The Cheeger problem in abstract measure spaces

Abstract

We consider nonnegative $$\sigma$$-finite measure spaces coupled with a proper functional $$P$$ that plays the role of a perimeter. We introduce the Cheeger problem in this framework and extend many classical results on the Cheeger constant and on Cheeger sets to this setting, requiring minimal assumptions on the pair measure space perimeter. Throughout the paper, the measure space will never be asked to be metric, at most topological, and this requires the introduction of a suitable notion of Sobolev spaces, induced by the coarea formula with the given perimeter.

1 INTRODUCTION

In the Euclidean framework, the Cheeger constant of a given set $$\Omega \subset \mathbb {R}^n$$ is defined as

$$$$\begin{equation*} h(\mathrm{\Omega})=\inf\left\{\frac{P(E)}{{\mathcal{L}}^{n}(E)}:E\subset \mathrm{\Omega},{\mathcal{L}}^{n}(E)>0\right\}, \end{equation*}$$$$

where $$\mathcal {L}^n(E)$$ and $$P(E)$$, respectively, denote the $$n$$-dimensional Lebesgue measure of $$E$$ and the variational perimeter of $$E$$. The constant was first introduced by Jeff Cheeger in a Riemannian setting as a way to bound from below the first eigenvalue of the Laplace–Beltrami operator [62]. The argument proposed is sound and robust, as noticed even earlier by Maz'ya [102, 103] (an English translation is available in [80]).

In the past decades, the Cheeger constant has been extensively studied in view of its many applications: the lower bounds on the first eigenvalue of the Dirichlet $$p$$-Laplacian operator [88] and the equivalence of such an inequality with Poincaré's one [104] (up to some convexity assumptions); the relation with the torsion problem [32, 33]; the existence of sets with prescribed mean curvature [5, 95]; the existence of graphs with prescribed mean curvature [77, 94]; the reconstruction of noisy images [46, 60, 73, 113]; the minimum flow-maximum cut problem [79, 117]; and its medical applications [16]. In addition, the Cheeger constant has been employed in elastic-plastic models of plate failure [90] and (its Euclidean-weighted variant) has found applications to Bingham fluids [83] and landslide models [84]. Moreover, the Cheeger constant of a square has been recently used to provide an elementary proof of the Prime Number Theorem [17]. For more literature and a general overview of the problem, we refer the interested reader to the two surveys [91, 112].

Because of its numerous applications, several authors have been drawn to the subject and started to investigate the constants and the above-mentioned links with other problems in several frameworks: weighted Euclidean spaces [7, 26, 96, 114]; finite-dimensional Gaussian spaces [51, 86]; anisotropic Euclidean and Riemannian spaces [8, 21, 49, 89]; the fractional perimeter [28] or nonsingular nonlocal perimeter functionals [100]; Carnot groups [108]; $$\mathsf {RCD}$$-spaces [70, 71]; and lately smooth metric-measure spaces [1].

In the settings mentioned above, the proofs mostly follow those available in the usual Euclidean space. In the present paper, we are interested in pinpointing the minimal assumptions needed on the space and on the perimeter functional in order to establish the fundamental properties of Cheeger sets, as well as the links to other problems. In the following, we shall be interested in nonnegative $$\sigma$$-finite measure spaces endowed with a perimeter functional satisfying some suitable assumptions.

Our approach fits into a broader current of research that has gained popularity in the past decade, aiming to study variational problems, well known in the Euclidean setting, in general spaces under the weakest possible assumptions. Quite often, the ambient space is a (metric) measure spaces. For example, such a general point of view has been adopted for the variational mean curvature of a set [19], for shape optimization problems [37], for Anzellotti–Gauss–Green formulas [78], for the total variation flow [34, 35], and, very recently, for the existence of isoperimetric clusters [111].

1.1 Structure of the paper and results

In Section 2, we introduce the basic setting of perimeter $$\sigma$$-finite measure spaces, that is, nonnegative $$\sigma$$-finite measure spaces $$(X, \mathcal {A}, \mathfrak {m})$$ endowed with a proper functional $$P\colon \mathcal {A}\rightarrow [0,+\infty]$$ possibly satisfying suitable properties (P.1)–(P.7) that we shall require from time to time.

A considerable effort goes toward defining $$BV$$ functions in measure spaces, where a metric is not available. Indeed, usually, the perimeter functional is induced by the metric. In our setting, instead, only a perimeter functional is at disposal, so we use it to define $$BV$$ functions by defining the total variation via the coarea formula with the given perimeter.

To properly define Sobolev functions, we need a slightly richer structure, requiring the measure space to be endowed with a topology, and the perimeter functional $$P(\cdot)$$ to arise from a relative (with respect to open sets $$A$$) perimeter functional $$P(\,\cdot \,; A)$$. By using the relative perimeter, we refine the notion of $$BV$$ function by requiring that the variation is a finite measure. This, in turn, allows us to define $$W^{1,1}$$ functions as $$BV$$ functions whose variation is absolutely continuous with respect to the reference measure. Afterward, via relaxation, we can define $$W^{1,p}$$ functions for any $$p\in (1,+\infty)$$. For a more detailed overview, we refer the reader to Section 1.1.3 and Section 1.1.4.

Once the general framework is set, we then shall start to tackle the problem of our interest.

1.1.1 Definition and existence

In Section 3, we define the Cheeger constant of a set $$\Omega$$ in terms of the ratio of the perimeter functional and the measure the space is endowed with. Actually, in a more general vein similar to that of [48], we shall define the $$N$$-Cheeger constant as

$$$$\begin{equation*} {h}_{N}(\mathrm{\Omega})=\inf\left\{\underset{i=1}{\sum ^{N}}\frac{P(\mathcal{E}(i))}{\mathfrak{m}(\mathcal{E}(i))}:\mathcal{E}=\{\mathcal{E}(i)\}_{i=1}^{N}\subset \mathrm{\Omega}\text{ is an }N\text{-cluster}\right\}, \end{equation*}$$$$

where, as usual, an $$N$$-cluster is an $$N$$-tuple of pairwise disjoint subsets of $$\Omega$$, called chambers, each of which with positive finite measure and finite perimeter.

In Theorem 3.6, we provide a general existence result. Unsurprisingly, the key assumptions on the perimeter are the lower semicontinuity and the compactness of its sublevels with respect to the $$L^1$$ norm, besides an isoperimetric-type property that prevents minimizing sequences to converge toward sets with null $$\mathfrak {m}$$-measure.

Further, we provide inequalities between the $$N$$-Cheeger and $$M$$-Cheeger constants and prove some basic properties of $$N$$-Cheeger sets, with a particular attention to the case $$N=1$$.

1.1.2 Link to sets with prescribed mean curvature

In Section 4, we introduce the notion of $$P$$-mean curvature in the spirit of [19]. With this notion at our disposal, we show that any 1-Cheeger set has $$h_1(\Omega)$$ as one of its $$P$$-mean curvatures, see Corollary 4.3. An analogous result holds for the chambers of an $$N$$-cluster minimizing $$h_N(\Omega)$$, see Corollary 4.4.

In Theorem 4.5, we investigate the link between $$h_1(\Omega)$$ and the existence of nontrivial minimizers of the prescribed $$P$$-curvature functional

$$$$\begin{equation*} {\mathcal{J}}_{\kappa}[F]=P(F)-\kappa \mathfrak{m}(F), \end{equation*}$$$$

where $$\kappa$$ is a fixed positive constant, among subsets $$F\subset \Omega$$. Such a functional, again requiring the lower semicontinuity and the $$L^1$$ compactness of sublevel sets of the perimeter, has minimizers. If, additionally, one assumes that the perimeter functional satisfies $$P(\emptyset)=0$$, then $$h_1(\Omega)$$ acts as a threshold for the existence of nontrivial minimizers, that is, for $$\kappa <h_1(\Omega)$$, negligible sets are the only minimizers, while for $$\kappa >h_1(\Omega)$$, nontrivial minimizers exist.

1.1.3 Link to the first eigenvalue of the Dirichlet 1-Laplacian

In the Euclidean space, one defines the first eigenvalue of the Dirichlet 1-Laplacian in a variational way as the infimum

$$$$\begin{equation} {\lambda}_{1,1}(\mathrm{\Omega})=\inf\left\{\frac{\displaystyle \int_{\mathrm{\Omega}}|\nabla u|\mathrm{d}x}{\Vert u{\Vert}_{1}}:u\in {\mathrm{C}}_{c}^{1}(\mathrm{\Omega}),\Vert u{\Vert}_{1}>0\right\}. \end{equation}$$$$(1.1)

In Section 5, we investigate the relation between the 1-Cheeger constant and a suitable reformulation of the constant $$\lambda _{1,1}(\Omega)$$ in our abstract context.

In the Euclidean setting [88] and, actually, in the more general anisotropic central-symmetric Euclidean setting [89], the constant $$\lambda _{1,1}(\Omega)$$ coincides with $$h_1(\Omega)$$, provided that the boundary of the set $$\Omega$$ is sufficiently smooth (e.g., Lipschitz regular). In particular, one can equivalently consider either smooth functions or $$BV$$-regular functions. Moreover, because of the smoothness of the boundary of $$\Omega$$, it holds that $$BV(\Omega)=BV_0(\Omega)$$, where

$$$$\begin{equation*} B{V}_{0}(\mathrm{\Omega})=\left\{u\in \textit{BV}({\mathbb{R}}^{n}):u=0\text{ a.e. on }{\mathbb{R}}^{n}\setminus \mathrm{\Omega}\right\}, \end{equation*}$$$$

see, for example, [48, Rem. 1.1] or [46]. Thus, under some regularity assumptions on $$\Omega$$, one can equivalently restate the problem in (1.1) as

$$$$\begin{equation} {\lambda}_{1,1}(\mathrm{\Omega})=\inf\left\{\frac{\displaystyle \int_{{\mathbb{R}}^{n}}\mathrm{d}|\textit{Du}|}{\Vert u{\Vert}_{1}}:u\in B{V}_{0}({\mathbb{R}}^{n}),\Vert u{\Vert}_{1}>0\right\}. \end{equation}$$$$(1.2)

On a general set $$\Omega$$ in the Euclidean space, the infimum in (1.2) is less than or equal to that in (1.1), since one only has the inclusion $$BV(\Omega)\subset BV_0(\Omega)$$.

In a (possibly nonmetric) perimeter-measure space, the constant $$\lambda _{1,1}(\Omega)$$ has to be suitably defined, since neither a notion of derivative (needed to state (1.1)) nor integration-by-parts formulas (needed to define $$BV$$ functions and thus state (1.2)) are at disposal. To overcome this difficulty, we adopt the usual point of view [59, 122, 123] and define the total variation of a function via the (generalized) coarea formula

$$$$\begin{equation} \operatorname{Var}(u) = \displaystyle \int _\mathbb {R}P(\lbrace u>t\rbrace)\,\mathrm{d}t, \end{equation}$$$$(1.3)

provided that the function $$t\mapsto P(\lbrace u>t\rbrace)$$ is $$\mathcal L^1$$-measurable, and define the relevant $$BV$$ space as that of those $$L^1$$ functions with finite total variation. For more details, we refer the reader to our Section 2.2.

This approach allows us to consider problem (1.2) without any underlying metric structure. In addition, no regularity of the set $$\Omega$$ is required, since there is no need for the problem (1.2) to be equivalent to its regular counterpart (1.1) that, in the present abstract framework, cannot be even formally stated.

With this notion of total variation at hand, we prove that the constant $$\lambda _{1,1}(\Omega)$$ coincides with the 1-Cheeger constant $$h_1(\Omega)$$ under minimal assumptions on the perimeter, that is, we require that the perimeter of negligible sets and of the whole space is zero, the perimeter is lower semicontinuous with respect to the $$L^1$$ norm, and that the perimeter of a set coincides with that of its complement set, see Theorem 5.4. Moreover, we prove some inequalities relating the $$N$$-Cheeger constant $$h_N(\Omega)$$ with a cluster counterpart of (1.2). As observed in Remark 5.9, if one slightly modifies the definition of $$\lambda _{1,1}(\Omega)$$ by considering nonnegative functions as the only competitors, then one can obtain the relation with the Cheeger constant even for perimeter functionals that are not symmetric with respect to the complement-set operation.

1.1.4 Link to the Dirichlet $$p$$-Laplacian and the $$p$$-torsion

In the Euclidean space, the 1-Cheeger constant comes into play in estimating some quantities related to the Laplace equation and to the torsional creep equation. More precisely, it provides lower bounds on the first eigenvalue of the Dirichlet $$p$$-Laplacian for $$p>1$$ and to the $$L^1$$ norm of the $$p$$-torsional creep function. In Section 6, we extend these results to our more general framework.

Both these problems require an extensive preliminary work to define Sobolev spaces in our general (nonmetric) context. In order to do so, we need a little more structure on the perimeter-measure space: we require it to be endowed with a topology, we require the class of measurable sets to be that of Borel sets, and we require the perimeter $$P(\cdot)$$ to stem from a relative perimeter when evaluated relatively to the whole space $$X$$.

We here quickly sketch how we construct these Sobolev spaces, and we refer the interested reader to Section 2.3. A relative perimeter functional allows, again via the relative coarea formula in a similar fashion to (1.3), to define the relative variation of an $$L^1$$ function $$u$$ with respect to a measurable set. When this happens to define a measure, we shall say that the function is in $$\mathsf {BV}(X, \mathfrak {m})$$, and this extends the notion briefly discussed in Section 1.1.3 and formally introduced in Section 2.2. When this measure happens to be absolutely continuous with respect to $$\mathfrak {m}$$, we shall say that the function is in $$\mathsf {W}^{1,1}(X,\mathfrak {m})$$ and that the density of the measure with respect to $$\mathfrak {m}$$ is the 1-slope of $$u$$. Via approximation arguments, one can then define the $$p$$-slope of a function and the associated $$\mathsf {W}^{1,p}(X,\mathfrak {m})$$ spaces. In turn, the approximation properties allow to define the Sobolev space $$\mathsf {W}^{1,p}_0(\Omega, \mathfrak {m})$$, refer to Definition 6.1.

Summing up, Sobolev spaces can be built as induced by a relative perimeter on the topological perimeter-measure space. Once this notion is available, one can define the first eigenvalue of the Dirichlet $$p$$-Laplacian for $$p>1$$ in an analogous manner to the standard, Euclidean one. In the classical setting, similarly to (1.1), one defines

$$$$\begin{equation} {\lambda}_{1,p}(\mathrm{\Omega})=\inf\left\{\frac{\displaystyle \int_{\mathrm{\Omega}}|\nabla u{|}^{p}\mathrm{d}x}{\Vert u{\Vert}_{p}^{p}}:u\in {\mathrm{C}}_{c}^{1}(\mathrm{\Omega}),\Vert u{\Vert}_{p}>0\right\}. \end{equation}$$$$(1.4)

In our setting, we cannot directly consider (1.4), since no notion of derivative is available. However, the natural space of competitors of such a problem is the classical space of $$W^{1,p}_0(\Omega)$$ functions, and we do have an analogous notion of Sobolev space at our disposal, and thus, such a way is viable.

In Euclidean settings [88, 89], it is known that the inequality

$$$$\begin{equation*} {\left(\frac{h_1(\Omega)}{p}\right)}^p \leqslant \lambda _{1,p}(\Omega) \end{equation*}$$$$

holds. In Theorem 6.3 and Corollary 6.4, we prove that this inequality naturally extends to our general framework, provided that the relative perimeter satisfies some general assumptions.

Finally, we recall that the $$p$$-torsional creep function is the solution of the PDE with homogeneous Dirichlet boundary datum

$$$$\begin{equation} {\begin{cases} \begin{aligned} -\Delta _p u &= 1, \qquad &&\text{in $\Omega $,} \\ u&=0, &&\text{on $\partial \Omega $,} \end{aligned} \end{cases}} \end{equation}$$$$(1.5)

where $$-\Delta _p$$ is the $$p$$-Laplace operator. It is known [32] that the solution $$w_p$$ of the PDE (1.5) satisfies

$$$$\begin{equation} {h}_{1}(\mathrm{\Omega})\leqslant p{\left(\frac{{\mathcal{L}}^{n}(\mathrm{\Omega})}{\Vert {w}_{p}{\Vert}_{1}}\right)}^{\frac{p-1}{p}}. \end{equation}$$$$(1.6)

As usual, we cannot directly consider (1.5), but we can work with the underlying Euler–Lagrange energy among functions in the Sobolev spaces we defined. In particular, we can prove that minimizers of the energy, if they exist, satisfy (1.6) up to a slightly worse prefactor of $$p^{1+\frac{1}{p}}$$, refer to Theorem 6.5, provided that the relative perimeter satisfies some very general properties.

1.1.5 Examples

In the last section of the paper, we collect several examples of spaces that meet our hypotheses. In particular, our very general approach basically covers all results known so far about the existence of Cheeger sets in finite-dimensional spaces, and the relation of the constant with the first eigenvalue of the Dirichlet $$p$$-Laplacian in numerous contexts. In some of the frameworks presented in Section 7, the results are new, up to our knowledge.

Unfortunately, our approach does not cover the case of the infinite-dimensional Wiener space. In this case, one can suitably define the Cheeger constant and prove the existence of Cheeger sets. Nonetheless, this requires ad hoc notions of $$BV$$ function and of perimeter that are quite different from the ones adopted in the present paper. We refer the interested reader to [51, Sect. 6] for a more detailed exposition about this specific framework.

2 PERIMETER-MEASURE SPACES

The basic setting is that of nonnegative $$\sigma$$-finite measure spaces $$(X,\mathcal {A}, \mathfrak {m})$$. We set that, for any $$A,B\in \mathcal {A}$$, by $$A\subset B$$, we mean that $$\mathfrak {m}(A\setminus B)=0$$. We also let $$L^0(X,\mathfrak {m})$$ be the vector space of $$\mathfrak {m}$$-measurable functions, and, for $$p\geqslant 1$$, we let $$L^p(X,\mathfrak {m})$$ be the usual space of $$p$$-integrable functions, that is,

$$$$\begin{equation*} L^p(X,\mathfrak {m})={\left\lbrace u\in L^0(X,\mathfrak {m}): \displaystyle \int _X|u|^p\;\mathrm{d}\mathfrak {m}<+\infty \right\rbrace}. \end{equation*}$$$$

As usual, we identify $$\mathfrak {m}$$-measurable functions coinciding $$\mathfrak {m}$$-a.e. on $$X$$. In case $$X$$ is endowed with a topology $$\mathcal T\subset \mathcal P(X)$$, we let $$\mathcal B(X)$$ be the Borel $$\sigma$$-algebra generated by $$\mathcal T$$ and, in this case, we shall assume that $$\mathcal A=\mathcal B(X)$$.

2.1 Perimeter functional

In the same spirit of [19, Sect. 3], we introduce the following definition.

Definition 2.1.A perimeter functional $$P(\cdot)$$ is any map

$$$$\begin{equation} P\colon \mathcal {A}\rightarrow [0,+\infty], \end{equation}$$$$(2.1)

which is proper, that is, $$P(A)<+\infty$$ for some $$A\in \mathcal A$$. In this case, we call $$(X,\mathcal A,\mathfrak {m},P)$$ a perimeter-measure space.

Throughout the paper, we will assume that the perimeter will satisfy some of the following properties:

• (P.1) $$P(\emptyset)=0$$;
• (P.2) $$P(X)=0$$;
• (P.3) $$P(E\cap F)+P(E\cup F)\leqslant P(E)+P(F)$$ for all $$E,F\in \mathcal {A}$$;
• (P.4) $$P$$ is lower semicontinuous with respect to the $$L^1(X,\mathfrak {m})$$ convergence;
• (P.5) for any $$\Omega \in \mathcal {A}$$ with $$\mathfrak {m}(\Omega)<+\infty$$, the family
  $$$$\begin{equation*} {\left\lbrace \chi _E: E\in \mathcal {A},\ E\subset \Omega,\ P(E)\leqslant c \right\rbrace} \end{equation*}$$$$
  is compact in $$L^1(X,\mathfrak {m})$$ for all $$c\geqslant 0$$;
• (P.6) there exists a function $$f\colon (0,+\infty)\rightarrow (0,+\infty)$$ such that
  $$$$\begin{equation*} \lim \limits _{\varepsilon \rightarrow 0^+}f(\varepsilon)=+\infty \end{equation*}$$$$
  with the following property: if $$\varepsilon >0$$ and $$E\in \mathcal {A}$$ with $$\mathfrak {m}(E)\leqslant \varepsilon$$, then $$P(E)\geqslant f(\varepsilon)\,\mathfrak {m}(E)$$;
• (P.7) $$P(E)=P(X\setminus E)$$ for all $$E\in \mathcal {A}$$.

Assuming property (P.7) true, properties (P.1) and (P.2) become equivalent. Throughout the paper, we often refer to (P.6) as an isoperimetric property. Notice that, in case an isoperimetric inequality $$P(E)\geqslant C \mathfrak {m}(E)^{\frac{Q-1}{Q}}$$ holds true for suitable $$Q>1$$ and $$C>0$$, and for all $$E\in \mathcal {A}$$ with $$\mathfrak {m}(E)<+\infty$$, then (P.6) clearly follows. Depending on the situation, it could be more convenient to prove (P.6) directly or to rely on a finer isoperimetric-type inequality, see Section 7. We remark that all the properties listed above will appear every now and then throughout the paper, but they are not enforced throughout — every statement will precisely contain the bare minimum for its validity.

Remark 2.2. ($$P$$ is invariant under $$\mathfrak {m}$$-negligible modifications)Let property (P.4) be in force. If $$A,B\in \mathcal {A}$$ are such that $$\mathfrak {m}(A\bigtriangleup B)=0$$, then $$P(A)=P(B)$$. To see this, consider any measurable set $$E$$ and any $$\mathfrak {m}$$-negligible set $$N$$, look at the constant sequence $$\lbrace E\cup N\rbrace _k$$ converging to $$E$$ in $$L^1(X,\mathfrak {m})$$, and at the constant one $$\lbrace E\rbrace _k$$ converging to $$E\cup N$$ and exploit (P.4).

Remark 2.3.Let property (P.6) be in force. If $$P(E)=0$$, then the set $$E$$ is $$\mathfrak {m}$$-negligible, that is, $$\mathfrak {m}(E)=0$$. Conversely, if $$\mathfrak {m}(E)>0$$, then $$P(E)\in (0,+\infty]$$. Thus, property (P.6) says that the only sets with finite measure that could possibly have zero perimeter are $$\mathfrak {m}$$-negligible sets. Moreover, if properties (P.1) and (P.4) are in force as well, then $$\mathfrak {m}$$-negligible sets have zero perimeter, thanks to Remark 2.2.

2.2 Variation and $$BV$$ functions

We define the variation of a function $$u\in L^0(X,\mathfrak {m})$$ as

$$$$\begin{equation} \operatorname{Var}(u) = {\begin{cases} \displaystyle \displaystyle \int _\mathbb {R}P(\lbrace u> t\rbrace) \mathrm{d}t, &\text{if $t\mapsto P(\lbrace u> t\rbrace)$ is $\mathcal L^1$-measurable,} \\[12pt] +\infty, &\text{otherwise.} \end{cases}} \end{equation}$$$$(2.2)

With this notation at hand, we let

$$$$\begin{equation} BV(X,\mathfrak {m})= {\left\lbrace u\in L^1(X, \mathfrak {m}): \operatorname{Var}(u)<+\infty \right\rbrace} \end{equation}$$$$(2.3)

be the set of $$L^1$$ functions with bounded variation.

We begin with the following result, proving that assuming the validity of properties (P.1) and (P.2), the variation coincides with the perimeter functional on characteristic functions.

Lemma 2.4. (Total variation of sets)Let properties (P.1) and (P.2) be in force. If $$E\in \mathcal A$$, then $$\operatorname{Var}(\chi _E)=P(E)$$.

Proof.By definition, (P.1), and (P.2), the function

$$$$\begin{equation*} t\mapsto P({\left\lbrace \chi _E>t \right\rbrace}) = {\begin{cases} P(X), & t\leqslant 0, \\ P(E), & 0<t\leqslant 1, \\ P(\emptyset), & t>1, \end{cases}} \end{equation*}$$$$

is $$\mathcal L^1$$-measurable, so that

$$$$\begin{equation*} \operatorname{Var}(\chi _E)=\displaystyle \int _{\mathbb {R}}P({\left\lbrace \chi _E>t \right\rbrace})\,\mathrm{d}t = \displaystyle \int _0^1 P(E)\,\mathrm{d}t = P(E), \end{equation*}$$$$

in virtue of (P.1) and (P.2).$$\Box$$

Remark 2.5.As an immediate consequence of Lemma 2.4, if (P.1) and (P.2) are in force, then $$\operatorname{Var}\colon L^0(X,\mathfrak {m})\rightarrow [0,+\infty]$$ is a proper functional and $$\chi _E\in BV(X,\mathfrak {m})$$ whenever $$E\in \mathcal A$$ is such that $$\mathfrak {m}(E)<+\infty$$ and $$P(E)<+\infty$$. In particular, $$0\in BV(X,\mathfrak {m})$$ with $$\operatorname{Var}(0)=0$$.

The following result rephrases [59, Prop. 3.2] in the present context.

Lemma 2.6. (Basic properties of total variation)The following hold:

• (i) $$\operatorname{Var}(\lambda u)=\lambda \operatorname{Var}(u)$$ for all $$\lambda >0$$ and $$u\in L^0(X,\mathfrak {m})$$;
• (ii) $$\operatorname{Var}(u+c)=\operatorname{Var}(u)$$ for all $$c\in \mathbb {R}$$ and $$u\in L^0(X,\mathfrak {m})$$;
• (iii) if (P.1) and (P.2) are in force, then $$\operatorname{Var}(c)=0$$ for all $$c\in \mathbb {R}$$;
• (iv) if (P.4) is in force, then $$\operatorname{Var}\colon L^1(X,\mathfrak {m})\rightarrow [0,+\infty]$$ is lower semicontinuous with respect to the (strong) convergence in $$L^1(X,\mathfrak {m})$$.

Proof.The proofs of the first three points are natural consequences of the definition.

Proof of (iv). Let $$u_k,u\in L^1(X,\mathfrak {m})$$ be such that $$u_k\rightarrow u$$ in $$L^1(X,\mathfrak {m})$$ as $$k\rightarrow +\infty$$. Without loss of generality, we can assume that

$$$$\begin{equation*} \liminf _{k\rightarrow +\infty}\operatorname{Var}(u_k)<+\infty, \end{equation*}$$$$

so that, up to possibly passing to a subsequence (which we do not relabel for simplicity), we have $$\operatorname{Var}(u_k)<+\infty$$ for all $$k\in \mathbb {N}$$. Following [98, Rem. 13.11], one has

$$$$\begin{equation*} \Vert u_k-u\Vert _1 = \displaystyle \int _\mathbb {R}\mathfrak {m}({\left\lbrace u_k>t \right\rbrace} \bigtriangleup {\left\lbrace u>t \right\rbrace})\,\mathrm{d}t, \end{equation*}$$$$

thus, we immediately deduce that $$\chi _{\lbrace u_k>t \rbrace}\rightarrow \chi _{\lbrace u>t \rbrace}$$ in $$L^1(X,\mathfrak {m})$$ as $$k\rightarrow +\infty$$ for $$\mathcal L^1$$-a.e. $$t\in \mathbb {R}$$. Thanks to property (P.4), we have that

$$$$\begin{equation*} P({\left\lbrace u>t \right\rbrace}) \leqslant \liminf _{k\rightarrow +\infty} P({\left\lbrace u_k>t \right\rbrace}) \end{equation*}$$$$

for $$\mathcal L^1$$-a.e. $$t\in \mathbb {R}$$, and the map $$t\mapsto P(\lbrace u>t \rbrace)$$ is also $$\mathcal L^1$$-measurable. Therefore, by Fatou's lemma, we conclude that

$$$$\begin{align*} \operatorname{Var}(u) &= \displaystyle \int _\mathbb {R}P({\left\lbrace u>t \right\rbrace})\,\mathrm{d}t \leqslant \displaystyle \int _\mathbb {R}\liminf _{k\rightarrow +\infty} P({\left\lbrace u_k>t \right\rbrace})\,\mathrm{d}t \\ &\leqslant \liminf _{k\rightarrow +\infty} \operatorname{Var}(u_k) <+\infty, \end{align*}$$$$

proving (iv).$$\Box$$

The following result, which can be proved as in [59] up to minor modifications, states that the variation functional is convex as soon as the perimeter functional is sufficiently well behaved.

Proposition 2.7. (Convexity of variation)Let properties (P.1), (P.2), (P.3), and (P.4) be in force. Then, $$ \operatorname{Var}\colon L^1(X,\mathfrak {m})\rightarrow [0,+\infty]$$ is convex. As a consequence, $$BV(X,\mathfrak {m})$$ is a convex cone in $$L^1(X,\mathfrak {m})$$.

2.3 Relative perimeter and relative variation

In this subsection, we assume that the set $$X$$ is endowed with a topology $$\mathcal T$$ such that $$\mathcal A=\mathcal B(X)$$, the Borel $$\sigma$$-algebra generated by $$\mathcal T$$.

Definition 2.8.A relative perimeter functional $$\mathsf {P}$$ is any map

$$$$\begin{equation} \mathsf {P}\colon \mathcal B(X)\times \mathcal B(X)\rightarrow [0,+\infty]. \end{equation}$$$$(2.4)

Throughout the paper, we will assume that a relative perimeter will satisfy some of the following properties:

• (RP.1) $$\mathsf {P}(\emptyset;A)=0$$ for all $$A\in \mathcal T$$;
• (RP.2) $$\mathsf {P}(X;A)=0$$ for all $$A\in \mathcal T$$;
• (RP.3) $$\mathsf {P}(E\cap F;A)+\mathsf {P}(E\cup F;A)\leqslant \mathsf {P}(E;A)+\mathsf {P}(F;A)$$ for all $$E,F\in \mathcal B(X)$$ and $$A\in \mathcal T$$;
• (RP.4) for each $$A\in \mathcal T$$, $$\mathsf {P}(\cdot \,;A)$$ is lower semicontinuous with respect to the (strong) convergence in $$L^1(X,\mathfrak {m})$$.

We stress that in the properties above, the perimeter is relative to an open set $$A$$, and not to a general element of the Borel $$\sigma$$-algebra.

Moreover, following the same idea of Section 2.2, we let

$$$$\begin{equation} \mathsf {Var}(u;A) = {\begin{cases} \displaystyle \displaystyle \int _{\mathbb {R}} \mathsf {P}({\left\lbrace u>t \right\rbrace};A)\,\mathrm{d}t, & \text{if}\ t\mapsto \mathsf {P}({\left\lbrace u>t \right\rbrace};A)\ \text{is $\mathcal L^1$-meas.}, \\[3pt] +\infty, & \text{otherwise}, \end{cases}} \end{equation}$$$$(2.5)

be the variation of $$u\in L^0(X,\mathfrak {m})$$ relative to $$A\in \mathcal B(X)$$. In analogy with the approach developed in the previous sections, for each $$A\in \mathcal T$$, one can regard the map

$$$$\begin{equation*} \mathsf {P}(\cdot \,;A)\colon \mathcal B(X)\rightarrow [0,+\infty] \end{equation*}$$$$

as a particular instance of the perimeter functional introduced in (2.1). Specifically, we use the notation

$$$$\begin{equation} P(E)=\mathsf {P}(E;X), \qquad \operatorname{Var}(u)=\mathsf {Var}(u;X), \end{equation}$$$$(2.6)

for all $$E\in \mathcal B(X)$$ and $$u\in L^0(X,\mathfrak {m})$$, and we consider $$P(E)$$ as the perimeter of $$E$$ in the sense of Section 2.1. Analogously, $$\operatorname{Var}(u)$$ stands as the variation of $$u$$ in the sense of Section 2.2. Consequently, the space

$$$$\begin{equation*} BV(X,\mathfrak {m})={\left\lbrace u\in L^1(X,\mathfrak {m}):\mathsf {Var}(u; X)<+\infty \right\rbrace} \end{equation*}$$$$

is the space defined in (2.3).

Below, we rephrase Lemma 2.4, Lemma 2.6, and Proposition 2.7 in the present setting. Their proofs are omitted, because they are similar to those already given or referred to.

Lemma 2.9. (Relative variation of sets)Let properties (RP.1) and (RP.2) be in force. If $$E\in \mathcal B(X)$$, then $$\mathsf {Var}(\chi _E;A)=\mathsf {P}(E;A)$$ for all $$A\in \mathcal T$$.

Lemma 2.10. (Basic properties of relative variation)The following hold:

• (i) $$\mathsf {Var}(\lambda u;A)=\lambda \mathsf {Var}(u;A)$$ for all $$\lambda >0$$, $$A\in \mathcal T$$, and $$u\in L^0(X,\mathfrak {m})$$;
• (ii) $$\mathsf {Var}(u+c;A)=\mathsf {Var}(u;A)$$ for all $$A\in \mathcal T$$, $$c\in \mathbb {R}$$, and $$u\in L^0(X,\mathfrak {m})$$;
• (iii) if (RP.1) and (RP.2) are in force, then $$\mathsf {Var}(c;A)=0$$ for all $$c\in \mathbb {R}$$ and $$A\in \mathcal T$$;
• (iv) if (RP.4) is in force, then, for each $$A\in \mathcal T$$, the relative variation $$\mathsf {Var}(\cdot \,;A)\colon L^1(X,\mathfrak {m})\rightarrow [0,+\infty]$$ is lower semicontinuous with respect to the (strong) convergence in $$L^1(X,\mathfrak {m})$$.

Proposition 2.11. (Convexity of relative variation)Let properties (RP.1), (RP.2), (RP.3), and (RP.4) be in force. Then, for each $$A\in \mathcal T$$, the functional $$\mathsf {Var}(\cdot \,;A)\colon L^1(X,\mathfrak {m})\rightarrow [0,+\infty]$$ is convex.

2.3.1 Variation measure

We now define the perimeter and variation measures by rephrasing the validity of the relative coarea formula (2.5) in a measure-theoretic sense.

Definition 2.12. (Perimeter and variation measures)We say that a set $$E\in \mathcal B(X)$$ has finite perimeter measure if its relative perimeter

$$$$\begin{equation*} \mathsf {P}(E;\,\cdot \,)\colon \mathcal B(X)\rightarrow [0,+\infty] \end{equation*}$$$$

defines a finite outer regular Borel measure on $$X$$. We hence say that a function $$u\in L^0(X,\mathfrak {m})$$ has finite variation measure if the set $$\lbrace u>t \rbrace$$ has finite perimeter for $$\mathcal L^1$$-a.e. $$t\in \mathbb {R}$$ and its relative variation

$$$$\begin{equation*} \mathsf {Var}(u;\,\cdot \,) \colon \mathcal B(X)\rightarrow [0,+\infty) \end{equation*}$$$$

defines a finite outer regular Borel measure on $$X$$.

Adopting the usual notation, if $$E\in \mathcal B(X)$$ has finite perimeter measure, then we write $$\mathsf {P}(E;A)=|\mathsf {D}\chi _E|(A)$$ for all $$A\in \mathcal B(X)$$. Similarly, if $$u\in L^0(X,\mathfrak {m})$$ has finite variation measure, then we write $$\mathsf {Var}(u;A)=|\mathsf {D}u|(A)$$ for all $$A\in \mathcal B(X)$$.

It is worth noticing that Definition 2.12 is well posed in the following sense. As soon as properties (RP.1) and (RP.2) are in force, if $$E\in \mathcal B(X)$$ has finite perimeter measure, then $$\chi _E\in L^0(X,\mathfrak {m})$$ has finite variation measure with $$\mathsf {Var}(\chi _E;\,\cdot \,)=\mathsf {P}(E;\,\cdot \,)$$, since they are outer regular Borel measures on $$X$$ agreeing on open sets. This is a simple consequence of Lemma 2.9.

By Definition 2.12, if $$u\in L^0(X,\mathfrak {m})$$ has finite variation measure, then, for each $$A\in \mathcal B(X)$$, we have $$\mathsf {Var}(u;A)<+\infty$$, and thus,

$$$$\begin{equation*} t\mapsto \mathsf {P}(\lbrace u>t \rbrace;A) \in L^1(\mathbb {R}), \end{equation*}$$$$

so that we can write

$$$$\begin{equation*} |\mathsf {D}u|(A)=\mathsf {Var}(u;A)=\displaystyle \int _\mathbb {R}\mathsf {P}({\left\lbrace u>t \right\rbrace};A)\,\mathrm{d}t = \displaystyle \int _\mathbb {R}|\mathsf {D}\chi _{{\left\lbrace u>t \right\rbrace}}|(A)\, \mathrm{d}t. \end{equation*}$$$$

In more general terms, we get the following extension of the relative coarea formula (2.5). Its proof follows from a routine approximation argument (see [12], e.g.) and is thus omitted.

Corollary 2.13. (Generalized coarea formula)If $$u\in L^0(X,\mathfrak {m})$$ has finite variation measure, then

$$$$\begin{equation*} \displaystyle \int _A\varphi \,\mathrm{d}|\mathsf {D}u| = \displaystyle \int _\mathbb {R}\displaystyle \int _A\varphi \,\mathrm{d}|\mathsf {D}\chi _{{\left\lbrace u>t \right\rbrace}}|\,\mathrm{d}t \end{equation*}$$$$

for all $$\varphi \in L^0(X,\mathfrak {m})$$ and $$A\in \mathcal B(X)$$.

Keeping the same notation used in the previous sections, we let

$$$$\begin{equation*} \mathrm{BV}(X,\mathfrak{m})=\left\{u\in {L}^{1}(X,\mathfrak{m}):u\ \text{has finite variation measure}\right\}. \end{equation*}$$$$

Notice that although $$\mathsf {BV}(X,\mathfrak {m})\subset BV(X,\mathfrak {m})$$ and $$BV(X,\mathfrak {m})$$ is a convex cone in $$L^1(X,\mathfrak {m})$$, the set $$\mathsf {BV}(X,\mathfrak {m})$$ may not be a convex cone in $$L^1(X,\mathfrak {m})$$ as well, since the validity of the implication

$$$$\begin{equation*} u,v\in \mathsf {BV}(X,\mathfrak {m}) \Rightarrow u+v\ \text{has finite variation measure} \end{equation*}$$$$

is not automatically granted. For an example of such a phenomenon, we refer the interested reader to the variation of intrinsic maps between subgroups of sub-Riemannian Carnot groups [116, Rem. 4.2], but we will not enter into the details of this issue because it is out of the scope of the present paper.

This being said, we introduce the following additional property for the relative perimeter $$\mathsf {P}$$ in (2.4) requiring the closure of $$\mathsf {BV}(X,\mathfrak {m})$$ with respect to the sum of functions:

• (RP.+) $$u,v\in \mathsf {BV}(X,\mathfrak {m})\Rightarrow u+v\in \mathsf {BV}(X,\mathfrak {m})$$.

We now outline some consequences of Lemma 2.10 and Proposition 2.11, and leave the simple proofs of these statements to the interested reader, see also the proof of Lemma 2.6.

Corollary 2.14. (Basic properties of variation measure)Let properties (RP.1), (RP.2), (RP.3), and (RP.4) be in force. The following hold:

• (i) if $$u\in L^0(X,\mathfrak {m})$$ has finite variation measure, then $$\lambda u$$ has finite variation measure, with $$|\mathsf {D}(\lambda u)|=\lambda|\mathsf {D}u|$$, for all $$\lambda >0$$;
• (ii) if $$u\in L^0(X,\mathfrak {m})$$ has finite variation measure, then $$u+c$$ has finite variation measure, with $$|\mathsf {D}(u+c)|=|\mathsf {D}u|$$, for all $$c\in \mathbb {R}$$;
• (iii) constant functions have finite variation measure and $$|\mathsf {D}c|=0$$ for all $$c\in \mathbb {R}$$ (in particular, $$0\in \mathsf {BV}(X,\mathfrak {m})$$);
• (iv) if $$\lbrace u_k\rbrace _{k\in \mathbb {N}}\subset \mathsf {BV}(X,\mathfrak {m})$$ and $$u_k\rightarrow u$$ in $$L^1(X,\mathfrak {m})$$ as $$k\rightarrow +\infty$$ for some $$u\in \mathsf {BV}(X,\mathfrak {m})$$, then
  $$$$\begin{equation*} |\mathsf {D}u|(A)\leqslant \liminf _{k\rightarrow +\infty}|\mathsf {D}u_k|(A) \end{equation*}$$$$
  for all $$A\in \mathcal T$$;
• (v) if also property (RP.+) is in force and $$u,v\in \mathsf {BV}(X,\mathfrak {m})$$, then
  $$$$\begin{equation*} |\mathsf {D}(u+v)|\leqslant|\mathsf {D}u|+|\mathsf {D}v| \end{equation*}$$$$
  as outer regular Borel measures on $$X$$.

2.3.2 Chain rule

We now establish a chain rule for the variation measure of continuous functions. To this aim, we need to assume the following locality property of the relative perimeter functional in (2.4):

• (RP.L) $$E\in \mathcal{T}\Rightarrow \mathrm{P}(E;A)=0$$ for all $$A\in \mathcal B(X)$$ with $$\mathrm{P}(E;A\cap \partial E)=0$$.

Loosely speaking, property (RP.L) states that, for any open set $$E\subset X$$, the relative perimeter functional $$A\mapsto P(E;A)$$ is supported (in a measure-theoretic sense) on the topological boundary $$\partial E$$ of the set $$E$$.

Theorem 2.15. (Chain rule)Let properties (RP.1), (RP.2), and (RP.L) be in force and let $$\varphi \in \mathrm{C}^1(\mathbb {R})$$ be a strictly increasing function. If $$u\in \mathrm{C}^0(X)$$ has finite variation measure, then also $$\varphi (u)\in \mathrm{C}^0(X)$$ has finite variation measure, with

$$$$\begin{equation} |\mathsf {D}\varphi (u)|=\varphi ^{\prime}(u)|\mathsf {D}u| \end{equation}$$$$(2.7)

as finite outer regular Borel measures on $$X$$.

Proof.Since $$\varphi$$ is strictly increasing, its inverse function $$\varphi ^{-1}\colon \varphi (\mathbb {R})\rightarrow \mathbb {R}$$ is well defined, continuous and strictly increasing, and we can write

$$$$\begin{equation*} {\left\lbrace \varphi (u)>t \right\rbrace} = {\begin{cases} X, &\text{if}\ t\leqslant \inf \varphi (\mathbb {R}), \\ {\left\lbrace u>\varphi ^{-1}(t) \right\rbrace}, \quad & \text{if}\ t\in \varphi (\mathbb {R}), \\ \emptyset, & \text{if}\ t\geqslant \sup \varphi (\mathbb {R}). \end{cases}} \end{equation*}$$$$

Therefore, the set $$\lbrace \varphi (u)>t \rbrace$$ has finite perimeter measure for $$\mathcal L^1$$-a.e. $$t\in \mathbb {R}$$, with

$$$$\begin{equation*} |\mathsf {D}\chi _{{\left\lbrace \varphi (u)>t \right\rbrace}}| = {\begin{cases} |\mathsf {D}\chi _{{\left\lbrace u>\varphi ^{-1}(t) \right\rbrace}}|, \quad & \text{if}\ t\in \varphi (\mathbb {R}), \\ 0, &\text{if}\ t\notin \varphi (\mathbb {R}). \end{cases}} \end{equation*}$$$$

Hence, given $$A\in \mathcal B(X)$$, we have

$$$$\begin{equation*} t\mapsto|\mathsf {D}\chi _{{\left\lbrace \varphi (u)>t \right\rbrace}}|(A) =|\mathsf {D}\chi _{{\left\lbrace u>\varphi ^{-1}(t) \right\rbrace}}|(A)\,\chi _{\varphi (\mathbb {R})}(t) \in L^1(\mathbb {R}), \end{equation*}$$$$

and so,

$$$$\begin{equation*} \mathsf {Var}(\varphi (u);A) = \displaystyle \int _\mathbb {R}|\mathsf {D}\chi _{{\left\lbrace \varphi (u)>t \right\rbrace}}|(A)\,\mathrm{d}t = \displaystyle \int _{\varphi (\mathbb {R})}|\mathsf {D}\chi _{{\left\lbrace u>\varphi ^{-1}(t) \right\rbrace}}|(A)\,\mathrm{d}t. \end{equation*}$$$$

Performing a change of variables, we can write

$$$$\begin{equation*} \displaystyle \int _{\varphi (\mathbb {R})}|\mathsf {D}\chi _{{\left\lbrace u>\varphi ^{-1}(t) \right\rbrace}}|(A)\,\mathrm{d}t = \displaystyle \int _\mathbb {R}|\mathsf {D}\chi _{{\left\lbrace u>s \right\rbrace}}|(A)\,\varphi ^{\prime}(s)\,\mathrm{d}s. \end{equation*}$$$$

Now, since $$u\in \mathrm{C}^0(X)$$, we know that $$\lbrace u>s \rbrace \in \mathcal T$$ and $$\partial \lbrace u>s \rbrace \subset \lbrace u=s \rbrace$$ for all $$s\in \mathbb {R}$$. Therefore, because of (RP.L), we have $$|\mathsf {D}\chi _{\lbrace u>s \rbrace}|(B)=0$$ for all $$B\in \mathcal B(X)$$ such that $$|\mathsf {D}\chi _{\lbrace u>s \rbrace}|(B\cap \lbrace u=s \rbrace)=0$$.

Thus, letting $$B=A\cap \lbrace u\ne s \rbrace$$, we have that $$|\mathsf {D}\chi _{\lbrace u>s \rbrace}|(A\cap \lbrace u\ne s \rbrace)=0$$ for all $$s\in \mathbb {R}$$; hence, the following equalities hold:

$$$$\begin{align*} \displaystyle \int _\mathbb {R}|\mathsf {D}\chi _{{\left\lbrace u>s \right\rbrace}}|(A)\,\varphi ^{\prime}(s)\,\mathrm{d}s &= \displaystyle \int _\mathbb {R}\varphi ^{\prime}(s)\displaystyle \int _A \mathrm{d}|\mathsf {D}\chi _{{\left\lbrace u>s \right\rbrace}}|\,\mathrm{d}s \\ &= \displaystyle \int _\mathbb {R}\displaystyle \int _A \varphi ^{\prime}(u)\,\mathrm{d}|\mathsf {D}\chi _{{\left\lbrace u>s \right\rbrace}}|\,\mathrm{d}s. \end{align*}$$$$

By Corollary 2.13, we can write

$$$$\begin{equation*} \displaystyle \int _\mathbb {R}\displaystyle \int _A \varphi ^{\prime}(u)\,\mathrm{d}|\mathsf {D}\chi _{{\left\lbrace u>s \right\rbrace}}|\,\mathrm{d}s = \displaystyle \int _A\varphi ^{\prime}(u)\,\mathrm{d}|\mathsf {D}u|, \end{equation*}$$$$

so that, by combining all the above equalities, we conclude that

$$$$\begin{equation*} \mathsf {Var}(\varphi (u);A) = \displaystyle \int _A\varphi ^{\prime}(u)\,\mathrm{d}|\mathsf {D}u| \end{equation*}$$$$

for all $$A\in \mathcal B(X)$$, proving (2.7) and completing the proof.$$\Box$$

2.3.3 $$p$$-Slope and Sobolev functions

As customary, we let

$$$$\begin{equation*} {\mathrm{W}}^{1,1}(X,\mathfrak{m})=\left\{u\in \mathrm{BV}(X,\mathfrak{m}):|\mathrm{D}u|\ll \mathfrak{m}\right\} \end{equation*}$$$$

be the set of Sobolev $$\mathsf {W}^{1,1}$$ functions on $$X$$.

If $$u\in \mathsf {W}^{1,1}(X,\mathfrak {m})$$, then we let $$|\nabla u|\in L^1(X,\mathfrak {m})$$, $$|\nabla u|\geqslant 0$$ $$\mathfrak {m}$$-a.e. in $$X$$, be the 1-slope of $$u$$, that is, the unique $$L^1(X,\mathfrak {m})$$ function such that

$$$$\begin{equation*} |\mathsf {D}u|(A) = \displaystyle \int _A|\nabla u|\,\mathrm{d}\mathfrak {m}\qquad \text{for all}\ A\in \mathcal B(X). \end{equation*}$$$$

From Corollary 2.14, we immediately deduce the following simple properties of 1-slopes of $$\mathsf {W}^{1,1}$$ functions.

Corollary 2.16. (Basic properties of 1-slope)Let properties (RP.1), (RP.2), (RP.3), (RP.4), and (RP.+) be in force. The following hold:

• (i) if $$u\in \mathsf {W}^{1,1}(X,\mathfrak {m})$$, then $$\lambda u\in \mathsf {W}^{1,1}(X,\mathfrak {m})$$, with $$|\nabla (\lambda u)|=\lambda|\nabla u|$$, for all $$\lambda >0$$;
• (ii) $$0\in \mathsf {W}^{1,1}(X,\mathfrak {m})$$, with $$|\nabla 0|=0$$;
• (iii) if $$u,v\in \mathsf {W}^{1,1}(X,\mathfrak {m})$$, then $$u+v\in \mathsf {W}^{1,1}(X,\mathfrak {m})$$ with
  $$$$\begin{equation*} |\nabla (u+v)|\leqslant|\nabla u|+|\nabla v|. \end{equation*}$$$$

As a consequence, $$\mathsf {W}^{1,1}(X,\mathfrak {m})$$ is a convex cone in $$L^1(X,\mathfrak {m})$$.

Having the notion of 1-slope at our disposal, following the standard approach about slopes (see [13], e.g.), we can introduce the notion of $$p$$-relaxed 1-slope, for $$p\in (1,+\infty)$$.

Definition 2.17. ($$p$$-Relaxed 1-slope)Let $$p\in (1,+\infty)$$. We shall say that a function $$g\in L^p(X,\mathfrak {m})$$ is a $$p$$-relaxed 1-slope of $$u\in L^p(X,\mathfrak {m})$$ if there exist a function $$\tilde{g}\in L^p(X,\mathfrak {m})$$ and a sequence $$\lbrace u_k\rbrace _{k\in \mathbb {N}}\subset \mathsf {W}^{1,1}(X,\mathfrak {m})\cap L^p(X,\mathfrak {m})$$ such that:

• (i) $$u_k\rightarrow u$$ in $$L^p(X,\mathfrak {m})$$;
• (ii) $$|\nabla u_k|\in L^p(X,\mathfrak {m})$$ for all $$k\in \mathbb {N}$$ and $$|\nabla u_k|\rightharpoonup \tilde{g}$$ weakly in $$L^p(X,\mathfrak {m})$$;
• (iii) $$\tilde{g}\leqslant g$$ $$\mathfrak {m}$$-a.e. in $$X$$.

Clearly, according to Definition 2.17 and thanks to the sequential compactness of weak topologies, if $$\lbrace u_k\rbrace _{k\in \mathbb {N}}\subset \mathsf {W}^{1,1}(X,\mathfrak {m})\cap L^p(X,\mathfrak {m})$$ is such that

$$$$\begin{equation*} \sup _{k\in \mathbb {N}}\displaystyle \int _X|\nabla u_k|^p\,\mathrm{d}\mathfrak {m}<+\infty, \end{equation*}$$$$

then any $$L^p(X,\mathfrak {m})$$-limit of $$\lbrace u_k\rbrace _{k\in \mathbb {N}}$$ has at least one $$p$$-relaxed 1-slope. Given any $$u\in L^p(X,\mathfrak {m})$$, we define

$$$$\begin{equation*} {\mathrm{Slope}}_{p}(u)=\left\{g\in {L}^{p}(X,\mathfrak{m}):g\ \text{is a}\ p\text{-relaxed 1-slope of}\ u\right\}. \end{equation*}$$$$

Following the point of view of [13], one can prove the following basic properties of $$p$$-relaxed 1-slopes that will be useful in the sequel.

Lemma 2.18. (Basic properties of $$p$$-relaxed 1-slope)Let properties (RP.1), (RP.2), (RP.3), (RP.4), and (RP.+) be in force and let $$p\in (1,+\infty)$$. The following hold:

• (i) $$\mathsf {Slope}_p(u)$$ is a convex subset (possibly empty) for all $$u\in L^p(X,\mathfrak {m})$$;
• (ii) if $$u\in L^p(X,\mathfrak {m})$$ and $$g\in \mathsf {Slope}_p(u)$$, then there exist a sequence $$\lbrace u_k\rbrace _k\subset \mathsf {W}^{1,1}(X,\mathfrak {m})\cap L^p(X,\mathfrak {m})$$, a sequence $$\lbrace g_k\rbrace _k \subset L^p(X,\mathfrak {m})$$, and a function $$\tilde{g}\in L^p(X,\mathfrak {m})$$, such that $$u_k\rightarrow u$$ and $$g_k\rightarrow \tilde{g}$$ both in $$L^p(X,\mathfrak {m})$$, with $$|\nabla u_k|\leqslant g_k$$ for all $$k\in \mathbb {N}$$ and $$\tilde{g}\leqslant g$$;
• (iii) if $$\lbrace u_k\rbrace _k$$ and $$\lbrace g_k\rbrace$$ are sequences in $$L^p(X,\mathfrak {m})$$, with $$g_k\in \mathsf {Slope}_p(u_k)$$ for all $$k\in \mathbb {N}$$, such that $$u_k\rightharpoonup u$$ and $$g_k\rightharpoonup g$$ weakly in $$L^p(X,\mathfrak {m})$$, then $$g\in \mathsf {Slope}_p(u)$$.

Under the assumptions of the above Lemma 2.18, for each $$u\in L^p(X,\mathfrak {m})$$, the set $$\mathsf {Slope}_p(u)$$ is a (possibly empty) closed convex subset of $$L^p(X,\mathfrak {m})$$, and thus, the following definition is well posed.

Definition 2.19. (Weak $$p$$-slope)Let $$p\in (1,+\infty)$$ and let properties (RP.1), (RP.2), (RP.3), (RP.4), and (RP.+) be in force. If $$u\in L^p(X,\mathfrak {m})$$ is such that $$\mathsf {Slope}_p(u)\ne \emptyset$$, we let $$|\nabla u|_p$$ be the element of $$\mathsf {Slope}_p(u)$$ of minimal $$L^p(X,\mathfrak {m})$$-norm and we call it the weak $$p$$-slope of $$u$$. Finally, we let

$$$$\begin{equation*} {\mathrm{W}}^{1,p}(X,\mathfrak{m})=\left\{u\in {L}^{p}(X,\mathfrak{m}):\exists |\nabla u{|}_{p}\in {L}^{p}(X,\mathfrak{m})\right\}. \end{equation*}$$$$

Following the same line of [13], one can show that the weak $$p$$-slope can be actually approximated in $$L^p(X,\mathfrak {m})$$ in the strong sense.

Corollary 2.20. (Strong approximation of weak $$p$$-slope)Let $$p\in (1,+\infty)$$ and let properties (RP.1), (RP.2), (RP.3), (RP.4), and (RP.+) be in force. If $$u\in \mathsf {W}^{1,p}(X,\mathfrak {m})$$, then there exists a sequence $$\lbrace u_k\rbrace _{k}\subset \mathsf {W}^{1,1}(X,\mathfrak {m})\cap L^p(X,\mathfrak {m})$$ such that $$|\nabla u_k|\in L^p(X,\mathfrak {m})$$ for all $$k\in \mathbb {N}$$ and

$$$$\begin{equation*} {u}_{k}\to u\text{ and }|\nabla {u}_{k}|\to |\nabla u{|}_{p}\text{ both in }{L}^{p}(X,\mathfrak{m})\text{ as }k\to +\infty . \end{equation*}$$$$

3 CHEEGER SETS IN PERIMETER-MEASURE SPACES

In this section, we work in a measure space endowed with a perimeter functional as in Section 2.1.

3.1 $$N$$-Cheeger constant and $$N$$-Cheeger sets

We begin by introducing the central notions of the present paper.

Definition 3.1.Let $$N\in \mathbb {N}$$. An $$N$$-cluster $$\mathcal {E}$$ is a collection of $$N$$ measurable sets $$\mathcal {E}=\lbrace \mathcal {E}(i)\rbrace _{i=1}^N \subset \mathcal {A}$$ satisfying:

• $$0<\mathfrak {m}(\mathcal {E}(i))<+\infty$$ for all $$i=1,\ldots , N$$;
• $$\mathfrak {m}(\mathcal {E}(i)\cap \mathcal {E}(j))=0$$ for all $$i,j=1,\ldots , N$$ with $$i\ne j$$;
• $$P(\mathcal {E}(i))<+\infty$$ for all $$i=1,\ldots , N$$.

Each of the $$\mathcal {E}(i)$$, $$i=1,\dots,N$$, is called a chamber.

Definition 3.2. ($$N$$-admissible set)Let $$N\in \mathbb {N}$$. We say that $$\Omega \in \mathcal {A}$$ is N-admissible if there exists an $$N$$-cluster $$\mathcal {E}=\lbrace \mathcal {E}(i)\rbrace _{i=1}^N\subset \Omega$$.

Remark 3.3.Let $$N\in \mathbb {N}$$. Trivially, if $$\Omega \in \mathcal {A}$$ is $$N$$-admissible, then it is $$M$$-admissible for all integers $$M\leqslant N$$.

Definition 3.4. ($$N$$-Cheeger constant and $$N$$-Cheeger sets)Let $$N\in \mathbb {N}$$ and let $$\Omega \in \mathcal {A}$$ be an $$N$$-admissible set. The $$N$$-Cheeger constant of $$\Omega$$ is

$$$$\begin{equation*} {h}_{N}(\mathrm{\Omega})=\inf\left\{\sum _{i=1}^{N}\frac{P(\mathcal{E}(i))}{\mathfrak{m}(\mathcal{E}(i))}:\ \mathcal{E}=\{\mathcal{E}(i)\}_{i=1}^{N}\subset \mathrm{\Omega}\ \text{is an}\ N\text{-cluster}\right\}. \end{equation*}$$$$

If $$\mathcal {C}=\lbrace \mathcal {C}(i)\rbrace _{i=1}^N$$ is an $$N$$-cluster realizing the above infimum, we call it an $$N$$-Cheeger set (or cluster) of $$\Omega$$. We let $$\mathcal {C}_N(\Omega)$$ be the collection of all $$N$$-Cheeger sets of $$\Omega$$.

Remark 3.5.By definition, as $$\Omega$$ is required to be $$N$$-admissible, the $$N$$-Cheeger constant of $$\Omega$$ is finite. Moreover, by Remark 3.3, so it is $$h_M(\Omega)$$ for all integers $$M$$ such that $$M\leqslant N$$. We also refer to Proposition 3.10.

3.1 Existence of $$N$$-Cheeger sets

We prove that the existence of $$N$$-Cheeger clusters of $$\Omega$$ is ensured whenever the perimeter functional possesses properties (P.4), (P.5), and (P.6), and the set $$\Omega \in \mathcal {A}$$ is $$N$$-admissible with finite $$\mathfrak {m}$$-measure. These requests are not necessary though, as some examples at the end of this section show.

Theorem 3.6.Let properties (P.4), (P.5), and (P.6) be in force. Let $$N\in \mathbb {N}$$, and let $$\Omega \in \mathcal {A}$$ be an $$N$$-admissible set with $$\mathfrak {m}(\Omega)\in (0,+\infty)$$. Then there exists an $$N$$-Cheeger set of $$\Omega$$.

Proof.On the one hand, since $$\Omega$$ is $$N$$-admissible, there exists an $$N$$-cluster $$\mathcal {E}\subset \Omega$$, which immediately implies that $$h_N(\Omega)<+\infty$$. On the other hand, for any $$N$$-cluster $$\mathcal {E}=\lbrace \mathcal {E}(i)\rbrace _{i=1}^N$$ of $$\Omega$$, property (P.6) gives

$$$$\begin{equation*} \sum _{i=1}^N\frac{P(\mathcal {E}(i))}{\mathfrak {m}(\mathcal {E}(i))}\geqslant N f(\mathfrak {m}(\Omega))\,, \end{equation*}$$$$

hence

$$$$\begin{equation*} h_N(\Omega)\geqslant Nf(\mathfrak {m}(\Omega))>0\,. \end{equation*}$$$$

Now let $$\lbrace \mathcal {E}_k\rbrace _{k\in \mathbb {N}}\subset \Omega$$ be a minimizing sequence, that is,

$$$$\begin{equation*} \underset{k\to +\infty}{\lim}\sum _{i=1}^{N}\frac{P({\mathcal{E}}_{k}(i))}{\mathfrak{m}({\mathcal{E}}_{k}(i))}={h}_{N}(\mathrm{\Omega}). \end{equation*}$$$$

Clearly, for any $$k\in \mathbb {N}$$ sufficiently large and any $$i=1,\ldots , N$$, we have

$$$$\begin{equation*} P(\mathcal {E}_k(i))\leqslant \mathfrak {m}(\Omega)\sum _{i=1}^N\frac{P(\mathcal {E}_k(i))}{\mathfrak {m}(\mathcal {E}_k(i))}\leqslant 2\mathfrak {m}(\Omega)h_N(\Omega), \end{equation*}$$$$

and thus,

$$$$\begin{equation*} \sup _{k}{\left\lbrace \max _i{\left\lbrace P(\mathcal {E}_k(i)) \right\rbrace} \right\rbrace} \leqslant 2\mathfrak {m}(\Omega)h_N(\Omega), \end{equation*}$$$$

which is finite, having assumed $$\mathfrak {m}(\Omega)<+\infty$$.

By (P.4) and (P.5) (recall also Remark 2.2), possibly passing to a subsequence, for each $$i=1,\dots, N$$, there exists $$\mathcal {E}(i)\in \mathcal {A}$$ such that $$\mathcal {E}(i)\subset \Omega$$, with $$\mathfrak {m}(\mathcal {E}(i))\in [0,\mathfrak {m}(\Omega)]$$, $$P(\mathcal {E}(i))\leqslant 2\mathfrak {m}(\Omega)h_N(\Omega)$$, and $$\mathfrak {m}(\mathcal {E}_k(i)\bigtriangleup \mathcal {E}(i))\rightarrow 0^+$$ as $$k\rightarrow +\infty$$. Now, using (P.6), for all $$k\in \mathbb {N}$$ sufficiently large and any $$i\in \lbrace 1,\ldots , N\rbrace$$, we get

$$$$\begin{equation} f(\mathfrak {m}(\mathcal {E}_k(i)))\leqslant \frac{P(\mathcal {E}_k(i))}{\mathfrak {m}(\mathcal {E}_k(i))}. \end{equation}$$$$(3.1)

The behavior of $$f$$ near zero prescribed by (P.6) immediately implies that $$\mathfrak {m}(\mathcal {E}(i))\ne 0$$ for all $$i\in \lbrace 1,\ldots , N\rbrace$$, as otherwise a contradiction would arise with $$h_N(\Omega)<+\infty$$. Indeed, on the one hand, being $$\lbrace \mathcal {E}_k(i)\rbrace _k$$ a minimizing sequence, and owing to (3.1), there exists $$\bar{k}\gg 1$$ such that for all $$k\geqslant \bar{k}$$, we have

$$$$\begin{equation*} f(\mathfrak {m}(\mathcal {E}_k(i)))\leqslant 2h_N(\Omega). \end{equation*}$$$$

On the other hand, the isoperimetric property (P.6) implies that there exists $$\delta >0$$ such that $$f(x)>2h_N(\Omega)$$ for all $$x\leqslant \delta$$. Hence, we deduce that $$\mathfrak {m}(\mathcal {E}_k(i)) \geqslant \delta$$ for all $$i=1,\dots, N$$ and all $$k\geqslant \bar{k}$$.

It remains to be proved that $$\mathcal {E}=\lbrace \mathcal {E}(i)\rbrace _{i=1}^N$$ is an $$N$$-cluster contained in $$\Omega$$, that is, that the chambers $$\mathcal {E}(i)$$ are pairwise disjoint, and the reader can easily check it on its own.

Consequently, thanks to (P.4), we find that

$$$$\begin{align*} h_N(\Omega)\leqslant \sum _{i=1}^N\frac{P(\mathcal {E}(i))}{\mathfrak {m}(\mathcal {E}(i))}\leqslant \sum _{i=1}^N\liminf _{k\rightarrow +\infty}\frac{P(\mathcal {E}_k(i))}{\mathfrak {m}(\mathcal {E}_k(i))}\leqslant \liminf _{k\rightarrow +\infty} \sum _{i=1}^N\frac{P(\mathcal {E}_k(i))}{\mathfrak {m}(\mathcal {E}_k(i))}=h_N(\Omega), \end{align*}$$$$

and the conclusion follows.$$\Box$$

Let us point out that properties (P.4), (P.5), and (P.6) are all crucial in the above proof. Among them (P.6) looks as the “most artificial”; nevertheless, it is essential in the reasoning: an example where existence fails when (P.6) is missing is given in Example 3.7 below. It is also relevant to point out that these properties provide a sufficient but in no way a necessary condition, as Example 3.8 and Example 3.9 show.

Example 3.7.Consider the measure space $$(X,\mathcal{A},\mathfrak{m})=({\mathbb{R}}^{2},\mathcal{B}({\mathbb{R}}^{2}),w{\mathcal{L}}^{2})$$, where $$\mathcal{B}({\mathbb{R}}^{2})$$ denotes the Borel $$\sigma$$-algebra, $$w\in L^1(\mathbb {R}^2)$$ is defined by

$$$$\begin{equation*} w(x)=\left\{ \def\eqcellsep{&}\begin{array}{ll}{\Vert x{\Vert}^{-\frac{3}{2}}}& {\Vert x\Vert \leqslant 1,}\\ {{e}^{-\Vert x\Vert}}& {\Vert x\Vert >1,}\end{array} \right. \end{equation*}$$$$

and $$P(\cdot)$$ is the Euclidean perimeter. In this setting, properties (P.1) through (P.5) hold, but (P.6) does not.

Within this framework, one has $$h_1(\Omega)=0$$, for any set $$\Omega$$ containing an open neighborhood of the origin. Indeed, it is enough to consider the sequence of balls centered at the origin $$B_r \subset \Omega$$ (for $$r$$ sufficiently small), for which we have

$$$$\begin{equation*} P(B_r) = 2\pi r, \end{equation*}$$$$

and

$$$$\begin{equation*} \mathfrak {m}(B_r) = \displaystyle \int _{B_r} w(x)\, \mathrm{d}x = 2\pi \displaystyle \int _0^r \varrho ^{-\frac{3}{2}} \varrho \, \mathrm{d}\varrho = 4 \pi r^{\frac{1}{2}}. \end{equation*}$$$$

Were to exist $$E\in \mathcal {C}_1(\Omega)$$, then $$P(E)=0$$, and by the Euclidean isoperimetric inequality, we would have $$|E|=0$$. Being the weight $$w\in L^1(\mathbb {R}^2)$$, this would eventually lead to

$$$$\begin{equation*} \mathfrak {m}(E)=\displaystyle \int _{E} w(x)\, \mathrm{d}x=0, \end{equation*}$$$$

contradicting the fact that $$\mathfrak {m}(E)>0$$. This shows that Cheeger sets do not exist. In more generality, the same happens in any measure space $$(X, \mathcal {A}, \mathfrak {m})$$ and in any 1-admissible set $$\Omega \in \mathcal {A}$$ such that $$h_1(\Omega)=0$$ and the only measurable subsets $$E$$ of $$\Omega$$ with $$P(E)=0$$ have zero $$\mathfrak {m}$$-measure.

For the sake of completeness, we shall note that, in the situation depicted in this remark, $$N$$-Cheeger sets exist in any open set $$\Omega$$ not containing the origin, since the weight $$w$$ would be $$L^\infty (\Omega)$$, refer to [19, Prop. 3.3] or to [114, Prop. 3.2].

We now present two simple examples in which the existence of Cheeger sets is ensured even if properties (P.5) and (P.6) do not hold.

Example 3.8.Consider any nonnegative ($$\sigma$$-finite) measure space $$(X,\mathcal {A}, \mathfrak {m})$$, and consider $$P(E)=\mathfrak {m}(E)$$, for all $$E\in \mathcal {A}$$, as perimeter functional. For this choice, while (P.4) holds, neither property (P.5) nor (P.6) hold, the latter because any isoperimetric function $$f$$ is bounded from above by 1. Nevertheless, fixed any $$\Omega \in \mathcal {A}$$, we have $$h_N(\Omega) = N$$, for any integer $$N$$, and any $$N$$-cluster is an $$N$$-Cheeger set.

Example 3.9.Consider any nonnegative ($$\sigma$$-finite) measure space $$(X,\mathcal {A}, \mathfrak {m})$$, and consider $$P(E)=0$$, for all $$E\in \mathcal {A}$$, as perimeter functional. While (P.4) holds, neither property (P.5) nor (P.6) hold. Nevertheless, fixed any $$\Omega \in \mathcal {A}$$, we have $$h_N(\Omega) = 0$$, for any integer $$N$$, and any $$N$$-cluster is an $$N$$-Cheeger set.

3.2 Inequalities between the $$N$$- and $$M$$-Cheeger constants

Proposition 3.10.Let $$\Omega \in \mathcal {A}$$ be an $$N$$-admissible set. Then, for all $$M\in \mathbb {N}$$ with $$M<N$$, one has

$$$$\begin{equation} h_M(\Omega) + h_{N-M}(\Omega) \leqslant h_N(\Omega). \end{equation}$$$$(3.2)

Proof.Let $$M$$ and $$N$$ be fixed integers, with $$M<N$$. Let $$\mathcal {E}$$ be any fixed $$N$$-cluster. For any subset $$J_M$$ of $$\lbrace 1,\dots, N\rbrace$$ of cardinality $$M$$, the $$M$$-cluster $$\lbrace \mathcal {E}(i)\rbrace _{i\in J_M}$$ provides an upper bound to $$h_M(\Omega)$$, whereras the $$(N-M)$$-cluster $$\lbrace \mathcal {E}(i)\rbrace _{i\notin J_M}$$ to $$h_{N-M}(\Omega)$$.

Hence, no matter how we choose $$J_M$$, we have

$$$$\begin{equation*} \sum _{i=1}^N \frac{P(\mathcal {E}(i))}{\mathfrak {m}(\mathcal {E}(i))} = \sum _{i\in J_M} \frac{P(\mathcal {E}(i))}{\mathfrak {m}(\mathcal {E}(i))} + \sum _{i\notin J_M} \frac{P(\mathcal {E}(i))}{\mathfrak {m}(\mathcal {E}(i))} \geqslant h_M(\Omega) + h_{N-M}(\Omega). \end{equation*}$$$$

By taking the infimum among all $$N$$-clusters, the desired inequality follows.$$\Box$$

Corollary 3.11.Let $$\Omega \in \mathcal {A}$$ be an $$N$$-admissible set. Then, for all $$M\in \mathbb {N}$$ such that for some integer $$k$$ one has $$N=kM$$, one has

$$$$\begin{equation} kh_M(\Omega) \leqslant h_N(\Omega). \end{equation}$$$$(3.3)

Remark 3.12.The inequalities (3.2) and (3.3) hold as equalities in some cases, as, for instance, it happens anytime a set has multiple disjoint 1-Cheeger sets. A trivial example of this behavior is given by $$N$$ disjoint and equal balls in the usual Euclidean space.

One can also build connected sets that have this feature. For $$N=2$$, it is enough to consider a standard dumbbell in the usual two-dimensional Euclidean space, that is, the set given by two disjoint equal balls, spaced sufficiently far apart, and connected via a thin tube. Such a set has two connected 1-Cheeger sets $$\mathcal {E}(1)$$ and $$\mathcal {E}(2)$$ given by small perturbations of the two balls, and the 2-cluster $$\mathcal {E}=\lbrace \mathcal {E}(i)\rbrace$$ is necessarily a 2-Cheeger set, refer, for instance, to [93, Ex. 4.5].

An easy connected example for $$N>2$$ is instead given by an $$(N+2)$$-dumbbell in the usual two-dimensional Euclidean space, that is, a set formed by $$N+2$$ disjoint equal balls and linked by a thin tube, say

$$$$\begin{equation*} \bigcup _{i=1}^{N+2} B_1((4i, 0)) \cup {\left((4, 4(N+2))\times (-\varepsilon, +\varepsilon)\right)}, \end{equation*}$$$$

where $$B_1((4i, 0))$$ denotes the two-dimensional Euclidean ball of radius 1 centered at the point $$(4i,0)\in \mathbb {R}^2$$. For $$\varepsilon$$ sufficiently small, and arguing as in [93, Ex. 4.5], it can be shown that such a set has $$N$$ connected and disjoint 1-Cheeger sets, each corresponding to a small perturbation of the $$N$$ balls with two neighboring ones.

3.3 $$M$$-subclusters of $$N$$-Cheeger sets

Given an $$N$$-Cheeger set of $$\Omega$$, consider any of its $$M$$-subcluster. It is natural to imagine that such an $$M$$-cluster is an $$M$$-Cheeger set in the ambient space given by $$\Omega$$ minus the $$N-M$$ chambers not belonging to the subcluster. In this short section, we prove that this is true.

For the sake of clarity of notation, we let $$|J|\in \mathbb {N}\cup \lbrace 0 \rbrace \cup \lbrace +\infty \rbrace$$ be the cardinality of a set $$J\subset \mathbb {N}$$.

Proposition 3.13.Let $$\Omega \in \mathcal {A}$$ be an $$N$$-admissible set, and assume that it has an $$N$$-Cheeger set $$\mathcal {E} = \lbrace \mathcal {E}(i)\rbrace _{i=1}^N \in \mathcal {C}_N(\Omega)$$. For any proper subset $$J\subset \lbrace 1,\dots, N\rbrace$$, let

$$$$\begin{equation} {\mathrm{\Omega}}_{J}=\mathrm{\Omega}\setminus \bigcup _{j\notin J}\mathcal{E}(j), \end{equation}$$$$(3.4)

and let $$\mathcal {E}_J$$ be the $$|J|$$-cluster given by

$$$$\begin{equation*} {\mathcal{E}}_{J}=\{\mathcal{E}(j)\}_{j\in J}. \end{equation*}$$$$

Then, $$\mathcal {E}_J$$ is a $$|J|$$-Cheeger set of $$\Omega _J$$.

Proof.It is enough to prove the claim for a subset $$J$$ of cardinality $$N-1$$, and then to reason by induction. In particular, up to relabeling, we can assume $$J$$ to be the proper subset $$\lbrace 1,\dots, N-1\rbrace$$.

As both $$\Omega$$ and $$\mathcal {E}(N)$$ are measurable, so it is the set $$\Omega _{J}$$. Moreover, this latter is $$(N-1)$$-admissible because there exists at least the $$(N-1)$$-cluster $$\lbrace \mathcal {E}(i)\rbrace _{i=1}^{N-1}$$.

By contradiction, assume that $$\lbrace \mathcal {E}(i)\rbrace _{i=1}^{N-1}$$ is not an $$(N-1)$$-Cheeger set of $$\Omega _J$$. Then, for $$\varepsilon$$ small enough, we find a different $$(N-1)$$-cluster $$\lbrace \mathcal {F}(i)\rbrace _{i=1}^{N-1}$$ with

$$$$\begin{equation*} \sum _{i=1}^{N-1}\frac{P(\mathcal {F}(i))}{\mathfrak {m}(\mathcal {F}(i))} < h_{N-1}(\Omega)+\varepsilon < \sum _{i=1}^{N-1}\frac{P(\mathcal {E}(i))}{\mathfrak {m}(\mathcal {E}(i))}. \end{equation*}$$$$

It is then immediate that the $$N$$-cluster

$$$$\begin{equation*} \{\mathcal{F}(i)\}_{i=1}^{N}=\{\mathcal{F}(1),\ldots ,\mathcal{F}(N-1),\mathcal{E}(N)\} \end{equation*}$$$$

contradicts the minimality of the $$N$$-cluster $$\lbrace \mathcal {E}(i)\rbrace _{i=1}^{N}$$ in $$\Omega$$.$$\Box$$

3.4 Properties of $$N$$-Cheeger sets

Proposition 3.14. (Basic properties of $$N$$-Cheeger sets)Let $$\lbrace \Omega _k\rbrace _k\subset \mathcal {A}$$ be a collection of $$N$$-admissible sets. The following hold for all integers $$M\leqslant N$$:

• (i) if $$\Omega _1\subset \Omega _2$$, then $$h_M(\Omega _1)\geqslant h_M(\Omega _2)$$;
• (ii) if (P.6) is in force, and $$\mathfrak {m}(\Omega _k)\rightarrow 0^+$$, then $$h_M(\Omega _k)\rightarrow +\infty$$;
• (iii) if (P.4), (P.5), and (P.6) are in force, and $$\Omega _k\rightarrow \Omega$$ in $$L^1(X, \mathfrak {m})$$, with $$\mathfrak {m}(\Omega)\in (0,+\infty)$$, then
  $$$$\begin{equation*} h_M(\Omega)\leqslant \liminf _k h_M(\Omega _k). \end{equation*}$$$$
  Moreover, if also (P.3) is in force, $$P(\Omega)$$ is finite, and $$P(\Omega _k) \rightarrow P(\Omega)$$, then
  $$$$\begin{equation*} h_M(\Omega)= \lim _k h_M(\Omega _k). \end{equation*}$$$$

Proof.Recall that an $$N$$-admissible set $$\Omega$$ is also $$M$$-admissible for all integers $$M\leqslant N$$, see Remark 3.3.

Proof of (i). For any two fixed $$N$$-admissible sets with $$\Omega _1\subset \Omega _2$$, any $$M$$-cluster of $$\Omega _1$$ is also an $$M$$-cluster of $$\Omega _2$$. The inequality immediately follows by definition of $$M$$-Cheeger constant.

Proof of (ii). In virtue of (3.3) and the positivity of $$h_M(\Omega)$$, it is enough to prove the claim for $$M=1$$. Fix $$\varepsilon >0$$, and for all $$k$$, let $$C_k \subset \Omega _k$$ be such that

$$$$\begin{equation*} h_1(\Omega _k) + \varepsilon \geqslant \frac{P(C_k)}{\mathfrak {m}(C_k)}. \end{equation*}$$$$

Then, by (P.6), we have

$$$$\begin{equation*} h_1(\Omega _k) +\varepsilon \geqslant f(\mathfrak {m}(C_k)), \end{equation*}$$$$

and the claim follows by the monotonicity of the measure paired with the hypothesis that the $$\mathfrak {m}$$-measure of $$\Omega _k$$ vanishes, that is, $$\mathfrak {m}(C_k)\leqslant \mathfrak {m}(\Omega _k) \rightarrow 0$$, and the behavior of $$f$$ prescribed by (P.6).

Proof of (iii). Without loss of generality, we can assume that there exists a constant $$C_1<+\infty$$ independent of $$k$$ such that

$$$$\begin{equation*} \liminf _k h_M(\Omega _k)\leqslant C_1, \end{equation*}$$$$

as otherwise there is nothing to prove. Let us consider the (not relabeled) sequence realizing the $$\liminf$$. Since $$\Omega _k$$ is converging in $$L^1(X, \mathfrak {m})$$ to a set of finite $$\mathfrak {m}$$-measure, we can also assume that $$\mathfrak {m}(\Omega _k)$$ is equibounded, independent of $$k$$, that is,

$$$$\begin{equation*} \mathfrak {m}(\Omega _k)\leqslant C_2. \end{equation*}$$$$

Thus, by Theorem 3.6 for each $$k$$, there exists an $$M$$-Cheeger set $$\mathcal {E}_k$$ for $$\Omega _k$$. Moreover, for any $$k$$, we have

$$$$\begin{equation} \sum _{i=1}^M P(\mathcal {E}_k(i)) \leqslant h_M(\Omega _k) \mathfrak {m}(\Omega _k) \leqslant C. \end{equation}$$$$(3.5)

Hence, by (P.5), we can extract a (not relabeled) subsequence $$\lbrace \mathcal {E}_k(i)\rbrace _k$$ such that for all indexes $$i$$, the chamber $$\mathcal {E}_k(i)$$ converges in $$L^1(X, \mathfrak {m})$$ to a limit set $$\mathcal {E}(i)$$ necessarily contained in $$\Omega$$ up to $$\mathfrak {m}$$-null sets. Moreover, by (P.6), one necessarily has $$\mathfrak {m}(\mathcal {E}(i))>0$$ as otherwise a contradiction with the finiteness of $$\liminf _k h_M(\Omega _k)$$ would arise. Hence, $$\mathcal {E}$$ is an $$M$$-cluster of $$\Omega$$. Thus, owing to (P.4) and the fact that $$\mathcal {E}_k$$ is an $$M$$-Cheeger cluster of $$\Omega _k$$, we have

$$$$\begin{equation*} h_M(\Omega) \leqslant \sum _{i=1}^M \frac{P(\mathcal {E}(i))}{\mathfrak {m}(\mathcal {E}(i))} \leqslant \liminf _k \sum _{i=1}^M \frac{P(\mathcal {E}_k(i))}{\mathfrak {m}(\mathcal {E}_k(i))} = \liminf _k h_M(\Omega _k), \end{equation*}$$$$

that is, the first part of the claim.

To show the second part, let us pick an $$M$$-Cheeger cluster $$\mathcal {E}$$ of $$\Omega$$, which exists since we are under the assumptions of Theorem 3.6. Let us consider the collections

$$$$\begin{equation*} \lbrace \mathcal {E}_k(i) = \mathcal {E}(i) \cap \Omega _k\rbrace, \end{equation*}$$$$

which are $$M$$-clusters of $$\Omega _k$$ for $$k$$ sufficiently large. Clearly, for each fixed $$i$$, we have that $$\mathcal {E}_k(i)$$ converges in $$L^1(X, \mathfrak {m})$$ to $$\mathcal {E}(i)$$, while $$\mathcal {E}_k(i) \cup \Omega _k$$ to $$\Omega$$. Therefore, by (P.3), for each $$i$$, we have

$$$$\begin{align*} P(\mathcal {E}_k(i)) \leqslant P(\mathcal {E}(i)) + P(\Omega _k) - P(\mathcal {E}(i)\cup \Omega _k). \end{align*}$$$$

Taking the $$\limsup _k$$, using the assumption of the convergence of $$P(\Omega _k)$$, we have

$$$$\begin{equation*} \limsup _k P(\mathcal {E}_k(i)) \leqslant P(\mathcal {E}(i)) + P(\Omega) - \liminf _k P(\mathcal {E}(i)\cup \Omega _k) \leqslant P(\mathcal {E}(i)). \end{equation*}$$$$

Together with (P.4) this implies that, for each $$i$$, $$\lim _k P(\mathcal {E}_k(i))$$ exists and equals $$P(\mathcal {E}(i))$$. Combining this fact with the minimality of $$\mathcal {E}$$ for $$h_M(\Omega)$$ and the first part of the claim, we conclude the proof.$$\Box$$

Remark 3.15.Notice that to prove point (iii), all requests come into play. Indeed, a priori, one can work with an “almost-infimizing” $$M$$-cluster for $$h_M(\Omega _k)$$ and find an analogous of (3.5) up to an additive factor $$\varepsilon \mathfrak {m}(\Omega _k)$$. Then, the compactness granted by (P.5) is needed, but in order to ensure that the limiting collection