$}|$
$\mathfrak{m}$-a.e. in $\Omega$ (by the locality). Moreover, for every
$f\in\mathit{W}^{1,2}_{0}(\Omega)$ we have $f=0$ $\mathfrak{m}$-a.e. in
$X\setminus\Omega$ and thus $\|f\lower
3.0pt\hbox{$|_{\Omega}$}\|_{\mathit{W}^{1,2}(\Omega)}=\|f\|_{\mathit{W}^{1,2}({\rm
X})}$, which shows that the map

$T:\mathit{W}^{1,2}_{0}(\Omega)\rightarrow\mathit{W}^{1,2}(\Omega),\qquad
T(f):=f\lower 3.0pt\hbox{$|_{\Omega}$}$ (2.3)

is a linear isometry. For these reasons, with a little abuse of notation,
sometimes we identify $\mathit{W}^{1,2}_{0}(\Omega)$ with
$T(\mathit{W}^{1,2}_{0}(\Omega))\subset\mathit{W}^{1,2}(\Omega)$ and think to
$f\in\mathit{W}^{1,2}_{0}(\Omega)$ as an element of $L^{2}(\Omega)$.

If we choose $\Omega={\rm X}$, then $\mathit{W}^{1,2}({\rm
X})=\mathit{W}^{1,2}({\rm X},{\sf d},\mathfrak{m})$ with the same norm and
minimal w.u.g., so the notation is consistent with the one given above.

######  Remark 2.3.

It can be shown (see e.g. [6, Remark 2.15]) that $\mathit{W}^{1,2}(\Omega)$
coincides, up to $\mathfrak{m}$-a.e. equivalence of functions, with the
Newtonian Sobolev space $N^{1,2}(\Omega,{\sf d},\mathfrak{m}\lower
3.0pt\hbox{$|_{\Omega}$})$ defined in [85, 31] (see also [19]). The norms of
the two spaces coincide as well thanks to the equivalence proved in [5]
between the various notions of minimal weak upper gradients.  $\blacksquare$

Following [48] we say that $({\rm X},{\sf d},\mathfrak{m})$ is
_infinitesimally Hilbertian_ if $\mathit{W}^{1,2}({\rm X})$ is a Hilbert space
or equivalently if the Cheeger energy satisfies the parallelogram identity:

${\rm Ch}(f+g)+{\rm Ch}(f-g)=2{\rm Ch}(f)+2{\rm Ch}(g),\quad\forall
f,g\in\mathit{W}^{1,2}({\rm X}).$ (2.4)

If $({\rm X},{\sf d},\mathfrak{m})$ is infinitesimally Hilbertian, then
$\mathit{W}^{1,2}(\Omega)$ is a Hilbert space as well for every
$\Omega\subset{\rm X}$ open (see e.g. [27, Remark A.3]). Moreover, we can give
a notion of _scalar product between gradients_ of functions
$f,g\in\mathit{W}^{1,2}(\Omega)$ by setting

$L^{1}(\Omega)\ni\nabla f\cdot\nabla
g\coloneqq\frac{1}{2}\left(|D(f+g)|^{2}-|Df|^{2}-|Dg|^{2}\right),$ (2.5)

which is bilinear and satisfies

$\begin{split}&|\nabla f\cdot\nabla
g|\leq|Df||Dg|,\quad\mathfrak{m}\text{-a.e.,}\quad\forall\,f,g\in\mathit{W}^{1,2}(\Omega),\\\
&|\nabla f\cdot\nabla
f|=|Df|^{2},\quad\mathfrak{m}\text{-a.e.,}\quad\forall\,f\in\mathit{W}^{1,2}(\Omega).\end{split}$
(2.6)

Under the infinitesimally Hilbertian assumption we can define a notion of
Laplacian via integration by parts.

######  Definition 2.4 (Neumann Laplacian).

Let $({\rm X},{\sf d},\mathfrak{m})$ be an infinitesimally Hilbertian metric
measure space and $\Omega\subset{\rm X}$ be open. We say that
$f\in\mathit{W}^{1,2}(\Omega)$ belongs to the domain of the Neumann Laplacian,
and we write $f\in{\sf D}(\Delta_{\mathcal{N}},\Omega)$, if there exists $h\in
L^{2}(\Omega)$ such that

$\int_{\Omega}hg\,{\mathrm{d}}\mathfrak{m}=-\int_{\Omega}\nabla f\cdot\nabla
g\,{\mathrm{d}}\mathfrak{m}\,,\quad\forall\,g\in\mathit{W}^{1,2}(\Omega).$
(2.7)

If $f\in{\sf D}(\Delta_{\mathcal{N}},\Omega)$ then the function $h$ is unique
and is denoted by $\Delta_{\mathcal{N}}f.$

######  Definition 2.5 (Dirichlet Laplacian).

Let $({\rm X},{\sf d},\mathfrak{m})$ be an infinitesimally Hilbertian metric
measure space and $\Omega\subset{\rm X}$ be open. Then
$f\in\mathit{W}^{1,2}_{0}(\Omega)$ belongs to the domain of the Dirichlet
Laplacian, and we write $f\in{\sf D}(\Delta_{\mathcal{D}},\Omega)$, if there
exists $h\in L^{2}(\Omega)$ such that

$\int_{\Omega}hg\,{\mathrm{d}}\mathfrak{m}=-\int_{\Omega}\nabla f\cdot\nabla
g\,{\mathrm{d}}\mathfrak{m}\,,\quad\forall\,g\in\mathit{W}^{1,2}_{0}(\Omega).$
(2.8)

If $f\in{\sf D}(\Delta_{\mathcal{D}},\Omega)$ then the function $h$ is unique
and is denoted by $\Delta_{\mathcal{D}}f.$

Since under the infinitesimally Hilbertian assumption Lipschitz and bounded
functions are dense in $\mathit{W}^{1,2}({\rm X})$ (see [5]) we have
$\mathit{W}^{1,2}_{0}({\rm X})=\mathit{W}^{1,2}({\rm X})$ and so the Dirichlet
and Neumann Laplacian coincide for $\Omega={\rm X}$. In this situation we
simply write $\Delta=\Delta_{\mathcal{N}}=\Delta_{\mathcal{D}}$ and call it
simply Laplacian operator and write $f\in{\sf D}(\Delta)$ in place of
$f\in{\sf D}(\Delta_{\mathcal{N}},{\rm X})$ or $f\in{\sf
D}(\Delta_{\mathcal{D}},{\rm X})$.

######  Definition 2.6 (Eigenfunctions).

Let $({\rm X},{\sf d},\mathfrak{m})$ be an infinitesimally Hilbertian metric
measure space and $\Omega\subset{\rm X}$ be open. We say that a non-null
$f\in{\sf D}(\Delta_{\mathcal{D}},\Omega)$ (resp. ${\sf
D}(\Delta_{\mathcal{N}},\Omega)$) is a Dirichlet (resp. Neumann)
_eigenfunction of the Laplacian_ in $\Omega$ of _eigenvalue_
$\lambda\in\mathbb{R}$ if $\Delta_{\mathcal{D}}f=-\lambda f$ (resp.
$\Delta_{\mathcal{N}}f=-\lambda f$). In the case $\Omega={\rm X}$ we simply
write that $f$ is an eigenfunction of the Laplacian of eigenvalue $\lambda.$

######  Remark 2.7 (Compatibility with Euclidean Laplacian).

If $({\rm X},{\sf d},\mathfrak{m})=(\mathbb{R}^{N},|\cdot|,{\mathscr{L}}^{N})$
and $\Omega\subset\mathbb{R}^{N}$ is open, the spaces
$\mathit{W}^{1,2}(\Omega)$ and $\mathit{W}^{1,2}_{0}(\Omega)$ coincide with
the usual ones, also with the same norms, as shown in [85, Theorem 4.5] (see
also [19, Theorem A.2 and Corollary A.4] or [51, Section 2.1.5]). In
particular, by polarization, the right-hand side of both (2.7) and (2.8)
coincides with the integral of the scalar product between weak gradients in
the classical sense. This shows that the definition of eigenfunction (and
eigenvalue) of the Neumann or Dirichlet Laplacian in $\Omega$ given above
coincides with the usual one in the Euclidean case. $\blacksquare$

For later use we observe that, whenever Lipschitz functions are dense in
$\mathit{W}^{1,2}({\rm X})$, for every bounded and open set $\Omega\subset{\rm
X}$ it holds that

$\\{f\in\mathit{W}^{1,2}({\rm X})\ :\ {\sf d}(\mathop{\rm
supp}\nolimits(f),{\rm
X}\setminus\Omega)>0\\}\subset\mathit{W}^{1,2}_{0}(\Omega).$ (2.9)

Indeed there exist $\eta\in\mathsf{LIP}({\rm X})$ such that $\eta=1$ in
$\mathop{\rm supp}\nolimits(f)$ and $\mathop{\rm
supp}\nolimits(\eta)\subset\Omega$ and a sequence $f_{n}\in\mathsf{LIP}({\rm
X})$ with $f_{n}\to f$ in $\mathit{W}^{1,2}({\rm X})$, by density. Then $\eta
f_{n}\in\mathsf{LIP}_{bs}(\Omega)$ and $\eta f_{n}\to f$ in
$\mathit{W}^{1,2}({\rm X})$, which shows that
$f\in\mathit{W}^{1,2}_{0}(\Omega).$

We state in the next lemma an inequality between Neumann and Dirichlet
eigenvalues that will play a key role in the sequel. Note that in the
statement by $\mathit{W}^{1,2}_{0}(\Omega)\hookrightarrow L^{2}(\Omega)$ we
mean, more precisely, that $T(\mathit{W}^{1,2}_{0}(\Omega))\hookrightarrow
L^{2}(\Omega)$ where $T$ is defined in (2.3).

######  Lemma 2.8.

Let $({\rm X},{\sf d},\mathfrak{m})$ be an infinitesimally Hilbertian metric
measure space and let $\Omega\subset{\rm X}$ be open. Let us suppose that
$\mathit{W}^{1,2}_{0}(\Omega)\hookrightarrow L^{2}(\Omega)$ with compact
inclusion. Then $-\Delta_{\mathcal{D}}$ has discrete spectrum, i.e. the
eigenvalues form a diverging sequence (counted with multiplicity) that we
denote by

$0\leq\lambda_{1}^{\mathcal{D}}(\Omega)\leq\lambda_{2}^{\mathcal{D}}(\Omega)\leq\dots\lambda_{k}^{\mathcal{D}}(\Omega)\leq\dots\rightarrow+\infty\,.$
(2.10)

If moreover $\mathit{W}^{1,2}(\Omega)\hookrightarrow L^{2}(\Omega)$ with
compact inclusion, then also $-\Delta_{\mathcal{N}}$ has discrete spectrum
denoted by

$0=\lambda_{1}^{\mathcal{N}}(\Omega)\leq\lambda_{2}^{\mathcal{N}}(\Omega)\leq\dots\lambda_{k}^{\mathcal{N}}(\Omega)\leq\dots\rightarrow+\infty\,,$
(2.11)

and it holds

$\lambda_{k}^{\mathcal{N}}(\Omega)\leq\lambda_{k}^{\mathcal{D}}(\Omega)\,,\quad\forall
k\in\mathbb{N}.$ (2.12)

###### Proof.

Let us introduce the local Cheeger energies

$\displaystyle{\sf
Ch}_{{\mathcal{D}}}^{\Omega}:L^{2}(\Omega)\rightarrow[0,+\infty],\ \qquad{\sf
Ch}_{{\mathcal{D}}}^{\Omega}(f):=\begin{cases}\int_{\Omega}|Df|^{2}{\mathrm{d}}\mathfrak{m}\qquad&\textrm{if}\
f=g\lower 3.0pt\hbox{$|_{\Omega}$}\,\textrm{for some}\ g\in
W_{0}^{1,2}(\Omega)\,,\\\ +\infty&\textrm{otherwise},\end{cases}$
$\displaystyle{\sf
Ch}_{{\mathcal{N}}}^{\Omega}:L^{2}(\Omega)\rightarrow[0,+\infty],\qquad{\sf
Ch}_{{\mathcal{N}}}^{\Omega}(f):=\begin{cases}\int_{\Omega}|Df|^{2}{\mathrm{d}}\mathfrak{m}\qquad&\textrm{if}\
f\in\mathit{W}^{1,2}(\Omega)\,,\\\ +\infty&\textrm{otherwise},\end{cases}$

and notice that they define two Dirichlet forms, i.e. two densely defined,
Markovian, closed, quadratic forms [24, 46]. To check this, it is sufficient
to recall the calculus rules given in (2.1) and, for the $L^{2}$-lower
semicontinuity, the equivalent definition through relaxation (see [27] for all
the details). We denote by $L_{{\mathcal{D}}}$ (resp. $L_{{\mathcal{N}}}$) the
infinitesimal generator of ${\sf Ch}_{{\mathcal{D}}}^{\Omega}$ (resp. ${\sf
Ch}_{{\mathcal{N}}}^{\Omega}$) with its associated domain ${\sf
D}(L_{{\mathcal{D}}})$ (resp. ${\sf D}(L_{{\mathcal{N}}})$). Notice that, by
the very definition, ${\sf D}(\Delta_{\mathcal{N}},\Omega)={\sf
D}(L_{{\mathcal{N}}})$ with $\Delta_{\mathcal{N}}=L_{{\mathcal{N}}}$.
Regarding the Dirichlet Laplacian, we have $f\in{\sf
D}(\Delta_{\mathcal{D}},\Omega)$ if and only if $f\lower
3.0pt\hbox{$|_{\Omega}$}\in{\sf D}(L_{{\mathcal{D}}})$ with
$\Delta_{\mathcal{D}}f=L_{{\mathcal{D}}}(f\lower 3.0pt\hbox{$|_{\Omega}$})$.
In particular, $\lambda$ is an eigenvalue of $-\Delta_{\mathcal{D}}$ (resp.
$-\Delta_{\mathcal{N}}$) if and only if it is and eigenvalue of
$-L_{{\mathcal{D}}}$ (resp. $-L_{{\mathcal{N}}}$).

From the classical theory of Dirichlet forms [24, 46] we know that
$-L_{\mathcal{D}}$ and $-L_{\mathcal{N}}$ are non-negative, densely defined,
linear, self-adjoint operators on $L^{2}(\Omega)$. Under these assumptions, it
is well known (see e.g. [39]) that the compactness of the embedding of
$\mathit{W}^{1,2}_{0}(\Omega)$ (resp. $\mathit{W}^{1,2}(\Omega)$) in
$L^{2}(\Omega)$ implies the discreteness of the spectrum of
$-L_{{\mathcal{D}}}$ (resp. $-L_{{\mathcal{N}}}$) and thus of
$-\Delta_{\mathcal{D}}$ (resp. $-\Delta_{\mathcal{N}}$).

Since $T(\mathit{W}^{1,2}_{0}(\Omega))\subset\mathit{W}^{1,2}(\Omega)$ as
Hilbert spaces, we also know that whenever
$\mathit{W}^{1,2}(\Omega)\hookrightarrow L^{2}(\Omega)$ with compact inclusion
both the spectra are discrete.

We also have at disposal the variational characterization of the eigenvalues ,
see e.g. [39, Theorems 4.5.1, 4.5.3]. More precisely, defined

$\displaystyle\lambda^{{\mathcal{N}}}(\Omega)[M]:=\sup\\{{\sf
Ch}_{{\mathcal{N}}}^{\Omega}(f)\ :\ f\in M,\|f\|_{L^{2}(\Omega)}=1\\},$ (2.13)
$\displaystyle\lambda^{{\mathcal{D}}}(\Omega)[M]:=\sup\\{{\sf
Ch}_{{\mathcal{D}}}^{\Omega}(f)\ :\ f\in M,\|f\|_{L^{2}(\Omega)}=1\\},$

we know that for every $k\in\mathbb{N}$

$\displaystyle\lambda_{k}^{\mathcal{N}}(\Omega)=\inf\\{\lambda^{{\mathcal{N}}}(\Omega)[M]\
:\ M\subset\mathit{W}^{1,2}(\Omega),\ \dim(M)=k\\},$ (2.14)
$\displaystyle\lambda_{k}^{\mathcal{D}}(\Omega)=\inf\\{\lambda^{{\mathcal{D}}}(\Omega)[M]\
:\ M\subset T(\mathit{W}^{1,2}_{0}(\Omega)),\ \dim(M)=k\\},$

The inequality (2.12) thus follows immediately from (2.14) since the infimum
is taken on a larger set and ${\sf Ch}_{{\mathcal{N}}}^{\Omega}(f)={\sf
Ch}_{{\mathcal{D}}}^{\Omega}(f)$ for every $f\in
T(\mathit{W}^{1,2}_{0}(\Omega)).$ ∎

We will use in the sequel the notation introduced in the previous lemma, i.e.
whenever $-\Delta_{\mathcal{D}}$ (resp. $-\Delta_{\mathcal{N}}$) has discrete
spectrum in $\Omega$ we will denote by
$\\{\lambda_{k}^{\mathcal{D}}(\Omega)\\}_{k\in\mathbb{N}}$ (resp.
$\\{\lambda_{k}^{\mathcal{N}}(\Omega)\\}_{k\in\mathbb{N}}$) the sequence of
its eigenvalues. In the case $\Omega={\rm X}$, assuming the discreteness of
the spectrum of $-\Delta$, we will simply write $\lambda_{k}$ in place of
$\lambda_{k}^{{\mathcal{N}}}({\rm X})$.

For an arbitrary m.m.s. $({\rm X},{\sf d},\mathfrak{m})$ and any
$\Omega\subset{\rm X}$ open subset we also introduce

$\lambda_{1}(\Omega)\coloneqq\inf\left\\{\frac{\int|Du|^{2}{\mathrm{d}}\mathfrak{m}}{\int
u^{2}{\mathrm{d}}\mathfrak{m}}:u\in\mathsf{LIP}_{bs}(\Omega),\ u\not\equiv
0\right\\}$ (2.15)

and we call $\lambda_{1}(\Omega)$ the _first eigenvalue of the Laplacian on
$\Omega$ with zero Dirichlet boundary conditions_. Recalling the definition
definition of $\mathit{W}^{1,2}_{0}(\Omega)$ we have the following
characterization of $\lambda_{1}(\Omega)$:

$\lambda_{1}(\Omega)=\inf\left\\{\frac{\int|Du|^{2}{\mathrm{d}}\mathfrak{m}}{\int
u^{2}{\mathrm{d}}\mathfrak{m}}:u\in\mathit{W}^{1,2}_{0}(\Omega),\ u\not\equiv
0\right\\}.$ (2.16)

Note that differently from $\lambda_{1}^{\mathcal{D}}(\Omega)$, which we
defined only when the inclusion $\mathit{W}^{1,2}_{0}(\Omega)\hookrightarrow
L^{2}(\Omega)$ is compact, $\lambda_{1}(\Omega)$ is always defined.
Nevertheless, even if not needed, we stress that whenever
$\lambda^{{\mathcal{D}}}_{1}(\Omega)$ exists we do have

$\lambda_{1}(\Omega)=\lambda^{{\mathcal{D}}}_{1}(\Omega),$

as follows by (2.16) and (2.14).

###  2.2 Sets of finite perimeter

Let $f\in L^{1}_{\mathsf{loc}}({\rm X},\mathfrak{m})$ and let $U\subset{\rm
X}$ be open. Following [73, 4] we define

$|{\bf D}f|(U):=\inf\left\\{\varliminf_{n\to\infty}\int_{U}{\rm
lip}(f_{n})\,{\mathrm{d}}\mathfrak{m}\ :\
f_{n}\in\mathsf{LIP}_{\mathsf{loc}}(U),\ f_{n}\to f\ \textrm{in}\
L^{1}_{\mathsf{loc}}(U,\mathfrak{m})\right\\}\,,$ (2.17)

and we say that $f$ is of locally bounded variation if $|{\bf D}f|(U)<+\infty$
for every $U$ open and bounded. We also set

$|{\bf D}f|(A)\coloneqq\inf\\{|{\bf D}f|(U)\ :\ U\subset{\rm X}\text{ open,
}A\subset U\\},\quad\forall\,\text{$A\subset{\rm X}$ Borel}$

(note that this coincides with (2.17) if $A$ is open). For every Borel set
$E\subset{\rm X}$ and $A\subset{\rm X}$ Borel we define
${\rm{Per}}(E,A):=|{\bf D}\chi_{E}|(A)<+\infty$, where
$\chi_{E}:X\to\\{0,1\\}$ denotes the characteristic function of $E$. We say
that $E$ is of finite perimeter if ${\rm{Per}}(E)\coloneqq{\rm{Per}}(E,{\rm
X})<+\infty$.

When $f$ is of locally bounded variation (respectively, $E$ is a set of finite
perimeter), the map $A\mapsto|{\bf D}f|(A)$ (respectively,
$A\mapsto{\rm{Per}}(E,A)$) defines a Borel measure (see [4, 73]). Every
$f\in\mathsf{LIP}({\rm X})$ is of locally bounded variation and $|{\bf
D}f|\leq{\rm lip}(f)\mathfrak{m}$ (see [4, Remark 5.1]).

From the definitions it immediately follows that $E$ is of finite perimeter if
and only if $E^{c}$ is of finite perimeter, in which case
${\rm{Per}}(E,\cdot)={\rm{Per}}(E^{c},\cdot)$ holds.

In the sequel we will take advantage of the following coarea-type inequality.

######  Proposition 2.9.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a metric measure space and fix
$x\in{\rm X}.$ Then for a.e. $r>0$ the ball $B_{r}(x)$ has finite perimeter
and for every Borel set $A\subset{\rm X}$ it holds

$\int_{0}^{R}{\rm{Per}}(B_{r}(x),A)\,{\mathrm{d}}r\leq\mathfrak{m}(B_{R}(x)\cap
A),\quad\forall\,R>0.$ (2.18)

###### Proof.

Since the function ${\sf d}_{x}(\cdot)\coloneqq{\sf d}(x,\cdot)$ is
1-Lipschitz, it is of locally bounded variation and $|{\bf D}{\sf
d}_{x}|\leq{\rm lip}({\sf d}_{x})\mathfrak{m}\leq\mathfrak{m}.$ Then by the
coarea formula (see Proposition 4.2 in [73]) we get directly that
$B_{r}(x)=\\{{\sf d}_{x}(\cdot)<r\\}$ has finite perimeter for a.e. $r>0$ and
that

$\int_{0}^{+\infty}{\rm{Per}}(B_{r}(x),E){\mathrm{d}}r=|{{\bf D}{\sf
d}_{x}}|(E)\leq\mathfrak{m}(E),\quad\forall\,\,E\subset{\rm X}\text{ Borel}.$

Then (2.18) follows taking $E\coloneqq B_{R}(x)\cap A$ and observing that by
the very definition in (2.17) it holds ${\rm{Per}}(B_{r}(x),B_{R}(x))=0$ for
every $r>R$. ∎

Let $({\rm X},{\sf d},\mathfrak{m})$ be a metric measure space. Given a Borel
set $E\subset{\rm X}$ we define the upper and lower densities at $x$ as

$\overline{D}(E,x)\coloneqq\varlimsup_{r\to
0^{+}}\frac{\mathfrak{m}(B_{r}(x)\cap
E)}{\mathfrak{m}(B_{r}(x))},\quad\underline{D}(E,x)\coloneqq\varliminf_{r\to
0^{+}}\frac{\mathfrak{m}(B_{r}(x)\cap E)}{\mathfrak{m}(B_{r}(x))}.$

Clearly, if $x\in{\rm X}$ is such that $\overline{D}(E,x)>0$, by definition of
limit superior we have $x\in\overline{E}$ (since every open ball with center
$x$ must intersect $E$).

The _essential boundary_ and the _essential interior_ are given respectively
by

$\displaystyle\partial^{e}E\coloneqq\\{x\in{\rm X}\ :\
\overline{D}(E,x)>0,\,\overline{D}(E^{c},x)>0\\},$ $\displaystyle
E^{(1)}\coloneqq\\{x\in{\rm X}\ :\ \overline{D}(E,x)=\underline{D}(E,x)=1\\},$

which are both Borel sets. As a direct consequence of the definition of these
sets, notice that if $E\subset F$, then $E^{(1)}\subset F^{(1)}$. Moreover,
$\partial^{e}E=\partial^{e}(E^{c})$.

We collect in the next lemma all the elementary facts that we will need about
the essential boundary and the essential interior.

######  Lemma 2.10.

Let $({\rm X},{\sf d})$ be a metric space and $E,F\subset{\rm X}$ be Borel
sets. We have the following:

  1. (i)

If $E$ is open, $E\subset E^{(1)}$.

  2. (ii)

$\partial^{e}E\subset\partial E$.

  3. (iii)

$(E\cap F)^{(1)}\subset E^{(1)}\cap F^{(1)}$.

  4. (iv)

$(E^{c})^{(1)}\subset(E^{(1)})^{c}$. In particular, if $E$ is open then
$(E^{c})^{(1)}\subset E^{c}$.

  5. (v)

If $E$ and $F$ are disjoint, then also $E^{(1)}$ and $F^{(1)}$ are disjoint.

  6. (vi)

$\partial^{e}(E\cap F)\cup\partial^{e}(E\cup
F)\subset\partial^{e}E\cup\partial^{e}F$.

###### Proof.

  1. (i)

Let $x\in E$. Since $E$ is open, $B_{r}(x)\subset E$ for sufficiently small
$r>0$, thus $\underline{D}(E,x)=1$ and $x\in E^{(1)}$.

  2. (ii)

As we have already observed, if $x\in\partial^{e}E$ it holds
$x\in\overline{E}$ and $x\in\overline{E^{c}}$, thus $x\in\partial E$.

  3. (iii)

The result is a direct consequence of the fact that $\underline{D}(E\cap
F,x)\leq\underline{D}(E,x)\leq 1$ for every $x\in{\rm X}$.

  4. (iv)

Let $x\in(E^{c})^{(1)}$, i.e. $\underline{D}(E^{c},x)=1$. In particular,

$\frac{2}{3}\mathfrak{m}(B_{r}(x))<\mathfrak{m}(B_{r}(x)\cap
E^{c})=\mathfrak{m}(B_{r}(x))-\mathfrak{m}(B_{r}(x)\cap E)$

for sufficiently small $r>0$. Thus $\mathfrak{m}(B_{r}(x)\cap
E)<\frac{1}{3}\mathfrak{m}(B_{r}(x))$ for $r>0$ small enough, which implies
$x\notin E^{(1)}$. The second conclusion follows from what we have just proven
and point $({i})$.

  5. (v)

The assumption $E\cap F=\emptyset$ is equivalent to $E\subset F^{c}$. Passing
to the essential interior it holds $E^{(1)}\subset(F^{c})^{(1)}$ and using
$({iv})$ one deduces that $E^{(1)}\subset(F^{(1)})^{c}$ which gives the
desired conclusion.

  6. (vi)

This is proven e.g. in [23, Prop. 1.16] (note that the doubling assumption on
$\mathfrak{m}$ is not used in that statement).

∎

We conclude this part with the following elementary and well known result.
Since we could not find it stated exactly in this form in the literature, we
include a proof.

######  Lemma 2.11.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a metric measure space and let
$C\subset{\rm X}$ be closed. Then for every $E\subset C$ Borel satisfying
${\sf d}(E,{\rm X}\setminus C)>0$ it holds

${\rm{Per}}(E)={\rm{Per}}_{C}(E),$

where ${\rm{Per}}_{C}(E)$ denotes the perimeter of $E$ computed in the metric
measure space $(C,{\sf d}\lower 3.0pt\hbox{$|_{C}$},\mathfrak{m}\lower
3.0pt\hbox{$|_{C}$}).$

###### Proof.

First observe that for every $f\in\mathsf{LIP}_{\mathsf{loc}}({\rm X})$ it
holds $f\lower 3.0pt\hbox{$|_{C}$}\in\mathsf{LIP}_{\mathsf{loc}}(C)$ and ${\rm
lip}_{C}(f)\leq{\rm lip}(f)$, where ${\rm lip}_{C}(f)$ denotes the slope of
$f$ computed in the metric space $(C,{\sf d}\lower 3.0pt\hbox{$|_{C}$})$. This
and the definitions implies ${\rm{Per}}_{C}(E)\leq{\rm{Per}}(E).$ For the
other inequality it is sufficient to find a sequence
$f_{n}\in\mathsf{LIP}_{\mathsf{loc}}(C,{\sf d}\lower 3.0pt\hbox{$|_{C}$})$
such that ${\sf d}(\mathop{\rm supp}\nolimits(f_{n}),{\rm X}\setminus C)>0$,
$f_{n}\to{\raise 1.29167pt\hbox{$\chi$}}_{E}$ in $L^{1}(\mathfrak{m})$ and
$\int_{C}{\rm lip}_{C}(f_{n}){\mathrm{d}}\mathfrak{m}\to{\rm{Per}}_{C}(E)$.
Indeed extending $f_{n}$ by zero to the whole ${\rm X}$ we have $\int_{\rm
X}{\rm lip}(f_{n}){\mathrm{d}}\mathfrak{m}=\int_{C}{\rm
lip}_{C}(f_{n}){\mathrm{d}}\mathfrak{m}$, so that

${\rm{Per}}(E)\leq\lim_{n\to+\infty}\int_{X}{\rm
lip}(f_{n}){\mathrm{d}}\mathfrak{m}=\lim_{n\to+\infty}\int_{C}{\rm
lip}_{C}(f_{n}){\mathrm{d}}\mathfrak{m}={\rm{Per}}_{C}(E).$

To produce such sequence we consider a sequence
$g_{n}\in\mathsf{LIP}_{\mathsf{loc}}(C,{\sf d}\lower 3.0pt\hbox{$|_{C}$})$
such that $g_{n}\to{\raise 1.29167pt\hbox{$\chi$}}_{E}$ in
$L^{1}(C,\mathfrak{m}\lower 3.0pt\hbox{$|_{C}$})$ and $\int_{C}{\rm
lip}_{C}(g_{n}){\mathrm{d}}\mathfrak{m}\to{\rm{Per}}_{C}(E)$, which exists by
definition. Then we take any $\eta\in\mathsf{LIP}_{\mathsf{loc}}(C)$
satisfying ${\sf d}(\mathop{\rm supp}\nolimits(\eta),{\rm X}\setminus C)>0$,
$\eta=1$ in a neighbourhood of $E$, $0\leq\eta\leq 1$ and set
$f_{n}\coloneqq\eta g_{n}\in\mathsf{LIP}_{\mathsf{loc}}(C,{\sf d}\lower
3.0pt\hbox{$|_{C}$}).$ Clearly $f_{n}\to{\raise 1.29167pt\hbox{$\chi$}}_{E}$
in $L^{1}(C;\mathfrak{m}\lower 3.0pt\hbox{$|_{C}$})$. Moreover by the Leibniz
rule for the slope ${\rm lip}_{C}(f_{n})\leq g_{n}{\rm lip}_{C}(\eta)+{\rm
lip}_{C}(g_{n})$. In addition ${\rm lip}_{C}(\eta)=0$ in $E$. Therefore

$\displaystyle{\rm{Per}}_{C}(E)\leq\varliminf_{n}\int_{C}{\rm
lip}_{C}(f_{n}){\mathrm{d}}\mathfrak{m}\leq\varlimsup_{n}\int_{C}{\rm
lip}_{C}(f_{n}){\mathrm{d}}\mathfrak{m}$
$\displaystyle\leq\varlimsup_{n}\int_{C}{\rm
lip}_{C}(g_{n}){\mathrm{d}}\mathfrak{m}+{\rm lip}_{C}(\eta)\int_{{\rm
X}\setminus E}g_{n}{\mathrm{d}}\mathfrak{m}={\rm{Per}}_{C}(E),$

where the second term vanishes because $g_{n}\to{\raise
1.29167pt\hbox{$\chi$}}_{E}$ in $L^{1}(C;\mathfrak{m}\lower
3.0pt\hbox{$|_{C}$})$. ∎

###  2.3 PI spaces

Most of the arguments along the note will be carried out in the general
setting of locally doubling m.m. spaces supporting a Poincaré inequality, also
called PI spaces. We refer to [19, 57] and references therein for a thorough
introduction on this topic and recall here only the properties of these spaces
that will be used in this note.

######  Definition 2.12 (PI space).

A metric measure space $({\rm X},{\sf d},\mathfrak{m})$ is said to be a _PI
space_ if:

  1. i)

it is _uniformly locally doubling_ , i.e. if there exists a function
$C_{D}\,:(0,\infty)\to(0,\infty)$ such that

$\mathfrak{m}\big{(}B_{2r}(x)\big{)}\leq
C_{D}(R)\,\mathfrak{m}\big{(}B_{r}(x)\big{)}\,,\quad\text{for every
}0<r<R\text{ and }x\in{\rm X},$

  2. ii)

supports a _weak local $(1,1)$-Poincaré inequality_, i.e there exists a
constant $\lambda\geq 1$ and a function $C_{P}\,:(0,\infty)\to(0,\infty)$ such
that for any $f\in\mathsf{LIP}_{\mathsf{loc}}(X)$ it holds

$\fint_{B_{r}(x)}\bigg{|}f-\fint_{B_{r}(x)}f\,{\mathrm{d}}\mathfrak{m}\bigg{|}\,{\mathrm{d}}\mathfrak{m}\leq
C_{P}({R})\,r\fint_{B_{\lambda r}(x)}{\rm
lip}(f)\,{\mathrm{d}}\mathfrak{m}\,,\quad\text{for every }0<r<R\text{ and
}x\in{\rm X}.$

Observe that the uniformly locally doubling assumption implies that PI spaces
are proper. Additionally PI spaces are connected and locally connected (see
e.g. [19, Theorem 4.32] or [20, Prop. 4.8]).

We will need the following approximation result, which is a variation of [74,
Lemma 3.6].

######  Lemma 2.13 (Approximation with non-vanishing slope).

Let $({\rm X},{\sf d},\mathfrak{m})$ be a bounded PI space. Then for every
open subset $\Omega\subset{\rm X}$ and any non-negative
$u\in\mathsf{LIP}_{c}(\Omega)$ there exists a sequence of non-negative
$u_{n}\in\mathsf{LIP}_{c}(\Omega)$ satisfying ${\rm lip}(u_{n})\neq 0$
$\mathfrak{m}$-a.e. in $\\{u_{n}>0\\}$ and such that $u_{n}\to u$ in
$\mathit{W}^{1,2}({\rm X}).$

###### Proof.

Since $({\rm X},{\sf d},\mathfrak{m})$ is a bounded PI space it admits a
geodesic distance $\tilde{\sf d}$ that is bi-Lipschitz equivalent to ${\sf
d},$ i.e. $L^{-1}\tilde{\sf d}\leq{\sf d}\leq L\tilde{\sf d}$ for some
constant $L\geq 1$ (see e.g. [57, Corollary 8.3.16]). Hence we can apply [74,
Lemma 3.6] to deduce that the conclusion of the lemma holds in the m.m.s.
$({\rm X},\tilde{\sf d},\mathfrak{m})$. However
$u\in\mathsf{LIP}_{c}(\Omega;{\sf d})$ if and only if
$\mathsf{LIP}_{c}(\Omega;\tilde{\sf d})$, with ${\rm lip}(u)\geq
L^{-1}\tilde{\rm lip}(u)$, where $\tilde{\rm lip}(\cdot)$ denotes the slope
computed in the metric space $({\rm X},\tilde{\sf d})$. Hence if $\tilde{\rm
lip}(u)\neq 0$ $\mathfrak{m}$-a.e. in $\\{u>0\\}$, then ${\rm lip}(u)\neq 0$
in $\\{u>0\\}$ $\mathfrak{m}$-a.e.. Moreover, since
$\mathsf{LIP}_{\mathsf{loc}}({\rm X},{\sf d})=\mathsf{LIP}_{\mathsf{loc}}({\rm
X},\tilde{\sf d})$ and ${\rm lip}(\cdot)\leq L\tilde{\rm lip}(\cdot)$ we have
that $\|u\|_{\mathit{W}^{1,2}({\rm X},{\sf d},\mathfrak{m})}\leq
L\|u\|_{\mathit{W}^{1,2}({\rm X},\tilde{\sf d},\mathfrak{m})}$ for all
$u\in\mathsf{LIP}_{bs}({\rm X},{\sf d})$. Therefore the conclusion holds also
for the m.m.s. $({\rm X},{\sf d},\mathfrak{m}).$ ∎

The above lemma allows to give the following characterization of
$\lambda_{1}(\Omega)$.

######  Lemma 2.14 (Characterization of $\lambda_{1}(\Omega)$ via functions
with non-vanishing slope).

Let $({\rm X},{\sf d},\mathfrak{m})$ be a bounded PI space and
$\Omega\subset{\rm X}$ be open. Then

$\lambda_{1}(\Omega)=\inf\left\\{\frac{\int|Du|^{2}{\mathrm{d}}\mathfrak{m}}{\int
u^{2}{\mathrm{d}}\mathfrak{m}}:u\in\mathsf{LIP}_{c}(\Omega),\ u\not\equiv 0,\
u\geq 0,\ {\rm lip}(u)\neq 0\text{ $\mathfrak{m}$-a.e.\ in
$\\{u>0\\}$}\right\\}\,.$ (2.19)

###### Proof.

For every for every $u\in\mathsf{LIP}_{bs}(\Omega)$, setting
$\tilde{u}\coloneqq|u|$ we have $\tilde{u}\in\mathsf{LIP}_{bs}(\Omega)$ and by
the chain rule (see (2.1)) also that $|D\tilde{u}|=|Du|$ $\mathfrak{m}$-a.e..
This shows that (2.19) holds if we remove the requirement that ${\rm
lip}(u)\neq 0$ $\mathfrak{m}$-a.e. in $\\{u>0\\}$. From this, to get the
validity of the full (2.19) it is sufficient to apply Lemma 2.13. ∎

We recall the following deep result proved in [31], relating the notions of
minimal weak upper gradient and slope, in the setting of PI spaces .

######  Theorem 2.15.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a PI space. Then

${\rm lip}(f)=|Df|,\quad\mathfrak{m}\text{-a.e., for every
$f\in\mathsf{LIP}_{bs}({\rm X})$.}$ (2.20)

The following is a consequence of the Rellich-Kondrachov compactness theorem
in PI spaces.

######  Theorem 2.16 ([54, Theorem 8.3]).

Let $({\rm X},{\sf d},\mathfrak{m})$ be a PI space and $\Omega\subset{\rm X}$
be open and bounded. Then the embedding
$\mathit{W}^{1,2}_{0}(\Omega)\hookrightarrow L^{2}(\Omega)$ is compact.

From Theorem 2.16 and the discussion in Section 2.1 we deduce that

The Dirichlet Laplacian has discrete spectrum on any bounded open subset of an
infinitesimally Hilbertian PI space. (2.21)

The following result is well known. In particular the proof can be achieved by
a standard Moser iteration scheme (see e.g. [52, Theorem 8.24]), which is
available in a PI space. Indeed, as firstly observed in [84] (see also [54]),
a Poincaré inequality and a doubling condition together imply a Sobolev
inequality, which is then sufficient to perform the Moser scheme (see, for
example, [19, Chapter 8] or [21]).

######  Theorem 2.17 (Continuity of eigenfunctions).

Let $({\rm X},{\sf d},\mathfrak{m})$ be an infinitesimally Hilbertian PI
space, $\Omega\subset{\rm X}$ be open and $u$ be a Dirichlet or Neumann
eigenfunction of the Laplacian in $\Omega$. Then $u$ is locally Hölder
continuous in $\Omega.$

We now pass to the properties of sets of finite perimeter in the setting of PI
spaces . As the measure $\mathfrak{m}$ is locally doubling, the Lebesgue’s
differentiation theorem holds (see e.g. [57, Section 3.4]) hence we have
$\mathfrak{m}(E\triangle E^{(1)})=0$ for every Borel set $E$, where
$E\triangle E^{(1)}:=(E\setminus E^{(1)})\cup(E^{(1)}\setminus E)$ denotes the
symmetric difference between $E$ and $E^{(1)}$. Moreover by [2, Theorem 5.3]
we have that for every set of finite perimeter $E\subset{\rm X}$ the measure
$P(E,\cdot)$ is concentrated on $\partial^{e}E.$ In particular we get

${\rm{Per}}(B^{c},.)={\rm{Per}}(B,.)={\rm{Per}}(B,.)\lower
3.0pt\hbox{$|_{B^{c}}$},$ (2.22)

for every ball $B=B_{r}(x)\subset{\rm X}$ having finite perimeter, having used
$\partial^{e}B_{r}(x)\subset\partial B_{r}(x)\subset(B_{r}(x))^{c}$ (see
$(ii)$ in Lemma 2.10).

It is well known that every PI space admits an isoperimetric inequality (see
[2, 54, 73]). We report in the following statement a simplified version
sufficient to our purposes.

######  Proposition 2.18 (Isoperimetric inequality for small volumes).

Let $({\rm X},{\sf d},\mathfrak{m})$ be a bounded PI space satisfying for some
constant $s>1$ and $c>0$

$\frac{\mathfrak{m}(B_{r}(x))}{\mathfrak{m}(B_{R}(x))}\geq
c\left(\frac{r}{R}\right)^{s},\quad\forall\,x\in{\rm X},\quad\forall\,0<r<R.$
(2.23)

Then there exist constants $w_{0}=w_{0}({\rm X})>0$ and $C_{I}=C_{I}({\rm
X},s)>0$ such that

${\rm{Per}}(E)\geq
C_{I}\mathfrak{m}(E)^{\frac{s-1}{s}},\quad\forall\,E\subset{\rm X}\text{ Borel
such that $\mathfrak{m}(E)\leq w_{0}$}.$ (2.24)

###### Proof.

By [2, Theorem 4.3] there exist constants $\sigma=\sigma({\rm X})\geq 1$ and
$C=C({\rm X},s)>0$ such that

${\rm{Per}}(E,B_{\sigma r}(x))\geq
C\frac{\mathfrak{m}(B_{r}(x))^{\frac{1}{s}}}{r}\min\left(\mathfrak{m}(B_{r}(x)\cap
E),\mathfrak{m}(B_{r}(x)\setminus E)\right)^{\frac{s-1}{s}},\quad\forall
x\in{\rm X},\quad\forall r>0.$

Taking $r\coloneqq\text{diam}({\rm X})$ and $w_{0}\coloneqq\mathfrak{m}({\rm
X})/2$ the result follows (recall that $\mathfrak{m}({\rm X})<+\infty$ since
${\rm X}$ is bounded). ∎

We conclude this part reporting the following technical result.

######  Proposition 2.19 ([12, Lemma 2.6]).

Let $({\rm X},{\sf d},\mathfrak{m})$ be a PI space. Let $E,F\subseteq{\rm X}$
be sets of finite perimeter with $P(E,\partial^{e}F)=0$. Then

$P(E\cap F,\cdot)\leq P(E,\cdot)\lower
3.0pt\hbox{$|_{F^{(1)}}$}+P(F,\cdot)\lower 3.0pt\hbox{$|_{E^{(1)}}$}.$

###  2.4 Pólya-Szegő inequality in metric measure spaces

Here we report a version of the Pólya-Szegő rearrangement inequality for
metric measure spaces, following [74, 76]. The main difference with the
classical version in the Euclidean space [80] is that, even if the initial
function lives in a metric space, the symmetrization will be defined in
$\mathbb{R}^{N}$. The result is a generalization of the Pólya-Szegő inequality
introduced in [17] in the case of Riemannian manifolds.

######  Definition 2.20 (Distribution function).

Let $({\rm X},{\sf d},\mathfrak{m})$ be a metric measure space,
$\Omega\subseteq{\rm X}$ be an open set with $\mathfrak{m}(\Omega)<+\infty$
and $u:\Omega\to[0,+\infty)$ be a non-negative Borel function. We define
$\mu:[0,+\infty)\to[0,\mathfrak{m}(\Omega)]$, the distribution function of
$u$, as

$\mu(t):=\mathfrak{m}(\\{u>t\\}).$ (2.25)

For $u$ and $\mu$ as above, we let $u^{\\#}$ be the generalized inverse of
$\mu$, defined by

$u^{\\#}(s):=\begin{cases*}{\rm ess}\sup u&\text{if $s=0$},\\\
\inf\left\\{t:\mu(t)<s\right\\}&\text{if $s>0$}.\end{cases*}$

It can be checked that $u^{\\#}$ is non-increasing and left-continuous.

Next, we define the _Euclidean monotone rearrangement_ into the Euclidean
space $(\mathbb{R}^{N},|\cdot|,{\mathscr{L}}^{N})$, where ${\mathscr{L}}^{N}$
is the $N$-dimensional Lebesgue measure. From now on, we denote by
$\omega_{N}\coloneqq{\mathscr{L}}^{N}(B_{1}(0))$ the Lebesgue measure of the
unit ball in the Euclidean space $\mathbb{R}^{N}$.

######  Definition 2.21 (Euclidean monotone rearrangement).

Let $({\rm X},{\sf d},\mathfrak{m})$ be a metric measure space and
$\Omega\subset{\rm X}$ be open with $\mathfrak{m}(\Omega)<+\infty$ and
$N\in\mathbb{N}.$ For any Borel function
$u\colon\Omega\rightarrow\mathbb{R}^{+}$, we define
$\Omega^{*}:=B_{r}(0)\subset\mathbb{R}^{N}$, choosing $r>0$ so that
${\mathscr{L}}^{N}(B_{r}(0))=\mathfrak{m}(\Omega)$ (i.e.
$r^{N}=\omega_{N}^{-1}\mathfrak{m}(\Omega)$) and the monotone rearrangement
$u^{*}_{N}:\Omega^{*}\rightarrow\mathbb{R}^{+}$ by

$u^{*}_{N}(x):=u^{\\#}({\mathscr{L}}^{N}(B_{|x|}(0)))=u^{\\#}(\omega_{N}|x|^{N}),\qquad\forall
x\in\Omega^{*}.$

In particular $u$ and $u_{N}^{*}$ are equimeasurable, i.e.
$\mathfrak{m}(\\{u>t\\})={\mathscr{L}}^{N}(\\{u^{*}_{N}>t\\})$ for all $t>0.$
In the sequel, whenever we fix $\Omega$ and $u\colon\Omega\to[0,\infty)$, the
set $\Omega^{*}$ and the rearrangement $u^{*}_{N}$ are automatically defined
as above. Observe also that, given $u\in L^{2}(\Omega)$, its monotone
rearrangement must be defined by fixing a Borel representative of $u$.
However, this choice does not affect the outcome object $u^{*}_{N}$, as
clearly the distribution function $\mu(t)$ of $u$ is independent of the
representative.

The following result is essentially contained in [76], see in particular [76,
Remark 3.7], (see also [74] for a similar result), since the only difference
is that here the rearrangement is defined in $\mathbb{R}^{N}$ instead that on
an interval. Nevertheless we include a short argument outlining the main
points of the proof.

######  Theorem 2.22 (Euclidean Pólya-Szegő inequality).

Let $({\rm X},{\sf d},\mathfrak{m})$ be a bounded PI space,
$\Omega\subsetneq{\rm X}$ be open and fix $N\in\mathbb{N}\setminus\\{1\\}.$
Suppose there exists a constant $\tilde{C}>0$ such that

${\rm{Per}}(E)\geq\tilde{C}\mathfrak{m}(E)^{\frac{N-1}{N}}\,,\quad\forall\,E\subset\Omega\text{
Borel.}$ (2.26)

Then:

  1. i)

For every $u\in\mathsf{LIP}_{c}(\Omega)$ non-negative, $u\not\equiv 0$, with
${\rm lip}(u)\neq 0$ $\mathfrak{m}$-a.e. in $\\{u>0\\}$, then
$u^{*}_{N}\in\mathsf{LIP}_{c}(\Omega^{*})$ and it holds

$\int_{\\{u\leq
s\\}}|Du|^{2}\,{\mathrm{d}}\mathfrak{m}\geq\int_{0}^{s}\left(\frac{{\rm{Per}}(\\{u>t\\})}{N\omega_{N}^{\frac{1}{N}}\mu(t)^{\frac{N-1}{N}}}\right)^{2}\int_{\mathbb{R}^{N}}|Du_{N}^{*}|{\mathrm{d}}{\rm{Per}}(\\{u^{*}_{N}>t\\})\,{\mathrm{d}}t,\quad\forall
s\in(0,\max u].$ (2.27)

  2. ii)

The Euclidean-rearrangement maps $\mathit{W}^{1,2}_{0}(\Omega)$ to
$\mathit{W}^{1,2}_{0}(\Omega^{*})$ and

$\int_{\Omega}|{Du}|^{2}{\mathrm{d}}\mathfrak{m}\geq\Big{(}\frac{\tilde{C}}{N\omega_{N}^{1/N}}\Big{)}^{2}\int_{\Omega^{*}}|{Du_{N}^{*}}|^{2}{\mathrm{d}}{\mathscr{L}}^{N},\quad\forall\,u\in\mathit{W}^{1,2}_{0}(\Omega).$
(2.28)

###### Proof.

It is enough to prove $i)$, since $ii)$ then follows by approximation with
Lipschitz functions using Lemma 2.13 as in [76, Theorem 3.6] (see also [74]).

We fix $u\in\mathsf{LIP}_{c}(\Omega)$, $u\not\equiv 0$, with ${\rm lip}(u)\neq
0$ $\mathfrak{m}$-a.e. $\\{u>0\\}$. Set $M\coloneqq\sup u.$ Under these
assumptions $\mu$ is strictly monotone, absolutely continuous (hence
differentiable almost everywhere) and

$\int_{\\{u\leq
s\\}}|Du|^{2}\,{\mathrm{d}}\mathfrak{m}\geq\int_{0}^{s}\frac{{\rm{Per}}(\\{u>t\\})^{2}}{-\mu^{\prime}(t)}\,{\mathrm{d}}t,\quad\forall
s\in(0,M].$ (2.29)

This can be seen arguing exactly as in the proof of [74, Prop. 3.12 and
(3.23)] (see also [76]), recalling also that $|{\bf D}u|\leq{\rm
lip}(u)\mathfrak{m}=|Du|\mathfrak{m}$ (see (2.20)). Next we claim that
$u^{*}_{N}\in\mathsf{LIP}_{c}(\Omega^{*}).$ Recall that by definition
$\Omega^{*}=B_{r}(0)\subset\mathbb{R}^{N}$, where $r>0$ satisfies
${\mathscr{L}}^{N}(B_{r}(0))=\mathfrak{m}(\Omega).$ From the definitions
$u^{*}_{N}(x)=\tilde{u}^{*}_{N}(|x|)$, where
$\tilde{u}^{*}_{N}:[0,r]\to\mathbb{R}^{+}$ is the rearrangement into the space
$([0,\infty),|.|,N\omega_{N}t^{N-1}{\mathrm{d}}t)$ as defined in [76,
Definition 3.1]. Then the fact that $u^{*}_{N}\in\mathsf{LIP}(\Omega^{*})$
follows directly from $\tilde{u}^{*}_{N}\in\mathsf{LIP}[0,r]$ which is proved
in [76, Prop. 3.4] under the same assumptions on $u$ and $\Omega$. Finally
$\mathop{\rm supp}\nolimits(u^{*}_{N})\subset\Omega^{*}$. Indeed $\mathop{\rm
supp}\nolimits(u)\subsetneq\Omega$, otherwise $\Omega$ would be closed and
would coincide with ${\rm X}$ (as ${\rm X}$ is connected). This implies that
${\mathscr{L}}^{N}(\\{u_{N}^{*}>t\\})\leq\mathfrak{m}(\mathop{\rm
supp}\nolimits(u))<\mathfrak{m}(\Omega)={\mathscr{L}}^{N}(\Omega^{*}),$ for
all $t>0,$ because $\mathfrak{m}(\Omega\setminus\mathop{\rm
supp}\nolimits(u))>0$, as non-empty open sets in ${\rm X}$ have positive
measure. Since $u_{N}^{*}$ is a radial function centered at the origin this
shows $\mathop{\rm supp}\nolimits(u^{*}_{N})\subset\Omega$. Next we observe
that $\tilde{u}_{N}^{*}$ is strictly decreasing in
$(0,\mathfrak{m}(\mathop{\rm supp}\nolimits(u)))$ (since $\mu(t)$ is
continuous) and in particular $\\{u^{*}_{N}>t\\}=B_{r_{t}}(0)$ (and
$\\{u^{*}_{N}=t\\}=\partial B_{r_{t}}(0)$) for some
$r_{t}\in[0,\mathfrak{m}(\Omega)]$, for every $t\in(0,M)$. Note that $r_{t}$
can be computed explicitly to be $r_{t}=(\omega_{N}^{-1}\mu(t))^{1/N}$, which
also shows that $(0,M)\ni t\mapsto r_{t}$ is a strictly monotone and locally
absolutely continuous map. In particular

$\mathcal{H}^{N}(\partial
B_{r_{t}}(0))=N\omega_{N}^{\frac{1}{N}}\mu(t)^{\frac{N-1}{N}}.$ (2.30)

Combining these observations with the expression for the derivative of $\mu$
given in [74, Lemma 3.10] (see also [76, Lemma 3.5]) we have

$-\mu^{\prime}(t)=\int_{\partial B_{r_{t}}(0)}({\rm
lip}(u_{N}^{*}))^{-1}{\mathrm{d}}\mathcal{H}^{N-1}=\frac{N\omega_{N}^{\frac{1}{N}}\mu(t)^{\frac{N-1}{N}}}{{\rm
lip}(\tilde{u}_{N}^{*})(r_{t})}\,\quad\text{for a.e.\ $t\in(0,M)$},$ (2.31)

where we have used (2.30) and that ${\rm lip}(u_{N}^{*})(x)={\rm
lip}(\tilde{u}_{N}^{*})(|x|)$ which easily follows from the identity
$u^{*}_{N}(x)=\tilde{u}^{*}_{N}(|x|)$. Plugging the above in (2.29) we reach

$\displaystyle\int_{\\{u\leq s\\}}|Du|^{2}\,{\mathrm{d}}\mathfrak{m}$
$\displaystyle\geq\int_{0}^{s}\frac{{\rm{Per}}(\\{u>t\\})^{2}{\rm
lip}(\tilde{u}_{N}^{*})(r_{t})}{N\omega_{N}^{\frac{1}{N}}\mu(t)^{\frac{N-1}{N}}}\,{\mathrm{d}}t$
$\displaystyle=\int_{0}^{s}\left(\frac{{\rm{Per}}(\\{u>t\\})}{N\omega_{N}^{\frac{1}{N}}\mu(t)^{\frac{N-1}{N}}}\right)^{2}{\rm
lip}(\tilde{u}_{N}^{*})(r_{t})N\omega_{N}^{\frac{1}{N}}\mu(t))^{\frac{N-1}{N}}\,{\mathrm{d}}t$
$\displaystyle=\int_{0}^{s}\left(\frac{{\rm{Per}}(\\{u>t\\})}{N\omega_{N}^{\frac{1}{N}}\mu(t)^{\frac{N-1}{N}}}\right)^{2}\int_{\partial
B_{r_{t}}(0)}{\rm
lip}(u_{N}^{*}){\mathrm{d}}\mathcal{H}^{N-1}\,{\mathrm{d}}t\,,$

where for the last step we argue as in (2.31). This concludes the proof. ∎

###  2.5 RCD spaces

For brevity we do not recall the definition of $\mathrm{RCD}(K,N)$ spaces
(with $N\in[1,\infty)$ and $K\in\mathbb{R}$) since it will not be directly
used in this note, instead we recall here all the properties of these spaces
that will be needed. For further details on the definition and on the theory
of metric measure spaces with synthetic Ricci curvature lower bound we refer
to the surveys [3, 49] and references therein.

First recall that every $\mathrm{RCD}$ space is infinitesimally Hilbertian
from the very definition. In every $\mathrm{RCD}(K,N)$ space the _Bishop-
Gromov inequality_ holds [87], that is

$\frac{\mathfrak{m}(B_{r}(x))}{v_{K,N}(r)}\geq\frac{\mathfrak{m}(B_{R}(x))}{v_{K,N}(R)},\quad\forall
x\in{\rm X},\,\forall 0<r<R,$ (2.32)

where the quantities $v_{K,N}(r)$ coincides, for $N\in\mathbb{N}$, with the
volume of the ball of radius $r$ in the model space of curvature $K$ and
dimension $N$. For the definition of $v_{K,N}$ for non integer $N$ see [87],
however in the results of this note only the case $N\in\mathbb{N}$ will be
relevant.

As a consequence of (2.32) we obtain that for every $R_{0}>0$, $N<+\infty$ and
$K\in\mathbb{R}$ there exists a constant $C_{R_{0},K,N}$ such that for every
$\mathrm{RCD}(K,N)$ space $({\rm X},{\sf d},\mathfrak{m})$ it holds:

$\frac{\mathfrak{m}(B_{r}(x))}{\mathfrak{m}(B_{R}(x))}\geq
C_{R_{0},K,N}\left(\frac{r}{R}\right)^{N},\quad\forall x\in{\rm X},\,\forall
0<r<R\leq R_{0}.$ (2.33)

Taking $R=2r$ this shows also that every $\mathrm{RCD}(K,N)$ space with
$N<+\infty$ is uniformly locally doubling (recall Definition 2.12). It is also
proved in [82, 81] that every $\mathrm{RCD}(K,N)$ space supports also a weak
local (1,1)-Poincaré inequality. Combining the last two observations we
conclude that every $\mathrm{RCD}(K,N)$ space with $N<+\infty$ is an
infinitesimally Hilbertian PI space.

We recall the following embedding result.

######  Proposition 2.23 ([50, Theorem 6.3, $ii)$]).

Let $({\rm X},{\sf d},\mathfrak{m})$ be an $\mathrm{RCD}(K,N)$ space and let
$\Omega\subset{\rm X}$ be bounded. Then the inclusion
$\mathit{W}^{1,2}_{0}(\Omega)\hookrightarrow L^{2}(\mathfrak{m})$ is compact.

Given an $\mathrm{RCD}(K,N)$ space $({\rm X},{\sf d},\mathfrak{m})$ we define
the _Bishop-Gromov_ density function $\theta_{N}:{\rm X}\to(0,+\infty]$ by

$\theta_{N}(x)\coloneqq\lim_{r\to
0^{+}}\frac{\mathfrak{m}(B_{r}(x))}{\omega_{N}r^{N}}=\lim_{r\to
0^{+}}\frac{\mathfrak{m}(B_{r}(x))}{v_{N,K}(r)}\,,$ (2.34)

where the existence of the limits is ensured by (2.32) (see [41, Def. 1.9]).
As shown in [41, Lemma 2.2] the function $\theta_{N}$ is lower-semicontinuous
in ${\rm X}.$

A key property that we will need is the validity of a local almost-Euclidean
isoperimetric inequality. We will use the following version essentially proved
in [76] (see also [29, 11] for similar results in the setting of
$\mathrm{RCD}$ spaces and [17] for the Riemannian setting).

######  Theorem 2.24 (Local almost-Euclidean isoperimetric inequality).

Let $({\rm X},{\sf d},\mathfrak{m})$ be an $\mathrm{RCD}(K,N)$ space for some
$N\in(1,\infty),K\in\mathbb{R}$. Then for every $x\in{\rm X}$ with
$\theta_{N}(x)<+\infty$ and every $\varepsilon\in(0,\theta_{N}(x))$ there
exists $\rho=\rho(\varepsilon,x,N)$ such that

${\rm{Per}}(E)\geq\mathfrak{m}(E)^{\frac{N-1}{N}}N\omega_{N}^{\frac{1}{N}}(\theta_{N}(x)-\varepsilon)^{\frac{1}{N}}(1-\varepsilon),\quad\forall\,E\subset
B_{\rho}(x)\text{ Borel.}$ (2.35)

###### Proof.

It is sufficient to prove the statement with
$\varepsilon\in(0,\theta_{N}(x)/2\wedge 1/2)$. From [76, Theorem 3.9] there
exists $\bar{R}=\bar{R}(\varepsilon,K,N)$ such that for every $x\in{\rm X}$,
$R\in(0,\bar{R}]$ it holds

${\rm{Per}}(E)\geq\mathfrak{m}(E)^{\frac{N-1}{N}}N\omega_{N}^{\frac{1}{N}}\theta_{N,R}(x)^{\frac{1}{N}}(1-(2C^{1/N}_{\varepsilon,R}(x)+1)\varepsilon-\varepsilon),\quad\forall\,E\subset
B_{\varepsilon R}(x),$

where
$\theta_{N,\rho}(x)\coloneqq\frac{\mathfrak{m}(B_{\rho}(x))}{\omega_{N}\rho^{N}}$
and $C_{\varepsilon,R}(x)\coloneqq\frac{\theta_{N,\varepsilon
R}(x)}{\theta_{N,R}(x)}.$ Since $\theta_{N}(x)=\lim_{\rho\to
0}\theta_{N,\rho}(x)<+\infty$ there exists $\bar{r}=\bar{r}(x,\varepsilon)$ so
that
$\theta_{N,\rho}(x)\in(\theta_{N}(x)-\varepsilon,\theta_{N}(x)+\varepsilon)$
for all $\rho\leq\bar{r}$. Moreover since $\varepsilon<\theta_{N}(x)/2$ we
have that for every $R\leq\bar{r}$ it holds
$C_{\varepsilon,R}(x)\leq\frac{\frac{3}{2}\theta_{N}(x)}{\frac{1}{2}\theta_{N}(x)}=3.$
Hence choosing
$\rho=\rho(x,\varepsilon,K,N)\coloneqq\varepsilon(\bar{r}(\varepsilon,x)\wedge\bar{R}(\varepsilon,K,N))$,
we have that

${\rm{Per}}(E)\geq\mathfrak{m}(E)^{\frac{N-1}{N}}N\omega_{N}^{\frac{1}{N}}(\theta_{N}(x)-\varepsilon)^{\frac{1}{N}}(1-8\varepsilon),\quad\forall\,E\subset
B_{\rho}(x),$

from which the conclusion follows. ∎

Next we introduce the subclass of non-collapsed $\mathrm{RCD}(K,N)$ space.

######  Definition 2.25 ([41]).

An $\mathrm{RCD}(K,N)$ space $({\rm X},{\sf d},\mathfrak{m})$ is said to be
_non-collapsed_ if $\mathfrak{m}=\mathcal{H}^{N}$, where $\mathcal{H}^{N}$
denotes the $N$-dimensional Hausdorff measure on $({\rm X},{\sf d})$.

After the works [60, 26] this definition is known to be equivalent to one
given in [67]. As showed in [41, Theorem 1.12] if $({\rm X},{\sf
d},\mathfrak{m})$ is a non-collapsed $\mathrm{RCD}(K,N)$ space, then
$N\in\mathbb{N}.$ Moreover by [41, Corollary 1.7] (see also [8, Theorem 1.4])
it holds that

$\begin{split}&\theta_{N}(x)=1,\quad\mathfrak{m}\text{-a.e.\ $x\in{\rm
X}$,}\\\ &\theta_{N}(x)\leq 1,\quad\forall x\in{\rm X}.\end{split}$ (2.36)

######  Remark 2.26 (Consistency with the smooth setting).

It is worth to recall that $\mathrm{RCD}$ spaces are compatible with the
smooth setting in the following sense. Any $N$-dimensional Riemannian manifold
$(M,g)$ with Ricci curvature bounded below by a number $K\in\mathbb{R}$, i.e.
${\rm Ric}_{g}\geq Kg$, endowed with the Riemannian distance and volume
measure is a non-collapsed $\mathrm{RCD}(K,N)$ metric measure space [90, 34].
In particular the metric measure space
$(\mathbb{R}^{N},|\cdot|,\mathcal{H}^{N})$, where $\mathcal{H}^{N}$ is the
$N$-dimensional Hausdorff measure is a non-collapsed $\mathrm{RCD}(0,N)$
space.  $\blacksquare$

We conclude recalling the validity of the Weyl law in non-collapsed setting
proved in [8, 92].

######  Theorem 2.27 (Weyl law in $\mathrm{RCD}$ spaces).

Let $({\rm X},{\sf d},\mathcal{H}^{N})$ be an $\mathrm{RCD}(K,N)$ space and
$\Omega\subset{\rm X}$ be open and bounded. Then

$\lim_{k\to+\infty}\frac{k}{\lambda_{k}^{\mathcal{D}}(\Omega)^{N/2}}=\frac{\omega_{N}}{(2\pi)^{N}}\mathcal{H}^{N}(\Omega),$
(2.37)

where $\\{\lambda_{k}^{\mathcal{D}}(\Omega)\\}_{k\in\mathbb{N}}$ denotes the
spectrum of the Dirichlet Laplacian in $\Omega$ defined in (2.10).

###### Proof.

By the results in [8, 92] it holds that

$\lim_{\lambda\to+\infty}\frac{N(\lambda)}{\lambda^{N/2}}=\frac{\omega_{N}}{(2\pi)^{N}}\mathcal{H}^{N}(\Omega),$
(2.38)

where $N(\lambda)\coloneqq\\#\\{k\in\mathbb{N}\
:\lambda_{k}^{\mathcal{D}}(\Omega)\leq\lambda\\}$. This implies (2.37). To see
this, set $N^{-}(\lambda)\coloneqq\\#\\{k\in\mathbb{N}\
:\lambda_{k}^{\mathcal{D}}(\Omega)<\lambda\\}$, and observe

$N(\lambda_{k}^{\mathcal{D}}(\Omega)-1)\leq
N^{-}(\lambda_{k}^{\mathcal{D}}(\Omega))\leq k\leq
N(\lambda_{k}^{\mathcal{D}}(\Omega)),\quad\forall k\in\mathbb{N}.$

Note that in [92] formula (2.38) is stated with the further assumption that
$\text{diam}(\Omega)<\text{diam}({\rm X})$ (when $\Omega\neq{\rm X}$), however
this assumption is needed in [92] only to ensure the discreteness of the
spectrum, for which we know by Proposition 2.23 that the boundedness of
$\Omega$ is sufficient. ∎

##  3 Sobolev spaces and Neumann eigenfunctions in uniform domains

Our method to deal with Neumann eigenfunctions in domains with irregular
boundary in $\mathbb{R}^{N}$ (or in more abstract $\mathrm{RCD}$ spaces) will
be to translate the problem to a global one, by viewing the domain as a metric
measure space. The idea is that if the boundary satisfies an appropriate
regularity condition, then the resulting m.m. space is a PI space and in
particular has good analytic properties, like the isoperimetric inequality and
embedding theorems (recall Section 2.3). We stress that the possibility of
using abstract metric spaces to deal with Neumann and mixed boundary value
problems in irregular domains of the Euclidean space was noted before (see
e.g. [19, Page 33]).

The key notion that we will use is the one of _uniform domain_ that we now
introduce.

######  Definition 3.1 (Uniform domains).

A bounded open subset $\Omega$ of a metric space $({\rm X},{\sf d})$ is called
a _uniform domain_ if there exists a constant $C>1$ such that every pair of
points $x,y\in\Omega$ can be joined by a rectifiable curve
$\gamma:[0,1]\to\Omega$ such that $l(\gamma)\leq C{\sf d}(x,y)$ and

${\sf d}(\gamma(t),\partial\Omega)\geq C^{-1}\min\left(l(\gamma\lower
3.0pt\hbox{$|_{[0,t]}$}),l(\gamma\lower
3.0pt\hbox{$|_{[t,1]}$})\right),\quad\forall t\in[0,1].$

Uniform domains were introduced in [72] and [65] (see also [89]) and are
central in the theory of BV and Sobolev extension domains (see [65, 47, 58,
59, 22, 68]). Uniform domains are also equivalent to one-sided non
tangentially accessible (1-sided NTA) domains (see e.g. [13, Theorem 2.15] or
the Appendix in [55]). They include Lipschitz domains, but also more irregular
domains such as the quasi-disks, i.e. images of the unit ball under a global
quasi-conformal maps (see [72, Theorem 2.15], [47], [63, Section 3] or also
[88, Remarks 2.1]). In particular the interior of the Koch snowflake is an
example of uniform domain. It has also been proved recently that in every
doubling quasi-convex metric space (in particular any bounded
$\mathrm{RCD}(K,N)$ space) any bounded open set can be approximated from
inside and outside by uniform domains, see [83] for the precise statement.

It easily follows from the definition that every uniform domain is both
connected and locally connected (see e.g. [71]).

The main goal of this section is to prove the following theorem. It says that
a uniform domain in a non-collapsed $\mathrm{RCD}(K,N)$ space (and in
particular in $\mathbb{R}^{N}$) when viewed as a m.m. space admits an almost
Euclidean isoperimetric inequality near almost every point (point (ii)) and
the eigenvalues of the Laplacian satisfy a weak version of the Weyl law (point
(iii)). Item (i) is a technical condition that we will need to apply the
Faber-Krahn inequality in Proposition 5.1. We remark that, differently from
the preceding sections, in the following statement we will denote by ${\rm Y}$
the ambient space, while the notation $({\rm X},{\sf d},\mathfrak{m})$ is
reserved to the metric measure space associated to the closure of the domain
$\Omega\subset{\rm Y}$.

######  Theorem 3.2.

Let $({\rm Y},\tilde{{\sf d}},\mathcal{H}^{N})$ be an $\mathrm{RCD}(K,N)$
space and let $\Omega\subset{\rm Y}$ be a uniform domain. Then the metric
measure space $({\rm X},{\sf
d},\mathfrak{m})\coloneqq(\overline{\Omega},\tilde{{\sf d}}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$},\mathcal{H}^{N}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})$ is an infinitesimally Hilbertian PI
space and satisfies the following properties:

  1. (i)

there exists a constant $c>0$ such that

$\frac{\mathfrak{m}(B^{{\rm X}}_{r}(x))}{\mathfrak{m}(B^{{\rm X}}_{R}(x))}\geq
c\left(\frac{r}{R}\right)^{N},\quad\forall\,x\in{\rm X},\,\forall\,0<r<R.$

  2. (ii)

For every $\varepsilon>0$ there exists a closed set
$C_{\varepsilon}\subset{\rm X}$ with $\mathfrak{m}(C_{\varepsilon})=0$ such
that for every $x\in{\rm X}\setminus C_{\varepsilon}$ there exists a constant
$\rho=\rho(x,N,\varepsilon)>0$ satisfying

${\rm{Per}}_{\rm
X}(E)\geq(1-\varepsilon)N\omega_{N}^{\frac{1}{N}}\mathfrak{m}(E)^{\frac{N-1}{N}},\quad\forall\,E\subset
B^{\rm X}_{\rho}(x)\,\text{ Borel,}$ (3.1)

where ${\rm{Per}}_{\rm X}$ and $B^{\rm X}_{\rho}(x)$ are respectively the
perimeter and the metric ball in the space $({\rm X},{\sf d},\mathfrak{m}).$

  3. (iii)

denoted by $\\{\lambda_{k}\\}_{k}$ the spectrum of the Laplacian in $({\rm
X},{\sf d},\mathfrak{m})$ (recall (2.21)) it holds
$\lambda_{k}=\lambda_{k}^{{\mathcal{N}}}(\Omega)$ for all $k\in\mathbb{N}$ and

$\varlimsup_{k\to+\infty}\frac{\lambda_{k}^{N/2}}{k}\leq\frac{(2\pi)^{N}}{\omega_{N}\mathfrak{m}({\rm
X})}.$ (3.2)

######  Remark 3.3 (The ‘bad’ set $C_{\varepsilon}$).

If the ambient space $({\rm Y},\tilde{{\sf d}},\mathcal{H}^{N})$ is the
Euclidean space then item $ii)$ in Theorem 3.2 is immediate by taking
$C_{\varepsilon}=\partial\Omega$ and by the isoperimetric inequality (indeed
$\partial\Omega$ is negligible as we will show in Lemma 3.6). More generally
if $({\rm Y},\tilde{{\sf d}},\mathcal{H}^{N})$ is a Riemannian manifold with
$\tilde{{\sf d}}$ the geodesic distance then item $ii)$ follows taking again
$C_{\varepsilon}=\partial\Omega$ and applying the local almost-Euclidean
isoperimetric inequality in [17, Appendice C]. In the general case, as will be
shown in the proof, the set $C_{\varepsilon}$ in Theorem 3.2 can be taken to
be

$C_{\varepsilon}\coloneqq\\{x\in\overline{\Omega}\ :\ \theta_{N}(x)\leq
1-\varepsilon\\}\cup\partial\Omega,$

where $\theta_{N}:{\rm X}\to(0,+\infty]$ is the Bishop-Gromov density function
defined in (2.34). In other words the set $C_{\varepsilon}$ contains
$\partial\Omega$ plus a subset of the singular points, the latter being the
points where $\theta_{N}<1$. Note that $\theta_{N}\equiv 1$ if $({\rm
Y},\tilde{{\sf d}},\mathcal{H}^{N})$ is a Riemannian manifold, so that
$C_{\varepsilon}$ reduces to $\partial\Omega,$ in accordance to what we said
above.  $\blacksquare$

We start with some basic properties of uniform domains.

######  Definition 3.4 (Corkscrew-condition).

A bounded open subset $\Omega$ of a metric space $({\rm X},{\sf d})$ satisfies
the corkscrew-condition if there exists a constant $\varepsilon>0$ such that
for every point $x\in\overline{\Omega}$ and all $0<r\leq\text{diam}(\Omega)$,
the set $\Omega\cap B_{r}(x)$ contains a ball of radius $\varepsilon r.$

In Definition 3.4 it is equivalent to require only $x\in\partial\Omega.$

For the proof of the following well known fact see e.g. [22, Lemma 4.2].

######  Lemma 3.5.

Every uniform domain satisfies the corkscrew-condition.

######  Lemma 3.6.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a uniformly locally doubling metric
measure space and let $\Omega\subset{\rm X}$ satisfy the corkscrew-condition.
Then $\mathfrak{m}(\partial\Omega)=0.$ In particular this holds if
$\Omega\subset{\rm X}$ is a uniform domain.

###### Proof.

Thanks to the corkscrew-condition there exists a constant $\varepsilon>0$ such
that for every $x\in\partial\Omega$ and $r>0$ there exists $B_{\varepsilon
r}(y)\subset B_{r}(x)\cap\Omega$. Then by the uniformly locally doubling
assumption we have

$\mathfrak{m}(B_{r}(x)\cap\Omega)\geq\mathfrak{m}(B_{\varepsilon r}(y))\geq
C_{\varepsilon}\mathfrak{m}(B_{2r}(y))\geq
C_{\varepsilon}\mathfrak{m}(B_{r}(x)),\quad\forall r\in(0,1),$ (3.3)

where $C_{\varepsilon}>0$ is a constant depending only on $\varepsilon$.
Therefore no point of $x\in\partial\Omega$ can be a one-density point for
$\partial\Omega$. The conclusion follows by the Lebesgue differentiation
theorem for locally doubling metric measure spaces (see e.g. [57, Section
3.4]). The result applies to uniform domains, since by Lemma 3.5 they satisfy
they corkscrew-condition. ∎

The following lemma gives a lower bound on the measure of balls in domain
satisfying the corkscrew-condition.

######  Lemma 3.7.

Fix $s>0$. Let $({\rm X},{\sf d},\mathfrak{m})$ be a metric measure space such
that for every $R_{0}>0$ there exists a constant $c_{0}>0$ such that

$\frac{\mathfrak{m}(B_{r}(x))}{\mathfrak{m}(B_{R}(x))}\geq
c_{0}\left(\frac{r}{R}\right)^{s},\quad\forall x\in{\rm X},\,\,\forall
0<r<R\leq R_{0}.$ (3.4)

Then for every bounded domain $\Omega\subset{\rm X}$ satisfying the corkscrew-
condition there exists a constant $C>0$ such that

$\frac{\mathfrak{m}(B_{r}(x)\cap\Omega)}{\mathfrak{m}(B_{R}(x)\cap\Omega)}\geq
C\left(\frac{r}{R}\right)^{s},\quad\forall x\in\overline{\Omega},\,\,\forall
0<r<R.$ (3.5)

In particular this holds if $\Omega\subset{\rm X}$ is a uniform domain.

###### Proof.

Taking $R=2r$ in (3.4) shows that $({\rm X},{\sf d},\mathfrak{m})$ is
uniformly locally doubling. Then as in (3.3) we have the existence of a
constant $\tilde{C}$ such that for every $x\in\overline{\Omega}$

$\displaystyle\mathfrak{m}(B_{r}(x)\cap\Omega)\geq\tilde{C}\mathfrak{m}(B_{r}(x))\overset{\eqref{eq:dim
doubling}}{\geq}c_{0}\tilde{C}\mathfrak{m}(B_{R}(x))\left(\frac{r}{R}\right)^{s}$
$\displaystyle\geq
c_{0}\tilde{C}\mathfrak{m}(B_{R}(x)\cap\Omega)\left(\frac{r}{R}\right)^{s},\quad\forall
0<r<R\leq\text{diam}(\Omega).$

This proves (3.5) for $0<r<R\leq\text{diam}(\Omega)$. If instead
$r>\text{diam}(\Omega)$ we have

$\frac{\mathfrak{m}(B_{r}(x))\cap\Omega}{\mathfrak{m}(B_{R}(x)\cap\Omega)}=\frac{\mathfrak{m}(\Omega)}{\mathfrak{m}(\Omega)}=1\geq\frac{r^{s}}{R^{s}},\quad\forall
x\in\overline{\Omega},\quad\forall R\geq r.$

This shows (3.5) also for $\text{diam}(\Omega)<r<R$ and concludes the proof. ∎

The next step is to show that the Sobolev space on a uniform domain coincides
with the Sobolev space on its closure. In what follows, given a m.m. space
$({\rm X},{\sf d},\mathfrak{m})$ and an open subset $\Omega\subset{\rm X}$, we
denote by $(\overline{\Omega},{\sf d}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$},\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})$ the m.m. space obtained by endowing
$\overline{\Omega}$ with the restriction distance ${\sf d}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$}\coloneqq{\sf d}\lower
3.0pt\hbox{$|_{\overline{\Omega}\times\overline{\Omega}}$}$ and measure
$\mathfrak{m}\lower 3.0pt\hbox{$|_{\overline{\Omega}}$}$, obtained by
restricting $\mathfrak{m}$ to the induced Borel $\sigma$-algebra on
$\overline{\Omega}.$ Note that by definition $(\overline{\Omega},{\sf d}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$},\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})$ is a complete and separable metric
measure space with $\mathop{\rm supp}\nolimits(\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})=\overline{\Omega}.$ Moreover for a
function $u\in L^{2}(\overline{\Omega},\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})$ we will denote by $u\lower
3.0pt\hbox{$|_{\Omega}$}\in L^{2}(\Omega,\mathfrak{m})$ the function which
agrees $\mathfrak{m}$-a.e. with $u$ in $\Omega.$

######  Theorem 3.8 (Equivalence between
$\mathit{W}^{1,2}(\overline{\Omega},{\sf d}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$},\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})$ and $\mathit{W}^{1,2}(\Omega)$).

Let $({\rm X},{\sf d},\mathfrak{m})$ be a PI space and $\Omega\subset{\rm X}$
be a uniform domain. Then $\mathit{W}^{1,2}(\overline{\Omega},{\sf d}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$},\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})=\mathit{W}^{1,2}(\Omega)$ as function
spaces. More precisely, for every $u\in\mathit{W}^{1,2}(\overline{\Omega},{\sf
d}\lower 3.0pt\hbox{$|_{\overline{\Omega}}$},\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})$ it holds that $\Phi(u)\coloneqq u\lower
3.0pt\hbox{$|_{\Omega}$}\in\mathit{W}^{1,2}(\Omega)$ and the map
$\Phi:\mathit{W}^{1,2}(\overline{\Omega},{\sf d}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$},\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})\to\mathit{W}^{1,2}(\Omega)$ is a
surjective isometry. Moreover if $({\rm X},{\sf d},\mathfrak{m})$ is
infinitesimally Hilbertian, so is $(\overline{\Omega},{\sf d}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$},\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})$ and

$\int_{\overline{\Omega}}\nabla u\cdot\nabla
v\,{\mathrm{d}}\mathfrak{m}=\int_{\Omega}\nabla(u\lower
3.0pt\hbox{$|_{\Omega}$})\cdot\nabla(v\lower
3.0pt\hbox{$|_{\Omega}$})\,{\mathrm{d}}\mathfrak{m},\quad\forall
u,v\in\mathit{W}^{1,2}(\overline{\Omega},{\sf d}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$},\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$}).$ (3.6)

###### Proof.

The result is well known, but for convenience of the reader we include a short
argument which is a combination of results already present in literature. Fix
$u\in\mathit{W}^{1,2}(\overline{\Omega},{\sf d}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$},\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})$. The fact that $\Phi(u)=u\lower
3.0pt\hbox{$|_{\Omega}$}\in\mathit{W}^{1,2}(\Omega)$ is checked e.g. in [6,
Remark 2.15]. Moreover by [9, Prop. 6.4] for every
$\eta\in\mathsf{LIP}_{bs}(\Omega)$, since ${\sf d}(\mathop{\rm
supp}\nolimits(\eta u),{\rm X}\setminus\Omega)>0,$ it holds that $\eta
u\in\mathit{W}^{1,2}({\rm X})$ (when extended by zero in the whole ${\rm X}$)
and also that $|D(\eta u)|_{\rm X}=|D(\eta u)|_{\overline{\Omega}}$
$\mathfrak{m}$-a.e. in $\Omega$, where $|D(\cdot)|_{\overline{\Omega}}$ is the
w.u.g. in $\mathit{W}^{1,2}(\overline{\Omega},{\sf d}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$},\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})$ and $|D(\cdot)|_{{\rm X}}$ is the w.u.g.
in $\mathit{W}^{1,2}({\rm X})$. Hence by the arbitrariness of $\eta$ and by
locality we deduce that $|Du|_{\overline{\Omega}}=|D(u\lower
3.0pt\hbox{$|_{\Omega}$})|_{\Omega}$ $\mathfrak{m}$-a.e. in $\Omega$, where
$|D(\cdot)|_{\Omega}$ is the w.u.g. in $\mathit{W}^{1,2}(\Omega)$ (as defined
in (2.2)). This together with the fact that $\mathfrak{m}(\partial\Omega)=0$
(recall Lemma 3.6) shows that $\Phi(u)$ preserves the norm. It remains to show
that $\Phi$ is surjective (note that everything we said up to now holds for an
arbitrary open set $\Omega$ with $\mathfrak{m}(\partial\Omega)$=0). Now since
$\Omega$ is a uniform domain, by [22, Prop. 5.9], $\Omega$ is an extension
domain and in particular for every $u\in\mathit{W}^{1,2}(\Omega)$ there exists
$\tilde{u}\in\mathit{W}^{1,2}({\rm X})$ such that $\tilde{u}\lower
3.0pt\hbox{$|_{\Omega}$}=u$. By density of Lipschitz functions (see [85] or
[19, Theorem 5.1]) there exists a sequence $u_{n}\in\mathsf{LIP}_{bs}({\rm
X})$ such that $u_{n}\to\tilde{u}$ in $\mathit{W}^{1,2}({\rm X}).$ In
particular it holds $u_{n}\lower 3.0pt\hbox{$|_{\Omega}$}\to u$ in
$L^{2}(\overline{\Omega};\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})$. Moreover $u_{n}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$}\in\mathsf{LIP}(\overline{\Omega},{\sf
d}\lower 3.0pt\hbox{$|_{\overline{\Omega}}$})$ and by definition of slope we
have ${\rm lip}_{\overline{\Omega}}(u_{n}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})(x)\leq{\rm lip}(u_{n})(x)$ for every
$x\in\overline{\Omega}$, where ${\rm lip}_{\overline{\Omega}}$ denotes the
slope with respect to the metric space $(\overline{\Omega},{\sf d}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$}).$ Therefore
$\sup_{n}\int_{\overline{\Omega}}|{\rm lip}_{\overline{\Omega}}(u_{n}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})|^{2}{\mathrm{d}}\mathfrak{m}\leq\sup_{n}\int_{{\rm
X}}|{\rm lip}(u_{n})|^{2}{\mathrm{d}}\mathfrak{m}<+\infty$, which by
definition proves that $\tilde{u}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$}\in\mathit{W}^{1,2}(\overline{\Omega},{\sf
d}\lower 3.0pt\hbox{$|_{\overline{\Omega}}$},\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$}).$ Since $\tilde{u}\lower
3.0pt\hbox{$|_{\Omega}$}=u$ and by the arbitrariness of
$u\in\mathit{W}^{1,2}(\Omega)$ this shows that $\Phi$ is surjective. Finally
if $({\rm X},{\sf d},\mathfrak{m})$ is infinitesimally Hilbertian then
$\mathit{W}^{1,2}(\Omega)$ is a Hilbert space and since $\Phi$ is an isometry,
then $\mathit{W}^{1,2}(\overline{\Omega},{\sf d}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$},\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})$ is a Hilbert space as well and so
$(\overline{\Omega},{\sf d}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$},\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})$ is infinitesimally Hilbertian. Then
identity (3.6) follows by polarization using the definition of scalar product
between gradients. ∎

######  Theorem 3.9.

Let $({\rm X},{\sf d},\mathfrak{m})$ be an infinitesimally Hilbertian PI space
and let $\Omega\subset{\rm X}$ be a uniform domain. Then the metric measure
space $(\overline{\Omega},{\sf d}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$},\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})$ is an infinitesimally Hilbertian PI
space.

###### Proof.

The fact that $(\overline{\Omega},{\sf d}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$},\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})$ is a PI space is proved in [22, Theorem
4.4] (see also [1, Prop. 7.1] and [19, Theorem A.21]), recalling also that
$\mathfrak{m}(\partial\Omega)=0$. The infinitesimal Hilbertianity follows from
Theorem 3.8 (alternatively we could apply [48, Prop. 4.22]). ∎

From the previous results about the compatibility of the two Sobolev spaces
$\mathit{W}^{1,2}(\Omega)$ and $\mathit{W}^{1,2}(\overline{\Omega},{\sf
d}\lower 3.0pt\hbox{$|_{\overline{\Omega}}$},\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})$, we can deduce a compatibility between
eigenfunctions and eigenvalues in $\Omega$ and in $(\overline{\Omega},{\sf
d}\lower 3.0pt\hbox{$|_{\overline{\Omega}}$},\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})$.

######  Corollary 3.10 (Equivalence between
$\lambda_{k}^{\mathcal{N}}(\Omega)$ and $\lambda_{k}(\overline{\Omega},{\sf
d}\lower 3.0pt\hbox{$|_{\overline{\Omega}}$},\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})$).

Let $({\rm X},{\sf d},\mathfrak{m})$ be an infinitesimally Hilbertian PI space
and let $\Omega\subset{\rm X}$ be a uniform domain. Then:

  1. i)

$u\in\mathit{W}^{1,2}(\Omega)$ is an eigenfunction for the Neumann Laplacian
in $\Omega$ of eigenvalue $\lambda$ if and only there exists an eigenfunction
$\tilde{u}\in\mathit{W}^{1,2}(\overline{\Omega},{\sf d}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$},\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})$ of the Laplacian in the metric measure
space $(\overline{\Omega},{\sf d}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$},\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})$ of eigenvalue $\lambda$ and satisfying
$\tilde{u}\lower 3.0pt\hbox{$|_{\Omega}$}=u$. In particular, if this is the
case, then $u$ has a Hölder continuous representative in $\Omega.$

  2. ii)

the embedding $\mathit{W}^{1,2}(\Omega)\hookrightarrow L^{2}(\Omega)$ is
compact and in particular the Neumann Laplacian in $\Omega$ has a discrete
spectrum $\\{\lambda_{k}^{\mathcal{N}}(\Omega)\\}_{k\in\mathbb{N}}$ (counted
with multiplicity) satisfying

$0=\lambda_{1}^{\mathcal{N}}(\Omega)\leq\lambda_{2}^{\mathcal{N}}(\Omega)\leq\dots\lambda_{k}^{\mathcal{N}}(\Omega)\leq\dots\rightarrow+\infty.$

  3. iii)

denoted by $\\{\lambda_{k}\\}_{k}$ the spectrum for the Laplacian in
$(\overline{\Omega},{\sf d}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$},\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$}),$ it holds that

$\lambda_{k}^{\mathcal{N}}(\Omega)=\lambda_{k},\quad\forall k\in\mathbb{N}.$

###### Proof.

First observe that the statement makes sense since by Theorem 3.9 we have that
$(\overline{\Omega},{\sf d}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$},\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})$ is a bounded infinitesimally Hilbertian
PI space and so by (2.21) we have that the Laplacian in
$(\overline{\Omega},{\sf d}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$},\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})$ has a discrete spectrum
$\\{\lambda_{k}\\}_{k}.$ It is sufficient to show $i)$ and $ii)$, because then
$iii)$ would follow from the definitions. Suppose that
$u\in\mathit{W}^{1,2}(\Omega)$ is an eigenfunction for the Neumann Laplacian
in $\Omega$ of eigenvalue $\lambda$. Then by Theorem 3.8 there exists
$\tilde{u}\in\mathit{W}^{1,2}(\overline{\Omega},{\sf d}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$},\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})$ such that $\tilde{u}\lower
3.0pt\hbox{$|_{\Omega}$}=u.$ Moreover for every
$v\in\mathit{W}^{1,2}(\overline{\Omega},{\sf d}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$},\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})$, again by Theorem 3.8, we have that
$v\lower 3.0pt\hbox{$|_{\Omega}$}\in\mathit{W}^{1,2}(\Omega)$. Then applying
(3.6) we obtain

$\int_{\overline{\Omega}}\nabla\tilde{u}\cdot\nabla
v\,{\mathrm{d}}\mathfrak{m}=\int_{\Omega}\nabla u\cdot\nabla v\lower
3.0pt\hbox{$|_{\Omega}$}\,{\mathrm{d}}\mathfrak{m}=-\lambda\int_{\Omega}uv\lower
3.0pt\hbox{$|_{\Omega}$}{\mathrm{d}}\mathfrak{m}=-\lambda\int_{\overline{\Omega}}\tilde{u}v{\mathrm{d}}\mathfrak{m},$

where in the second identity we used the definition of eigenfunction and in
the last one that $\mathfrak{m}(\partial\Omega)=0,$ because $\Omega$ is a
uniform domain. This shows that $\tilde{u}$ is an eigenfunction of eigenvalue
$\lambda$ for the Laplacian in $(\overline{\Omega},{\sf d}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$},\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})$. Then by Theorem 2.17 $\tilde{u}$ has a
Hölder continuous representative in $\overline{\Omega}$, which implies that
$u$ has also a continuous representative in $\Omega$. Conversely suppose that
$\tilde{u}$ is an eigenfunction of eigenvalue $\lambda$ for the Laplacian in
$(\overline{\Omega},{\sf d}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$},\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})$. Then by Theorem 3.8 we have
$\tilde{u}\lower 3.0pt\hbox{$|_{\Omega}$}\in\mathit{W}^{1,2}(\Omega)$.
Moreover, again by Theorem 3.8, for every $v\in\mathit{W}^{1,2}(\Omega)$ there
exists $\tilde{v}\in\mathit{W}^{1,2}(\overline{\Omega},{\sf d}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$},\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})$ such that $\tilde{v}\lower
3.0pt\hbox{$|_{\Omega}$}=v.$ Therefore as above using (3.6)

$-\lambda\int_{\Omega}\tilde{u}\lower
3.0pt\hbox{$|_{\Omega}$}v{\mathrm{d}}\mathfrak{m}=-\lambda\int_{\overline{\Omega}}\tilde{u}\tilde{v}{\mathrm{d}}\mathfrak{m}=\int_{\overline{\Omega}}\nabla\tilde{u}\cdot\nabla\tilde{v}\,{\mathrm{d}}\mathfrak{m}=\int_{\Omega}\nabla\tilde{u}\lower
3.0pt\hbox{$|_{\Omega}$}\cdot\nabla v\,{\mathrm{d}}\mathfrak{m}.$

This shows that $\tilde{u}\lower 3.0pt\hbox{$|_{\Omega}$}$ is an eigenfunction
of eigenvalue $\lambda$ for the Neumann Laplacian in $\Omega$ and completes
the proof of $i).$ For $ii)$ recall that by Theorem 3.8 the map
$\Phi:\mathit{W}^{1,2}(\overline{\Omega},{\sf d}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$},\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})\to\mathit{W}^{1,2}(\Omega)$, given by
$\Phi(u)=u\lower 3.0pt\hbox{$|_{\Omega}$}$ is an isometry and that by Theorem
2.16 the inclusion $\iota:\mathit{W}^{1,2}(\overline{\Omega},{\sf d}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$},\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})\hookrightarrow
L^{2}(\overline{\Omega},\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})$ is compact. Let now
$u_{n}\in\mathit{W}^{1,2}(\Omega)$ be a sequence bounded in
$\mathit{W}^{1,2}(\Omega).$ Then the sequence
$\Phi^{-1}(u_{n})\in\mathit{W}^{1,2}(\overline{\Omega},{\sf d}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$},\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})$ is also bounded, hence it has a
converging subsequence in $L^{2}(\overline{\Omega},\mathfrak{m}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})$. However by definition
$\Phi^{-1}(u_{n})\lower 3.0pt\hbox{$|_{\Omega}$}=\Phi(\Phi^{-1}(u_{n})=u_{n}$
for every $n$. Hence $u_{n}$ has also a converging subsequence in
$L^{2}(\Omega,\mathfrak{m}\lower 3.0pt\hbox{$|_{\Omega}$})$, which shows that
the embedding $\mathit{W}^{1,2}(\Omega)\hookrightarrow
L^{2}(\Omega,\mathfrak{m}\lower 3.0pt\hbox{$|_{\Omega}$})$ is compact. This
completes the proof of $ii).$ ∎

######  Remark 3.11.

If the ambient space $({\rm X},{\sf d},\mathfrak{m})$ is an
$\mathrm{RCD}(K,N)$ space, with $N<+\infty$, then the Neumann and Dirichlet
eigenfunctions are actually locally Lipschitz in the interior of the domain,
as follows directly from [64, Theoem 1.1] (see also [7, Prop. 7.1]). Recall
that the continuity of eigenfunctions is crucial to define their nodal
domains.  $\blacksquare$

Next we show that for a uniform domain $\Omega$ there is a one to one
correspondence between the nodal domains in $\Omega$ and the nodal domains in
its closure.

######  Proposition 3.12 (Compatibility of nodal domains).

Let $({\rm X},{\sf d})$ be a metric space, $\Omega\subset{\rm X}$ be a uniform
domain and $f:\overline{\Omega}\to\mathbb{R}$ be a continuous function. Denote
by $\mathcal{C}$ (resp. $\overline{\mathcal{C}}$) the set all the connected
components of $\Omega\setminus\\{f=0\\}$ (resp.
$\overline{\Omega}\setminus\\{f=0\\}$). Then

$\overline{\mathcal{C}}=\\{U\cup(\partial
U\cap(\partial\Omega\setminus\\{f=0\\}))\ :\ U\in\mathcal{C}\\}.$ (3.7)

In particular the sets $\mathcal{C}$ and $\overline{\mathcal{C}}$ have the
same cardinality.

###### Proof.

For every $U\in\mathcal{C}$ we put $\varphi(U)\coloneqq U\cup(\partial
U\cap(\partial\Omega\setminus\\{f=0\\}))$ and we want to show that $\varphi$
defines a bijective map $\varphi:\mathcal{C}\to\overline{\mathcal{C}}$. We
note immediately that $\varphi$ is injective, because if
$\varphi(U)=\varphi(V)$ then $U=\varphi(U)\cap\Omega=\varphi(V)\cap\Omega=V$.

To conclude it is sufficient to prove that the sets
$\\{\varphi(U)\\}_{U\in\mathcal{C}}$ are open, closed and connected in the
topology of $\overline{\Omega}\setminus\\{f=0\\}$ and that their union is
$\overline{\Omega}\setminus\\{f=0\\}$. Indeed this would show that
$\\{\varphi(U)\\}_{U\in\mathcal{C}}$ are exactly the connected components of
$\overline{\Omega}\setminus\\{f=0\\}$, which would imply (3.7) and in
particular that $\varphi$ is surjective.

Note first that the elements of $\mathcal{C}$ are open in the topology of
$\Omega\setminus\\{f=0\\}$ (and thus also in the one of ${\rm X}$) because
$\Omega\setminus\\{f=0\\}$ is locally connected (recall Lemma 2.2) since it is
an open subset of $\Omega$ which is a uniform domain and thus locally
connected.

The key observation is that for every $x\in\partial\Omega\setminus\\{f=0\\}$
there exists $r>0$ and $U\in\mathcal{C}$ such that

$B_{r}(x)\cap\overline{\Omega}\subset\varphi(U)=U\cup(\partial
U\cap(\partial\Omega\setminus\\{f=0\\})).$ (3.8)

To prove this note that $f(x)\neq 0$. Then by continuity there exists
$r_{0}>0$ so that
$B_{r}(x)\cap\overline{\Omega}\subset\overline{\Omega}\setminus\\{f=0\\}$ for
every $r\in(0,r_{0}]$. In particular for every $r\in(0,r_{0}]$ the set
$B_{r}(x)\cap\Omega$ is contained in $\bigcup_{U\in\mathcal{C}}U$. Suppose
that for some $r>0$ the set $B_{r}(x)\cap\Omega$ intersects at at least two
distinct sets $U_{r},V_{r}\in\mathcal{C}$ and take two points $u_{r}\in
U_{r}\cap B_{r}(x),v_{r}\in V_{r}\cap B_{r}(x)$. Since $\Omega$ is a uniform
domain there exists a rectifiable curve $\gamma:[0,1]\to{\rm X}$ contained in
$\Omega$, connecting $u_{r}$ and $v_{r}$ and of length $l(\gamma)\leq C{\sf
d}(u_{r},v_{r})\leq 2Cr$, where $C>0$ is some constant independent of $r.$
Then $\gamma([0,1])$ must intersect $\\{f=0\\}$ otherwise
$\gamma([0,1])\subset\Omega\setminus\\{f=0\\}$ and $U_{r}\cup
V_{r}\cup\gamma([0,1])$ would be a connected subset of
$\Omega\setminus\\{f=0\\}$, which contradicts the fact that $U_{r},V_{r}$ are
distinct connected components of $\Omega\setminus\\{f=0\\}.$ Therefore, since
$\gamma([0,1])\subset B_{(2C+1)r}(x)$, it holds
$B_{(2C+1)r}(x)\cap\Omega\cap\\{f=0\\}\neq\emptyset$ and so by the choice of
$r_{0}$ we must have $(2C+1)r>r_{0}$. This proves that for $r$ small enough
$B_{r}(x)\cap\Omega$ intersects at most one set in $\mathcal{C}$. However, as
observed above, $B_{r}(x)\cap\Omega\subset\bigcup_{U\in\mathcal{C}}U$ for
every $r\in(0,r_{0}]$. Therefore for $r>0$ small enough we must have that
$B_{r}(x)\cap\Omega\subset U$ for some $U\in\mathcal{C}.$ Fix one such $r>0$
and fix $y\in B_{r}(x)\cap\partial\Omega$. Then for every $s>0$ small enough
$B_{s}(y)\cap\Omega\neq\emptyset$ and $B_{s}(y)\cap\Omega\subset
B_{r}(x)\cap\Omega\subset U$. This shows that $y\in\partial U$. Therefore for
every $r>0$ small enough $B_{r}(x)\cap\Omega\subset U$,
$B_{r}(x)\cap\partial\Omega\subset\partial U$ for some $U\in\mathcal{C}$ and
as observed above
$B_{r}(x)\cap\overline{\Omega}\subset\overline{\Omega}\setminus\\{f=0\\}$.
Combining these three facts proves (3.8).

Consider now any $U\in\mathcal{C}$ and note that $\varphi(U)$ is precisely the
closure of $U$ in the topology of $\overline{\Omega}\setminus\\{f=0\\}$.
Indeed

$\displaystyle\overline{U}\cap(\overline{\Omega}\setminus\\{f=0\\})$
$\displaystyle=\left(U\cap(\overline{\Omega}\setminus\\{f=0\\})\right)\cup\left(\partial
U\cap(\overline{\Omega}\setminus\\{f=0\\})\right)$
$\displaystyle=U\cup\left(\partial
U\cap(\partial\Omega\setminus\\{f=0\\})\right)\cup\left(\partial
U\cap(\Omega\setminus\\{f=0\\})\right)$ $\displaystyle=U\cup\left(\partial
U\cap(\partial\Omega\setminus\\{f=0\\})\right)=\varphi(U),$

where we used that $U\subset\Omega\setminus\\{f=0\\}$ and that $\partial
U\cap(\Omega\setminus\\{f=0\\})=\emptyset$ because $U$ is closed in the
topology of $\Omega\setminus\\{f=0\\}$, being a connected component, but also
open as observed above. Moreover $U$ is connected in
$\Omega\setminus\\{f=0\\}$ and thus also in
$\overline{\Omega}\setminus\\{f=0\\}$, hence $\varphi(U)$ is also connected in
the same topology, being the closure of a connected set. Additionally (3.8)
implies that $\varphi(U)$ is open in the topology of
$\overline{\Omega}\setminus\\{f=0\\}$. It remains to prove that the union of
the sets $\\{\varphi(U)\\}_{U\in\mathcal{C}}$ is
$\overline{\Omega}\setminus\\{f=0\\}$, which can be seen as follows

$\displaystyle\overline{\Omega}\setminus\\{f=0\\}$
$\displaystyle=(\Omega\setminus\\{f=0\\})\cup(\partial\Omega\setminus\\{f=0\\})\overset{\eqref{eq:key}}{\subset}(\Omega\setminus\\{f=0\\})\cup\left(\bigcup_{U\in\mathcal{C}}\varphi(U)\right)$
$\displaystyle\subset\left(\bigcup_{U\in\mathcal{C},}U\right)\cup\left(\bigcup_{U\in\mathcal{C}}\varphi(U)\right)=\bigcup_{U\in\mathcal{C}}\varphi(U).$

∎

We pass now to prove the main result of this section.

###### Proof of Theorem 3.2.

The fact that $({\rm X},{\sf d},\mathfrak{m})$ is an infinitesimally
Hilbertian PI space is contained in Theorem 3.9. Recall also that by Lemma 3.6
it holds $\mathcal{H}^{N}(\partial\Omega)=0.$ Item (i) follows immediately
combining (2.33) and Lemma 3.7 and recalling that $\Omega$ is bounded.

We pass to the proof of $\ref{it:isop}.$ Fix $\varepsilon\in(0,1)$ arbitrary.
We choose

$C_{\varepsilon}\coloneqq\\{x\in\overline{\Omega}\ :\ \theta_{N}(x)\leq
1-\varepsilon\\}\cup\partial\Omega$

(see (2.34) for the definition of $\theta_{N}$). From the lower semicontinuity
of the function $\theta_{N}(\cdot)$, it follows that $C_{\varepsilon}$ is a
closed subset of $\overline{\Omega}$. Moreover, since $\theta_{N}(x)=1$ for
$\mathcal{H}^{N}$-a.e. $x$ (recall (2.36)) and
$\mathcal{H}^{N}(\partial\Omega)=0,$ it follows
$\mathcal{H}^{N}(C_{\varepsilon})=0$. Note that by construction
$1-\varepsilon<\theta_{N}(x)\leq 1$ for all $x\in\overline{\Omega}\setminus
C_{\varepsilon}\subset\Omega$. Therefore we can apply the local almost-
Euclidean isoperimetric inequality given by Theorem 2.24 (recall also Lemma
2.11) and obtain that for every $x\in\overline{\Omega}\setminus
C_{\varepsilon}$ and every $\varepsilon\in(0,1/4)$ there exists
$\rho=\rho(x,N,\varepsilon)<\tilde{\sf d}(x,\Omega^{c})$ such that

${\rm{Per}}_{\rm
X}(E)\geq\mathfrak{m}(E)^{\frac{N-1}{N}}N\omega_{N}^{\frac{1}{N}}(1-2\varepsilon)^{\frac{1}{N}}(1-\varepsilon),\quad\forall\,E\subset
B_{\rho}(x)=B^{\rm X}_{\rho}(x)\text{ Borel},$

which shows (3.1). It remains to show (iii). Denote by $\lambda_{k}$,
$k\in\mathbb{N}$ the spectrum of the Laplacian in $({\rm X},{\sf
d},\mathfrak{m})$ (counted with multiplicity and in non-decreasing order).
Recall that by Corollary 3.10 $\lambda_{k}=\lambda_{k}^{\mathcal{N}}(\Omega)$
for every $k\in\mathbb{N}$, where $\lambda_{k}^{\mathcal{N}}(\Omega)$ is the
$k$-th Neumann eigenvalue of $\Omega$ (in non-decreasing order). Recalling
Lemma 2.8 we know that
$\lambda_{k}^{\mathcal{N}}(\Omega)\leq\lambda_{k}^{\mathcal{D}}(\Omega)$ for
every $k\in\mathbb{N}$, where $\lambda_{k}^{\mathcal{D}}(\Omega)$ is the
$k$-th Dirichlet Laplacian eigenvalue of $\Omega.$ Then (3.2) follows from the
Weyl law for the Dirichlet Laplacian (see Theorem 2.27):

$\varlimsup_{k\to+\infty}\frac{\lambda_{k}^{N/2}}{k}=\varlimsup_{k\to+\infty}\frac{(\lambda_{k}^{\mathcal{N}}(\Omega))^{N/2}}{k}\leq\lim_{k\to+\infty}\frac{(\lambda_{k}^{\mathcal{D}}(\Omega))^{N/2}}{k}=\frac{(2\pi)^{N}}{\omega_{n}\mathcal{H}^{N}(\Omega)}=\frac{(2\pi)^{N}}{\omega_{n}\mathfrak{m}({\rm
X})},$

having used again $\mathfrak{m}(\partial\Omega)=0.$ ∎

##  4 From local to global isoperimetric inequality

In this section we prove the following crucial result. Informally speaking, it
says that in a PI space satisfying an almost-Euclidean isoperimetric
inequality around almost-every point, the same isoperimetric inequality
extends to all sets having sufficiently small volume and avoiding a ‘bad’ but
small region of the space.

######  Theorem 4.1.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a bounded PI space and fix $N>1$.
Suppose that for every $\varepsilon>0$ there exists a closed set
$C_{\varepsilon}\subset{\rm X}$ with $\mathfrak{m}(C_{\varepsilon})=0$ such
that for every $x\in{\rm X}\setminus C_{\varepsilon}$ there exists a constant
$\rho=\rho(x,N,\varepsilon)>0$ satisfying

${\rm{Per}}(E)\geq(1-\varepsilon)N\omega_{N}^{\frac{1}{N}}\mathfrak{m}(E)^{\frac{N-1}{N}},\quad\forall\,E\subset
B_{\rho}(x)\,\text{ Borel.}$ (4.1)

Then for every $\varepsilon\in(0,1)$ and $\eta>0$ there exists an open set
$U_{\varepsilon,\eta}\subset{\rm X}$ with
$\mathfrak{m}(U_{\varepsilon,\eta})<\eta$ and constants $\beta=\beta({\rm
X},\varepsilon,N,\eta)>0$, $\beta^{\prime}=\beta^{\prime}(\varepsilon)>0$ such
that

${\rm{Per}}(E)\geq(1-\varepsilon)N\omega_{N}^{\frac{1}{N}}\mathfrak{m}(E)^{\frac{N-1}{N}},$
(4.2)

for every $E\subset{\rm X}$ Borel satisfying

$0<\mathfrak{m}(E)\leq\beta,\quad\frac{\mathfrak{m}(E\cap
U_{\varepsilon,\eta})}{\mathfrak{m}(E)}\leq\beta^{\prime}.$

Observe that assumption (4.1) is the same as item $ii)$ in Theorem 3.2.

The proof of Theorem 4.1 takes inspiration from the arguments in Appendix $C$
of [17] in the smooth setting, but requires also to deal with the technical
issues arising from working in a non-smooth metric space.

We start with an estimate for the perimeter of the complement of the union of
a finite number of balls.

######  Lemma 4.2.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a PI space. Suppose that
$B_{i}\coloneqq B_{r_{i}}(x_{i})\subset{\rm X}$, $i=1,...,k$,
$k\in\mathbb{N},$ have all finite perimeter and satisfy
${\rm{Per}}(B_{i},\partial^{e}B_{j})=0$ for all $i\neq j$. Then

${\rm{Per}}(B_{1}^{c}\cap B_{2}^{c}\cap...\cap
B_{k}^{c},.)\leq\sum_{i=1}^{k}{\rm{Per}}(B_{i},.)\lower
3.0pt\hbox{$|_{B_{1}^{c}\cap B_{2}^{c}\cap...\cap B_{k}^{c}}$}.$ (4.3)

###### Proof.

We argue by induction on $k$. By (2.22) we have that for every ball
$B\subset{\rm X}$ of finite perimeter

${\rm{Per}}(B^{c},.)={\rm{Per}}(B,.)={\rm{Per}}(B,.)\lower
3.0pt\hbox{$|_{B^{c}}$},$

which shows that the statement holds for $k=1$. Suppose that the statement is
true for some $k\in\mathbb{N}$ and let $B_{i}\coloneqq
B_{r_{i}}(x_{i})\subset{\rm X}$, $i=1,...,k+1$ be as in the statement. By a
repeated application of $(vi)$ in Lemma 2.10 we get

$\partial^{e}(B_{1}^{c}\cap B_{2}^{c}\cap...\cap
B_{k}^{c})\subset\partial^{e}B_{i}\cup...\cup\partial^{e}B_{k}.$ (4.4)

Using (4.4) and the assumption ${\rm{Per}}(B_{k+1},\partial^{e}B_{j})=0$ for
all $j\neq k+1$ gives

${\rm{Per}}(B_{k+1},\partial^{e}(B_{1}^{c}\cap B_{2}^{c}\cap...\cap
B_{k}^{c}))=0.$

Hence we can apply Proposition 2.19 and obtain

${\rm{Per}}(B_{1}^{c}\cap B_{2}^{c}\cap...\cap
B_{k+1}^{c},.)\leq{\rm{Per}}(B_{k+1},.)\lower 3.0pt\hbox{$|_{(B_{1}^{c}\cap
B_{2}^{c}\cap...\cap B_{k}^{c})^{(1)}}$}+{\rm{Per}}(B_{1}^{c}\cap
B_{2}^{c}\cap...\cap B_{k}^{c},.)\lower
3.0pt\hbox{$|_{(B_{k+1}^{c})^{(1)}}$}.$ (4.5)

Since balls are open sets, by using $(iii)$ and $(iv)$ in Lemma 2.10 it holds

$(B_{1}^{c}\cap B_{2}^{c}\cap...\cap B_{k}^{c})^{(1)}\subset B_{1}^{c}\cap
B_{2}^{c}\cap...\cap B_{k}^{c},$

that combined with (4.5) and the induction hypothesis gives

$\displaystyle{\rm{Per}}(B_{1}^{c}\cap B_{2}^{c}\cap...\cap B_{k+1}^{c},.)$
$\displaystyle\leq{\rm{Per}}(B_{k+1},.)\lower 3.0pt\hbox{$|_{B_{1}^{c}\cap
B_{2}^{c}\cap...\cap B_{k}^{c}}$}+{\rm{Per}}(B_{1}^{c}\cap
B_{2}^{c}\cap...\cap B_{k}^{c},.)\lower 3.0pt\hbox{$|_{(B_{k+1}^{c})^{(1)}}$}$
$\displaystyle\leq{\rm{Per}}(B_{k+1},.)\lower 3.0pt\hbox{$|_{B_{1}^{c}\cap
B_{2}^{c}\cap...\cap B_{k}^{c}}$}+{\rm{Per}}(B_{1}^{c}\cap
B_{2}^{c}\cap...\cap B_{k}^{c},.)\lower 3.0pt\hbox{$|_{B_{k+1}^{c}}$}$
$\displaystyle\leq{\rm{Per}}(B_{k+1},.)\lower 3.0pt\hbox{$|_{B_{1}^{c}\cap
B_{2}^{c}\cap...\cap B_{k}^{c}\cap
B_{k+1}^{c}}$}+\sum_{i=1}^{k}{\rm{Per}}(B_{i},.)\lower
3.0pt\hbox{$|_{B_{1}^{c}\cap B_{2}^{c}\cap...\cap B_{k}^{c}\cap
B_{k+1}^{c}}$},$

where in the second line we used that $(B_{k+1}^{c})^{(1)}\subset B_{k+1}^{c}$
(recall $(iv)$ in Lemma 2.10) and in the last line that
${\rm{Per}}(B_{k+1},.)={\rm{Per}}(B_{k+1},.)\lower
3.0pt\hbox{$|_{B_{k+1}^{c}}$}$ (recall (2.22)) for the first term and the
induction hypothesis for the second term. This concludes the proof. ∎

Combining the above estimate with a covering argument we can prove the
following proposition, from which Theorem 4.1 will easily follow.

######  Proposition 4.3 (From local-to-global isoperimetric inequality).

Let $({\rm X},{\sf d},\mathfrak{m})$ be a PI space. Suppose there exist
constants $\lambda>0$, $\alpha\in(0,1]$ and a compact set ${\sf K}\subset{\rm
X}$ such that for all $x\in{\sf K}$ there exists $\rho(x)>0$ so that

${\rm{Per}}(E)\geq\lambda\mathfrak{m}(E)^{\alpha},\quad\forall\,E\subset
B_{\rho(x)}(x)\,\text{ Borel.}$ (4.6)

Then there exists a constant $C=C({\sf K},\alpha,\lambda)$ such that

${\rm{Per}}(V)\geq\lambda\mathfrak{m}(V\cap{\sf
K})^{\alpha}-C\mathfrak{m}(V),\quad\forall\,V\subset{\rm X}\text{ Borel.}$
(4.7)

###### Proof.

We start by extracting once and for all a finite covering ${\sf
K}\subset\cup_{i=1}^{M}B_{\frac{\rho(x_{i})}{2}}(x_{i})$, with $x_{i}\in{\sf
K}$, and we set $\overline{\rho}\coloneqq\min_{i}\rho(x_{i})>0$. It is enough
to prove (4.7) for sets $V$ of finite perimeter. Fix one such set $V.$

We claim that there exist $r_{1},\dots,r_{M}$, with
$r_{i}\in\left(\frac{\rho(x_{i})}{2},\rho(x_{i})\right)$ such that the
following hold:

  1. a)

$B_{r_{i}}(x_{i})$ has finite perimeter, for every $i=1,...,M,$

  2. b)

${\rm{Per}}(V,\partial B_{r_{i}}(x_{i}))=0$, for every $i=1,...,M,$

  3. c)

${\rm{Per}}(B_{r_{i}}(x_{i}),V^{(1)})\leq\frac{3\mathfrak{m}(V)}{\rho(x_{i})}$,
for every $i=1,...,M$,

  4. d)

${\rm{Per}}(B_{r_{i}}(x_{i}),\partial B_{r_{j}}(x_{j}))=0$, for every
$i,j=1,...,M$ with $i\neq j.$

It is sufficient to prove that:

  1. (i)

for any $i\in\\{1,...,M\\}$, there exists
$A_{i}\subset\left(\frac{\rho(x_{i})}{2},\rho(x_{i})\right)$ with
$\mathcal{H}^{1}(A_{i})>0$ such that a),b) and c) holds for every $r_{i}\in
A_{i}$,

  2. (ii)

for every $r_{i}>0$ such that $B_{r_{i}}(x_{i})$ has finite perimeter, d)
holds for every $j\neq i$ and for a.e. $r_{j}>0$.

Indeed if these were true, up to removing from each $A_{i}$ a set of measure
zero, we would have that every choice $(r_{1},\dots,r_{M})\in
A_{i}\times\dots,\times A_{M}$ satisfies all a),b), c) and d). We start
proving (i). Fix $i\in\\{1,...,M\\}$. From Proposition 2.9 we have that
$B_{r}(x_{i})$ has finite perimeter for a.e. $r>0$ (i.e. a) holds for a.e.
$r_{i}>0$). Moreover since ${\rm{Per}}(V,.)$ is a finite measure and
$\\{\partial B_{r}(x_{i})\\}_{r>0}$ are pairwise disjoint sets, we must have
that ${\rm{Per}}(V,\partial B_{r_{i}}(x_{i}))=0$ for a.e. $r_{i}>0$ (i.e. b)
holds for a.e. $r_{i}>0$). By applying Proposition 2.9 with
$R\coloneqq\rho(x_{i})$ we get

$\int_{\frac{\rho(x_{i})}{2}}^{\rho(x_{i})}{\rm{Per}}(B_{r}(x_{i}),V^{(1)})\,{\mathrm{d}}r\leq\int_{0}^{\rho(x_{i})}{\rm{Per}}(B_{r}(x_{i}),V^{(1)})\,{\mathrm{d}}r\leq\mathfrak{m}(B_{\rho(x_{i})}(x_{i})\cap
V^{(1)})=\mathfrak{m}(B_{\rho(x_{i})}(x_{i})\cap V),$

and by the Markov inequality

$\mathcal{H}^{1}\left(\left\\{r\in\left(\frac{\rho(x_{i})}{2},\rho(x_{i})\right):\
{\rm{Per}}(B_{r}(x_{i}),V^{(1)})>3\mathfrak{m}(V\cap
B_{\rho(x_{i})}(x_{i}))/\rho(x_{i})\right\\}\right)\leq\frac{\rho(x_{i})}{3},$

which shows that the set

$\left\\{r\in\left(\frac{\rho(x_{i})}{2},\rho(x_{i})\right):\
{\rm{Per}}(B_{r}(x_{i}),V^{(1)})\leq 3\mathfrak{m}(V\cap
B_{\rho(x_{i})}(x_{i}))/\rho(x_{i})\right\\}$

has positive $\mathcal{H}^{1}$-measure, i.e. c) holds for every $r_{i}$ in a
subset of $\left(\frac{\rho(x_{i})}{2},\rho(x_{i})\right)$ of positive
$\mathcal{H}^{1}$-measure. Combining all the above observations gives (i). To
show (ii) fix $i,j\in\\{1,...,M\\}$, $i\neq j$ and $r_{i}>0$ such that
$B_{r_{i}}(x_{i})$ has finite perimeter. As above $\\{\partial
B_{r}(x_{j})\\}_{r>0}$ are pairwise disjoint sets, therefore we must have
${\rm{Per}}(B_{r_{i}}(x_{i}),\partial B_{r}(x_{j}))=0$ for a.e. $r>0.$ This
proves (ii) and completes the proof of the claim.

From now on we assume to have fixed $r_{1},\dots r_{M}$ such that a), b), c)
and d) above hold (note that this choice might depend on the set $V$) and we
set $B_{i}\coloneqq B_{r_{i}}(x_{i}).$ By construction
$K\subset\cup_{i=1}^{M}B_{i}.$ Consider the pairwise disjoints sets
$\\{U_{i}\\}_{i=1}^{M}$ defined inductively as follows:

$U_{1}\coloneqq B_{1},\quad U_{i}\coloneqq B_{i}\cap(B_{i-1}^{c}\cap...\cap
B_{1}^{c}),\quad\forall\,i=2,...,M.$

Clearly $\\{U_{i}\\}_{i=1}^{M}$ is a family of disjoint Borel sets which is a
covering of $K$. We claim that

${\rm{Per}}(V,\partial^{e}U_{i})=0,\quad\forall\,i=1,...,M.$ (4.8)

Indeed, from $(vi)$ and $(ii)$ of Lemma 2.10 one infers that

$\partial^{e}U_{i}\subset\partial^{e}B_{1}\cup\partial^{e}(B_{2}^{c})\cup...\cup\partial^{e}(B_{i-1}^{c})\subset\partial
B_{1}\cup...\cup\partial B_{i-1}.$

From this (4.8) follows recalling b). Thanks to (4.8) we are in position to
apply Proposition 2.19 to deduce that

${\rm{Per}}(V\cap
U_{i})\leq{\rm{Per}}(V,U_{i}^{(1)})+{\rm{Per}}(U_{i},V^{(1)}),\quad\forall\,i=1,...,M.$
(4.9)

The goal is now to give an upper bound on each term on the right-hand side of
(4.9). Since $U_{i}$ are pairwise disjoint by construction, by point $(v)$ of
Lemma 2.10 it follows that also the sets $U_{i}^{(1)}$ are pairwise disjoint,
hence

$\sum_{i=1}^{M}{\rm{Per}}(V,U_{i}^{(1)})\leq{\rm{Per}}(V).$ (4.10)

To estimate ${\rm{Per}}(U_{i},V^{(1)})$ we note that from d) it holds

${\rm{Per}}(B_{i},\partial^{e}(B_{1}^{c}\cap...\cap B_{i-1}^{c}))=0,$

indeed $\partial^{e}(B_{1}^{c}\cap...\cap B_{i-1}^{c})\subset\partial
B_{1}\cup...\cup\partial B_{i-1}$ (recall $(ii)$ in Lemma 2.10). Hence,
recalling that by construction $U_{i}=B_{i}\cap(B_{i-1}^{c}\cap...\cap
B_{1}^{c})$, we can apply again Proposition 2.19 to get

${\rm{Per}}(U_{i},V^{(1)})\leq{\rm{Per}}(B_{i},V^{(1)})+{\rm{Per}}(B_{1}^{c}\cap...\cap
B_{i-1}^{c},V^{(1)})\lower 3.0pt\hbox{$|_{B_{i}^{(1)}}$}.$

From this and Lemma 4.2

$\displaystyle{\rm{Per}}(U_{i},V^{(1)})$
$\displaystyle\overset{\eqref{eq:ballsper}}{\leq}{\rm{Per}}(B_{i},V^{(1)})+\sum_{j=1}^{i-1}{\rm{Per}}(B_{j},V^{(1)})\lower
3.0pt\hbox{$|_{B_{i}^{(1)}\cap B_{1}^{c}\cap...\cap B_{i-1}^{c}}$}$
$\displaystyle\leq{\rm{Per}}(B_{i},V^{(1)})+\sum_{j=1}^{i-1}{\rm{Per}}(B_{j},V^{(1)})\lower
3.0pt\hbox{$|_{B_{i}\cap B_{1}^{c}\cap...\cap B_{i-1}^{c}}$}$
$\displaystyle={\rm{Per}}(B_{i},V^{(1)})+\sum_{j=1}^{i-1}{\rm{Per}}(B_{j},V^{(1)})\lower
3.0pt\hbox{$|_{U_{i}}$},$

where in the second line we used that $B_{i}^{(1)}\subset B_{i}\cup\partial
B_{i}$ and that ${\rm{Per}}(B_{j},\partial B_{i})=0$ for all $j\neq i$, thanks
to d). Summing in $i$ and recalling that $U_{i}$ are disjoint

$\begin{split}\sum_{i=1}^{M}{\rm{Per}}(U_{i},V^{(1)})&\leq\sum_{i=1}^{M}{\rm{Per}}(B_{i},V^{(1)})+\sum_{i=1}^{M}\sum_{j=1}^{i-1}{\rm{Per}}(B_{j},V^{(1)})\lower
3.0pt\hbox{$|_{U_{i}}$}\\\
&\leq\sum_{i=1}^{M}{\rm{Per}}(B_{i},V^{(1)})+\sum_{j=1}^{M}\sum_{i=1}^{M}{\rm{Per}}(B_{j},V^{(1)})\lower
3.0pt\hbox{$|_{U_{i}}$}\\\ &\leq
2\sum_{i=1}^{M}{\rm{Per}}(B_{i},V^{(1)})\overset{\ref{it:good
balls2}}{\leq}6\sum_{i=1}^{M}\frac{\mathfrak{m}(V)}{\rho(x_{i})}\leq
6M\frac{\mathfrak{m}(V)}{\bar{\rho}}.\end{split}$ (4.11)

Combining (4.11), (4.10) and (4.9) we get

$\sum_{i=1}^{M}{\rm{Per}}(V\cap
U_{i})\leq{\rm{Per}}(V)+6M\frac{\mathfrak{m}(V)}{\bar{\rho}}.$ (4.12)

On the other hand $V\cap U_{i}\subset B_{i}\subset B_{\rho(x_{i})}(x_{i})$,
hence by assumption (4.6)

$\sum_{i=1}^{M}{\rm{Per}}(V\cap
U_{i})\geq\lambda\sum_{i=1}^{M}\mathfrak{m}(V\cap
U_{i})^{\alpha}\geq\lambda\left(\sum_{i=1}^{M}\mathfrak{m}(V\cap
U_{i})\right)^{\alpha}\geq\lambda\mathfrak{m}(V\cap{\sf K})^{\alpha},$

since the function $x\mapsto x^{\alpha}$ is subadditive and the sets
$\\{U_{i}\\}_{i=1}^{M}$ cover ${\sf K}$. This combined with (4.12) yields

${\rm{Per}}(V)\geq\lambda\mathfrak{m}(V\cap{\sf
K})^{\alpha}-6M\frac{\mathfrak{m}(V)}{\bar{\rho}}.$

The constants $M$ and $\bar{\rho}$ depend only on the initial choice of the
covering $B_{\frac{\rho(x_{i})}{2}}(x_{i})$ and thus depend only on $K,$
$\alpha$ and $\lambda,$ (and not on $V$). This concludes the proof. ∎

An application of Proposition 4.3 yields immediately the main result of this
section.

###### Proof of Theorem 4.1.

Fix $\varepsilon\in(0,1)$ and $\eta>0.$ Let $C_{\varepsilon/2}\subset{\rm X}$
be as in the statement. Since $C_{\varepsilon/2}$ is closed, by upper
regularity there exists an open set $U_{\varepsilon,\eta}$ containing
$C_{\varepsilon/2}$ and such that $\mathfrak{m}(U_{\varepsilon,\eta})<\eta.$
Set ${\sf K}_{\varepsilon,\eta}\coloneqq{\rm X}\setminus
U_{\varepsilon,\eta}$, which is compact (because it is closed and bounded and
${\rm X}$ is a proper, being a PI space). In particular (4.1) holds also for
every $x\in K_{\varepsilon,\eta}$. Therefore the hypotheses of Proposition 4.3
are satisfied with $K=K_{\varepsilon,\eta}$, $\alpha=\frac{N-1}{N}$ and
$\lambda=(1-\varepsilon)N\omega_{N}^{\frac{1}{N}}$ and we deduce that there
exists a constant $C$ depending only on $K_{\varepsilon,\eta},N$ and
$\varepsilon$ (and thus only on ${\rm X},$ $\varepsilon$, $N$ and $\eta$) such
that

$\displaystyle{\rm{Per}}(E)$
$\displaystyle\geq(1-\varepsilon/2)N\omega_{N}^{\frac{1}{N}}\mathfrak{m}(E\cap{\sf
K_{\varepsilon,\eta}})^{\frac{N-1}{N}}-C\mathfrak{m}(E)$
$\displaystyle=(1-\varepsilon/2)N\omega_{N}^{\frac{1}{N}}\mathfrak{m}(E)^{\frac{N-1}{N}}\left[\left(1-\frac{\mathfrak{m}(E\cap
U_{\varepsilon,\eta})}{\mathfrak{m}(E)}\right)^{\frac{N-1}{N}}-\hat{C}\mathfrak{m}(E)^{\frac{1}{N}}\right]$
$\displaystyle\geq(1-\varepsilon/2)N\omega_{N}^{\frac{1}{N}}\mathfrak{m}(E)^{\frac{N-1}{N}}\left[1-\frac{\mathfrak{m}(E\cap
U_{\varepsilon,\eta})}{\mathfrak{m}(E)}-\hat{C}\mathfrak{m}(E)^{\frac{1}{N}}\right],\quad\forall
E\subset{\rm X}\text{ Borel,}$

where $\hat{C}=C((1-\varepsilon/2)N\omega_{N}^{\frac{1}{N}})^{-1}$. From this
we obtain that the conclusion of theorem holds taking
$\beta\coloneqq\hat{C}^{-N}\delta_{\varepsilon}^{N}$ and
$\beta^{\prime}\coloneqq\delta_{\varepsilon},$ where
$\delta_{\varepsilon}\coloneqq\frac{1}{2}\frac{\varepsilon}{2-\varepsilon}.$ ∎

##  5 Almost Euclidean Faber-Krahn inequality for small volumes

Similarly to the classical Faber-Krahn inequality in $\mathbb{R}^{N}$,
combining the almost Euclidean isoperimetric inequality for small volumes in
Section 4 and the Pólya-Szegő inequality of Section 2.4, we deduce here an
almost-Euclidean Faber-Krahn inequality for small volumes similar to [17,
Lemme 16] in the case of Riemannian manifolds. However it is not possible to
apply directly the Pólya-Szegő inequality as in [17], because our
isoperimetric inequality applies only to sets that have small volumes _and
avoid_ a bad set with small measure. This technical difficulty will require a
more careful argument, which will eventually lead to a Faber-Krahn inequality
that applies only to sets that again avoid a portion of the space with small
measure (see Theorem 5.3).

We start by recalling the well known expression of the first Dirichlet
eigenvalue of a ball in the $N$-dimensional Euclidean space (see e.g. [17]):

$\lambda_{1}(B_{r}^{\mathbb{R}^{N}}(0))=\left(\frac{\omega_{N}}{{\mathscr{L}}^{N}(B_{r}^{\mathbb{R}^{N}}(0))}\right)^{2/N}j^{2}_{\frac{(N-2)}{N}},\qquad\forall\,r>0,$
(5.1)

where $j_{\frac{(N-2)}{N}}$ denotes the first positive zero of the Bessel
function (of the first kind) of index $\frac{(N-2)}{N}$.

Next we obtain a weaker Faber-Krahn inequality, i.e. with a rough constant,
but that applies to sets also with large volume.

######  Proposition 5.1 (Faber-Krahn inequality in PI spaces).

Let $({\rm X},{\sf d},\mathfrak{m})$ be a bounded PI space satisfying for some
$N\in\mathbb{N}$ and some constant $c>0$

$\frac{\mathfrak{m}(B_{r}(x))}{\mathfrak{m}(B_{R}(x))}\geq
c\left(\frac{r}{R}\right)^{N},\quad\forall\,x\in{\rm X},\,\forall\,0<r<R.$

Then there exist constants $v_{0}=v_{0}({\rm X})>0$ and $C=C({\rm X},N)>0$
such that

$\lambda_{1}(\Omega)\geq\frac{C}{\mathfrak{m}(\Omega)^{\frac{2}{N}}},\quad\forall\,\Omega\subset{\rm
X}\text{ open, }0<\mathfrak{m}(\Omega)\leq v_{0}.$ (5.2)

###### Proof.

By Proposition 2.18 there exist constants $w_{0}=w_{0}({\rm X})>0$ and
$C_{I}=C_{I}({\rm X},N)>0$ such that

${\rm{Per}}(E)\geq
C_{I}\mathfrak{m}(E)^{\frac{N-1}{N}},\qquad\forall\,E\subset{\rm X}\text{
Borel such that $\mathfrak{m}(E)\leq w_{0}$}$ (5.3)

In particular if $\mathfrak{m}(\Omega)\leq v_{0}\coloneqq w_{0}$, then (5.3)
holds for every $E\subset\Omega$ Borel. Moreover taking
$v_{0}<\mathfrak{m}({\rm X})$ we have $\Omega\neq{\rm X}.$ Therefore we can
apply the Pólya-Szegő inequality (2.28) and deduce

$\frac{\int_{\Omega}|{Du}|^{2}{\mathrm{d}}\mathfrak{m}}{\int_{\Omega}|u|^{2}{\mathrm{d}}\mathfrak{m}}\geq\Big{(}\frac{C_{I}}{N\omega_{N}^{1/N}}\Big{)}^{2}\frac{\int_{\Omega^{*}}|{Du_{N}^{*}}|^{2}{\mathrm{d}}{\mathscr{L}}^{N}}{\int_{\Omega^{*}}|u_{N}^{*}|^{2}{\mathrm{d}}{\mathscr{L}}^{N}}\geq\frac{C}{\mathfrak{m}(\Omega)^{\frac{2}{N}}},\quad\forall
u\in\mathit{W}^{1,2}_{0}(\Omega),\,u\geq 0,\,u\not\equiv 0,$ (5.4)

where we have also used that
$\|u\|_{L^{2}(\Omega,\mathfrak{m})}=\|u_{N}^{*}\|_{L^{2}(\Omega^{*},{\mathscr{L}}^{N})}$,
$\mathfrak{m}(\Omega)={\mathscr{L}}^{N}(\Omega^{*})$, and in the last passage
the identity (5.1) and the fact that
$u_{N}^{*}\in\mathit{W}^{1,2}_{0}(\Omega^{*})$. By taking the infimum with
respect to all the possible $u$ in the characterization (2.16), we get the
result. ∎

From the previous proposition we can deduce the following version of the
Faber-Krahn in the Euclidean setting. Even if it will be not used in this
note, we think it is worth to be isolated in a separate statement. Indeed it
has been pointed out repeatedly in the previous literature that one of the
major difficulties in counting nodal domains for non-Dirichlet boundary
condition in subset of $\mathbb{R}^{N}$ with irregular boundary is the absence
of a suitable Faber-Krahn inequality for subdomains close to the boundary (see
e.g. discussions in [56, Section 2], [69, Section 1.2], [53, Section 1.2],
[15, Section 1.1]). This was one of the main issues faced [69] which also
forced the assumption of a $C^{1,1}$ boundary (see also [56]). Here we show
precisely that a Faber-Krahn-type inequality does hold in any uniform domain,
no matter how close is the support of the function to the boundary.

######  Corollary 5.2 (Faber-Krahn inequality for uniform domains).

Let $\Omega\subset\mathbb{R}^{N}$ be a uniform domain. Then there exist
constants $v_{0}\in(0,{\mathscr{L}}^{N}(\Omega))$ and $C>0$, depending only on
$\Omega,$ such that

$\frac{\int_{\Omega}|\nabla
u|^{2}{\mathrm{d}}{\mathscr{L}}^{N}}{\int_{\Omega}u^{2}{\mathrm{d}}{\mathscr{L}}^{N}}\geq\frac{C}{({\mathscr{L}}^{N}(\mathop{\rm
supp}\nolimits(u)))^{\frac{2}{N}}},\quad\forall
u\in\mathit{W}^{1,2}(\Omega),u\not\equiv 0,\ \text{such that
}{\mathscr{L}}^{N}(\mathop{\rm supp}\nolimits(u))\leq v_{0}.$

###### Proof.

Let $u\in\mathit{W}^{1,2}(\Omega)$. Then by Theorem 3.8 there exists
$\tilde{u}\in L^{2}(\overline{\Omega},{\mathscr{L}}^{N})$ such that
$\tilde{u}=u$ ${\mathscr{L}}^{N}$-a.e. in $\Omega$ and such that
$\tilde{u}\in\mathit{W}^{1,2}(\overline{\Omega},{\sf d}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$},{\mathscr{L}}^{N}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})$ where ${\sf d}$ denotes the Euclidean
distance (recall also Remark 2.7). Moreover by Theorem 3.2 the m.m.s. $({\rm
X},{\sf d},\mathfrak{m})\coloneqq(\overline{\Omega},{\sf d}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$},{\mathscr{L}}^{N}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})$ satisfies the hypotheses of Proposition
5.1. Finally by Lemma 3.6 we have ${\mathscr{L}}^{N}(\partial\Omega)=0.$ Let
$v_{0}<\mathfrak{m}({\rm X})={\mathscr{L}}^{N}(\Omega)$ and $C>0$ be the
constants given by Proposition 5.1, which depend only on $\Omega.$ Consider
the open set $U_{\varepsilon}\coloneqq(\mathop{\rm
supp}\nolimits(\tilde{u}))^{\varepsilon}$, $\varepsilon>0$. Then
${\mathscr{L}}^{N}(U_{\varepsilon})\to{\mathscr{L}}^{N}(\mathop{\rm
supp}\nolimits(\tilde{u}))={\mathscr{L}}^{N}(\mathop{\rm supp}\nolimits(u))$
as $\varepsilon\to 0,$ having also used that
${\mathscr{L}}^{N}(\partial\Omega)=0.$ Hence assuming that
${\mathscr{L}}^{N}(\mathop{\rm supp}\nolimits(u))<v_{0}({\rm X})$ we have that
$\mathfrak{m}(U_{\varepsilon})\leq v_{0}$ for $\varepsilon$ small enough.
Clearly $\tilde{u}\in\mathit{W}^{1,2}_{0}(U_{\varepsilon})$ (recall (2.9)),
hence we can apply (5.2) and obtain

$\frac{\int_{\overline{\Omega}}|\nabla\tilde{u}|^{2}{\mathrm{d}}{\mathscr{L}}^{N}}{\int_{\overline{\Omega}}\tilde{u}^{2}{\mathrm{d}}{\mathscr{L}}^{N}}\geq\frac{C}{\mathfrak{m}(U_{\varepsilon})^{\frac{2}{n}}}.$

Letting $\varepsilon\to 0$ and recalling that
$\int_{\overline{\Omega}}|\nabla\tilde{u}|^{2}{\mathrm{d}}{\mathscr{L}}^{N}=\int_{\Omega}|\nabla
u|^{2}{\mathrm{d}}{\mathscr{L}}^{N}$ concludes the proof. ∎

We pass to the statement of our main Faber-Krahn inequality for small volumes.

######  Theorem 5.3 (Almost Euclidean Faber-Krahn inequality for small
volumes).

Let $({\rm X},{\sf d},\mathfrak{m})$ be a bounded PI space satisfying for some
$N\in\mathbb{N}$ and some constant $c>0$

$\frac{\mathfrak{m}(B_{r}(x))}{\mathfrak{m}(B_{R}(x))}\geq
c\left(\frac{r}{R}\right)^{N},\quad\forall\,x\in{\rm X},\,\forall 0<r<R.$

Suppose that for every $\varepsilon>0$ there exists a closed set
$C_{\varepsilon}\subset{\rm X}$ with $\mathfrak{m}(C_{\varepsilon})=0$ such
that for every $x\in{\rm X}\setminus C_{\varepsilon}$ there exists a constant
$\rho=\rho(x,N,\varepsilon)>0$ satisfying

${\rm{Per}}(E)\geq(1-\varepsilon)N\omega_{N}^{\frac{1}{N}}\mathfrak{m}(E)^{\frac{N-1}{N}},\quad\forall\,E\subset
B_{\rho}(x)\,\text{ Borel,}$ (5.5)

Then for every $\varepsilon\in(0,1)$ and $\eta>0$ there exists an open set
$U_{\varepsilon,\eta}$ with $\mathfrak{m}(U_{\varepsilon,\eta})<\eta$ and
constants $\delta=\delta({\rm X},\varepsilon,N,\eta)>0$,
$\delta^{\prime}=\delta^{\prime}({\rm X},\varepsilon,N)>0$ such that for every
$\Omega\subset{\rm X}$ open satisfying

$\mathfrak{m}(\Omega)\leq\delta,\quad\frac{\mathfrak{m}(\Omega\cap
U_{\varepsilon,\eta})}{\mathfrak{m}(\Omega)}\leq\delta^{\prime},$

and, denoted by $\Omega^{*}\coloneqq B_{r}(0)\subset\mathbb{R}^{N}$ the ball
satisfying $\mathfrak{m}(\Omega)={\mathscr{L}}^{N}(\Omega^{*})$, it holds

$\lambda_{1}(\Omega)\geq(1-\varepsilon){\lambda_{1}}(\Omega^{*}).$ (5.6)

The key point of Theorem 5.3 is that the constant $\delta^{\prime}$ _does not
depend_ on $\eta$. This will be crucial in the proof of the main result, which
will be done in Section 6. Indeed we will eventually need to get rid of
$\delta^{\prime}$ by sending $\eta\to 0$ (see in particular (6.8)).

######  Remark 5.4.

Thanks to Theorem 3.2 we know that, given $({\rm Y},\tilde{\sf
d},\mathcal{H}^{N})$ an $\mathrm{RCD}(K,N)$ space and $\Omega\subset{\rm Y}$ a
uniform domain, the metric measure space $({\rm X},{\sf
d},\mathfrak{m})\coloneqq(\overline{\Omega},\tilde{\sf d}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$},\mathcal{H}^{N}\lower
3.0pt\hbox{$|_{\overline{\Omega}}$})$ satisfies the hypotheses of the above
Theorem 5.3. This will actually be the way in which we will apply this result
in the sequel.  $\blacksquare$

In the proof of Theorem 5.3 we will make use of the following elementary
observation.

######  Lemma 5.5.

Let $({\rm X},{\sf d},\mathfrak{m})$ be a m.m.s. and
$f\in\mathsf{LIP}_{c}(X)$, $f\not\equiv 0$, $f\geq 0$. It holds

$\frac{\int_{X}|Df|^{2}\,{\mathrm{d}}\,\mathfrak{m}}{\int_{X}|f|^{2}\,{\mathrm{d}}\mathfrak{m}}\geq\lambda_{1}(\\{f>0\\}),$

where $\\{f>0\\}\coloneqq\\{x\in{\rm X}\colon f(x)>0\\}$.

###### Proof.

Let $\psi_{n}\coloneqq(f-\frac{1}{n})^{+}$. Then
$\psi_{n}\in\mathsf{LIP}_{c}(\\{f>0\\})$ and for $n\in\mathbb{N}$ sufficiently
large, $\psi_{n}\not\equiv 0$. Then for $n$ large $\psi_{n}$ is a competitor
in (2.15), so

$\lambda_{1}(\\{f>0\\})\leq\varliminf_{n\to+\infty}\frac{\int_{X}|D\psi_{n}|^{2}\,{\mathrm{d}}\mathfrak{m}}{\int_{X}|\psi_{n}|^{2}\,{\mathrm{d}}\mathfrak{m}}=\varliminf_{n\to+\infty}\frac{\int_{\\{f\geq
1/n\\}}|Df|^{2}\,{\mathrm{d}}\mathfrak{m}}{\int_{\\{f\geq
1/n\\}}|f-1/n|^{2}\,{\mathrm{d}}\mathfrak{m}}=\frac{\int_{X}|Df|^{2}\,{\mathrm{d}}\mathfrak{m}}{\int_{X}|f|^{2}\,{\mathrm{d}}\mathfrak{m}},$

where the first equality follows from the locality of the weak upper gradient.
∎

We are now ready to prove the Faber-Krahn inequality for small volumes.

###### Proof of Theorem 5.3.

Fix $\varepsilon\in(0,1)$ and $\eta>0$. Let $\beta=\beta({\rm