On the Relaxation of Gauss’s Capillarity Theory Under Spanning Conditions

Abstract

We study a variational model for soap films in which the films are represented by sets with fixed small volume rather than surfaces. In this problem, a minimizing sequence of completely “wet" films, or sets of finite perimeter spanning a wire frame, may converge to a film containing both wet regions of positive volume and collapsed (dry) surfaces. When collapsing occurs, these limiting objects lie outside the original minimization class and instead are admissible for a relaxed problem. Here we show that the relaxation and the original formulation are equivalent by approximating the collapsed films in the relaxed class by wet films in the original class.

1 Introduction

We analyze a model for soap films based on the classical Gauss free energy functional from capillarity theory [14, Ch. 1.4]. In this model, which was proposed by King-Maggi-Stuvard in [20], one minimizes the surface tension energy among sets with small volume v that satisfy a spanning condition with respect to a wire frame ${\textbf{W}}\subset {\mathbb {R}}^{n+1}$, the complement of which is the container accessible to the soap. Informally, the problem is

$$\begin{aligned} \inf \{{\mathcal {H}}^n(\partial E \setminus {\textbf{W}}) : E \subset {\mathbb {R}}^{n+1}\setminus {\textbf{W}}\,,\,\,|E|=v\,, \partial E\ \text{ spans }\ {\textbf{W}}\}\,. \end{aligned}$$

(1.1)

In this paper we use the notion of “spanning" from [26], which generalizes the idea of Harrison-Pugh [15]; see Section 2 for a complete discussion. The addition of the volume constraint adds a length scale to the minimal surface model, which is recovered in the vanishing volume limit. The soap film capillarity model is thus well-suited for describing some features of real films that cannot be captured by minimal surfaces. Such features include the thickened tubes of liquid “wetting" a line of Y-point singularities, which play an important role in behaviors such as drainage and are known in the physical literature as Plateau borders; see e.g. [22, 23] and the books [32] and [6, Ch. 2].

An important aspect of the model is the phenomenon of collapsing, which occurs in regions where it is energetically convenient for the (multiplicity one) boundaries of a minimizing sequence $\{E_j\}_j$ to collapse onto a multiplicity two surface; see Fig. 1 below. Collapsing presents mathematical challenges stemming from the fact that the limit of a minimizing sequence for (1.1), or generalized minimizer, may not belong to the original class, but rather to a relaxed class also including collapsed competitors. This complicates the study of properties such as regularity and the wetting of singularities, since the convergence of the minimizing sequence to the generalized minimizer only enables one to test the minimality amongst objects which also arise as limits of sets of finite perimeter and not against the entire relaxed class.

Our main goal is to show that (1.1) and its relaxation to collapsed films are in fact equivalent minimization problems. Concise statements of the main results in this article are as follows:

• 1. every collapsed competitor in the relaxed class can be approximated by a sequence of non-collapsed competitors with the same volume (Theorem 2.3);

• 2. generalized minimizers arising as limits in (1.1) minimize the relaxed energy functional among the entire relaxed class (Theorem 2.4).

The strengthened minimality of (ii) simplifies the mathematical investigation of collapsing and wetting phenomena in the soap film capillarity model.

The paper is organized as follows. Section 2 contains the necessary background and precise statements of our results. After collecting some preliminaries in Sect. 3, we then prove Theorems 2.3-2.4 in Sect. 4.

Fig. 1

On the left is a non-collapsed set $E'$ approximating the collapsed minimizer on the right. $K\setminus \partial ^* E$ is the three black segments

2 Statements

2.1 Spanning and the Plateau Problem

In order to formulate any Plateau-type problem, one must make mathematically rigorous the notion that a soap film “spans a given wire frame.” The approach taken here is based on a generalization by F. Maggi, D. Restrepo, and the author [26] of the idea of Harrison-Pugh [15]. Among the various versions of Plateau’s problem, a key feature of the Harrison-Pugh version is that it leads to minimizers that exhibit the physical singularities predicted by Plateau and validated mathematically in [31]. We mention that [15] has spurred much recent progress on the Plateau problem. More generally, Harrison-Pugh [17] proved an existence result in arbitrary dimension/co-dimension for the anisotropic Plateau problem in a large class of ambient spaces and encompassing several different spanning conditions inside a general axiomatic framework. We refer the reader also to [7,8,9,10,11,12, 16, 17] and the references therein for additional works in this vein.

Let $n\ge 0$, fix closed ${\textbf{W}}\subset {\mathbb {R}}^{n+1}$ (the wire frame), and set $\Omega ={\mathbb {R}}^{n+1}\setminus {\textbf{W}}$. Following the presentation of the Harrison-Pugh spanning condition from [8], we define a spanning class ${\mathcal {C}}$ to be a non-empty family of smooth embeddings of ${\mathbb {S}}^1$ into $\Omega$ which is closed by homotopy in $\Omega$.

Definition 2.1

(Homotopic spanning for closed sets) A relatively closed subset S of $\Omega$ is ${\mathcal {C}}$ -spanning ${\textbf{W}}$ if $S \cap \gamma \ne \varnothing$ for all $\gamma \in {\mathcal {C}}$.

The corresponding formulation of the Plateau problem is

$$\begin{aligned} \ell := \inf \{{\mathcal {H}}^n(S): S\subset \Omega \ \text{ is } \text{ relatively } \text{ closed } \text{ and }\ {\mathcal {C}}\text{-spanning }\ {\textbf{W}}\}. \end{aligned}$$

(2.1)

Compactness and lower-semicontinuity for minimizing sequences yield the existence of a minimizer [8, 15]. In these arguments, the asymptotic minimality of the sequence is utilized to make energy comparisons that yield a limiting surface satisfying Definition 2.1. In scenarios such as the proof of the approximation item (i) from the introduction or the Allen-Cahn setting of [25], it is useful to have a definition of spanning compatible with energy-bounded but non-minimizing sequences. Such a framework was developed in [26], which we now recall.

If ${\mathcal {C}}$ is a spanning class for a closed set ${\textbf{W}}$, we define the tubular spanning class ${\mathcal {T}}({\mathcal {C}})$ associated to ${\mathcal {C}}$ to be the family of triples $(\gamma , \Phi , T)$ such that $\gamma \in {\mathcal {C}}$, $T = \Phi ({\mathbb {S}}^1 \times B_1^n)$, and

$$\begin{aligned} \Phi :{\mathbb {S}}^1 \times \textrm{cl}\,B_1^n \rightarrow \Omega \ \text {is a diffeomorphism with}\ {\left. \Phi \right| _{{\mathbb {S}}^1\times \{0\}} }= \gamma . \end{aligned}$$

When $(\gamma , \Phi , T)\in {\mathcal {T}}({\mathcal {C}})$, the slice of T defined by $s\in {\mathbb {S}}^1$ is

$$\begin{aligned} T[s]=\Phi (\{s\}\times B_1^n). \end{aligned}$$

We will need the measure theoretic notion of connectedness from [4, 5]. For a Borel set G, we let $G^{(t)}$, $t\in [0,1]$, be the points of Lebesgue density t and $\partial ^e G= (G^{(1)} \cup G^{(0)})^c$. For Borel sets S, G, $G_1$ and $G_2$ in ${\mathbb {R}}^{n+1}$, S essentially disconnects G into $\{G_1,G_2\}$ if

$$\begin{aligned} |G\Delta (G_1\cup G_2)|=0,\quad |G_1||G_2|>0,\quad \text {and}\quad G^{(1)}\cap \partial ^e G_1 \cap \partial ^e G_2\overset{{\mathcal {H}}^n}{\subset }S . \end{aligned}$$

Here $|\cdot |$ is the Lebesgue measure ${\mathcal {L}}^{n+1}$ and $A\overset{{\mathcal {H}}^n}{\subset }B$ means ${\mathcal {H}}^n(A \setminus B)=0$; in words, “A is ${\mathcal {H}}^n$-contained in B."

Definition 2.2

(Measure-theoretic spanning) Given a closed set ${\textbf{W}}$ and a spanning class ${\mathcal {C}}$ for ${\textbf{W}}$, a Borel set $S \subset \Omega$ is ${\mathcal {C}}$-spanning ${\textbf{W}}$ if for each $(\gamma ,\Phi , T)\in {\mathcal {T}}({\mathcal {C}})$, ${\mathcal {H}}^1$-a.e. $s\in {\mathbb {S}}^1$ has the following property:

$$\begin{aligned}{} & {} \text{ for }\ {\mathcal {H}}^n\text{-a.e. }\ x\in T[s], \exists \ \text{ a } \text{ partition }\ \{T_1,T_2\}\ \text{ of }\ T\ \text{ with }\ x\in \partial ^e T_1 \cap \partial ^e T_2,\\{} & {} \quad \text{ and } \text{ such } \text{ that }\ S \cup T[s]\ \text{ essentially } \text{ disconnects }\ T\ \text{ into }\ \{T_1,T_2\}. \end{aligned}$$

Associated to this concept is the Plateau problem with measure-theoretic spanning

$$\begin{aligned} \ell _{\textrm{B}} := \inf \{{\mathcal {H}}^n(S): S\subset \Omega \ \text{ is } \text{ Borel } \text{ and }\ {\mathcal {C}}\text{-spanning }\ {\textbf{W}}\}\,. \end{aligned}$$

Definition 2.2 is an acceptable generalization of Definition 2.1 for two reasons.

• Equivalence of spanning definitions: If S is relatively closed in $\Omega$, then S satisfies Definition 2.1 if and only if it satisfies Definition 2.2 [26, Theorem A.1].

• Equivalence of Plateau problems: If $\ell <\infty$, then $\ell _{\textrm{B}}=\ell$, and any minimizer for $\ell _{\textrm{B}}$ is ${\mathcal {H}}^n$-equivalent to some relatively closed minimizer for $\ell$ [26, Theorem 6.1].

2.2 Soap Film Capillarity Model from [19,20,21]

Building upon [27], in which soap films were studied as regions of small volume, for a compact set ${\textbf{W}}\subset {\mathbb {R}}^{n+1}$, in [20] King-Maggi-Stuvard formulated the problem:

$$\begin{aligned} \psi (v)&= \inf \{{\mathcal {H}}^n(\partial E \cap \Omega ): E\in {\mathcal {E}}, |E|=v, \Omega \cap \partial E\ \text{ is }\ {\mathcal {C}}\text{-spanning }\ {\textbf{W}}\}\,, \end{aligned}$$

(2.2)

where

$$\begin{aligned} {\mathcal {E}}=\{E \subset \Omega : E\ \text{ is } \text{ open } \text{ and }\ \partial E\ \text{ is }\ {\mathcal {H}}^n\text{-rectifiable } \}\,. \end{aligned}$$

As explained in the introduction, one cannot expect a minimizing sequence in (2.2) to converge to an admissible competitor for (2.2), so they also consider the relaxed class

$$\begin{aligned} {\mathcal {K}}:= \{(K,E): {E\subset \Omega }&\text{ is } \text{ open } \text{ with } \Omega \cap \textrm{cl}\,(\partial ^* E) = \Omega \cap \partial E\subset K\nonumber \\&K\subset \Omega \ \text{ is } \text{ relatively } \text{ closed } \text{ and }\ {\mathcal {H}}^n\text{-rectifiable }\}\,. \end{aligned}$$

(2.3)

with the corresponding energy

$$\begin{aligned} {\mathcal {F}}_{\textrm{bd}}(K,E) = {\mathcal {H}}^n( \Omega \cap \partial ^* E ) + 2 \,{\mathcal {H}}^n (K \setminus \partial ^* E)\,; \end{aligned}$$

see below for an explanation of the subscript “bd". The main results of [20] are two-fold.

• Existence: There exist generalized minimizers, that is, there exists $(K,E)\in {\mathcal {K}}$ such that K is ${\mathcal {C}}$-spanning ${\textbf{W}}$, $|E|=v$, and $\psi (v)={\mathcal {F}}_{\textrm{bd}}(K,E)$ [20, Theorem 1.4].

• Convergence to the Plateau problem: $\psi (v)\rightarrow 2\,\ell$ with the corresponding subsequential convergence of minimizers in the sense of Radon measures [20, Theorem 1.9].

When $K \setminus \partial E \ne \varnothing$, the generalized minimizer (K, E) is “collapsed” and does not belong to the original admissible class from (2.2). A natural conjecture therefore is that generalized minimizers for (2.2) are minimal among the relaxed class ${\mathcal {K}}$ of possibly collapsed soap films [20, Remark 1.7].

2.3 Formulation and Previous Results from [26]

Following [26, Section 1.4], we use Definition 2.2 to expand the soap film capillarity model to finite perimeter sets. We are actually able to treat two versions of the problem which differ in the choice of spanning set:

$$\begin{aligned} \psi _{\textrm{bd}}(v)&= \inf \{P(E; \Omega ): E\subset \Omega ,\, P(E;\Omega )<\infty , \nonumber \\ {}&\quad |E|=v, \,\Omega \cap \partial ^* E\ {\mathcal {C}}\text{-spans }\ {\textbf{W}}\},\hspace{.5cm}\text{ and } \end{aligned}$$

(2.4)

$$\begin{aligned} \psi _{\textrm{bk}}(v)&= \inf \{P(E; \Omega ):E\subset \Omega ,\,P(E;\Omega )<\infty ,\nonumber \\ {}&\quad |E|=v,\, E^{\scriptscriptstyle {(1)}}\cup (\Omega \cap \partial ^* E)\ {\mathcal {C}}\text{-spans }\ {\textbf{W}}\}. \end{aligned}$$

(2.5)

Here $P(E;\Omega )$, |E|, and $\partial ^* E$ denote the relative perimeter of E in $\Omega$ and the Lebesgue measure and reduced boundary of E, respectively. The subscripts “bd" and “bk" stand for boundary and bulk to emphasize the particular spanning set. The relaxed class is

$$\begin{aligned} {\mathcal {K}}_{\textrm{B}}&:= \big \{(K,E) :K,E\subset \Omega \ \text{ are } \text{ Borel, }\ P(E;F)<\infty \forall F \subset \subset \Omega , \text{ and }\ \Omega \cap \partial ^* E \overset{{\mathcal {H}}^n}{\subset }K\big \}\,, \end{aligned}$$

(2.6)

and the corresponding energies and variational problems are

$$\begin{aligned} {\mathcal {F}}_{\textrm{bd}}(K,E)&= {\mathcal {H}}^n( \Omega \cap \partial ^* E ) + 2 \,{\mathcal {H}}^n (K \setminus \partial ^* E)\,, \\ {\mathcal {F}}_{\textrm{bk}}(K,E)&= {\mathcal {H}}^n( \Omega \cap \partial ^* E ) + 2 \,{\mathcal {H}}^n (K \cap E^{\scriptscriptstyle {(0)}})\,, \end{aligned}$$

and

$$\begin{aligned} \Psi _{\textrm{bd}}(v)&= \inf \{{\mathcal {F}}_{\textrm{bd}}(K,E) : (K,E)\in {\mathcal {K}}_{\textrm{B}}, \, |E|=v,\, K\ \text{ is }\ {\mathcal {C}}\text{-spanning }\ {\textbf{W}}\}\, \end{aligned}$$

(2.7)

$$\begin{aligned} \Psi _{\textrm{bk}}(v)&= \inf \{{\mathcal {F}}_{\textrm{bk}}(K,E) : (K,E)\in {\mathcal {K}}_{\textrm{B}}, \, |E|=v,\, E^{\scriptscriptstyle {(1)}}\cup K\ \text{ is }\ {\mathcal {C}}\text{-spanning }\ {\textbf{W}}\}\,. \end{aligned}$$

(2.8)

Each problem (2.4) and (2.5) has some advantages: for example the lower-dimensional spanning set $\Omega \cap \partial ^* E$ in (2.4) is closer mathematically to the spanning surfaces in the Plateau problem $\ell$, while the bulk spanning problem (2.5) arises in an asymptotic limit of Allen-Cahn problems [25] with spanning constraint. In any case, we suspect that (generalized) minimizers for both problems coincide, at least for small volumes. For the bulk problem $\Psi _{\textrm{bk}}(v)$, in [26], F. Maggi, D. Restrepo, and the author proved:

• Existence: There exists $(K,E)\in {\mathcal {K}}\subset {\mathcal {K}}_{\textrm{B}}$ such that (K, E) is minimal for $\Psi _{\textrm{bk}}(v)$.

• Regularity: There exists $\Sigma$ of Hausdorff dimension at most $n-7$ such that $(\Omega \cap \partial ^* E){\setminus } \Sigma$ is smooth with constant mean curvature and $K {\setminus } (\partial E\cup \Sigma )$ is smooth with zero mean curvature. Also, $\Gamma =(\Omega \cap \partial E) {\setminus } (\partial ^* E\cup \Sigma )$ is locally ${\mathcal {H}}^{n-1}$-rectifiable, and, for any $x\in \Gamma$, there is $r>0$ such that $K\cap B_r(x)$ is a union of two $C^{1,1}$ hypersurfaces touching tangentially at x.

• Convergence to the Plateau problem: $\Psi _{\textrm{bk}}(v)\rightarrow 2\,\ell$ with the corresponding subsequential convergence of minimizers in the sense of Radon measures.

Note that any admissible E for $\psi _{\textrm{bd}}(v)$ or $\psi _{\textrm{bk}}(v)$ corresponds to a pair $(\partial ^*E \cap \Omega ,E)\in {\mathcal {K}}_{\textrm{B}}$ which is admissible for $\Psi _\textrm{bd}(v)$ or $\Psi _{\textrm{bk}}(v)$, respectively. Therefore $\Psi _\textrm{bd}(v) \le \psi _{\textrm{bd}}(v)$, and similarly $\Psi _{\textrm{bk}}(v) \le \psi _{\textrm{bk}}(v)$, but equality is unclear. This leads to the following question:

Are the soap film capillarity model and its relaxation equivalent minimization problems, i.e. does $\varvec{\Psi }_{\textbf{bd}}\varvec{(v)}=\varvec{\psi }_{{\textbf{bd}}}\varvec{(v)}$ and $\varvec{\Psi }_{ \textbf{bk}}\varvec{(v)}=\varvec{\psi }_{{\textbf{bk}}}\varvec{(v)}$?

2.4 Main Results

In our main results, we answer this question. We obtain the equivalence as a corollary of an approximation theorem which we state first.

Theorem 2.3

(Approximation of Collapsed Competitors) Let ${\textbf{W}}\subset {\mathbb {R}}^{n+1}$ be closed and ${\mathcal {C}}$ be a spanning class for ${\textbf{W}}$ such that $\ell <\infty$. If $(K,E)\in {\mathcal {K}}_{\textrm{B}}$, ${\mathcal {H}}^n(K)<\infty$, and $\delta >0$, then there exists a set $E'\subset \Omega$ of finite perimeter in $\Omega$ such that $|E|=|E'|$,

$$\begin{aligned} |E' \Delta E|&\le \delta \,, \end{aligned}$$

(2.9)

$$\begin{aligned} {\mathcal {H}}^n(\partial ^* E' \cap \Omega )&\le {\mathcal {F}}(K,E) + \delta \,, \end{aligned}$$

(2.10)

and, if K is ${\mathcal {C}}$-spanning ${\textbf{W}}$, so is $\Omega \cap \partial ^* E'$, while if $K \cup E^{\scriptscriptstyle {(1)}}$ is ${\mathcal {C}}$-spanning ${\textbf{W}}$, so is $(E')^{\scriptscriptstyle {(1)}}\cup (\Omega \cap \partial ^* E')$.

A consequence is that collapsed and non-collapsed formulations of the soap film capillarity problem are equivalent, and that limits of minimizing sequences for $\psi _{\textrm{bd}}(v)$ or $\psi _{\textrm{bk}}(v)$ are, up to resolving possible volume loss at infinity, minimizers for $\Psi _{\textrm{bd}}(v)$ or $\Psi _{\textrm{bk}}(v)$, respectively.

Theorem 2.4

(Equivalence of collapsed and non-collapsed problems) If ${\textbf{W}}\subset {\mathbb {R}}^{n+1}$ is closed and ${\mathcal {C}}$ is a spanning class for ${\textbf{W}}$ with $\ell <\infty$, then $\Psi _\textrm{bd}(v)=\psi _{\textrm{bd}}(v)$ and $\Psi _{\textrm{bk}}(v)=\psi _{\textrm{bk}}(v)$. Furthermore, if in addition ${\textbf{W}}$ is compact and $\{E_j\}_j$ is a minimizing sequence of finite perimeter sets for $\psi _{\textrm{bd}}(v)$ or $\psi _{\textrm{bk}}(v)$, then up to a subsequence, there exists $(K,E)\in {\mathcal {K}}_{\textrm{B}}$ such that $E_j\rightarrow E$ locally in ${\mathbb {R}}^{n+1}$,

and there exists $B_r(x)\subset (K \cup E \cup {\textbf{W}})^c$ such that $|B_r(x)| = v - |E|$ and $(K \cup \partial B_r(x),E \cup B_r(x))$ is minimal for $\Psi _{\textrm{bd}}(v)$ or $\Psi _{\textrm{bk}}(v)$, respectively.

Remark 2.5

(Loss of volume at infinity) The presence of the ball $B_r(x)$ accounts for possible volume loss at infinity for a minimizing sequence. In the bulk problem $\Psi _{\textrm{bk}}(v)$, this cannot happen for a minimizing sequence [26, Appendix B], and we expect the same to hold for $\Psi _{\textrm{bd}}(v)$.

One benefit of the complete variational theory for $\psi _\textrm{bd}(v)$ and $\psi _{\textrm{bk}}(v)$ provided by Theorem 2.4 is in the study wetting and collapsing, which are expected whenever the minimizer for Plateau’s problem $\ell$ admits a minimizer with Plateau-type singularities (as recently shown to be optimal in a capillarity model for planar soap bubbles [29, Theorem 1.6]; see also [2, 3, 18]). In particular, Theorem 2.4 shows that the limiting pair (K, E) for a minimizing sequence $\{E_j\}_j$ of $\psi _{\textrm{bd}}(v)$ or $\psi _{\textrm{bk}}(v)$ is a proper minimizer in the relaxed problem $\Psi _{\textrm{bd}}(v)$ or $\Psi _{\textrm{bk}}(v)$, respectively. As a consequence, the regularity statements proved for the bulk problem $\Psi _{\textrm{bk}}(v)$ in [26] automatically hold for (K, E) when $\{E_j\}_j$ are admissible for $\psi _{\textrm{bk}}(v)$. In the boundary spanning problem, Theorem 2.3 provides an alternative to the approximation arguments used in [19, 21] to study the Lagrange multipliers and singular sets of generalized minimizers in the presence of collapsing.

3 Notation and Preliminaries

3.1 Notation

As mentioned in the introduction, we will use the notation $A\overset{{\mathcal {H}}^k}{\subset } B$ to signify that ${\mathcal {H}}^k(A \setminus B)=0$. Similarly, $A \overset{{\mathcal {H}}^k}{=} B$ means that ${\mathcal {H}}^k(A \Delta B)=0$. For a Radon measure $\mu$, the k-dimensional lower density of $\mu$ at a point x, denoted by $\theta _*^k(\mu )(x)$, is computed via

$$\begin{aligned} \theta _*^k(\mu )(x) = \liminf _{r\rightarrow 0^+}\frac{\mu (\textrm{cl}\,B_r(x))}{\omega _k r^k}\,, \end{aligned}$$

where $\omega _k$ is the k-dimensional volume of the unit ball in ${\mathbb {R}}^k$. A Borel set $S\subset {\mathbb {R}}^{n+1}$ is locally ${\mathcal {H}}^k$-rectifiable if it can be covered up to an ${\mathcal {H}}^k$-null set by a countable union of Lipschitz images of maps from ${\mathbb {R}}^k$ to ${\mathbb {R}}^{n+1}$. Lastly, for a locally ${\mathcal {H}}^k$-finite set S, we use the notation ${\mathcal {R}}(S)$ to denote the (locally) ${\mathcal {H}}^k$-rectifiable part of S, where, following [30, 13.1], any locally ${\mathcal {H}}^k$-finite set S can be uniquely partitioned up to ${\mathcal {H}}^k$-null sets into a locally ${\mathcal {H}}^k$-rectifiable portion ${\mathcal {R}}(S)$ and a purely ${\mathcal {H}}^k$-unrectifiable portion ${\mathcal {P}}(S)$. The definition and properties of ${\mathcal {P}}(S)$ will not be relevant here.

3.2 Preliminaries

In this section we quote several results from [26]. First we have a fact showing that the rectifiable part of a ${\mathcal {H}}^n$-finite ${\mathcal {C}}$-spanning set is itself ${\mathcal {C}}$-spanning and the compactness theorem mentioned in the introduction.

Proposition 3.1

[26, Lemma 2.2] If ${\textbf{W}}\subset {\mathbb {R}}^{n+1}$ is closed, ${\mathcal {C}}$ is a spanning class for ${\textbf{W}}$, S is ${\mathcal {C}}$-spanning ${\textbf{W}}$, and is a Radon measure in $\Omega$, then ${\mathcal {R}}(S)$ is ${\mathcal {C}}$-spanning ${\textbf{W}}$. Moreover, the sets $T_1$ and and $T_2$ appearing in Definition 2.2 are sets of finite perimeter.

Theorem 3.2

[26, Theorem 1.4] Let ${\textbf{W}}$ be a closed set in ${\mathbb {R}}^{n+1}$, ${\mathcal {C}}$ be a spanning class for ${\textbf{W}}$, and $\{(K_j,E_j)\}_j$ be a sequence in ${\mathcal {K}}_{\textrm{B}}$ such that $\sup _j\,{\mathcal {H}}^n(K_j)<\infty$, and let a Borel set E and Radon measures $\mu _{\textrm{bk}}$ and $\mu _{\textrm{bd}}$ in $\Omega$ be such that $E_j\rightarrow E$ locally in $L^1(\Omega )$ and

(3.1)

(3.2)

as $j\rightarrow \infty$. Then the sets

$$\begin{aligned} K_{\textrm{bk}}:= & {} \big (\Omega \cap \partial ^* E\big ) \cup \Big \{x\in \Omega \cap E^{\scriptscriptstyle {(0)}}: \theta ^n_*(\mu _{\textrm{bk}})(x)\ge 2 \Big \}, \end{aligned}$$

(3.3)

$$\begin{aligned} K_{\textrm{bd}}:= & {} \big (\Omega \cap \partial ^* E\big ) \cup \Big \{x\in \Omega \setminus \partial ^*E: \theta ^n_*(\mu _\textrm{bd})(x)\ge 2 \Big \}, \end{aligned}$$

(3.4)

are such that $(K_{\textrm{bk}},E),(K_{\textrm{bd}},E)\in {\mathcal {K}}_{\textrm{B}}$ and

(3.5)

(3.6)

with

$$\begin{aligned} \liminf _{j\rightarrow \infty }{\mathcal {F}}_{\textrm{bk}}(K_j,E_j)\ge {\mathcal {F}}_\textrm{bk}(K_{\textrm{bk}},E),\qquad \liminf _{j\rightarrow \infty }{\mathcal {F}}_\textrm{bd}(K_j,E_j)\ge {\mathcal {F}}_{\textrm{bd}}(K_{\textrm{bd}},E). \end{aligned}$$

(3.7)

Finally, if $K_j\cup E_j^{\scriptscriptstyle {(1)}}$ (resp. $K_j$) is ${\mathcal {C}}$-spanning ${\textbf{W}}$ for every j, then $K_{\textrm{bk}}\cup E^{\scriptscriptstyle {(1)}}$ (resp. $K_\textrm{bd}$) is ${\mathcal {C}}$-spanning ${\textbf{W}}$.

In addition to Theorem 3.2, we will also use some further tools originating from its proof, for which we will need some terminology. If $U\subset {\mathbb {R}}^{n+1}$ is Lebesgue measurable, $\{U_i\}_i$ is a Lebesgue-partition of U if $\{U_i\}_i$ is a finite or countable Borel partition of $U\setminus U'$ where $|U'|=0$. Two Lebesgue-partitions $\{U_i\}_i$ and $\{\tilde{U}_j\}_j$ of U are Lebesgue-equivalent if there is a bijection $\sigma$ such that $|U_i\Delta \tilde{U}_{\sigma (i)}|=0$ for every i. If S is a Borel set, a Lebesgue partition $\{U_i\}_i$ of U induced by S must satisfy

$$\begin{aligned} {U^{\scriptscriptstyle {(1)}}}\cap \partial ^eU_i\overset{{\mathcal {H}}^n}{\subset }S\qquad \forall i. \end{aligned}$$

(3.8)

Lastly, $\{U_i\}_i$ is called an essential partition of U induced by S if it is a Lebesgue partition of U induced by S and, for every i, S does not essentially disconnect $U_i$.

Theorem 3.3

[26, Theorem 2.1] If U is a bounded set of finite perimeter, and if S is a Borel set with ${\mathcal {H}}^n(S \cap {U^{\scriptscriptstyle {(1)}}})<\infty$, then there exists an essential partition $\{U_i\}_{i\in I}$ of U induced by S such that each $U_i$ is a set of finite perimeter and

$$\begin{aligned} \sum _{i\in I}P(U_i;{U^{\scriptscriptstyle {(1)}}}) \le 2\,{\mathcal {H}}^n(S \cap {U^{\scriptscriptstyle {(1)}}}). \end{aligned}$$

(3.9)

Moreover: (a): if $S^*$ is a Borel set with ${\mathcal {H}}^n(S^* \cap {U^{\scriptscriptstyle {(1)}}})<\infty$, $S^*$ is ${\mathcal {H}}^n$-contained in S, $\{U_j^*\}_{j\in J}$ is a Lebesgue partition of U induced by $S^*$, and $\{U_i\}_i$ is an essential partition of U induced by S, then

$$\begin{aligned} \bigcup _{j\in J}\partial ^*U^*_j\overset{{\mathcal {H}}^n}{\subset }\bigcup _{i\in I}\partial ^*U_i; \end{aligned}$$

(3.10)

(b): if S and $S^*$ are ${\mathcal {H}}^n$-finite sets in ${U^{\scriptscriptstyle {(1)}}}$, and either $S^*={\mathcal {R}}(S)$ or $S^*$ is ${\mathcal {H}}^n$-equivalent to S, then S and $S^*$ induce ${\mathcal {L}}^{n+1}$-equivalent essential partitions of U.

Theorem 3.4

[26, Theorem 3.1, Remark 3.2] If ${\textbf{W}}\subset {\mathbb {R}}^{n+1}$ is a closed set in ${\mathbb {R}}^{n+1}$, ${\mathcal {C}}$ is a spanning class for ${\textbf{W}}$, K is a Borel set locally ${\mathcal {H}}^n$-finite in $\Omega$, E has locally finite perimeter in $\Omega$, and $\Omega \cap \partial ^*E\overset{{\mathcal {H}}^n}{\subset }K$, then the set $S=K\cup E^{{\scriptscriptstyle {(1)}}}$ is ${\mathcal {C}}$-spanning ${\textbf{W}}$ if and only if, for every $(\gamma ,\Phi , T)\in {\mathcal {T}}({\mathcal {C}})$ and ${\mathcal {H}}^1$-a.e. $s\in {\mathbb {S}}^1$, denoting by $\{U_i\}$ the essential partition of T induced by ${\mathcal {R}}(K)\cup T[s]$,

$$\begin{aligned}{} & {} {T[s]\cap E^{\scriptscriptstyle {(0)}}\overset{{\mathcal {H}}^n}{\subset }\cup _i \partial ^* U_i}. \end{aligned}$$

(3.11)

Remark 3.5

We point out that for any $(K,E)\in {\mathcal {K}}_{\textrm{B}}$ and $(\gamma ,\Phi ,T)\in {\mathcal {T}}({\mathcal {C}})$ and $s\in {\mathbb {S}}^1$,

$$\begin{aligned} \partial ^* E\cap T\ \text{ is }\ {\mathcal {H}}^n\ \text{ contained } \text{ in }\ \cup _i \partial ^* U_i\,, \end{aligned}$$

(3.12)

where $\{U_i\}$ is the essential partition of T induced by ${\mathcal {R}}(K) \cup T[s]$. Indeed, $\{E \cap T,T\setminus E\}$ is a Lebesgue partition of T induced by ${\mathcal {R}}(K) \cup T[s]$ since $T \cap {\mathcal {R}}(K) \overset{{\mathcal {H}}^n}{\supset }T \cap \partial ^* E$. By Theorem 3.3(a), we deduce (3.12).

Remark 3.6

(Consequences of $\ell <\infty$) If ${\mathcal {C}}$ is a spanning class for ${\textbf{W}}$ such that $\ell <\infty$, then no $\gamma \in {\mathcal {C}}$ is homotopic in $\Omega$ to a point - if $\gamma \in {\mathcal {C}}$ were homotopic to a point, then the only ${\mathcal {C}}$-spanning set is $\Omega$, which has infinite ${\mathcal {H}}^n$-measure.

4 Proofs of the Approximation and Minimality Theorems

4.1 Proof of Theorem 2.3

The proof in the boundary spanning case proceeds in several steps and is based on an iteration procedure. The most delicate part of the iteration is ensuring that the reduced boundary of our approximating set will be ${\mathcal {C}}$-spanning. Each stage involves “replacing" a portion of the ${\mathcal {H}}^n$-rectifiable set ${\mathcal {R}}(K)\setminus \partial ^* E$ with a small set of finite perimeter ${\mathcal {D}}_i$. By choosing each ${\mathcal {D}}_i$ carefully to satisfy certain properties, we will be able to ensure that $\partial ^* E\cup \cup _i \partial ^* {\mathcal {D}}_i$ is ${\mathcal {C}}$-spanning and that

$$\begin{aligned} \partial ^* \big [E \Delta \bigcup _i {\mathcal {D}}_i \big ] \overset{{\mathcal {H}}^n}{=}\partial ^* E \cup \bigcup _i \partial ^* {\mathcal {D}}_i\,. \end{aligned}$$

Together, these two properties will allow us to conclude that $\Omega \cap \partial ^* (E \Delta (\cup _i {\mathcal {D}}_i))$ is ${\mathcal {C}}$-spanning, and from here we fix the volumes to obtain $E'$. The bulk spanning case is significantly simpler, since we only have to ensure that $(E')^{\scriptscriptstyle {(1)}}\cup (\Omega \cap \partial ^* E')$ is ${\mathcal {C}}$-spanning and not the (much) smaller set $\Omega \cap \partial ^* E'$.

Proof of Theorem 2.3for $\Psi _{\textrm{bd}}$ An outline is as follows. After making a preliminary reduction of the problem, we prove two necessary facts about measure-theoretic spanning. After that, in step two, we record a basic property of locally ${\mathcal {H}}^n$-rectifiable sets. Then, we devise a technical tool for the iteration in step three. We perform the iteration in step four, and use Theorem 3.2 in steps five and six to take a limit and arrive at our approximating set $E'$.

Step zero: Before making the approximation, we simplify the problem. We claim that to prove the theorem, it is enough to choose $\delta _j \rightarrow 0$ and produce a sequence of sets $\{E_j\}$ with ${\mathcal {C}}$-spanning reduced boundaries such that

$$\begin{aligned} |E_j\Delta E|\le \delta _j\quad \text {and}\quad P(E_j;\Omega ) \le {\mathcal {F}}_{\textrm{bd}}(K,E) + \delta _j\,. \end{aligned}$$

(4.1)

To see why this is sufficient, by the $L^1$-convergence to E and the volume-fixing variations lemma [24, Section 29.6], there exists $C_0>0$, $C_1>0$ such that for all large j, we may obtain $G_j=g_j(E_j)$ for some diffeomorphism $g_j$ which is identity away from finitely many balls, and with

$$\begin{aligned}&|G_j|=|E|\,,\quad |G_j \Delta E_j| \le C_0\Big ||E|-|E_j| \Big |\,,\\ {}&\quad \text {and}\quad |P(G_j;\Omega )-P(E_j;\Omega )| \le C_1\Big ||E|-|E_j| \Big |\,. \end{aligned}$$

Given some $\delta >0$, by choosing j large enough so that $\max \{C_0,C_1\}\delta _j<\delta$, $G_j$ satisfies the conclusions of Theorem 2.3. In particular, the ${\mathcal {C}}$-spanning requirement is fulfilled since $\partial ^* E_j$ is ${\mathcal {C}}$-spanning and the diffeomorphisms $g_j$ preserve the property of being ${\mathcal {C}}$-spanning (as can be seen from Definition 2.2 and the fact that, if B is Borel, $g_j(B^{\small (t)}) = (g_j(B))^{\small (t)}$ if $t=0,1$ and $g_j(\partial ^e B) = \partial ^e (g_j(B))$). Therefore, for the rest of the proof, we consider fixed small $\delta :=\delta _j$ and construct $E_j$ with ${\mathcal {C}}$-spanning reduced boundary such that (4.1) holds.

Step one: Here we prove two claims. The second will ensure that our approximating sets are ${\mathcal {C}}$-spanning.

Claim one: If $(\gamma ,\Phi ,T)\in {\mathcal {T}}({\mathcal {C}})$, $x\in T[s_0]$, $B_r(x){\setminus } T[s_0]$ consists of two connected components $B_r^+(x)$ and $B_r^-(x)$ with $\partial B_r^+(x) \cap \partial B_r^-(x)=T[s_0]\cap B_r(x)$, and there exists an open set D and ball $B_s(y)$ with $B_r(x)\subset D\subset B_s(y)\subset \Omega$, then $B_r^+(x)$ and $B_r^-(x)$ are subsets of two distinct open, connected components $A^+$ and $A^-$ of $(D \cap T){\setminus } T[s_0]$.

The argument relies on the same construction [26, Proof of Theorem A.1] that is behind the equivalence of Definition A and Definition B among closed sets. Crucial to the argument is the inclusion $D\subset B_s(y)$, which ensures that D is contractible in $\Omega$.

To prove the claim, suppose for contradiction that $B_r^+(x)$ and $B_r^-(x)$ both were subsets of a single open, connected component A of $(D \cap T) \setminus T[s_0]$. We are going to use this to construct a curve in ${\mathcal {C}}$ homotopic to a point, which will be a contradiction. Since open connected sets in Euclidean space are path connected, for some $x_+\in B_r^+(x)$ and $x_-\in B_r^-(x)$, we can find a smooth curve $\gamma _1\subset (D \cap T) {\setminus } T[s_0]$ with endpoints $x_+$ and $x_-$. By Sard’s theorem [8, Proof of Lemma 10, Step 2], $\gamma _1$ can be deformed so that it meets $\partial B_{r}(x)$ transversally at a finite set of points, in which case $\gamma _1([0,1]) \setminus \textrm{cl}\,B_{r}(x)$ consists of a finite disjoint union of arcs $\gamma _1((a_i,b_i))$ with $(a_i,b_i)\subset [0,1]$. Since $\gamma _1\cap T[s_0]=\varnothing$ and $T[s_0]$ disconnects $B_r(x)$, we may find i such that $\gamma _1(a_i)\in \textrm{cl}\,B_r^+(x) \cap \partial B_{r}(x)$ and $\gamma _1(b_i)\in \textrm{cl}\,B_r^-(x) \cap \partial B_{r}(x)$ (modulo reversing the orientation of $\gamma _1$). Set $\gamma _2$ to be $\gamma _1$ restricted to $[a_i,b_i]$ so that $\gamma _2([a_i,b_i])\cap B_r(x)=\varnothing$. Now let $\gamma _3:[0,1]\rightarrow B_r(x)$ be an embedding with $\gamma _3(0)=\gamma _2(a_i)$, $\gamma _3(1)=\gamma _2(b_i)$ and such that $\gamma _2([0,1])\cap T[s]\cap B_{r(x)}(x)$ is one point $x_{3}=\gamma _3(t)$ and

$$\begin{aligned} \gamma _3'(t)\ne 0\,. \end{aligned}$$

(4.2)

We may further arrange $\gamma _3$ so that the concatenation $\gamma _*$ of $\gamma _2$ and $\gamma _3$ is smooth. Letting ${\textbf{p}}_{{\mathbb {S}}^1}$ denote the canonical projection map from ${\mathbb {S}}^1\times B_1^n$ to ${\mathbb {S}}^1$, we find from (4.2) and $\gamma _2\cap T[s_0]=\varnothing$ that the ${\mathbb {S}}^1$-valued curve ${\textbf{p}}_{{\mathbb {S}}^1}(\Phi ^{-1}(\gamma _*))$ has Brouwer degree either $+1$ or $-1$ [28, pg. 27]. If its degree is $-1$, let us set $\gamma _{**}(s)=\gamma _{*}({\overline{s}})$ (where the bar denotes complex conjugation) so that $\gamma _{**}$ has the same image as $\gamma _*$ and the degree of ${\textbf{p}}_{{\mathbb {S}}^1}(\Phi ^{-1}(\gamma _{**}))=+1$. If the degree is $+1$, simply set $\gamma _{**}=\gamma _*$. Now the ${\mathbb {S}}^1$-valued curves ${\textbf{p}}_{{\mathbb {S}}^1}(\Phi ^{-1}(\gamma _{**}))$ and ${\textbf{p}}_{{\mathbb {S}}^1}(\Phi ^{-1}(\gamma ))$ both have winding number 1 and are therefore homotopic to one another. Since $B_1^n$ is convex and $\Phi$ is a diffeomorphism, then $\gamma$ is homotopic to $\gamma _{**}$. But this is impossible, since $\gamma _{**} \subset D \subset B_s(y)\subset \Omega$, implying the contractibility of D in $\Omega$, and, by $\ell <\infty$, ${\mathcal {C}}$ does not contain any curves homotopic in $\Omega$ to a point (see Remark (3.6)). This gives the desired contradiction and thus concludes the proof of claim one.

Claim two: If $(\gamma ,\Phi ,T)\in {\mathcal {T}}({\mathcal {C}})$ and D is an open bounded set of finite perimeter with $D\subset B_s(y)\subset \Omega$ for some ball $B_s(y)$, then for ${\mathcal {H}}^n$-a.e. $x\in T[s_0] \cap D$, there exists an element $V_i$ of the essential partition of T induced by $\partial ^* D \cup T[s_0]$ such that $x\in \partial ^* V_i$.

To see why this second claim follows from the first, fix any $B_r(x) \subset D \cap T$ centered at $x\in T[s_0] \cap D$ small enough so that $T[s_0]$ divides $B_r(x)$ into two open connected components, $B_r^+(x)$ and $B_r^-(x)$ with common boundary $T[s_0] \cap B_r(x)$. By claim one, $B_r^+(x)$ and $B_r^-(x)$ are subsets of distinct connected components $A_+$ and $A_-$ of $(D\cap T)\setminus T[s_0]$. Moreover, $A_+$ and $A_-$ are sets of finite perimeter by [13, 4.5.11] since $\partial ^e A_{\pm } \subset T[s_0]\cup \partial ^e D \cup \partial T$. Then setting $B = (D \cap T){\setminus } (A_+ \cup A_-)$ and $C= T {\setminus } D$ (note that $C\ne \varnothing$ since if it were we would have $T\subset B_s(y)$ so that $\gamma$ is homotopic to a point), we find that $\{A_+,A_-,B,C\}$ is a non-trivial Lebesgue partition of T into sets of finite perimeter such that, by standard facts about unions/intersections of sets of finite perimeter [24, Ch. 16],

$$\begin{aligned} T \cap ( \partial ^* A_+ \cup \partial ^* A_- \cup \partial ^* B \cup \partial ^* C) \text{ is }\ {\mathcal {H}}^n\text{-contained } \text{ in }\ \partial ^* D \cap T[s_0]\,. \end{aligned}$$

(4.3)

In words, it is a Lebesgue partition of T induced by $\partial ^* D \cap T[s_0]$, and so therefore Theorem 3.3(a) implies that $\partial ^* A_+ \cap T$ is ${\mathcal {H}}^n$-contained in $\cup _i \partial ^* V_i$, where $\{V_i\}_i$ is the essential partition of T induced by $\partial ^* D \cap T[s_0]$. Since $T[s_0] \cap B_r(x) \subset \partial ^* A_+ \cap T$ and $x\in T[s_0] \cap D$ was arbitrary, this finishes the claim.

Step two: Here we show that if $R\subset \Omega$ is a locally ${\mathcal {H}}^n$-rectifiable set with finite ${\mathcal {H}}^n$-measure, then up to an ${\mathcal {H}}^n$-null set, it can be decomposed into a countable union of pairwise disjoint compact Lipschitz graphs, each of which is contained in some ball disjoint from ${\textbf{W}}$. More precisely, we can write

$$\begin{aligned} R = A_0 \cup \bigcup _{m=1}^\infty f_m(A_m) \end{aligned}$$

(4.4)

where ${\mathcal {H}}^n(A_0)=0$, and, for each $m\ge 1$, there exist $v_m\in {\mathbb {S}}^1$, Lipschitz function $F_m:v_m^\perp \rightarrow {\mathbb {R}}$, and compact $A_m \subset v_m^\perp$ such that

$$\begin{aligned} f_m(y) =y+ F(y)v_m \quad \forall y\in A_m. \end{aligned}$$

and, for some $B_{t_m}(x_m) \subset \subset \Omega$,

$$\begin{aligned} f_m(A_m) \subset B_{t_m}(x_m)\,. \end{aligned}$$

(4.5)

To see this, recall that any locally ${\mathcal {H}}^n$-rectifiable set can be covered by a family of Lipschitz graphs [1, Proposition 2.76] and can therefore be decomposed up to an ${\mathcal {H}}^n$-null set $B_0$ into a countable union of pairwise disjoint Lipschitz graphs $f_m(B_m)$, where each $B_m$ is a Borel subset of some n-dimensional plane. Since each such $B_m$ is ${\mathcal {H}}^n$-equivalent to a countable union of disjoint compact sets, which we make take to be small enough so that their images under $f_m$ are each contained in some ball disjoint from ${\textbf{W}}$, the desired decomposition follows. In the next step, we will use the notation $\partial _{v_m^\perp }$ to denote the natural notion of boundary for subsets of $v_m^\perp$ that arises by identifying $v_m^\perp$ with ${\mathbb {R}}^n$. Let also denote by ${\textbf{p}}_{v_m^\perp }$ the projection onto $v_m^\perp$.

Step three: In this step we prove the following claim, which is the main technical tool for the iteration.

Claim: If $F\subset \Omega$ is relatively closed, $E \subset \Omega$ is a set of locally finite perimeter in $\Omega$, $R\subset \Omega \setminus F$ is a locally ${\mathcal {H}}^n$-rectifiable set with finite ${\mathcal {H}}^n$-measure, $\{v_m\}$, $\{f_m\}$, $\{A_m\}$, and $\{B_{t_m}(x_m)\}$ are vectors, functions, compact sets, and balls corresponding to the decomposition of R from step two, and $\beta \in (0,1)$, then there exists $M\in {\mathbb {N}}$ and open sets of finite perimeter $D_1, \dots , D_M$ with the following properties:

$$\begin{aligned}&{\mathcal {H}}^n\Big (R \setminus \bigcup _{m=1}^M \textrm{cl}\,D_m \Big ) < \frac{{\mathcal {H}}^n(R)}{2}\,, \end{aligned}$$

(4.6)

$$\begin{aligned}&\textrm{cl}\,(D_m) \cap \textrm{cl}\,(D_{m'})=\varnothing \quad \text{ for } m\ne m', 1\le m, m' \le M\,, \end{aligned}$$

(4.7)

$$\begin{aligned}&\textrm{dist}( D_m , F)>0\quad \forall m\le M\,, \end{aligned}$$

(4.8)

$$\begin{aligned}&f_m(A_m)\subset D_m \subset \subset B_{t_m}(x_m)\subset \Omega \quad \forall m\le M\,, \end{aligned}$$

(4.9)

$$\begin{aligned}&\partial D_m\text { is Lipschitz }\text { with}\ {\mathcal {H}}^n(\partial D_m \setminus \partial ^* D_m)=0 \quad \forall m\le M\,, \end{aligned}$$

(4.10)

$$\begin{aligned}&\sum _{m=1}^M|D_m| \le \beta \,, \end{aligned}$$

(4.11)

$$\begin{aligned}&\Big |B_s(z) \cap \bigcup _{m=1}^M D_m \Big | < \beta |B_s(z)|\quad \forall z\in F \cap \Omega \,,\, s>0\,,\end{aligned}$$

(4.12)

$$\begin{aligned}&{\mathcal {H}}^n(\partial D_m \cap \partial ^* E )=0\quad \forall m\le M\,, \end{aligned}$$

(4.13)

$$\begin{aligned}&P(D_m)\le 2\,{\mathcal {H}}^n(f_m(A_m)) + \frac{\beta }{2^m}\quad \forall m \le M\,. \end{aligned}$$

(4.14)

Since there are many requirements, to aid the reader we will make our construction in an order that mirrors the order of (4.6)-(4.14), refining as we go so as to satisfy each successive requirement.

First, let us choose $M\in {\mathbb {N}}$ large enough so that

$$\begin{aligned} \sum _{m=M+1}^\infty {\mathcal {H}}^n(f_m(A_m)) < \frac{{\mathcal {H}}^n(R)}{2}\,. \end{aligned}$$

With M chosen as such, any open sets $D_m$ satisfying $f_m(A_m)\subset D_m$ will therefore give

$$\begin{aligned}&{\mathcal {H}}^n\Big (R \setminus \bigcup _{m=1}^M \textrm{cl}\,D_m \Big )=\sum _{m'=1}^\infty {\mathcal {H}}^n\Big (f_{m'}(A_{m'}) \setminus \bigcup _{m=1}^M \textrm{cl}\,D_m \Big ) \nonumber \\ {}&\quad \le \sum _{m=M+1}^\infty {\mathcal {H}}^n(f_m(A_m)) < \frac{{\mathcal {H}}^n(R)}{2}\,, \end{aligned}$$

(4.15)

which is (4.6). Next, using the fact that $f_m(A_m)$ are pairwise disjoint compact sets each of which is at positive distance from F (by virtue of $R\subset \Omega \setminus F$ and the relative closedness of F) and each other, we choose may choose open sets $A_m\subset T_m\subset v_m^\perp$ with smooth boundary such that for $m\le M$,

$$\begin{aligned} \min \{ \textrm{dist}( f_m(T_m), f_{m'}(T_{m'}))&,\,\textrm{dist}(f_m(T_m),F)\}>0\quad \forall m'\ne m\,, \end{aligned}$$

(4.16)

$$\begin{aligned} f_m(\textrm{cl}\,T_m)&\overset{(4.5)}{\subset } B_{t_m}(x_m)\,, \end{aligned}$$

(4.17)

$$\begin{aligned} {\mathcal {H}}^n(f_m(T_m))&< {\mathcal {H}}^n(f_m(A_m)) + \frac{\beta }{3\times 2^{m}}\,. \end{aligned}$$

(4.18)

Let us set $(T_m)_t=\{x\in T_m: \textrm{dist}(x,\partial _{v_m^\perp } T_m)>t\}$. Now, for each $m\in \{1,\dots ,M\}$, by (4.16)-(4.17), there exists $\tau _{m,0}>0$ such that for any $\tau _{m,i}<\tau _{m,0}$, $i=1,2$,

$$\begin{aligned} A_m \subset (T_m)_{\tau _{m,1}} \end{aligned}$$

and the cylindrical type sets over the graphs $f_m((T_m)_{\tau _{m,1}})$ defined by

$$\begin{aligned} C_{\tau _{m,1},\tau _{m,2}}^m:=\{z\in {\mathbb {R}}^{n+1}:{\textbf{p}}_{v_m^\perp }(z)\in (T_m)_{\tau _{m,1}},\, |z\cdot v_m - F_m({\textbf{p}}_{v_m^\perp }(z))|<\tau _{m,2} \}\,, \end{aligned}$$

satisfy

$$\begin{aligned} \min \{ \textrm{dist}(C_{\tau _{m,1},\tau _{m,2}}^m, C_{\tau _{m',1},\tau _{m',2}}^{m'})&,\,\textrm{dist}(C_{\tau _{m,1},\tau _{m,2}}^m,F)\}>0\quad \forall m'\ne m, \tau _{m',i}<\tau _{m',0}\,, \end{aligned}$$

(4.19)

$$\begin{aligned} f_m(\textrm{cl}\,C_{\tau _{m,1},\tau _{m,2}}^m)&\subset B_{t_m}(x_m)\,, \end{aligned}$$

(4.20)

that is, (4.7)-(4.9) with $C_{\tau _{m,1},\tau _{m,2}}^m=D_m$. Postponing our choice of $\tau _{m,i}$ until later on (we will need this choice for (4.13)), note that each $\partial C_{\tau _{m,1},\tau _{m,2}}^m$ is Lipschitz since it is the image through the Lipschitz map

$$\begin{aligned} x\mapsto {\textbf{p}}_{v_m^\perp }(x) + [F_m({\textbf{p}}_{v_m^\perp }(x))+x\cdot v_m]v_m \end{aligned}$$

of the Lipschitz cylinder $\partial _{v_m^\perp } (T_m)_{\tau _{m,1}} + \{tv_m:|t|\le \tau _{m,2}\}$, which is (4.10). Furthermore, it is clear that we may decrease the $\tau _{m,0}$’s if necessary to ensure (4.11). For (4.12), we first notice that

$$\begin{aligned} |C_{\tau _{m,1},\tau _{m,2}}^m| = 2\tau _{m,2} {\mathcal {H}}^n((T_m)_{\tau _{m,1}})\le 2\tau _{m,0} {\mathcal {H}}^n(T_m)\rightarrow 0\quad \text {as }\tau _{m,0}\rightarrow 0\,. \end{aligned}$$

(4.21)

Together with the fact that $C_{\tau _{m,1},\tau _{m,2}}^m$ is at positive distance from F (by (4.19)), (4.21) guarantees that we can decrease $\tau _{m,0}$ if necessary to ensure that for any $\tau _{m,1},\tau _{m,2}<\tau _{m,0}$,

$$\begin{aligned} \Big |B_s(z) \cap \bigcup _{m=1}^M C_{\tau _{m,1},\tau _{m,2}}^m \Big |&< \beta |B_s(z)|\quad \forall z\in F \cap \Omega \,,\, s>0\,, \end{aligned}$$

(4.22)

which is (4.12) with $C_{\tau _{m,1},\tau _{m,2}}^m=D_m$. To recap, every choice of $\tau _{m,1},\tau _{m,2}<\tau _{m,0}$ and corresponding $C_{\tau _{m,1},\tau _{m,2}}^m=D_m$ gives sets that satisfy (4.6)-(4.12), and so we must choose the small parameters so as to satisfy (4.13)-(4.14). By the smoothness of $\partial _{v_m^\perp } T_m$, we may decrease each $\tau _{m,0}$ so that for $\tau <\tau _{m,0}$, $\partial _{v_m^\perp } (T_m)_{\tau }$ is smooth. Now, aiming towards (4.13), we recall that since ${\mathcal {H}}^n(\partial ^* E )<\infty$, for any uncountable family of pairwise disjoint sets, $\partial ^*E$ can have ${\mathcal {H}}^n$-positive overlap with at mostly countably many. Therefore, there is an at most countable set ${\mathcal {T}}_1\subset (0,\min _m \tau _{m,0})$ such that

$$\begin{aligned} \sup _{1\le m\le M} \big \{{\mathcal {H}}^n(\partial ^* E \cap \{x\in {\mathbb {R}}^{n+1}: {\textbf{p}}_{v_m^\perp }(x)\in \partial _{v_m^\perp }(T_m)_\tau \})\big \}=0 \quad \forall \tau \notin {\mathcal {T}}_1\,. \end{aligned}$$

(4.23)

Let use choose $\tau _1\notin {\mathcal {T}}_1$ and set $\tau _{m,1}=\tau _1$ for $1\le m \le M$. Then by (4.23), the lateral boundaries of our sets $C_{\tau _{m,1},\tau _{m,2}}^m$ will have trivial ${\mathcal {H}}^n$-overlap with $\partial ^* E$. To now choose $\tau _{m,2}$, we compute

$$\begin{aligned} {\mathcal {H}}^n(\partial C_{\tau _{m,1},\tau _{m,2}}^m)&=2\,{\mathcal {H}}^n(f_m((T_m)_{\tau _1}))+ 2\,\tau _{m,2} {\mathcal {H}}^{n-1}(\partial _{v_m^\perp } (T_m)_{\tau _1}) \nonumber \\&\overset{(4.18)}{<} 2\,{\mathcal {H}}^n(f_m(A_m)) + \frac{2\beta }{3\times 2^{m}} + 2\,\tau _{m,2} {\mathcal {H}}^{n-1}(\partial _{v_m^\perp } (T_m)_{\tau _1}) \,. \end{aligned}$$

(4.24)

Next, by the exact same countability argument as leading to (4.23), there is an at most countable set ${\mathcal {T}}_2\subset (-\min _m \tau _{m,0},\min _m \tau _{m,0})$ such that

$$\begin{aligned} \sup _{1\le m\le M} \big \{{\mathcal {H}}^n(\partial ^* E \cap \{x\in {\mathbb {R}}^{n+1}: F({\textbf{p}}_{v_m^\perp }(x))+\tau = x\cdot v_m \})\big \}=0 \quad \forall \tau \notin {\mathcal {T}}_2\,. \end{aligned}$$

(4.25)

We choose $\tau _2$ with $\pm \tau _2\notin {\mathcal {T}}_2$ and small enough so that

$$\begin{aligned} \sup _{1\le m \le M}2\,\tau _{2} {\mathcal {H}}^{n-1}(\partial _{v_m^\perp } (T_m)_{\tau _1})< \frac{\beta }{3\times 2^m} \end{aligned}$$

and set $D_m = C_{\tau _{1},\tau _{2}}^m$. By (4.24) and the choice of $\tau _2$, we have

$$\begin{aligned} {\mathcal {H}}^n(\partial D_m) < 2\,{\mathcal {H}}^n(f_m(A_m)) + \frac{\beta }{2^m}\,, \end{aligned}$$

which is (4.14). Lastly, by the definitions of $D_m$ and ${\mathcal {T}}_1$ and ${\mathcal {T}}_2$, we have

$$\begin{aligned}&{\mathcal {H}}^n(\partial D_m\cap \partial ^* E)= {\mathcal {H}}^n(\partial C_{\tau _{1},\tau _{2}}^m\cap \partial ^* E) \\ {}&\le {\mathcal {H}}^n(\partial ^* E \cap \{x\in {\mathbb {R}}^{n+1}: {\textbf{p}}_{v_m^\perp }(x)\in \partial _{v_m^\perp }(T_m)_{\tau _1}\})\\&\quad + {\mathcal {H}}^n(\partial ^* E \cap \{x\in {\mathbb {R}}^{n+1}: F({\textbf{p}}_{v_m^\perp }(x))\pm \tau _2= x\cdot v_m \})\\&=0\,, \end{aligned}$$

which is (4.13).

Step four: For the rest of the proof, we fix $\delta _j$ and attempt to verify the reduction of the theorem laid out in step zero with this $\delta _j$. Here we will iteratively apply the previous step’s claim to ${\mathcal {R}}(K){\setminus } \partial ^* E$. To begin with, we apply the claim with $F=\varnothing$, $R={\mathcal {R}}(K){\setminus } \partial ^* E$, and $\beta =\min \{1/8,\delta _j/4\}$, yielding $M_0\in {\mathbb {N}}$ and open sets of finite perimeter $D_1^0,\dots , D_M^0$. If we set ${\mathcal {D}}_0=\cup _{m=1}^M D_m^0$, then by (4.7), $\partial {\mathcal {D}}_0$ is the disjoint union of $\partial D_m^0$, and is therefore Lipschitz with $P({\mathcal {D}}_0) = \sum _{m=1}^{M_0}P(D_m^0)$. By (4.6)-(4.14), our sets $D_m^0$ and ${\mathcal {D}}_0$ satisfy:

$$\begin{aligned}&{\mathcal {H}}^n \big ( {\mathcal {R}}(K)\setminus (\partial ^* E \cup \textrm{cl}\,{\mathcal {D}}_0) \big ) < \frac{{\mathcal {H}}^n({\mathcal {R}}(K)\setminus \partial ^* E)}{2} \end{aligned}$$

(4.26)

$$\begin{aligned}&f_m^0(A_m^0) \subset D_m^0 \subset \subset B_{t_{m,0}}(x_{m,0}) \subset \Omega \quad \forall m\le M_0 \end{aligned}$$

(4.27)

$$\begin{aligned}&|{\mathcal {D}}_0| \le \frac{\delta _j}{4}\,, \end{aligned}$$

(4.28)

$$\begin{aligned}&{\mathcal {H}}^n(\partial {\mathcal {D}}_0 \cap \partial ^* E) = 0 \,, \end{aligned}$$

(4.29)

$$\begin{aligned}&P({\mathcal {D}}_0)=\sum _{m=1}^{M_0}P(D_m^0) \le \sum _{m=1}^{M_0} 2{\mathcal {H}}^n(f_m^0(A_m)) + \frac{\delta _j}{4} \le 2{\mathcal {H}}^n({\mathcal {R}}(K) \cap \textrm{cl}\,{\mathcal {D}}_0 \setminus \partial ^* E) + \frac{\delta _j}{4}\,. \end{aligned}$$

(4.30)

With the initial step complete, now we iteratively apply the claim for $k=1,2,3,\dots$, with, at the k-th stage, $F=F_{k-1} = \textrm{cl}\,{\mathcal {D}}_{0}\cup \dots \cup \textrm{cl}\,{\mathcal {D}}_{k-1}$, $R=R_{k} = {\mathcal {R}}(K) {\setminus } (\partial ^* E \cup F_{k-1})$, and $\beta =\min \{2^{-k-3},\delta _j / 2^{k+2}\}$. We obtain a sequence of families of sets $D_1^{k},\dots , D_{M_k}^{k}$ with ${\mathcal {D}}_k = \cup _{m=1}^{M_k}D_m^k$ having Lipschitz boundary $\cup _{m=1}^{M_k} \partial D_m^k$ and satisfying, for each $k\ge 1$,

$$\begin{aligned}&{\mathcal {H}}^n \big ( {\mathcal {R}}(K)\setminus (\partial ^* E\cup F_{k-1} \cup \textrm{cl}\,{\mathcal {D}}_{k} )\big ) < \frac{{\mathcal {H}}^n({\mathcal {R}}(K)\setminus (\partial ^* E\cup F_{k-1}))}{2}\,, \end{aligned}$$

(4.31)

$$\begin{aligned}&\textrm{dist}( {\mathcal {D}}_k, F_{k-1}) > 0\,, \end{aligned}$$

(4.32)

$$\begin{aligned}&f_m^k(A_m^k) \subset D_m^k \subset \subset B_{t_{m,k}}(x_{m,k}) \subset \Omega \quad \forall m\le M_k\,, \end{aligned}$$

(4.33)

$$\begin{aligned}&|{\mathcal {D}}_k| \le \frac{\delta _j}{2^{k+2}}\,, \end{aligned}$$

(4.34)

$$\begin{aligned}&|B_s(z) \cap {\mathcal {D}}_k| < \frac{|B_s(z)|}{2^{k+3}}\quad \forall z\in \textrm{cl}\,{\mathcal {D}}_0 \cup \dots \cup \textrm{cl}\,{\mathcal {D}}_{k-1}\,,\, s>0\,, \end{aligned}$$

(4.35)

$$\begin{aligned}&{\mathcal {H}}^n(\partial {\mathcal {D}}_k \cap \partial ^* E) = 0 \,, \end{aligned}$$

(4.36)

$$\begin{aligned}&P({\mathcal {D}}_k)=\sum _{m=1}^{M_k}P(D_m^k) \le \sum _{m=1}^{M_k} 2{\mathcal {H}}^n(f_m^k(A_m^k)) + \frac{\delta _j}{2^{k+2}} \nonumber \\ {}&\le 2{\mathcal {H}}^n({\mathcal {R}}(K) \cap \textrm{cl}\,{\mathcal {D}}_k \setminus \partial ^* E) + \frac{\delta _j}{2^{k+2}}\,. \end{aligned}$$

(4.37)

We remark that a consequence of (4.32) and the fact that each ${\mathcal {D}}_k$ has Lipschitz boundary is

(4.38)

Now by repetitively using (4.31) and finally (4.26), we obtain

$$\begin{aligned} {\mathcal {H}}^n\big ({\mathcal {R}}(K) \setminus (\partial ^* E \cup F_k ) \big )< \frac{{\mathcal {H}}^n({\mathcal {R}}(K)\setminus \partial ^* E)}{2^{k+1}}\,. \end{aligned}$$

(4.39)

We also claim that for each $k\ge 0$, the ${\mathcal {H}}^n$-rectifiable sets

$$\begin{aligned}&S_k:= (\Omega \cap \partial ^* E)\cup \partial F_k \cup [{\mathcal {R}}(K) \setminus (F_k\cup \partial ^* E)] \nonumber \\&\quad \text{ are }\ {\mathcal {H}}^n\text{-equivalent } \text{ to } [\Omega \cap \partial ^*( E \Delta F_k)] \cup [{\mathcal {R}}(K) \setminus (F_k \cup \partial ^* E )] \end{aligned}$$

(4.40)

and are also ${\mathcal {C}}$-spanning. For the ${\mathcal {H}}^n$-equivalence, we first observe that since ${\mathcal {D}}_{k'}$ are each at mutual positive distance from each other for $0\le k' \le k$ and have Lipschitz boundaries,

$$\begin{aligned} \partial F_k \overset{{\mathcal {H}}^n}{=}\partial ^* F_k \overset{{\mathcal {H}}^n}{=} \cup _{k'=0}^k \partial {\mathcal {D}}_{k'}\,. \end{aligned}$$

(4.41)

Therefore, by the “non-overlapping” (4.29) and (4.36) of each $\partial {\mathcal {D}}_{k'}$ with $\partial ^* E$, we have

$$\begin{aligned} {\mathcal {H}}^n(\partial ^* E \cap \partial ^* F_k) = 0\,. \end{aligned}$$

(4.42)

As a consequence, the formula for the reduced boundary of the symmetric difference of two sets [24, Exercise 16.5] gives

$$\begin{aligned} \partial ^*(E \Delta F_k)\overset{{\mathcal {H}}^n}{=}(\partial ^* E \setminus \partial ^* F_k) \cup (\partial ^* F_k \setminus \partial ^* E) \overset{{\mathcal {H}}^n}{=} \partial ^* E \cup \partial ^* F_k \overset{{\mathcal {H}}^n}{=} \partial ^* E \cup \partial F_k\,, \end{aligned}$$

(4.43)

which immediately implies (4.40). Note that as a consequence of this equivalence,

$$\begin{aligned} (S_k,E \Delta F_k) \in {\mathcal {K}}_{\textrm{B}}\,, \end{aligned}$$

(4.44)

with (4.43), (4.40), and the ${\mathcal {H}}^n$-equivalence of $\partial F_k$ and $\partial ^* F_k$ giving

$$\begin{aligned} {\mathcal {R}}(S_k) \setminus \partial ^* (E \Delta F_k) \overset{{\mathcal {H}}^n}{=} S_k \setminus (\partial ^* E \cup \partial ^* F_k) \overset{{\mathcal {H}}^n}{=} {\mathcal {R}}(K) \setminus (F_k \cup \partial ^* E)\,. \end{aligned}$$

(4.45)

To prove that $S_k$ is ${\mathcal {C}}$-spanning, let us fix $(\gamma ,\Phi ,T)\in {\mathcal {T}}({\mathcal {C}})$. We first recall that ${\mathcal {R}}(K)$ is ${\mathcal {C}}$-spanning by Proposition 3.1. Therefore, by Theorem 3.4, there exists $J\subset {\mathbb {S}}^1$ of full ${\mathcal {H}}^1$-measure such that if $s\in J$, then, letting $\{U_i\}_i$ denote the essential partition of T induced by ${\mathcal {R}}(K) \cup T[s]$,

$$\begin{aligned}{} & {} T[s]\ \text{ is }\ {\mathcal {H}}^n\text{-contained } \text{ in }\ \cup _i \partial ^* U_i. \end{aligned}$$

(4.46)

Again by Theorem 3.4, to show that $S_k$ is ${\mathcal {C}}$-spanning, it is enough to show that for $s\in J$, letting $\{V_i\}_i$ denote the essential partition of T induced by ${\mathcal {R}}(S_k)\cup T[s]$,

$$\begin{aligned}{} & {} T[s]\ \text{ is }\ {\mathcal {H}}^n\text{-contained } \text{ in }\ \cup _i \partial ^* V_i. \end{aligned}$$

(4.47)

Towards demonstrating (4.47), we first claim that

$$\begin{aligned} {\mathcal {P}}_k:=\{W : W= U_i \setminus F_k \text{ or } W =T \cap D_m^{k'} \text{ for } \text{ some }\ k' \le k\ \text{ and }\ m\le M_{k'}\} \end{aligned}$$

is a Lebesgue partition of T induced by $S_k\cup T[s]$. Since $\{U_i\}$ is a Lebesgue partition of T and $F_k$ is the disjoint union of $D_m^{k'}$ for $k'\le k$ and $m\le M_{k'}$, the fact that this is a Lebesgue partition of T follows. Let us enumerate its (nontrivial) sets as $W_i$. To see that ${\mathcal {P}}_k$ is induced by $S_k\cup T[s]$, we begin by appealing to [24, Ch. 16] and the fact that $\{U_i\}$ is induced by ${\mathcal {R}}(K) \cup T[s]$ to see that

$$\begin{aligned}&T \cap \partial ^* (U_i \setminus F_k)\overset{{\mathcal {H}}^n}{\subset }T \cap [(\partial ^* U_i \cap F_k^{(0)})\nonumber \\ {}&\cup \partial ^* F_k ] \subset T \cap \big [\big (({\mathcal {R}}(K)\cup T[s]) \setminus F_k\big ) \cup \partial F_k\big ] \overset{{\mathcal {H}}^n}{\subset }S_k \cup T[s]\,, \end{aligned}$$

(4.48)

where the last inclusion follows by the definition of $S_k$. On the other hand, if $W_i=T \cap D_m^{k'}$, then

$$\begin{aligned} \partial ^* W_i \cap T \subset \partial D_m^{k'}\cap T\subset \partial F_k \cap T\subset S_k\,. \end{aligned}$$

(4.49)

Therefore, since ${\mathcal {P}}_k$ is induced by $S_k \cup T[s]$, we may employ Theorem 3.3(a) to conclude that

$$\begin{aligned} \cup _i \partial ^* W_i \overset{{\mathcal {H}}^n}{\subset }\cup _i \partial ^* V_i\,. \end{aligned}$$

(4.50)

Finally we check (4.47) in three cases. By (4.46), it is enough to verify among those $x\in T[s]$ such that $x\in \partial ^* U_i$ for some i. In the first case, suppose that $x\in T[s]$ is such that $\textrm{dist}(x,F_k)>0$ and $x\in \partial ^* U_i$ for some i. Then $x\in \partial ^* (U_i{\setminus } F_k)$, and thus ${\mathcal {H}}^n$-a.e. such point belongs to $S_k \cup T[s]$ as desired by (4.50). The second two cases both involve $x\in T[s]$ such that $\textrm{dist}(x,F_k)=0$. Suppose that $x\in \partial F_k$. We recall from Remark 3.5, which applies to the pair $(S_k,E\Delta F_k)$ by (4.44), that $\partial ^* (E \Delta F_k) \cap T$ is ${\mathcal {H}}^n$-contained in $\cup _i \partial ^* V_i$. Since $\partial ^* (E\Delta F_k)\overset{{\mathcal {H}}^n}{=}\partial ^* E \cup \partial F_k$ by (4.43), we find that ${\mathcal {H}}^n$-a.e. $x\in T[s]\cap \partial F_k$ belongs to some $\partial ^* V_i$ as desired. Lastly, let us consider those $x\in \textrm{int}\,F_k \cap T[s]$; it is for these x that we will use the result of step one. By the definition of $F_k$, every such x must belong to $D_m^{k'}$ for some $k' \le k$ and $m\le M_{k'}$. So we consider those x belonging to some fixed $D_m^{k'}$. According to (4.33), $D_m^{k'}\subset \subset B_{t_{m,k'}}(x_{m,k'})\subset \Omega$. Then since $D_m^{k'}$ has Lipschitz boundary, we are precisely in the position of being able to apply claim two from step one to deduce that ${\mathcal {H}}^n$-a.e. $x\in T[s] \cap D_m^{k'}$ belongs to $\partial ^* \tilde{V}_i$ for some $\tilde{V}_i$ in the essential partition of T induced by $\partial ^* D_m^{k'} \cup T[s]$. Since $\partial ^* D_m^{k'} \cup T[s] \subset S_k \cup T[s]$ by (4.49), Theorem 3.3(a) is in force, and we conclude that ${\mathcal {H}}^n$-a.e. $x\in T[s] \cap D_m^{k'}$ belongs to $\cup _i \partial ^* V_i$. This finishes the third case and therefore the proof of (4.47).

Step five: As a preliminary computation before taking the limit in k in the next step, we set $F = \cup _{k=0}^\infty {\mathcal {D}}_k$, and claim that it is a set of finite perimeter with

$$\begin{aligned} |F|&\le \frac{\delta _j}{2}\quad \text {and} \end{aligned}$$

(4.51)

$$\begin{aligned} \partial ^* F \cap \Omega&\text{ is }\ {\mathcal {H}}^n\text{-equivalent } \text{ to } \cup _k \partial ^* {\mathcal {D}}_k\,. \end{aligned}$$

(4.52)

Let us begin by observing that by (4.34) (the volume estimate on ${\mathcal {D}}_k$), we have

$$\begin{aligned} |F|\le \lim _{k\rightarrow \infty } \sum _{k'=0}^k \frac{\delta _j}{2^{k'+2}} \le \frac{\delta _j}{2}\,, \end{aligned}$$

which is (4.51), and also

$$\begin{aligned} |F \Delta F_k| \rightarrow 0\quad \text {as }k\rightarrow \infty \,. \end{aligned}$$

(4.53)

By (4.37) and the fact that the sets $\textrm{cl}\,{\mathcal {D}}_k$ are mutually pairwise disjoint, we have

$$\begin{aligned}&\limsup _{k\rightarrow \infty }P(F_k;\Omega ) = \limsup _{k\rightarrow \infty }\sum _{k'=0}^k P({\mathcal {D}}_k;\Omega ) \nonumber \\ {}&\le \limsup _{k\rightarrow \infty }\sum _{k'=0}^k 2{\mathcal {H}}^n({\mathcal {R}}(K) \cap \textrm{cl}\,{\mathcal {D}}_{k'} \setminus \partial ^* E) + \frac{\delta _j}{2^{k'+2}} \nonumber \\&\le 2{\mathcal {H}}^n({\mathcal {R}}(K)\setminus \partial ^* E) + \frac{\delta _j}{2}\,, \end{aligned}$$

(4.54)

which combined with the $L^1$ convergence of $F_k$ to F implies that F is a set of finite perimeter in $\Omega$. It remains to determine $\partial ^* F\cap \Omega$. Now since $F= \cup _k {\mathcal {D}}_k$ and those sets are open, we have

$$\begin{aligned} \Omega \cap \partial ^* F \subset \Omega \cap \partial F \subset \cup _k \partial {\mathcal {D}}_k \overset{{\mathcal {H}}^n}{\subset }\cup _k \partial ^* {\mathcal {D}}_k\,, \end{aligned}$$

(4.55)

where in the last containment we used the fact, recorded above (4.31), that ${\mathcal {D}}_k$ has Lipschitz boundary. To prove the reverse inclusion, we claim that it is enough to prove that

$$\begin{aligned} \frac{1}{2}\le \limsup _{r\rightarrow 0}\frac{|F \cap B_r(x)|}{|B_r(x)|}\le \frac{3}{4}\quad \text {for}\ {\mathcal {H}}^n\text {-a.e.}\ x\in \partial ^* {\mathcal {D}}_k, k\ge 0\,. \end{aligned}$$

(4.56)

Indeed, if (4.56) held, then since $\Omega \overset{{\mathcal {H}}^n}{\subset }\Omega \cap ( F^{(1)} \cup F^{(0)} \cup \partial ^* F)$ and $\Omega \cap \partial ^* F \overset{{\mathcal {H}}^n}{=}\Omega \cap \partial ^e F$ by a theorem of Federer (see e.g. [1, Theorem 3.61]), we would thus have

$$\begin{aligned} \cup _k \partial ^* {\mathcal {D}}_k \overset{{\mathcal {H}}^n}{\subset }\Omega \cap \partial ^* F. \end{aligned}$$

Combining this inclusion with (4.55) would complete the proof of (4.52). Now suppose $x\in \partial ^* {\mathcal {D}}_k$. By (4.32), there exists $r_x>0$ such that $B_{r_x}(x) \cap {\mathcal {D}}_{k'}=\varnothing$ for $k'<k$. Then by this avoidance and (4.35) applied for those ${\mathcal {D}}_{k'}$ with $k'>k$, we may estimate for $r<r_x$

$$\begin{aligned} \frac{|{\mathcal {D}}_k \cap B_r(x)|}{|B_r(x)|}&\le \frac{|F \cap B_r(x)|}{|B_r(x)|}\\&\le \frac{\sum _{k'\ge k}|{\mathcal {D}}_{k'}\cap B_r(x)|}{|B_r(x)|} \le \frac{|{\mathcal {D}}_k\cap B_r(x)|}{|B_r(x)|}+ \frac{\sum _{k'> k}2^{-k'-3}|B_r(x)|}{|B_r(x)|} \\ {}&\le \frac{|{\mathcal {D}}_k\cap B_r(x)|}{|B_r(x)|}+\frac{1}{4}\,. \end{aligned}$$

Taking $r\rightarrow 0$ and using the fact that $x\in \partial ^* {\mathcal {D}}_k$ gives (4.56).

Before moving on to the final step, we record for later use that by (4.52) and (4.36),

$$\begin{aligned} {\mathcal {H}}^n(\Omega \cap (\partial ^* F \cap \partial ^* E)) = 0\,. \end{aligned}$$

(4.57)

Again by [24, Exercise 16.5], this gives

$$\begin{aligned} \Omega \cap \partial ^* (E\Delta F) \overset{{\mathcal {H}}^n}{=}\Omega \cap [(\partial ^* E \setminus \partial ^* F)\cup (\partial ^* F \setminus \partial ^* E)] \overset{{\mathcal {H}}^n}{=}\Omega \cap (\partial ^* E \cup \partial ^* F)\,. \end{aligned}$$

(4.58)

Step six: Our goal now is to apply the compactness Theorem 3.2 to the pairs $(S_k,E \Delta F_k)$ and then verify (4.1). Now each $(S_k,E \Delta F_k)$ belongs to ${\mathcal {K}}_{\textrm{B}}$ by (4.44) and we have

$$\begin{aligned} \sup _j {\mathcal {H}}^n(S_k) \le P(E;\Omega ) + \sup _{k} P(F_k;\Omega ) + {\mathcal {H}}^n({\mathcal {R}}(K)) \overset{(4.54)}{<} \infty \,. \end{aligned}$$

Note that $E\Delta F_k\overset{L^1}{\rightarrow }\ E\Delta F$ by (4.53). Therefore, Theorem 3.2 applies and says that the pair $(S,E\Delta F)$ belongs to ${\mathcal {K}}_{\textrm{B}}$ and S is ${\mathcal {C}}$-spanning, where

$$\begin{aligned} S&= (\Omega \cap \partial ^* (E\Delta F)) \cup \{x\in \Omega \setminus \partial ^*(E\Delta F) : \theta _*^n(\mu )(x)\ge 2 \} \\&\overset{(4.58)}{=} (\Omega \cap (\partial ^* E\cup \partial ^* F)) \cup \{x\in \Omega \setminus (\partial ^*E\cup \partial ^* F) : \theta _*^n(\mu )(x)\ge 2 \}\,, \end{aligned}$$

where $\mu$ is the weak star limit of

Next, by (4.39), ${\mathcal {H}}^n({\mathcal {R}}(K){\setminus }(\partial ^* E \cup F_k) )\rightarrow 0$, which implies then that $\mu$ is in fact the weak star limit of the measures . Therefore, for any $B_r(x)\subset \subset \Omega$ with $\mu (\partial B_r(x))=0$, (4.42)-(4.43) and (4.38) give

$$\begin{aligned}&\mu (B_r(x)) =\lim _{k\rightarrow \infty }{\mathcal {H}}^n(\partial ^* E \cup \partial ^* F_k\cap B_r(x))\\&\quad = \lim _{k\rightarrow \infty } {\mathcal {H}}^n(\partial ^* E\cap B_r(x)) + \sum _{k'=0}^k {\mathcal {H}}^n(\partial ^* {\mathcal {D}}_{k'} \cap B_r(x)) \\&\quad = {\mathcal {H}}^n(\partial ^* E\cap B_r(x)) + \sum _{k'=0}^\infty {\mathcal {H}}^n(\partial ^* {\mathcal {D}}_{k'} \cap B_r(x))\,. \end{aligned}$$

But by (4.57)-(4.58) and (4.52), we also have

$$\begin{aligned} {\mathcal {H}}^n (B_r(x) \cap \partial ^* (E \Delta F)) ={\mathcal {H}}^n(B_r(x) \cap \partial ^* E)+ {\mathcal {H}}^n \Big (B_r(x) \cap \bigcup _{k=0}^\infty \partial ^* {\mathcal {D}}_k \Big )\,. \end{aligned}$$

(4.59)

We have thus shown that $\mu (B_r(x)) = {\mathcal {H}}^n (B_r(x) \cap \partial ^* (E \Delta F))$ for every $B_r(x)\subset \subset \Omega$ such that $\mu (\partial B_r(x))=0$. It follows that . Therefore, $S{\setminus } (\partial ^* (E \Delta F))\overset{{\mathcal {H}}^n}{=}\varnothing$, and so $\Omega \cap \partial ^* (E \Delta F)$ is ${\mathcal {C}}$-spanning since S is ${\mathcal {C}}$-spanning.

We now have a set of finite perimeter $E \Delta F=:E_j$ with ${\mathcal {C}}$-spanning reduced boundary, and we must check (4.1). For the volume estimate, we utilize (4.51) to compute

$$\begin{aligned} |E\Delta (E \Delta F)| = |E \cap F| + |F \setminus E| \le 2|F| \le \delta _j\,. \end{aligned}$$

For the perimeter estimate, by (4.58) and (4.52), (4.37), and the fact that the ${\mathcal {D}}_k$ are pairwise disjoint,

$$\begin{aligned} P(E\Delta F;\Omega )&= P(E;\Omega ) + \sum _{k=0}^\infty P({\mathcal {D}}_k)\\&\le P(E;\Omega )+\sum _{k=0}^\infty 2\,{\mathcal {H}}^n({\mathcal {R}}(K) \cap \textrm{cl}\,{\mathcal {D}}_k \setminus \partial ^* E) + \frac{\delta _j}{2^{k+2}} \\&\le P(E;\Omega ) + 2\,{\mathcal {H}}^n({\mathcal {R}}(K) \setminus \partial ^* E) + \frac{\delta _j}{2}\,, \end{aligned}$$

which is the desired perimeter bound in (4.1). $\square$

Proof of Theorem 2.3for $\Psi _{\textrm{bk}}$ The proof can be accomplished by a simplification of the arguments in the boundary spanning case. First, as in step zero previously, it is enough to choose $\delta _j \rightarrow 0$ and produce a sequence of sets $\{E_j\}$ with $E_j^{\scriptscriptstyle {(1)}}\cup (\Omega \cap \partial ^* E_j)$ ${\mathcal {C}}$-spanning such that

$$\begin{aligned} |E_j\Delta E|\le \delta _j\quad \text {and}\quad P(E_j;\Omega ) \le {\mathcal {F}}_{\textrm{bk}}(K,E) + \delta _j\,. \end{aligned}$$

(4.60)

Fix $\delta _j$, and let $A_0 \cup \cup _m f_m(A_m)$ be the decomposition of ${\mathcal {R}}(K)\cap E^{\scriptscriptstyle {(0)}}$ into an ${\mathcal {H}}^n$-null set and countably many compact Lipschitz graphs as in step two of the boundary case. Arguing as in step three, specifically the inequalities (4.11) and (4.14), we may choose open sets $D_m$ containing $f_m(A_m)$ such that

$$\begin{aligned} |D_m| \le 2^{-m}\delta _j\,,\quad P(D_m) \le 2{\mathcal {H}}^n(f_m(A_m)) + 2^{-m}\delta _j \,. \end{aligned}$$

We then set $E_j = E \cup \cup _m D_m$, so that the estimates

$$\begin{aligned}&|E_j \Delta E| \le \sum _m |D_m| \le \delta _j\,,\quad P(E_j;\Omega ) \le P(E;\Omega ) \\ {}&+ \sum _m P(D_m;\Omega )\le P(E;\Omega ) + \delta _j \end{aligned}$$

follow. To see that $E_j^{\scriptscriptstyle {(1)}}\cup (\Omega \cap \partial ^*E_j)$ is ${\mathcal {C}}$-spanning, we first note that, since supersets of ${\mathcal {C}}$-spanning sets are ${\mathcal {C}}$-spanning and property of being ${\mathcal {C}}$-spanning is stable under ${\mathcal {H}}^n$-null perturbations, it is enough to show that

$$\begin{aligned} E^{\scriptscriptstyle {(1)}}\cup (\Omega \cap \partial ^* E) \cup ({\mathcal {R}}(K) \cap E^{\scriptscriptstyle {(0)}}) \overset{{\mathcal {H}}^n}{\subset }E_j^{\scriptscriptstyle {(1)}}\cup (\Omega \cap \partial ^* E_j)\,, \end{aligned}$$

(4.61)

where the former set is ${\mathcal {C}}$-spanning by Theorem 3.4. Since $E_j$ is a superset of E, we have: first, that $E^{\scriptscriptstyle {(1)}}\subset E_j^{\scriptscriptstyle {(1)}}$; and second, that the Lebesgue density of $E_j$ at any $x\in \Omega \cap \partial ^* E$ is at least 1/2. Therefore, by Federer’s theorem, $E^{\scriptscriptstyle {(1)}}\cup (\Omega \cap \partial ^* E)\overset{{\mathcal {H}}^n}{\subset }E_j^{\scriptscriptstyle {(1)}}\cup (\Omega \cap \partial ^* E_j)$. The inclusion ${\mathcal {R}}(K)\cap E^{\scriptscriptstyle {(0)}}\overset{{\mathcal {H}}^n}{\subset }E_j^{\scriptscriptstyle {(1)}}\cup (\Omega \cap \partial ^* E_j)$ follows directly from the inclusions ${\mathcal {R}}(K) \cap E^{\scriptscriptstyle {(0)}}\overset{{\mathcal {H}}^n}{\subset }\cup _m f_m(A_m)$ and $f_m(A_m) \subset D_m^{\scriptscriptstyle {(1)}}\subset E_j^{\scriptscriptstyle {(1)}}$. $\square$

4.2 Proof of Theorem 2.4

We can now show the equivalence of the collapsed and non-collapsed minimization problems.

Proof of Theorem 2.4

For the equalities $\Psi _{\textrm{bd}}(v)=\psi _{\textrm{bd}}(v)$ and $\Psi _{\textrm{bk}}(v)=\psi _{\textrm{bk}}(v)$, first recall that $\Psi _{\textrm{bd}}(v)\le \psi _{\textrm{bd}}(v)$ and $\Psi _{\textrm{bk}}(v)\le \psi _{\textrm{bk}}(v)$, as noted in Sect. 2.3. The reverse inequalities are direct consequences of Theorem 2.3, since any minimizing sequence for either relaxed problem (2.7)/(2.8) can be approximated by a minimizing sequence for the corresponding soap film capillarity problem (2.4)/(2.5). It remains to show that, when ${\textbf{W}}$ is compact, given a minimizing sequence $\{E_j\}_j$ for $\psi _{\textrm{bd}}(v)$ or $\psi _{\textrm{bk}}(v)$, up to a subsequence we can extract $(K,E)\in {\mathcal {K}}_{\textrm{B}}$ and ball $B_r(x)\subset (K \cup E \cup {\textbf{W}})^c$ of volume $v-|E|$ such that $E_j\rightarrow E$ in locally in $L^1$,

and $(K \cup \partial B_r(x),E \cup B_r(x))$ is minimal for $\Psi _\textrm{bd}(v)$ or $\Psi _{\textrm{bk}}(v)$. The statement for the bulk problem $\psi _{\textrm{bk}}(v)$ can be deduced as follows. By $\Psi _{\textrm{bk}}(v) = \psi _{\textrm{bk}}(v)$ and the fact that $E_j$ are admissible for $\Psi _{\textrm{bk}}(v)$, $\{E_j\}_j$ is a minimizing sequence for $\Psi _{\textrm{bk}}(v)$, and so [26, Theorem 6.2], which extracts such a ball and pair $(K,E)\in {\mathcal {K}}_{\textrm{B}}$ out of an arbitrary minimizing sequence for $\Psi _{\textrm{bk}}(v)$ such as $\{E_j\}_j$, yields the desired result. For $\psi _{\textrm{bd}}(v)$, we can follow the same strategy. Let us sketch the argument, which, given the compactness in Theorem 3.2, consists of standard arguments to deal with volume loss at infinity. By Theorem 3.2 and compactness for sets of finite perimeter, up to a subsequence, we obtain a limiting pair (K, E) which is admissible for $\Psi _{\textrm{bd}}(|E|)$ (where $v=0$ is $2\ell _{\textrm{B}}$) such that K is ${\mathcal {C}}$-spanning and

(4.62)

Since ${\textbf{W}}$ is compact, the Euclidean isoperimetric inequality and (4.62) imply that the escaping mass contributes at least as much energy as a ball (see [26, Proof of Theorem 6.2, step three]), that is

$$\begin{aligned} \Psi _{\textrm{bd}}(v) \ge {\mathcal {F}}_{\textrm{bd}}(K,E) + P(B_r(x)) \end{aligned}$$

whenever $|B_r(x)|=v-|E|$. However, by a construction consisting of adding balls of volume $v-|E|$ to any admissible $(K',E')$ for $\Psi _{\textrm{bd}}(|E|)$ (see [26, Proof of Theorem 6.2, step two]), we also have the reverse inequality

$$\begin{aligned} \Psi _{\textrm{bd}}(v) \le \Psi _{\textrm{bd}}(|E|) + P(B_r(x))\le {\mathcal {F}}_{\textrm{bd}}(K,E) + P(B_r(x)) \end{aligned}$$

Combining these two, we find that (K, E) is minimal for $\Psi _\textrm{bd}(|E|)$, and so by Proposition 3.1 and $\Omega \cap \partial ^* E \overset{{\mathcal {H}}^n}{\subset }{\mathcal {R}}(K)$, we have $K\overset{{\mathcal {H}}^n}{=}{\mathcal {R}}(K)$. By a first variation argument [20, Appendix C], the integer multiplicity rectifiable varifold $V=\textbf{var}\,(K,\theta )$, where $\theta =1$ on $\partial ^* E\cap \Omega$ and $\theta =2$ on $K \setminus \partial ^* E$, has $L^\infty$-mean curvature in ${\mathbb {R}}^{n+1}\setminus {\textbf{W}}$. Therefore, by the monotonicity formula and the boundedness of ${\textbf{W}}$, $\textrm{spt}\, V$ is bounded, and so are ${\mathcal {R}}(K)$ and E. Since ${\textbf{W}}\cup K \cup E$ is bounded and $\Psi _{\textrm{bd}}(v) = {\mathcal {F}}_{\textrm{bd}}(K,E) + P(B_r(x))$ whenever $|B_r(x)| = v - |E|$, we may choose $B_r(x)\subset (K \cup E \cup {\textbf{W}})^c$ to add to (K, E) and conclude the argument. $\square$