We provide, in the setting of Gauss' capillarity theory, a rigorous derivation of the equilibrium law for the three dimensional structures known as Plateau borders which arise in “wet” soap films and foams. A key step in our analysis is a complete measure-theoretic overhaul of the homotopic spanning condition introduced by Harrison and Pugh in the study of Plateau's laws for two-dimensional area minimizing surfaces (“dry” soap films). This new point of view allows us to obtain effective compactness theorems and energy representation formulae for the homotopic spanning relaxation of Gauss' capillarity theory which, in turn, lead to prove sharp regularity properties of energy minimizers. The equilibrium law for Plateau borders in wet foams is also addressed as a (simpler) variant of the theory for wet soap films.

§ INTRODUCTION

§.§ Overview
Equilibrium configurations of soap films and foams are governed, at leading order, by the balance between surface tension forces and atmospheric pressure. This balance is expressed by the Laplace–Young law of pressures, according to which such systems can be decomposed into smooth interfaces with constant mean curvature equal to the pressure difference across them, and by the Plateau laws, which precisely postulate which arrangements of smooth interfaces joined together along lines of “singular” points are stable, and thus observable.

The physics literature identifies two (closely related) classes of soap films and foams, respectively labeled as “dry” and “wet”. This difference is either marked in terms of the amount of liquid contained in the soap film/foam <cit.>, or in terms of the scale at which the soap film/foam is described <cit.>.

In the dry case, Plateau laws postulates that (i) interfaces can only meet in three at a time forming 120-degree angles along lines of “$Y$-points”; and (ii) lines of $Y$-points can only meet in fours at isolated “$T$-points”, where six interfaces asymptotically form a perfectly symmetric tetrahedral angle; see, e.g. <cit.>.

In the wet case, small but positive amounts of liquid are bounded by negatively curved interfaces, known as Plateau borders, and arranged near ideal lines of $Y$-points or isolated $T$-points;  see Figure

(a) A Plateau border develops around a “wet” line of $Y$-points. The wet region is bounded by interfaces of negative constant mean curvature. The equilibrium condition which needs to hold across the transition lines (here depicted in bold) between the negatively curved interfaces of a Plateau border and the incoming dry interfaces is that these interfaces meet tangentially. In the case of soap films, where the dry interfaces have zero mean curvature, the jump in the mean curvature across the transition lines implies a discontinuity in the gradient of the unit normal. (b) An arrangement of Plateau borders near a tetrahedral singularity. The transition lines are again depicted in bold. The incoming dry interfaces are omitted for clarity.

<ref> and <cit.>. A “third Plateau law” is then postulated to hold across the transition lines between wet and dry parts of soap films/foams, and can be formulated as follows:

\begin{eqnarray}\label{third pl}
&&\mbox{{\it the unit normal to a soap film/foam changes continuously}}
\\\nonumber
&&\mbox{{\it across the transition lines between wet and dry interfaces}}\,;
\end{eqnarray}

see, e.g., <cit.> and <cit.>. It is important to recall that Plateau borders play a crucial role in determining the mechanical properties of the many physical and biological systems in which they are observed. As a sample of older and newer papers discussing Plateau borders, we mention here [38, 30, 7, 31, 37, 32, 27, 49]. Postulate (<ref>) is assumed in all these works.

The goal of this paper is answering the natural problem of rigorously deriving the equilibrium condition for Plateau borders (<ref>) in the context of Gauss' capillarity theory. Since the case of soap films is much harder and interesting from the mathematical viewpoint, we will postpone the discussion of foams until the very last section of this introduction. The main highlight is that, in addressing Plateau borders of soap films, we will develop a new “theory of spanning” for surfaces of geometric measure theory (GMT) which will find further applications in the two companion papers [42, 43]; see the closing of this overview for more details about these additional applications.

We now give an informal description of our approach. The starting point is [44], where the idea  is introduced  of modeling soap films as regions $E$ of positive volume $|E|=v$ contained in the complement $\Omega=\R^{n+1}\setminus\wire$ of a “wire frame” $\wire$ ($n=2$ is the physical case, although the planar case $n=1$ is also quite interesting in applications). We associate to $E$ the surface tension energy $\H^n(\Om\cap\pa E)$ (where $\H^n$ stands for $n$-dimensional (Hausdorff) measure, i.e., area when $n=2$ and length when $n=1$), and minimize $\H^n(\Om\cap\pa E)$ under the constraints that $|E|=v$ (for some given $v>0$) and

\begin{equation}
\label{meaning of}
\mbox{$\Om\cap\pa E$ is spanning $\wire$}\,.
\end{equation}

From the mathematical viewpoint the meaning assigned to (<ref>) is, of course, the crux of the matter. In the informal spirit of this overview, we momentarily leave the concept of “spanning” only intuitively defined.

As proved in [34], this minimization process leads to the identification of generalized minimizers in the form of pairs $(K,E)$ with $E\subset\Om$, $|E|=v$, and such that

\begin{equation}
\label{meaning of generalized}
\mbox{$\Om\cap\pa E\subset K$ and $K$ is spanning $\wire$}\,.
\end{equation}

These pairs are minimizing in the sense that

\begin{equation}
\label{kms minimizers}
\H^n(\Om\cap\pa E)+2\,\H^n(K\setminus\pa E)\le \H^n(\Om\cap\pa E')\,,
\end{equation}

whenever $E'\subset\Om$, $|E'|=v$ and $\Om\cap\pa E'$ is spanning $\wire$.

If $K=\Om\cap\pa E$, then generalized minimizers are of course minimizers in the proper sense. If not, the collapsed interface $K\setminus\pa E$ is a surface whose positive area has to be counted with a multiplicity factor $2$ (which arises from the asymptotic collapsing along $K\setminus\pa E$ of oppositely oriented boundaries in minimizing sequences $\{E_j\}_j$, see

Emergence of collapsing along a minimizing sequence $\{E_j\}_j$ for the minimization of $\H^n(\Om\cap\pa E)$ among sets $E\subset\Omega=\R^{n+1}\setminus\wire$ with $|E|=v$ and $\Om\cap\pa E$ spanning $\wire$, when $n=1$ and $\wire$ is the union of three disks in the plane. Notice that for this choice of $\wire$ the minimization of $\H^n(S)$ among $S\subset\Om$ such that $S$ is spanning $\wire$ is solved by three segments meeting at $Y$-point. Collapsing is intuitively related to the presence of $Y$-type and $T$-type singularities.

Figure <ref>). We expect collapsing to occur whenever the Plateau problem for $\wire$ admits one minimizer $S$ with Plateau-type singularities. Whenever this happens, a wetting conjecture is made: sequences $\{(K_{v_j},E_{v_j})\}_j$ of generalized minimizers with $|E_{v_j}|=v_j\to 0^+$ as $j\to\infty$ will be such that the set of Plateau's singularities $\Sigma(S)$ of $S$ is such that $\sup\{\dist(x,E_{v_j}):x\in\Sigma(S)\}\to 0$. Thus we expect that Plateau's singularities are never “left dry” in the small volume capillarity approximation of the Plateau problem.

A lot of information about generalized minimizers can be extracted from (<ref>), and this is the content of [34, 33, 35]. With reference to the cases when $n=1$ or $n=2$, one can deduce from (<ref>) that if $\H^n(K\setminus\pa E)>0$, then $K\setminus\pa E$ is a smooth minimal surface (a union of segments if $n=1$) and that $\pa E$ contains a regular part $\pa^*E$ that is a smooth constant mean curvature surface (a union of circular arcs if $n=1$) with negative curvature. This is of course strongly reminiscent of the behavior of Plateau borders, and invites to analyze the validity of (<ref>) in this context. A main obstacle is that, due to serious technical issues (described in more detail later on) related to how minimality is expressed in (<ref>), it turns out to be very difficult to say much about the “transition line”

\[
\pa E\setminus\pa^*E
\]

between the zero and the negative constant mean curvature interfaces in $K$, across which one should check the validity of (<ref>). More precisely, all that descends from (<ref>) and a direct application of Allard's regularity theorem [3] is that $\pa E\setminus\pa^*E$ has empty interior in $K$. Far from being a line in dimension $n=2$, or a discrete set of points when $n=1$, the transition line $\pa E\setminus\pa^*E$ could very well have positive $\H^n$-measure and be everywhere dense in $K$! With such poor understanding of $\pa E\setminus\pa^*E$, proving the validity of (<ref>) – that is, the continuity of the unit normals to $K\setminus\pa E$ and $\pa^*E$ in passing across $\pa E\setminus\pa^*E$ – is of course out of question.

We overcome these difficulties by performing a major measure-theoretic overhaul of the Harrison–Pugh homotopic spanning condition [28, 29] used in [44, 34, 33, 35] to give a rigorous meaning to (<ref>), and thus to formulate the homotopic spanning relaxation of Gauss' capillarity discussed above.

The transformation of this purely topological concept into a measure-theoretic one is particularly powerful. Its most important consequence for the problem discussed in this paper is that it allows us to upgrade the partial minimality property (<ref>) of $(K,E)$ into the full minimality property

\begin{equation}
\label{full minimizers}
\H^n(\Om\cap\pa E)+2\,\H^n(K\setminus\pa E)\le  \H^n(\Om\cap\pa E')+2\,\H^n(K'\setminus\pa E')
\end{equation}

whenever $E'\subset\Om$, $|E'|=v$, $\Om\cap\pa E'\subset K'$ and $K'$ is spanning $\wire$. The crucial difference between (<ref>) and (<ref>) is that the latter is much more efficient than the former when it comes to study the regularity of generalized minimizers $(K,E)$, something that is evidently done by energy comparison with competitors $(K',E')$. Such comparisons are immediate when working with (<ref>), but they are actually quite delicate to set up when we only have (<ref>). In the latter case, given a competitor $(K',E')$, to set up the energy comparison with $(K,E)$ we first need to find a sequence of non-collapsed competitors $\{E'_j\}_j$ (with $E'_j\subset\Om$, $|E'_j|=v$, and $\Om\cap\pa E'_j$ spanning $\wire$) such that $\H^n(\Om\cap\pa E'_j)\to\H^n(\Om\cap\pa E')+2\,\H^n(K'\setminus\pa E')$. Intuitively, $E_j'$ needs to be a $\de_j$-neighborhood of $K'\cup E'$ for some $\de_j\to 0^+$ and the energy approximation property has to be deduced from the theory of Minkowski content. But applying the theory of Minkowski content to $(K',E')$ (which is the approach followed, e.g., in [35]) requires $(K',E')$ to satisfy rectifiability and uniform density properties that substantially restrict the class of available competitors $(K',E')$.

In contrast, once the validity of (<ref>) is established, a suitable generalization (Theorem <ref>) of the partition theorem of sets of finite perimeter into indecomposable components <cit.> combined with a subtle variational argument (see Figure <ref>) allows us to show that, in any ball $B\cc\Omega$ with sufficiently small radius and for some sufficiently large constant $\Lambda$ (both depending just on $(K,E)$), the connected components $\{U_i\}_i$ of $B\setminus (K\cup E)$ satisfy a perturbed area minimizing property of the form

\begin{equation}
\label{lambda minimality}
\H^n(B\cap\pa U_i)\le\H^n(B\cap \pa V)+\Lambda\,|U_i\Delta V|\,,
\end{equation}

with respect to completely arbitrary perturbations $V\subset B$, $V\Delta U_i\cc B$. By a classical theorem of De Giorgi [14, 51], (<ref>) implies (away from a closed singular set of codimension at least $8$, which is thus empty if $n\le 6$) the $C^{1,\a}$-regularity of $B\cap\pa U_i$ for each $i$, and thus establishes the continuity of the normal stated in (<ref>). In fact, locally at each $x$ on the transition line, $K$ is the union of the graphs of two $C^{1,\a}$-functions $u_1\le u_2$ defined on an $n$-dimensional disk, having zero mean curvature above the interior of $\{u_1=u_2\}$, and opposite constant mean curvature above $\{u_1<u_2\}$. We can thus exploit the regularity theory for double-membrane free boundary problems devised in [47, 24] to deduce that the transition line $\pa E\setminus\pa^*E$ is indeed $(n-1)$-dimensional, and to improve the $C^{1,\a}$-regularity of $B\cap\pa U_i$ to $C^{1,1}$-regularity. Given the mean curvature jump across $\pa E\setminus\pa^*E$ we have thus established the sharp degree of regularity for minimizers of the homotopic spanning relaxation of Gauss' capillarity theory.

The measure-theoretic framework for homotopic spanning conditions laid down in this paper provides the starting point for additional investigations that would otherwise seem unaccessible. In two forthcoming companion papers we indeed establish (i) the convergence towards Plateau-type singularities of energy-minimizing diffused interface solutions of the Allen–Cahn equation [42], and (ii) some sharp convergence theorems for generalized minimizers in the homotopic spanning relaxation of Gauss' capillarity theory in the vanishing volume limit, including a proof of the above mentioned wetting conjecture [43].

The rest of this introduction is devoted to a rigorous formulation of the results presented in this overview. We begin in Section <ref> with a review of the Harrison and Pugh homotopic spanning condition in relation to the classical Plateau problem and to the foundational work of Almgren and Taylor [4, 52]. In Section <ref> we introduce the new measure-theoretic formulation of homotopic spanning and discuss its relation to the measure-theoretic notion of essential connectedness introduced by Cagnetti, Colombo, De Philippis and the first-named author in the study of symmetrization inequalities [10, 9]. In Section <ref> we introduce the bulk and boundary spanning relaxations of Gauss' capillarity theory, state a general closure theorem for “generalized soap films” that applies to both relaxed problems (Theorem <ref>). In Section <ref> we prove the existence of generalized soap film minimizers (Theorem <ref>) and their convergence in energy to solutions to the Plateau problem. A sharp regularity theorem (Theorem <ref>) for these minimizers, which validates (<ref>), is stated in Section <ref>. Finally, in Section <ref> we reformulate the above results in the case of foams, see in particular Theorem <ref>.

§.§ Homotopic spanning: from Plateau's problem to Gauss' capillarity
The theories of currents and of sets of finite perimeter, i.e. the basic distributional theories of surface area at the center of GMT, fall short in the task of modeling Plateau's laws. Indeed, two-dimensional area minimizing currents in $\R^3$ are carried by smooth minimal surfaces, and thus cannot model $Y$-type[Currents modulo $3$ are compatible with $Y$-type singularities, but not with $T$-type singularities.] and $T$-type singularities. This basic issue motivated the introduction of Almgren minimal sets as models for soap films in [4]: these are sets $S\subset\R^{n+1}$ that are relatively closed in a given open set $\Om\subset\R^{n+1}$, and satisfy $\H^n(S)\le\H^n(f(S))$ whenever $f:\Om\to\Om$ is a Lipschitz (not necessarily injective) map with $\{f\ne\id\}\cc\Om$. Taylor's historical result [52] validates the Plateau laws in this context, by showing that, when[Similar regularity assertions hold when $n=1$ (by elementary methods) and, in much more recent developments, when $n\ge 3$ [11].] $n=2$, Almgren minimal sets are locally $C^{1,\a}$-diffeomorphic either to planes, to $Y$-cones, or to $T$-cones.

The issue of proposing and solving a formulation of Plateau's problem whose minimizers are Almgren minimal sets, and indeed admit Plateau-type singularities, is quite elusive, as carefully explained in [13]. In this direction, a major breakthrough has been obtained by Harrison and Pugh in [28] with the introduction of a new spanning condition, which, following the presentation in [16], can be defined as follows:

Given a closed set $\wire \subset \mathbb{R}^{n+1}$ (the “wire frame”), a spanning class for $\wire$ is a family $\C$ of smooth embeddings of $\mathbb{S}^1$ into

\[
\Om=\mathbb{R}^{n+1}\setminus \wire
\]

that is closed under homotopies in $\Omega$, that is, if $\Phi:[0,1]\times\SS^1\to\Om$ is smooth family of embeddings $\Phi_t=\Phi(t,\cdot):\SS^1\to\Om$ with $\Phi_0\in\C$, then $\Phi_t\in\C$ for every $t\in(0,1]$. A set $S$, contained and relatively closed in $\Om$, is said to be $\C$-spanning $\wire$ if

\[
S \cap \gamma \neq \varnothing\,,\qquad \forall \gamma \in \C\,.
\]

Denoting by $\Sc(\C)$ the class of sets $S$ $\C$-spanning $\wire$, one can correspondingly formulate the Plateau problem (with homotopic spanning)

\begin{equation}
\label{hp problem intro}
\ell=\ell(\C):=\inf\big\{\H^n(S):S\in\Sc(\C)\big\}\,.
\end{equation}

Existence of minimizers of $\ell$ holds as soon as $\ell<\infty$, and minimizers $S$ of $\ell$ are Almgren minimal sets in $\Om$ [28, 16] that are indeed going to exhibit Plateau-type singularities (this is easily seen in the plane, but see also [6] for a higher dimensional example). Moreover, given a same $\wire$, different choices of $\C$ are possible and can lead to different minimizers, see
The dashed lines denote the embeddings of $\SS^1$ whose homotopy classes relative to $\Om$ generate different spanning classes $\C$, to which there correspond different minimizers of $\ell$.
Figure <ref>. Finally, the approach is robust enough to provide the starting point for several important extensions [20, 22, 29, 26, 15, 21], including higher codimension, anisotropic energies, etc.

The study of soap films as minimizers of Gauss's capillarity energy with small volume and under homotopic spanning conditions has been initiated in [44, 34], with the introduction of the model

\begin{equation}
\label{def of psi}
\psi(v):=\inf\Big\{\H^n(\Om\cap\pa E):|E|=v\,,\,\,\mbox{$\Om\cap\pa E$ is $\C$-spanning $\wire$}\Big\}\,,
\end{equation}

where $E\subset\Omega$ is an open set with smooth boundary. Without the spanning condition, at small volumes, minimizers of $\H^n(\Om\cap\pa E)$ would be small diffeomorphic images of half-balls [41]. However, the introduction of the $\C$-spanning constraint rules out small droplets, and forces the exploration of a different part of the energy landscape of $\H^n(\Om\cap\pa E)$. As informally discussed in Section <ref>, this leads to the emergence of generalized minimizers $(K,E)$. More precisely, in [34] the existence is proved of $(K,E)$ in the class

\begin{eqnarray} \label{def of K}
&&\K=\Big\{(K,E):\mbox{$K$ is relatively closed and $\H^n$-rectifiable in $\Om$, $E$ is open,}
\\\nonumber
&&\hspace{2.7cm}\mbox{$E$ has finite perimeter in $\Om$, and $\Om\cap\cl(\pa^*E)=\Om\cap\pa E\subset K$}\Big\}\,,
\end{eqnarray}

(where $\pa^* E$ denotes the reduced boundary of $E$) such that, for every competitor $E'$ in $\psi(v)$, it holds

\begin{equation}
\label{kms minimizers intro}
\H^n(\Om\cap\pa^*E)+2\,\H^n(\Om\cap(K\setminus\pa^*E))\le \H^n(\Om\cap\pa E')\,.
\end{equation}

Starting from (<ref>) one can apply Allard's regularity theorem [3] and various ad hoc comparison arguments [33, 35] to prove that $\Om\cap\pa^*E$ is a smooth hypersurface with constant mean curvature (negative if $\H^n(K\setminus\pa^*E)>0$), $\Om\cap(\pa E\setminus\pa^*E)$ has empty interior in $K$, and that $K\setminus(\Sigma\cup\pa E)$ is a smooth minimal hypersurface, where $\Sigma$ is a closed set with codimension at least $8$.

§.§ Measure theoretic homotopic spanning
In a nutshell, the idea behind our measure theoretic revision the Harrison–Pugh homotopic spanning condition is the following. Rather than asking that $S\cap\g(\SS^1)\ne\varnothing$ for every $\g\in\C$, as done in Definition <ref>, we shall replace $\g$ with an open “tube” $T$ containing $\g(\SS^1)$, and ask that $S$, with the help of a generic “slice” $T[s]$ of $T$, “disconnects” $T$ itself into two nontrivial regions $T_1$ and $T_2$; see
(a) Homotopic spanning according to Harrison–Pugh: $S$ must intersect every curve $\g\in\C$, in particular, the $\C$-spanning property may be lost by removing a single point from $S$; (b) Homotopic spanning based on essential connectedness: for a.e. section $T[s]$ of the tube $T$ around a curve $\g\in\C$, the union $T[s]\cup S$ (essentially) disconnects $T$ (i.e., divides $T$ into two non-trivial parts, depicted here with two different shades of gray).
Figure <ref>. The key to make this idea work is, of course, giving a proper meaning to the word “disconnects”.

To this end, we recall the  notion of essential connectedness introduced in [10, 9] in the study of the rigidity of equality cases in Gaussian and Euclidean perimeter symmetrization inequalities. Essential connectedness is the “right” notion to deal with such problems since it leads to the formulation of sharp rigidity theorems, and can indeed be used to address other rigidity problems (see [12, 46, 19]). This said, it seems remarkable that the very same notion of what it means for “one Borel set to disconnect another Borel set” proves to be extremely effective also in the context of the present paper, which is of course very far from the context of symmetrization theory.

Denoting by $T^{{\scriptscriptstyle{(t)}}}$ ($0\leq t \leq 1$) the points of density $t$ of a Borel set $T\subset\R^{n+1}$ (i.e., $x\in T^{{\scriptscriptstyle{(t)}}}$ if and only if $|T\cap B_r(x)|/\om_{n+1}\,r^{n+1}\to t$ as $r\to 0^+$, where $\om_k$ is the Lebesgue measure of the unit ball in $\R^k$), and by $\partial^e T= \mathbb{R}^{n+1}\setminus (T^{\zero} \cup T^{\one})$ the essential boundary of $T$, given Borel sets $S$, $T$, $T_1$ and $T_2$ in $\mathbb{R}^{n+1}$, and given $n\geq 0$, we say that $S$  essentially disconnects $T$ into $\{T_1,T_2\}$, if

\begin{equation}\label{def essential connectedness}
\begin{split}
&\mbox{$\{T_1, T_2\}$ is a non-trivial Borel partition of $T$}\,,
\\
&\mbox{and $T^{\one}\cap \partial^e T_1 \cap \partial^e T_2$ is $\H^n$-contained in $S$}\,.
\end{split}
\end{equation}

(For example, if $K$ is a set of full $\L^1$-measure in $[-1,1]$, then $S=K\times\{0\}$ essentially disconnects the unit disk in $\R^2$.)
Moreover, we say that $T$ is essentially connected[Whenever $T$ is of locally finite perimeter, being essentially connected is equivalent to being indecomposable.] if $\varnothing$ does not essentially disconnect $T$. The requirement that $\{T_1,T_2\}$ is a non-trivial Borel partition of $T$ means that $|T\Delta (T_1\cup T_2)|=0$ and $|T_1|\,|T_2|>0$. By saying that “$E$ is $\H^n$-contained in $F$” we mean that $\H^n(E\setminus F)=0$. We also notice that, in (<ref>), we have
$T^{\one}\cap \partial^e T_1 \cap \partial^e T_2=T^{\one}\cap \partial^e T_i$ ($i=1,2$), a fact that is tacitly and repeatedly considered in the use of (<ref>) in order to shorten formulas.

With this terminology in mind, we introduce the following definition:

Given a closed set $\wire$ and a spanning class $\C$ for $\wire$, the tubular spanning class $\mathcal{T}(\C)$ associated to $\C$ is the family of triples $(\gamma, \Phi, T)$ such that $\g\in\C$, $T = \Phi(\mathbb{S}^1 \times B_1^n)$, and[Here $B_1^n=\{x\in\R^n:|x|<1\}$ and $\SS^1=\{s\in\R^2:|s|=1\}$.]

\[
\textup{$\Phi:\mathbb{S}^1 \times \cl B_1^n \to  \Omega$ is a diffeomorphism with $\restr{\Phi}{\mathbb{S}^1\times \{0\}}= \gamma$}\,.
\]

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

\[
T[s]=\Phi(\{s\}\times B_1^n)\,.
\]

Finally, we say that a Borel set $S \subset\Omega$ is $\C$-spanning $\wire$ if for each $(\gamma,\Phi, T)\in \mathcal{T}(\C)$, $\H^1$-a.e. $s\in \mathbb{S}^1$ has the following property:

\begin{eqnarray}\nonumber
&&\mbox{for $\H^n$-a.e. $x\in T[s]$}
\\\label{spanning borel intro}
&&\mbox{$\exists$ a partition $\{T_1,T_2\}$ of $T$ s.t. $x\in\partial^e T_1 \cap \partial^e T_2$}\\ \nonumber
%&&\mbox{\color{blue}$\tn(\H^n\mres ( T[s]\cap \partial^e T_1 \cap\partial^e T_2))(x)=1$}\,,
&&\mbox{and s.t. $S \cup  T[s]$ essentially disconnects $T$ into $\{T_1,T_2\}$}\,.
\end{eqnarray}

Before commenting on (<ref>), we notice that the terminology of Definition <ref> is coherent with that of Definition <ref> thanks to the following theorem.

Given a closed set $\wire\subset\R^{n+1}$, a spanning class $\C$ for $\wire$, and a set $S$ relatively closed in $\Om$, then $S$ is $\C$-spanning $\wire$ in the sense of Definition <ref> if and only if $S$ is $\C$-spanning $\wire$ in the sense of Definition <ref>.

Theorem <ref> is proved in Appendix <ref>. There we also comment on the delicate reason why, in formulating (<ref>), the partition $\{T_1,T_2\}$ must be allowed to depend on specific points $x\in T[s]$. This would not seem necessary by looking at the simple situation depicted in Figure <ref>, but it is actually so when dealing with more complex situations; see Figure <ref>.

Homotopic spanning according to Definition <ref> is clearly stable under modifications of $S$ by $\H^n$-negligible sets, but there is more to it.

An example of induced essential partition. The union of the boundaries of the $U_i$'s (inside of $U$) is contained in $S$, and the containment may be strict. However, the part of $S$ not contained in $U\cap\bigcup_i\pa U_i$ is not such to disconnect any of the $U_i$'s. In particular, each $U_i$ is essentially connected.

Indeed, even a notion like “$\H^n(S\cap T)>0$ for every $T\in\T(\C)$” would be stable under modifications by $\H^n$-negligible sets, and would probably look more appealing in its simplicity. The catch, of course, is finding an extension of Definition <ref> for which compactness theorems, like Theorem <ref> below, hold true. This is evidently not the case, for example, if one tries to work with a notion like “$\H^n(S\cap T)>0$ for every $T\in\T(\C)$”.

The first key insight on Definition <ref> is that, if restricted to Borel sets $S$ that are locally $\H^n$-finite in $\Om$, then it can be reformulated in terms of partitions into indecomposable sets of finite perimeter. This is the content of the following theorem, whose case $S=\varnothing$ corresponds to the standard decomposition theorem for sets of finite perimeter <cit.>. For an illustration of this result, see Figure <ref>.

If $U\subset\R^{n+1}$ is a bounded set of finite perimeter and $S\subset\R^{n+1}$ is a Borel set with $\H^n(S \cap U^\one)<\infty$, then there exists a unique[Uniqueness is meant modulo relabeling and modulo Lebesgue negligible modifications of the $U_i$'s.] essential partition $\{U_i\}_i$ of $U$ induced by $S$, that is to say, $\{U_i\}_i$ is a countable partition of $U$ modulo Lebesgue negligible sets such that, for each $i$, $S$ does not essentially disconnect $U_i$.

Given $U$ and $S$ as in the statement of Theorem <ref> we can define[Uniquely modulo $\H^n$-null sets thanks to Federer's theorem recalled in (<ref>) below.] the union of the (reduced) boundaries (relative to $U$)  of the essential partition induced by $S$ on $U$ by setting[Given a Borel set $E$, we denote by $\pa^*E$ its reduced boundary relative to the maximal open set $A$ wherein $E$ has locally finite perimeter.]

\begin{equation}
\label{def of UBEP}
{\rm UBEP}(S;U)=U^\one\cap\bigcup_i\pa^*U_i\,.
\end{equation}

Two properties of ${\rm UBEP}$'s which well illustrate the concept are: first, if $\RR(S)$ denotes the rectifiable part of $S$, then ${\rm UBEP}(S;U)$ is $\H^n$-equivalent to ${\rm UBEP}(\RR(S);U)$; second, if $S^*$ is $\H^n$-contained in $S$, then ${\rm UBEP}(S;U)$ is $\H^n$-contained in ${\rm UBEP}(S;U)$; both properties are proved in Theorem <ref> (an expanded restatement of Theorem <ref>).

We can use the concepts just introduced to provide an alternative and technically more workable characterization of homotopic spanning in the measure theoretic setting. This is the content of our first main result, which is illustrated in Figure <ref>.

If $\wire\subset\R^{n+1}$ is a closed set in $\R^{n+1}$, $\C$ is a spanning class for $\wire$, and $S\subset\Om$ is locally $\H^n$-finite in $\Om$, then $S$ is $\C$-spanning $\wire$ if and only if for every $(\gamma,\Phi, T)\in \mathcal{T}(\C)$ we have that, for $\H^1$-a.e. $s\in\SS^1$,

\begin{eqnarray}\label{spanning and the S partition equation S case}
&&\mbox{$T[s]$ is $\H^n$-contained in ${\rm UBEP}(S\cup T[s];T)$}\,.
\end{eqnarray}

With $\wire$ consisting of two disks in the plane, and $T$ a test tube for the $\C$-spanning condition: (a) $S$ consists of a segment with a gap: since the gap is inside of $T$, the essential partition of $T$ induced by $S\cup T[s]$ consists of only one set, $U_1=T$, so that $T\cap\pa^*U_1=\varnothing$ and (<ref>) cannot hold; (b) $S$ consists of a full segment and in this case (with the possible exception of a choice of $s$ such that $T[s]$ is contained in $S$), the essential partition of $T$ induced by $S\cup T[s]$ consists of two sets $\{U_1,U_2\}$, such that $T[s]\subset T\cap\pa^*U_1\cap\pa^*U_2$; in this case (<ref>) holds.

§.§ Direct Method on generalized soap films and Gauss' capillarity
The most convenient setting for addressing the existence of minimizers in Gauss' capillarity theory is of course that of sets of finite perimeter [25, 39]. However, if the notion of homotopic spanning is limited to closed sets, as it is the case when working with Definition <ref>, then one cannot directly use homotopic spanning on sets of finite perimeter, and this is the reason behind the specific formulation (<ref>) of $\psi(v)$ used in  [44, 34]. Equipped with Definition <ref> we can now formulate Gauss' capillarity theory with homotopic spanning conditions directly on sets of finite perimeter. We shall actually consider two different possible formulations

\begin{eqnarray*}
&&\psi_{\rm bk}(v)=\inf\Big\{\H^n(\Om\cap\pa^*E):\mbox{$|E|=v$ and $\Om\cap(\pa^* E\cup E^\one)$ is $\C$-spanning $\wire$}\Big\}\,,
\\
&&\psi_{\rm bd}(v)=\inf\Big\{\H^n(\Om\cap\pa^*E):\mbox{$|E|=v$ and $\Om\cap \pa^*E$ is $\C$-spanning $\wire$}\Big\}\,,
\end{eqnarray*}

where the subscripts “bk” and “bd” stand to indicate that the spanning is prescribed via the bulk of $E$ (that is, in measure theoretic terms, via the set $\Om\cap(\pa^*E\cup E^\one)$ or via the (reduced) boundary of $E$. Inspired by the definition of the class $\K$ introduced in (<ref>), we also introduce the class $\K_{\rm B}$ of generalized soap films defined by

\begin{eqnarray}\label{def of KB}
&&\mathcal{K}_{\rm B}=\Big\{(K,E):\mbox{$K$ and $E$ are Borel subsets of $\Om$,}
\\\nonumber
&&\hspace{2.7cm}\mbox{$E$ has locally finite perimeter in $\Om$ and $\partial^*E\cap \Omega \shn K$}\Big\}\,.
\end{eqnarray}

Here the subscript “B” stands for “Borel”, and $\K_{\rm B}$ stands as a sort of measure-theoretic version of $\K$.

In the companion paper [45] the following relaxation formulas for problems $\psi_{\rm bk}$ and $\psi_{\rm bd}$ are proved,

\begin{equation}
\label{relaxation formulas}
\psi_{\rm bk}(v)=\Psi_{\rm bk}(v)\,,\qquad   \psi_{\rm bd}(v)=\Psi_{\rm bd}(v)\,,\qquad\forall v>0\,,
\end{equation}

where the following minimization problems on $\KK_{\rm B}$ are introduced

\begin{eqnarray}
\label{def of Psibk}
&&\Psi_{\rm bk}(v)=\inf\Big\{\F_{\rm bk}(K,E):(K,E)\in\mathcal{K}_{\rm B}\,,|E|=v\,,\mbox{$K\cup E^{\one}$ is $\C$-spanning $\wire$}\Big\}\,,\hspace{1cm}
\\
\label{def of Psibd}
&&\Psi_{\rm bd}(v)=\inf\Big\{\F_{\rm bd}(K,E):(K,E)\in\mathcal{K}_{\rm B}\,,|E|=v\,,\mbox{$K$ is $\C$-spanning $\wire$}\Big\}\,.
\end{eqnarray}

Here $\F_{\rm bk}$ and $\F_{\rm bd}$ are the relaxed energies defined for $(K,E)\in\KK_{\rm B}$ and $A\subset\Om$ as

\begin{eqnarray}
\label{def of Fb}
&&\F_{\rm bk}(K,E;A)=2\,\H^n(A\cap K\cap E^{\zero})+\H^n(A\cap\pa^*E)\,,
\\\label{def of F}
&&\F_{\rm bd}(K,E;A)=2\,\H^n(A\cap K\setminus\pa^*E)+\H^n(A\cap\pa^*E)\,,
\end{eqnarray}

(We also set, for brevity, $\F_{\rm bk}(K,E):=\F_{\rm bk}(K,E;\Om)$ and $\F_{\rm bd}(K,E):=\F_{\rm bd}(K,E;\Om)$.) We refer to these problems, respectively, as the “bulk-spanning” or “boundary-spanning” Gauss' capillarity models. In this paper we shall directly work with these relaxed models. In particular, the validity of (<ref>), although of definite conceptual importance, is not playing any formal role in our deductions.

A first remark concerning the advantage of working with the relaxed problems $\Psi_{\rm bk}$ and $\Psi_{\rm bd}$ rather than with their “classical” counterparts $\psi_{\rm bk}$ and $\psi_{\rm bd}$ is that while the latter two with $v=0$ are trivial (sets with zero volume have zero distributional perimeter), the problems $\Psi_{\rm bk}(0)$ and $\Psi_{\rm bd}(0)$ are actually non-trivial, equal to each other, and amount to a measure-theoretic version of the Harrison–Pugh formulation of Plateau's problem $\ell$ introduced in (<ref>): more precisely, if we set

\begin{equation}
\label{def of ellB}
\ell_{\rm B}:=\frac{\Psi_{\rm bk}(0)}2=\frac{\Psi_{\rm bd}(0)}2=\inf\Big\{\H^n(S):\mbox{$S$ is a Borel set $\C$-spanning $\wire$}\Big\}\,,
\end{equation}

then, by Theorem <ref>, we evidently have $\ell_{\rm B}\le\ell$; and, as we shall prove in the course of our analysis, we actually have that $\ell=\ell_{\rm B}$ as soon as $\ell<\infty$.

Our second main result concerns the applicability of the Direct Method on the competition classes of $\Psi_{\rm bk}(v)$ and $\Psi_{\rm bd}(v)$.

Let $\wire$ be a closed set in $\mathbb{R}^{n+1}$, $\C$ a spanning class for $\wire$, $\{(K_j,E_j)\}_j$ be a sequence in $\K_{\rm B}$ such that $\sup_j\H^n(K_j)<\infty$, and let a Borel set $E$ and Radon measures $\mu_{\rm bk}$ and $\mu_{\rm bd}$ in $\Om$ be such that $E_j\toloc E$ and

\begin{eqnarray*}
&&\H^n\mres (\Om\cap\pa^*E_j) + 2\,\H^n \mres (\mathcal{R}(K_j) \cap E_j^\zero) \weakstar \mu_{\rm bk}\,,
\\
&&\H^n\mres (\Om\cap\pa^*E_j) + 2\,\H^n \mres (\mathcal{R}(K_j) \setminus \pa^* E_j) \weakstar \mu_{\rm bd}\,,
\end{eqnarray*}

as $j\to\infty$. Then:

(i) Lower semicontinuity: the sets

\begin{eqnarray*}
K_{\rm bk}\!\! &:=&\!\!\big(\Om\cap\partial^* E\big) \cup \Big\{x\in \Omega \cap E^\zero : \theta^n_*(\mu_{\rm bk})(x)\geq 2 \Big\}\,,
\\
K_{\rm bd}\!\!&:=&\!\!\big(\Om\cap\partial^* E\big) \cup \Big\{x\in \Omega \setminus\pa^*E : \theta^n_*(\mu_{\rm bd})(x)\geq 2 \Big\}\,,
\end{eqnarray*}

are such that $(K_{\rm bk},E),(K_{\rm bd},E)\in\K_{\rm B}$ and

\begin{eqnarray*}
\mu_{\rm bk}\!\!&\ge&\!\! \H^n\mres (\Om\cap\pa^*E) + 2\,\H^n \mres (K_{\rm bk} \cap E^\zero)\,,
\\
\mu_{\rm bd}\!\!&\ge&\!\!\H^n\mres (\Om\cap\pa^*E) + 2\,\H^n \mres (K_{\rm bd} \setminus\pa^*E)\,,
\end{eqnarray*}

\begin{equation}\nonumber
\liminf_{j\to\infty}\F_{\rm bk}(K_j,E_j)\ge\F_{\rm bk}(K_{\rm bk},E)\,,\qquad
\liminf_{j\to\infty}\F_{\rm bd}(K_j,E_j)\ge\F_{\rm bd}(K_{\rm bd},E)\,.
\end{equation}

(ii) Closure: we have that

\begin{eqnarray*}
&&\mbox{if $K_j\cup E_j^\one$ is $\C$-spanning $\wire$ for every $j$,}
\\
&&\mbox{then $K_{\rm bk}\cup E^\one$ is $\C$-spanning $\wire$}\,,
\end{eqnarray*}

and that

\begin{eqnarray*}
&&\mbox{if $K_j$ is $\C$-spanning $\wire$ for every $j$,}
\\
&&\mbox{then $K_{\rm bd}$ is $\C$-spanning $\wire$}\,.
\end{eqnarray*}

The delicate part of Theorem <ref> is proving the closure statements. This will require first to extend the characterization of homotopic spanning from locally $\H^n$-finite sets to generalized soap films (Theorem <ref>), and then to discuss the behavior under weak-star convergence of the associated Radon measures of the objects appearing in conditions like (<ref>) (Theorem <ref>).

§.§ Existence of minimizers in $\Psi_{\rm bk}(v)$ and convergence to $\ell$
From this point onward, we focus our analysis on the bulk-spanning relaxation $\Psi_{\rm bk}(v)$ of Gauss' capillarity. There are a few important reasons for this choice: (i) from the point of view of physical modeling, working with the boundary or with the bulk spanning conditions seem comparable; (ii) the fact that $\Psi_{\rm bk}(0)=\Psi_{\rm bd}(0)$ suggest that, at small values of $v$, the two problems should actually be equivalent (have the same infima and the same minimizers); (iii) the bulk spanning variant is the one which is relevant for the approximation of Plateau-type singularities with solutions of the Allen–Cahn equations discussed in [42]; (iv) despite their similarities, carrying over the following theorems for both problems would require the repeated introduction of two versions of many arguments, with a significant increase in length, and possibly with at the expense of clarity.

The following theorem provides the starting point in the study of $\Psi_{\rm bk}(v)$.

If $\wire$ is a compact set in $\mathbb{R}^{n+1}$ and $\C$ is a spanning class for $\wire$ such that $\ell<\infty$, then

\begin{equation}
\label{ellb equals ell}
\ell_{\rm B}=\ell\,,
\end{equation}

and, moreover:

(i) Existence of minimizers and Euler–Lagrange equation: for every $v>0$ there exist minimizers $(K,E)$ of $\Psi_{\rm bk}(v)$ such that $(K,E)\in\KK$ and both $E$ and $K$ are bounded; moreover, there is $\l\in\R$ such that

\begin{align}\label{euler lagrange equation bulk}
\lambda \int_{\partial^* E} X \cdot \nu_{E} \,d\H^n = \int_{\partial^* E} \mathrm{div}^{K}\,X\,d\H^n + 2\int_{K \cap E^{\zero}} \mathrm{div}^{K}\, X\,d\H^n\,,
\end{align}

for every $X\in C^1_c(\mathbb{R}^{n+1};\mathbb{R}^{n+1})$ with $X\cdot \nu_\Omega =0$ on $\partial \Omega$;

(ii) Regularity from the Euler–Lagrange equations: if $(K,E)\in\KK$ is a minimizer of either $\Psi_{\rm bk}(v)$,

then there is a closed set $\Sigma\subset K$, with empty interior in $K$, such that $K\setminus\Sigma$ is a smooth hypersurface; moreover, $K\setminus(\Sigma\cup\pa E)$ is a smooth minimal hypersurface, $\Om\cap\pa^*E$ is a smooth hypersurface with mean curvature constantly equal to $\l$, and $\H^n(\Sigma\setminus\pa E)=0$; in particular, $\Om\cap(\pa E\setminus\pa^*E)$ has empty interior in $K$;

(iii) Convergence to the Plateau problem: if $(K_j,E_j)$ is a sequence of minimizers for $\Psi_{\rm bk}(v_j)$ with $v_j\to 0^+$, then there exists a minimizer $S$ of $\ell$ such  that, up to extracting subsequences, as Radon measures in $\Om$,

\begin{align}
\label{convergence as measures}
\H^n\mres(\partial^* E_j \cap \Omega) + 2\H^n\mres (K_j \cap E_j^{\zero}) \weakstar 2\H^n\mres S\,,
\end{align}

as $j\to\infty$;

In particular, $\Psi_{\rm bk}(v)\to 2\,\ell=\Psi_{\rm bk}(0)$ as $v\to 0^+$.

The conclusions of Theorem <ref> about $\Psi_{\rm bk}(v)$ can be read in parallel to the conclusions about $\psi(v)$ obtained in [34]. The crucial difference is that, in place of the “weak” minimality inequality (<ref>), which in this context would be equivalent to $\F_{\rm bk}(K,E)\le\H^n(\Om\cap\pa^*E')$ for every competitor $E'$ in $\psi_{\rm bk}(v)$, we now have the proper minimality inequality

\begin{equation}
\label{proper min}
\F_{\rm bk}(K,E)\le \F_{\rm bk}(K',E')
\end{equation}

for every competitor $(K',E')$ in $\Psi_{\rm bk}(v)$. Not only the final conclusion is stronger, but the proof is also entirely different: whereas [34] required the combination of a whole bestiary of specific competitors (like the cup, cone, and slab competitors described therein) with the full force of Preiss' theorem, the approach presented here seems more robust as it does not exploit any specific geometry, and it is squarely rooted in the basic theory of sets of finite perimeter.

§.§ Equilibrium across transition lines in wet soap films
We now formalize the validation of (<ref>) for soap films in the form of a sharp regularity theorem for minimizers $(K,E)$ of $\Psi_{\rm bk}(v)$.

The starting point to obtain this result is the connection between homotopic spanning and partitions into indecomposable sets of finite perimeter established in Theorem <ref>/Theorem <ref>. This connection hints at the possibility of showing that if $(K,E)$ is a minimizer of $\Psi_{\rm bk}(v)$, then the elements $\{U_i\}_i$ of the essential partition of $\Om$ induced by $K\cup E^\one$ are actually $(\Lambda,r_0)$-minimizers of the perimeter in $\Om$, i.e., there exist $\Lambda$ and $r_0$ positive constants such that

\[
P(U_i;B_r(x))\le P(V;B_r(x))+\Lambda\,|V\Delta U_i|\,,
\]

whenever $V\Delta U_i\cc \Om$ and $\diam(V\Delta U_i)<r_0$. The reason why this property is not obvious is that proving the $(\Lambda,r_0)$-minimality of $U_i$ requires working with arbitrary local competitors $V_i$ of $U_i$. However, when working with homotopic spanning conditions, checking the admissibility of competitors is the notoriously delicate heart of the matter – as reflected in the fact that only very special classes of competitors have been considered in the literature (see, e.g., the cup and cone competitors and the Lipschitz deformations considered in [16], the slab competitors and exterior cup competitors of [34], etc.).

The idea used to overcome this difficulty, which is illustrated in

On the left, a minimizer $(K,E)$ of $\Psi_{\rm bk}(v)$, and the essential partition induced by $(K,E)$ in a ball $B_r(x)$; the multiplicity $2$ part of $K\cap B_r(x)$ are depicted with bold lines, to distinguish them from the multiplicity one parts in $B_r(x)\cap\pa^*E$. On the right, a choice of $(K',E')$ that guarantees both the energy gap identity (<ref>) and the $\H^n$-containment (<ref>) needed to preserve homotopic spanning. The volume constraint can of course be restored as a lower order perimeter perturbation by taking a diffeomorphic image of $(K',E')$, an operation that trivially preserves homotopic spanning.

Figure <ref>, is the following. By Theorem <ref>, we can locally represent $\F_{\rm bk}(K,E;B_r(x))$ as the sum of perimeters $P(U_i;B_r(x))+P(U_j;B_r(x))+P(U_k;B_r(x))$. Given a local competitor $V_i$ for $U_i$ we can carefully define a competitor $(K',E')$ so that the elements of the essential partition induced by $K'\cup (E')^\one$ in $\Om$, that can be used to represent  $\F_{\rm bk}(K',E';B_r(x))$ as the sum $P(V_i;B_r(x))+P(V_j;B_r(x))+P(V_k;B_r(x))$, are such that

\begin{eqnarray}\label{energy difference}
\F_{\rm bk}(K',E';B_r(x))-\F_{\rm bk}(K,E;B_r(x))=P(V;B_r(x)) - P(U_i;B_r(x))\,.
\end{eqnarray}

The trick is that by suitably defining $K'$ and $E'$ we can recover the entirety of $B_r(x)\cap\pa^*U_j$ and $B_r(x)\cap\pa^*U_k$ by attributing different parts of these boundaries to different terms in the representation of $\F_{\rm bk}(K',E';B_r(x))$. In other words we are claiming that things can be arranged so that we still have

\begin{equation}
\label{still contains}
B_r(x)\cap\big(\pa^*U_j\cap\pa^*U_k)\shn K'\cup (E')^\one\,.
\end{equation}

The fact that we have been able to preserve all but one reduced boundary among those of the elements of the essential partition of $B_r(x)$ induced by $(K,E)$ is enough to shows that $K'\cup (E')^\one$ is still $\C$-spanning $\wire$ by means of Theorem <ref>/Theorem <ref>.

By the regularity theory of $(\Lambda,r_0)$-perimeter minimizers (see, e.g. <cit.>) we can deduce the $C^{1,\a}$-regularity of the elements of the partition (away from a closed singular set with area minimizing dimensional bounds). This is already sufficient to prove the continuity of the normal across $\Om\cap(\pa E\setminus\pa^*E)$, but it also allows us to invoke the regularity theory for free boundaries in the double membrane problem, and to obtain the following sharp regularity result, with which we conclude our introduction.

If $\wire$ is a compact set in $\mathbb{R}^{n+1}$, $\mathcal{C}$ is a spanning class for $\wire$ such that $\ell<\infty$, $v>0$, and $(K_*,E_*)$ is a minimizer of $\Psi_{\rm bk}(v)$, then there is $(K,E)\in\KK$ such that $K$ is $\H^n$-equivalent to $K_*$, $E$ is Lebesgue equivalent to $E_*$, $(K,E)$ is a minimizer of $\Psi_{\rm bk}(v)$, both $E$ and $K$ are bounded, $K\cup E$ is $\C$-spanning $\wire$, and

\begin{equation}
\label{no collapsing}
K\cap E^\one=\varnothing\,;
\end{equation}

in particular, $K$ is the disjoint union of $\Om\cap\pa^*E$, $\Om\cap(\pa E \setminus\pa^*E)$, and $K\setminus\pa E$.

Moreover, there is a closed set $\Sigma\subset K$ with the following properties:

(i): $\Sigma=\varnothing$ if $1\leq n \leq 6$, $\Sigma$ is locally finite in $\Omega$ if $n=7$, and $\mathcal{H}^{s}(\Sigma)=0$ for every $s>n-7$ if $n\geq 8$;

(ii): $(\Om\cap\pa^*E)\setminus\Sigma$ is a smooth hypersurface with constant mean curvature (denoted by $\l$ if computed with respect to $\nu_E$);

(iii): $(K\setminus\pa E)\setminus\Sigma$ is a smooth minimal hypersurface;

(iv): if $\Om\cap(\pa E\setminus\pa^*E)\setminus\Sigma\ne\varnothing$, then $\l<0$; moreover, for every $x\in\Om\cap(\pa E\setminus\pa^*E)\setminus\Sigma$, $K$ is the union of two $C^{1,1}$-hypersurfaces that detach tangentially at $x$; more precisely, there are $r>0$, $\nu\in\SS^n$, $u_1,u_2\in C^{1,1}(\DD_r^\nu(x))$ such that

\[
u_1(x)=u_2(x)=0\,,\qquad\mbox{$u_1\le u_2$ on $\DD_r^\nu(x)$}\,,
\]

with $\{u_1<u_2\}$ and ${\rm int}\{u_1=u_2\}$ both non-empty, and

\begin{eqnarray}\label{grafico K}
\CC_r^\nu(x)\cap K&=&\cup_{i=1,2}\big\{y+u_i(y)\,\nu:y\in\DD_r^\nu(x)\big\}\,,
\\\label{grafico pa E}
\CC_r^\nu(x)\cap \pa^*E&=&\cup_{i=1,2}\big\{y+u_i(y)\nu:y\in\{u_1<u_2\}\big\}\,,
\\\label{grafico E}
\CC_r^\nu(x)\cap E&=&\big\{y+t\,\nu:t\in\big(u_1(y),u_2(y)\big)\big\}\,.
\end{eqnarray}

\begin{eqnarray*}
\\
\end{eqnarray*}

(v): we have

\[
\Gamma:=\Om\cap(\pa E\setminus\pa^*E)=\Gamma_{\rm reg}\cup\Gamma_{\rm sing}\,,\qquad \Gamma_{\rm reg}\cap\Gamma_{\rm sing}=\varnothing\,,
\]

where: $\Gamma_{\rm reg}$ is relatively open in $\Gamma$ and for every $x\in\Gamma_{\rm reg}$ there are $r>0$ and $\beta\in(0,1)$ such that $\Gamma_{\rm reg}\cap B_r(x)$ is a $C^{1,\b}$-embedded $(n-1)$-dimensional manifold; $\Gamma_{\rm sing}$ is relatively closed in $\Gamma$ and can be partitioned into a family $\{\Gamma_{\rm sing}^k\}_{k=0}^{n-1}$ where, for each $k$, $\Gamma_{\rm sing}^k$ is locally $\H^k$-rectifiable in $\Om$.

§.§ Equilibrium across transition lines in wet foams
Based on the descriptions provided in [53, 8], an effective mathematical model for dry foams at equilibrium in a container is that of locally perimeter minimizing clusters, originating with different terminology in [4], and presented in
<cit.> as follows. Given an open set $\Om\subset\R^{n+1}$, a locally perimeter minimizing clusters is a finite Lebesgue partition $\{U_i\}_i$ of $\Om$ into sets of finite perimeter such that, for some $r_0>0$,

\begin{equation}
\label{dry foam}
\sum_i P(U_i;B)\le \sum_iP(V_i;B)
\end{equation}

whenever $B\cc\Om$ is a ball with radius less than $r_0$, and $\{V_i\}_i$ is a Lebesgue partition of $\Om$ with $V_i\Delta U_i\cc B$ and $|V_i|=|U_i|$ for every $i$. The previously cited results of Almgren and Taylor [4, 52] imply that, up to modification of the $U_i$'s by sets of zero Lebesgue measure, when $n=2$, $K=\Om\cap\bigcup_i\pa U_i$ is a closed subset of $\Om$ that is locally $C^{1,\a}$-diffeomorphic to a plane, a $Y$-cone, or a $T$-cone; moreover, the part of $K$ that is a surface is actually smooth and each of its connected component has constant mean curvature. Similar results holds when $n=1$ (by elementary methods) and when $n\ge 3$ (by exploiting [11]).

The theory for the relaxed capillarity energy $\F_{\rm bk}$ developed in this paper provides an option for modeling wet foams. Again based on the descriptions provided in [53, 8], the following seems to be a reasonable model for wet foams at equilibrium in a container. Given an open set $\Omega\subset\R^{n+1}$ we model wet foams by introducing the class

\[
\KK_{{\rm foam}}
\]

of those $(K,E)\in\KK_{\rm B}$ such that, for some positive constants $\Lambda_0$ and $r_0$,

\begin{equation}
\label{def of wet foams}
\F_{\rm bk}(K,E;B)\le \F_{\rm bk}(K',E';B)+\,\Lambda_0\,|E\Delta E'|
\end{equation}

whenever $B$ is a ball compactly contained in $\Om$ and with radius less than $r_0$, and $(K',E')\in\KK_{\rm B}$ is such that $(K\Delta K')\cup (E\Delta E')\cc B$ and there are finite Lebesgue partitions $\{U_i\}_i$ and $\{U'_i\}_i$ of $B$ induced, respectively, by $K\cup E^\one$ and by $K'\cup(E')^\one$, such that $|U_i|=|U_i'|$ for every $i$. Notice that inclusion of the term $\Lambda_0\,|E\Delta E'|$ in (<ref>) allows for the inclusion of energy perturbations due to gravity or other forces. Lemma <ref> will clarify that by taking $(K,E)\in\KK_{\rm foam}$ with $|E|=0$ we obtain a slightly more general notion of dry foam than the one proposed in (<ref>).

If $\Omega\subset\R^{n+1}$ is open and $(K_*,E_*)\in\KK_{\rm foam}$, then there is $(K,E)\in\KK\cap\KK_{\rm foam}$ such that $K$ is $\H^n$-equivalent to $K_*$, $E$ Lebesgue equivalent to $E_*$, $K\cap E^\one=\varnothing$, and such that, for every ball $B\cc\Om$, the open connected components $\{U_i\}_i$ of $B\setminus (K\cup E)$ are such that each $U_i$ is (Lebesgue equivalent to an) open set with $C^{1,\a}$-boundary in $B\setminus\Sigma$. Here $\Sigma$ is a closed subset of $\Om$ with $\Sigma=\varnothing$ if $1\leq n \leq 6$, $\Sigma$ locally finite in $\Omega$ if $n=7$, and $\mathcal{H}^{s}(\Sigma)=0$ for every $s>n-7$ if $n\geq 8$.

§.§ Organization of the paper
The sections of the paper contain the proofs of the main theorems listed above, as already specified in the statements. To these section we add three appendices. In Appendix <ref>, as already noted, we prove the equivalence of Definition <ref> and Definition <ref>. In Appendix <ref> we prove that, with some regularity of $\partial\Om$, every minimizing sequence of $\Psi_{\rm bk}(v)$ is converging to a minimizers, without need for modifications at infinity: this is, strictly speaking, not needed to prove Theorem <ref>, but it is a result of its own conceptual interest, it will be crucial for the analysis presented in [42], and it is easily discussed here in light of the proof of Theorem <ref>. Finally, Appendix <ref> contains an elementary lemma concerning the use of homotopic spanning in the plane that, to our knowledge, has not been proved in two dimensions.

§.§ Acknowledgements
We thank Guido De Philippis, Darren King, Felix Otto, Antonello Scardicchio, Salvatore Stuvard, and Bozhidar Velichkov for several interesting discussions concerning these problems. FM has been supported by NSF Grant DMS-2247544. FM, MN, and DR have been supported by NSF Grant DMS-2000034 and NSF FRG Grant DMS-1854344. MN has been supported by NSF RTG Grant DMS-1840314.

§.§ Notation
Sets and measures: We denote by $B_r(x)$ (resp., $B_r^k(x)$) the open ball of center $x$ and radius $r$ in $\R^{n+1}$ (resp., $\R^k$), and omit $(x)$ when $x=0$. We denote by $\cl(X)$, ${\rm int}(X)$, and $I_r(X)$ the closure, interior and open $\e$-neighborhood of $X\subset\R^k$. We denote by $\L^{n+1}$ and $\H^s$ the Lebesgue measure and the $s$-dimensional Hausdorff measure on $\R^{n+1}$, $s\in[0,n+1]$. If $E\subset\R^k$, we set $|E|=\L^k(E)$ and $\om_k=|B_1^k|$. We denote by $E^{{\scriptscriptstyle{(t)}}}$, $t\in[0,1]$, the points of density $t$ of a Borel set $E\subset\R^{n+1}$, so that $E$ is $\L^{n+1}$-equivalent to $E^{\one}$, and, for every pair of Borel sets $E,F\subset\R^{n+1}$,

\begin{equation}
\label{X cup Y zero}
(E\cup F)^\zero=E^\zero\cap F^\zero\,.
\end{equation}

We define by $\pa^eE=\R^{n+1}\setminus(E^{\zero}\cup E^{\one})$ the essential boundary of $E$. Given Borel sets $E_j,E\subset \Om$ we write

\[
E_j\to E\,,\qquad E_j\toloc E\,,
\]

when, respectively, $|E_j\Delta E|\to 0$ or $|(E_j\Delta E)\cap \Om'|\to 0$ for every $\Om'\cc\Om$, as $j\to\infty$. Given a Radon measure $\mu$ on $\R^{n+1}$, the $k$-dimensional lower density of $\mu$ is the Borel function $\theta^k_*(\mu):\R^{n+1}\to [0,\infty]$ defined by

\begin{align*}
\theta^k_*(\mu)(x) = \liminf_{r\to 0^+}\frac{\mu(\cl(B_r(x)))}{\omega_k r^k}\,.
\end{align*}

We repeatedly use the fact that, if $\theta^k_*(\mu)\ge\l$ on some Borel set $K$ and for some $\l\ge0$, then $\mu\ge\l\,\H^k\llcorner K$; see, e.g. <cit.>.

Rectifiable sets: Given an integer $0\le k\le n+1$, a Borel set $S\subset\R^{n+1}$ is locally $\H^k$-rectifiable in an open set $\Om$ if $S$ is locally $\H^k$-finite in $\Om$ and $S$ can be covered, modulo $\H^k$-null sets, by a countable union of Lipschitz images of $\R^k$ in $\R^{n+1}$. We say that $S$ is purely $\H^k$-unrectifiable if $\H^k(S\cap M)=0$ whenever $M$ is a Lipschitz image of $\R^k$ into $\R^{n+1}$. Finally, we recall that if $S$ is a locally $\H^k$-finite set in $\Om$, then there is a pair $(\RR(S),\P(S))$ of Borel sets, uniquely determined modulo $\H^k$-null sets, and that are thus called, with a slight abuse of language, the rectifiable part and the unrectifiable part of $S$, so that $\RR(S)$ is locally $\H^k$-rectifiable in $\Om$, $\P(S)$ is purely $\H^k$-unrectifiable, and $S=\RR(S)\cup\P(S)$; see, e.g. <cit.>.

Sets of finite perimeter: If $E$ is a Borel set in $\R^{n+1}$ and $D1_E$ is the distributional derivative of the characteristic function of $E$, then we set $\mu_E=-D1_E$. If $A$ is the largest open set of $\R^{n+1}$ such that $\mu_E$ is a Radon measure in $A$ (of course it could be $A=\varnothing$), then $E$ is of locally finite perimeter in $A$ and the reduced boundary $\pa^*E$ of $E$ is defined as the set of those $x\in A\cap\spt\mu_E$ such that $\mu_E(B_r(x))/|\mu_E|(B_r(x))$ has a limit $\nu_E(x)\in \SS^n$ as $r\to 0^+$. Moreover, we have the general identity (see <cit.>)

\begin{equation}
\label{sofp1}
A\cap\cl(\pa^*E)=A\cap\spt\mu_E=\Big\{x\in A:0<|E\cap B_r(x)|<|B_r(x)|\,\,\forall r>0\Big\}\subset A\cap\pa E\,.
\end{equation}

By De Giorgi's rectifiability theorem, $\pa^*E$ is locally $\H^n$-rectifiable in $A$, $\mu_E=\nu_E\,\H^n\mres(A\cap\pa^*E)$ on $A$, and $\pa^*E\subset A\cap E^\half\subset A\cap \pa^eE$, and

\begin{equation}
\label{degiorgi rect th}
(E-x)/r \toloc H_{E,x}:=\big\{y\in\R^{n+1}:y\cdot\nu_E(x)<0\big\}\,,\qquad\mbox{as $r\to 0^+$}\,.
\end{equation}

By a result of Federer,

\begin{equation}
\label{federer theorem}
\mbox{$A$ is $\H^n$-contained in $E^{\zero}\cup E^{\one}\cup \pa^*E$}\,;
\end{equation}

in particular, $\pa^*E$ is $\H^n$-equivalent to $A\cap\pa^eE$, a fact frequently used in the following. By Federer's criterion for finite perimeter, if $\Om$ is open and $E$ is a Borel set, then

\begin{equation}
\label{federer criterion}
\H^n(\Om\cap\pa^eE)<\infty\qquad\Rightarrow\qquad\mbox{$E$ is of finite perimeter in $\Om$}\,,
\end{equation}

see <cit.>. If $E$ and $F$ are of locally finite perimeter in $\Om$ open, then so are $E\cup F$, $E\cap F$, and $E\setminus F$, and by <cit.>, we have

\begin{eqnarray}
\label{E cup F}
\Om\cap\pa^*(E\cup F)\ehn \Om\cap\Big\{\big(E^\zero\cap\pa^*F\big)\cup\big(F^\zero\cap\pa^*E\big)\cup\{\nu_E=\nu_F\}\Big\}\,,
\\
\label{E cap F}
\Om\cap\pa^*(E\cap F)\ehn \Om\cap\Big\{\big(E^\one\cap\pa^*F\big)\cup\big(F^\one\cap\pa^*E\big)\cup\{\nu_E=\nu_F\}\Big\}\,,
\\
\label{E minus F}
\Om\cap\pa^*(E\setminus F)\ehn \Om\cap\Big\{\big(E^\one\cap\pa^*F\big)\cup\big(F^\zero\cap\pa^*E\big)\cup\{\nu_E=-\nu_F\}\Big\}\,,
\end{eqnarray}

where $\{\nu_E=\pm\nu_F\}:=\{x\in\pa^*E\cap\pa^*F:\nu_E(x)=\pm\nu_F(x)\}$. By exploiting Federer's theorem (<ref>), (<ref>), (<ref>), and (<ref>) we can also deduce (the details are left to the reader)

\begin{eqnarray}
\label{X cap Y zero}
(E\cap F)^\zero&\ehn&E^\zero\cup F^\zero\cup\{\nu_E=-\nu_F\}\,,
\\
\label{X minus Y zero}
(E\setminus F)^\zero&\ehn&E^\zero\cup F^\one\cup\{\nu_E=\nu_F\}\,.
\end{eqnarray}

Finally, combining (<ref>), (<ref>), and (<ref>), we find

\begin{equation}
\label{E delta F}
\pa^*(E\Delta F)\ehn(\pa^*E)\Delta(\pa^*F)\,.
\end{equation}

Partitions: Given a Radon measure $\mu$ on $\R^{n+1}$ and Borel set $U\subset\R^{n+1}$ we say that  $\{U_i\}_i$ is a $\mu$-partition of $U$ if $\{U_i\}_i$ is an at most countable family of Borel subsets of $U$ such that

\begin{equation}
\label{mu partition}
\mu\Big(U\setminus\bigcup_i U_i\Big)=0\,,\qquad \mu(U_i\cap U_j)=0\quad\forall i,j\,;
\end{equation}

and we say that $\{U_i\}_i$ is a monotone $\mu$-partition if, in addition to (<ref>), we also have $\mu(U_i)\ge\mu(U_{i+1})$ for every $i$. When $\mu=\L^{n+1}$ we replace “$\mu$-partition” with “Lebesgue partition”. When $U$ is a set of finite perimeter in $\R^{n+1}$, we say that $\{U_i\}_i$ is a Caccioppoli partition of $U$ if $\{U_i\}_i$ is a Lebesgue partition of $U$ and each $U_i$ is a set of finite perimeter in $\R^{n+1}$: in this case we have

\begin{eqnarray}
\label{caccioppoli partitions}
\pa^*U\shn\bigcup_i\pa^*U_i\,,\qquad 2\, \H^n\Big(U^\one\cap\bigcup_i\pa^*U_i\Big)=\sum_i\H^n(U^\one\cap\pa^*U_i)\,,
\end{eqnarray}

see, e.g., <cit.>; moreover,

\begin{equation}
\label{caccioppoli exactly two}
1\le \#\Big\{i:x\in\pa^*U_i\Big\}\le 2\,,\qquad\forall x\in \bigcup_i\pa^*U_i\,,
\end{equation}

thanks to (<ref>) and to the fact that there cannot be three disjoint half-spaces in $\R^{n+1}$.

§ INDUCED ESSENTIAL PARTITIONS (THEOREM <REF>)
Given a Borel set $S$, we say that a Lebesgue partition $\{U_i\}_i$ of $U$ is induced by $S$ if, for each $i$,

\begin{equation}
\label{induced partition}
\mbox{$U^\one\cap\pa^eU_i$ is $\H^n$-contained in $S$}\,.
\end{equation}

We say that $\{U_i\}_i$ is an essential partition of $U$ induced by $S$ if it is a Lebesgue partition of $U$ induced by $S$ such that, for each $i$,

\begin{equation}
\label{essential partition}
\mbox{$S$ does not essentially disconnect $U_i$}\,.
\end{equation}

The next theorem, which expands the statement of Theorem <ref>, shows that when $\H^n$-finite sets uniquely determine induced essential partitions on sets of finite perimeter.

If $U\subset\R^{n+1}$ is a bounded set of finite perimeter and $S\subset\R^{n+1}$ is a Borel set with $\H^n(S \cap U^\one)<\infty$, then there exists an essential partition $\{U_i\}_i$ of $U$ induced by $S$ such that each $U_i$ is a set of finite perimeter and

\begin{equation}
\label{essential partition perimeter bound}
\sum_iP(U_i;U^\one) \leq 2\,\H^n(S \cap U^\one)\,.
\end{equation}

Moreover: (a): if $S^*$ is a Borel set with $\H^n(S^* \cap U^\one)<\infty$, $S^*$ is $\H^n$-contained in $S$, $\{V_j\}_j$ is a Lebesgue partition[Notice that here we are not requiring that $S^*$ does not essentially disconnect each $V_j$, i.e., we are not requiring that $\{V_j\}_j$ is an essential partition induced by $S^*$. This detail will be useful in the applications of this theorem.] of $U$ induced by $S^*$, and $\{U_i\}_i$ is the essential partition of $U$ induced by $S$, then

\begin{equation}
\label{essential partitions are monotone}
\mbox{$\bigcup_j\,\pa^*V_j$ is $\H^n$-contained in $\bigcup_i\,\pa^*U_i$}\,;
\end{equation}

(b): if $S$ and $S^*$ are $\H^n$-finite sets in $U^\one$, and either[Here $\RR(S)$ denotes the $\H^n$-rectifiable part of $S$.] $S^*=\RR(S)$ or $S^*$ is $\H^n$-equivalent to $S$, then $S$ and $S^*$ induce $\mathcal{L}^{n+1}$-equivalent essential partitions of $U$.

Immediate consequence of Theorem <ref>.

The proof of Theorem <ref> follows the main lines of the proof of <cit.>, which is indeed the case $S=\varnothing$ of Theorem <ref>. We premise to this proof two lemmas that will find repeated applications in later sections too. To introduce the first lemma, we notice that, while it is evident that if $S$ is $\C$-spanning $\wire$ and $S$ is $\H^n$-contained into some Borel set $S^*$, then $S^*$ is also $\C$-spanning $\wire$, however, it is not immediately clear if the rectifiable part $\RR(S)$ of $S$ (which may not be $\H^n$-equivalent to $S$) retains the $\C$-spanning property.

If $\wire$ is compact, $\C$ is a spanning class for $\wire$, $S$ is $\C$-spanning $\wire$, and $\H^n\mres S$ is a Radon measure in $\Omega$, then $\mathcal{R}(S)$ is $\C$-spanning $\wire$. Moreover, the sets $T_1$ and $T_2$ appearing in (<ref>) are sets of finite perimeter.

We make the following claim: if $T$ is open, $T^{\one}\shn T$, $\H^n\mres Z$ is a Radon measure in an open neighborhood of $T$, and $Z$ essentially disconnects $T$ into $\{T_1,T_2\}$, then

\begin{eqnarray}\label{rect disc 1}
&&\mbox{$T_1$ and $T_2$ are of locally finite perimeter in $T$}\,,
\\\label{rect disc 2}
&&\mbox{$\RR(Z)$ essentially disconnects $T$ into $\{T_1,T_2\}$}\,.
\end{eqnarray}

Indeed: Since $T$ is open, we trivially have $T \subset T^{\one}$, and hence $T$ is $\H^n$-equivalent to $T^{\one}$. Taking also into account that $Z$ essentially disconnects $T$ into $\{T_1,T_2\}$, we thus find

\begin{align}\notag
T\cap\partial^e T_1 \cap \partial^e T_2 \ehn T^{\one}\cap \partial^e T_1 \cap \partial^e T_2  \shn Z \cap T^{\one} \shn Z \cap T\,.
\end{align}

By Federer's criterion (<ref>) and the $\H^n$-finiteness of $Z$ in an open neighborhood of $T$ we deduce (<ref>). By Federer's theorem (<ref>), $\partial^e T_i $ is $(\H^n\mres T)$-equivalent to $\partial^* T_i$, which combined with the $\H^n$-equivalence of $T^{\one}$ and $T$ gives

\begin{align}\notag
\partial^e T_1 \cap \partial^e T_2 \cap T^{\one} \ehn \partial^* T_1 \cap \partial^* T_2 \cap T\,. %\shn \mathcal{R}(Z) \cap T\,,
\end{align}

Since $\partial^* T_1 \cap \partial^* T_2 \cap T$ is $\H^n$-rectifiable and $\partial^e T_1 \cap \partial^e T_2 \cap T^{\one}\shn Z$, we conclude that $\H^n(\partial^e T_1 \cap \partial^e T_2 \cap T^{\one} \cap\mathcal{P}(Z))=0$. Hence,

\[
\partial^e T_1 \cap \partial^e T_2 \cap T^{\one}\shn \RR(Z)\,,
\]

and (<ref>) follows.

To prove the lemma: Let $(\g,\Phi,T)\in\T(\C)$, let $J$ be of full measure such that (<ref>) holds for every $s\in J$, so that, for every $s\in J$ one finds that for $\H^n$-a.e. $x\in T[s]$ there is a partition $\{T_1,T_2\}$ of $T$ with $x\in\partial^e T_1 \cap \partial^e T_2$ and such that $S \cup  T[s]$ essentially disconnects $T$ into $\{T_1,T_2\}$. By applying the claim with  $Z=S\cup T[s]$, we see that $\RR(S\cup T[s])$ essentially disconnects $T$ into $\{T_1,T_2\}$, and that $T_1$ and $T_2$ have locally finite perimeter in $T$. On noticing that $\RR(S\cup T[s])$ is $\H^n$-equivalent to $\RR(S)\cup T[s]$, we conclude the proof.

The second lemma is just a simple compactness statement for finite perimeter partitions.

If $U$ is a bounded open set and $\{\{U_i^j\}_{i=1}^\infty\}_{j=1}^\infty$ is a sequence of Lebesgue partitions of $U$ into sets of finite perimeter such that

\begin{eqnarray}\label{uni per part bound}
\sup_j\,\sum_{i=1}^\infty P(U_i^j) < \infty\,,
\end{eqnarray}

then, up to extracting a subsequence, there exists a Lebesgue partition $\{U_i\}_{i\in \mathbb{N}}$ of $U$ such that for every $i$ and every $A\subset U$ open,

\begin{align}\label{partition convergence}
\lim_{j\to\infty}|U_i^j\Delta U_i|=0\,,\qquad P(U_i;A) \leq \liminf_{j\to \infty} P(U_i^j;A)\,.
\end{align}

\begin{align}\label{partition volume with power s}
\lim_{i\to \infty}\limsup_{j\to \infty} \sum_{k=i+1}^\infty |U^j_k|^{s} =0\,,\qquad\forall s\in\Big(\frac{n}{n+1},1\Big)\,.
\end{align}

Up to a relabeling we can assume each $\{U_i^j\}_i$ is monotone. By (<ref>)  and the boundedness of $U$, a diagonal argument combined with standard lower semicontinuity and compactness properties of sets of finite perimeter implies that we can find a not relabeled subsequence in $j$ and a family $\{U_i\}_i$ of Borel subsets of $U$ with $|U_i|\ge |U_{i+1}|$ and $|U_i\cap U_j|=0$ for every $i\ne j$, such that (<ref>) holds. We are thus left to prove (<ref>) and

\begin{align}\label{is a partition}
\Big|U \setminus \bigcup_{i=1}^\infty U_i\Big|=0\,.
\end{align}

We start by noticing that for each $i$ there is $J(i)\in\N$ such that $|U_k^j|\le 2\,|U_k|$ for every $j\ge J(i)$ and $1\le k\le i$. Therefore if $k\ge i+1$ and $j\ge J(i)$ we find $|U_k^j|\le |U_i^j|\le 2\,|U_i|$, so that, if $j\ge J(i)$,

\begin{align}\label{bad thing take two}
\sum_{k=i+1}^\infty |U_k^{j}|^s \leq C(n)\,\sum_{k=i+1}^\infty P(U_{k}^{j})|U_k^{j}|^{s-(n/(n+1))}\le C\,|U_i|^{s-(n/(n+1))}\,,
\end{align}

where we have also used the isoperimetric inequality and (<ref>). Since $|U_{i}|\to 0$ as $i\to\infty$ (indeed, $\sum_i|U_i|\le|U|<\infty$), (<ref>) implies (<ref>). To prove (<ref>), we notice that if we set $M=|U \setminus \cup_i U_i|$, and we assume that $M$ is positive, then up to further increasing the value of $J(i)$ we can require that

\begin{align}\label{preliminary bad thing}
|U_k^j|\leq  |U_k| + \frac{M}{2^{k+2}}\,,\qquad\forall 1\leq k \leq i\,,\,\forall j\ge J(i)\,,
\end{align}

(in addition to $|U_k^j|\le 2\,|U_k|$). By (<ref>) we obtain that, if $j\ge J(i)$, then

\begin{align}\label{bad thing}
|U| - \sum_{k=i+1}^\infty |U_k^{j}|=\sum_{k=1}^i |U_k^{j}| \leq \sum_{k=1}^i  |U_k| + \frac{M}{2^{k+2}}\leq |U|-M+\sum_{k=1}^i  \frac{M}{2^{k+2}}\le |U|- \frac{M}{4}\,.
\end{align}

Rearranging (<ref>) and using the sub-additivity of $z\mapsto z^s$ we conclude that

\begin{align}\notag
(M/4)^s \leq \sum_{k=i+1}^\infty |U_k^{j}|^s\,.
\end{align}

We obtain a contradiction with $M>0$ by letting $i\to\infty$ and by using (<ref>).

Let $\mathcal{U}(S)$ be the set of all the monotone Lebesgue partitions of $U$ induced by $S$. We notice that $\mathcal{U}(S)\ne\varnothing$, since $\mathcal{U}(S)$ contains the trivial partition with $U_1=U$ and $U_i=\varnothing$ if $i\ge 2$. If $U_i\in\{U_i\}_i$ for $\{U_i\}_i\in \mathcal{U}(S)$, then $\partial^e U_i$ is $\H^n$-contained in $\partial^e U \cup (U^{(1)} \cap S)$, which, by Federer's criterion applied to $U$ and $\mathcal{H}^n(S \cap U^\one)<\infty$, has finite $\mathcal{H}^n$-measure; it follows then that $U_i$ is a set of finite perimeter due to Federer's criterion. We now fix $s\in(n/(n+1),1)$, and consider a maximizing sequence $\{\{U_i^j\}_i\}_j$ for

\[
m=\max\Big\{\sum_{i=1}^\infty |U_i|^{s}:\{U_i\}_i\in \mathcal{U}(S)\Big\}\,.
\]

By standard arguments concerning reduced boundaries of disjoint sets of finite perimeter (see, e.g. <cit.>), we deduce from (<ref>) that for every $j$,

\begin{eqnarray}\notag
\sum_{i=1}^\infty \mathcal{H}^n\mres \partial^*
U_i^j&=&\sum_{i=1}^\infty \mathcal{H}^n\mres (\partial^*U_i^j \cap U^{\one})+\sum_{i=1}^\infty \mathcal{H}^n\mres( \partial^*U_i^j\cap \partial^* U)\\ \label{actual perimeter bound 2}
&\leq&2\,\H^n\mres(S\cap U^\one) + \mathcal{H}^n\mres \partial^* U\,.
\end{eqnarray}

Also, due to the sub-additivity of $z\mapsto z^s$ and the general fact that $\partial^e (A \cap B) \subset \partial^e A \cup \partial^e B$, we can refine $\{U^j_i\}_i$ by replacing each $U_i^j$ with the disjoint family

\[
\big\{U_i^j \cap U^\ell_k:k\ge1\,,1\le\ell<j\big\}\,,
\]

thus obtaining a new sequence in $\mathcal{U}(S)$ which is still maximizing for $m$. As a consequence of this remark, we can assume without loss of generality that the considered maximizing sequence $\{\{U_i^j\}_i\}_j$ for $m$ has the additional property that

\begin{align}\label{nestedness}
U\cap\bigcup_i \partial^* U_i^j \subset U\cap\bigcup_i \partial^* U_i^{j+1}\,,\qquad\forall j\,.
\end{align}

Thanks to (<ref>) we can apply Lemma <ref> and, up to extracting a subsequence in $j$, we can find a Lebesgue partition $\{U_i\}_{i\in \mathbb{N}}$ of $U$ by sets of finite perimeter which satisfies (<ref>) and (<ref>). Moreover, after taking a subsequence, we may assume that $\mathcal{H}^n\mres \partial^* U_i^j\weakstar\mu_i$ for some Radon measures $\mu_i$ such that $\mathcal{H}^n\mres \partial^* U_i \leq \mu_i$ <cit.>. Therefore, by (<ref>), Federer's theorem for reduced boundaries, and by (<ref>) for $\{U^j_i\}_i$, we see that

\begin{align}\notag
\mathcal{H}^n&\mres ( \pa^* U)+\sum_{i=1}^\infty \mathcal{H}^n\mres (\partial^* U_i\cap U^\one )=\sum_{i=1}^\infty \mathcal{H}^n\mres (\partial^* U_i)\leq {\rm w}^*\lim_{j\to \infty} \sum_{i=1}^\infty \mathcal{H}^n\mres (\partial^* U_i^j)\\ \notag
&={\rm w}^*\lim_{j\to \infty}\mathcal{H}^n\mres (\pa^* U) + \sum_{i=1}^\infty \H^n\mres (\partial^e U_i^j \cap U^\one ) \leq \mathcal{H}^n\mres (\pa^* U)+ 2\H^n\mres (S\cap U^\one) \,.
\end{align}

By subtracting $\mathcal{H}^n\mres (\pa^* U)$ from both sides, we deduce (<ref>).

We now show, first, that $\{U_i\}_i\in\mathcal{U}(S)$ (i.e., we check the validity of (<ref>) on $\{U_i\}_i$), and then that $S$ does not essentially disconnect any of the $U_i$. This will complete the proof of the first part of the statement.

To prove that $U^\one \cap \partial^e U_i \shn S$, let us introduce the $\H^n$-rectifiable set $S_0$ defined by

\begin{align}\label{s prime def}
S_0 = U^\one\cap \bigcup_{i,j}\partial^* U_i^j\,.
\end{align}

By $\{U_i^j\}_i\in\mathcal{U}(S)$, $S_0$ is contained into $S$ modulo $\H^n$-null sets. Therefore, in order to prove (<ref>) it will be enough to show that

\begin{align}\label{reduced containment}
U^{\one}\cap \partial^* U_i\shn S_0\,,\qquad\forall i\,.
\end{align}

Should this not be the case, it would be $\H^n(U^\one \cap \partial^* U_i \setminus S_0)>0$ for some $i$. We could thus pick $x\in U^\one\cap \partial^* U_i$ such that

\begin{align}\label{1 density at x}
\theta^n(\H^n \mres (U^\one \cap \partial^* U_i \setminus S_0))(x) = 1\,.
\end{align}

Since $\theta^n(\H^n\mres\pa^*U_i)(x)=1$ and $S_0 \subset U^\one$ this implies $\H^n(S_0\cap B_r(x))={\rm o}(r^n)$, while $\pa^*U_i\subset U_i^\half$ gives $|U_i\cap B_r(x)|=(\om_{n+1}/2)\,r^{n+1}+{\rm o}(r^{n+1})$. Therefore, given $\de>0$ we can find $r>0$ such that

\[
\H^n( S_0 \cap B_r(x)) < \delta\, r^n\,,\qquad \min\big\{|U_i\cap B_r(x)|,|U_i\setminus B_r(x)|\big\}\ge \Big(\frac{\om_{n+1}}2-\de\Big)\,r^{n+1}\,,
\]

and then exploit the relative isoperimetric inequality and (<ref>) to conclude that

\begin{eqnarray*}
c(n)\, \Big[\Big(\frac{\om_{n+1}}2-\de\Big)\,r^{n+1}\Big]^{n/(n+1)}&\le& P(U_i;B_r(x))\le\liminf_{j\to\infty}P(U_i^j;B_r(x))
\\
&\le&\H^n(S_0\cap B_r(x))\le \de\,r^n\,,
\end{eqnarray*}

where in the next to last inequality we have used the definition (<ref>) of $S_0$. Choosing $\de>0$ small enough we reach a contradiction, thus deducing that $\{U_i\}_i\in\mathcal{U}(S)$.

Taking into account the subadditivity of $z\mapsto z^s$, in order to prove that $S$ does not essentially disconnect any $U_i$ it is sufficient to show that $\{U_i\}_i$ is a maximizer of $m$. To see this, we notice that $|U_i^j\Delta U_i|\to 0$ as $j\to\infty$ implies

\[
\]

so that, letting $k\to\infty$ and exploiting (<ref>), we conclude that

\begin{equation}
\label{maximal energy}
\end{equation}

This completes the proof of the first part of the statement (existence of essential partitions).

Let now $S$, $S^*$, $\{U_i\}_i$, and $\{U_j^*\}_j$  be as in statement (a) – that is, $S^*$ is $\H^n$-contained in $S$, $\{U_i\}_i$ is an essential partition of $U$ induced by $S$, and, for every $j$, $\{U_j^*\}_j$ is a Lebesgue partition of $U$ induced by $S^*$ – and set $Z=\cup_i\pa^*U_i$ and $Z^*=\cup_j\pa^*U_j^*$. Arguing by contradiction with (<ref>), let us assume $\H^n(Z^*\setminus Z)>0$. By the definition of Lebesgue partition we have that $Z\setminus U^\one$ and $Z^*\setminus U^\one$ are both $\H^n$-equivalent to $\pa^*U$. Therefore we have $\H^n((Z^*\setminus Z)\cap U^\one)>0$. Since $U^\one$ is $\H^n$-equivalent to the union of the $\{U_i^{\one}\cup \pa^* U_i\}_{i\in I}$ we can find $i\in I$ and $j\in J$ such that $\H^n(U_i^{\one}\cap\pa^*U_j^*)>0$. This implies that both $(U_i\cap U_j^*)^\half$ and $(U_i\setminus U_j^*)^\half$ are non-empty, and thus that $\{U_j^*\cap U_i,U_i\setminus U_j^*\}$ is a non-trivial Borel partition of $U_i$. Since

\[
U_i^{\one}\cap\pa^e(U_j^*\cap U_i)\shn U^\one\cap \pa^*U_j^*\shn S^*\,,
\]

we conclude that $S^*$ is essentially disconnecting $U_i$, against the fact that $S$ is not essentially disconnecting $U_i$ and the fact that $S^*$ is $\H^n$-contained in $S$.

We finally prove statement (b). Let $\{U_i\}_{i\in I}$, and $\{U_j^*\}_{j\in J}$ be essential partitions of $U$ induced by $S$ and $S^*$ respectively. Given $i\in I$ such that $|U_i|>0$, there is at least one $j\in J$ such that $|U_i\cap U_j^*|>0$. We claim that it must be $|U_i\setminus U_j^*|=0$. Should this not be the case, $\pa^*U_j^*$ would be essentially disconnecting $U_i$, thus implying that $S^*$ (which contains $\pa^*U_j^*$) is essentially disconnecting $U_i$. Now, either because we are assuming that $S^*$ is $\H^n$-equivalent to $S$, or because we are assuming that $S^*=\RR(S)$ and we have Lemma <ref>, the fact that $S^*$ is essentially disconnecting $U_i$ implies that $S$ is essentially disconnecting $U_i$, a contradiction. Having proved the claim, for each $i\in I$ with $|U_i|>0$ there is a unique $\s(i)\in J$ such that $|U_i\Delta U_{\s(j)}^*|=0$. This completes the proof.

§ HOMOTOPIC SPANNING ON GENERALIZED SOAP FILMS (THEOREM <REF>)
The goal of this section is proving Theorem <ref>, and, actually, to obtain an even more general result. Let us recall that the objective of Theorem <ref> was to reformulate the homotopic spanning property for a Borel set $S$, in the case when $S$ is locally $\H^n$-finite, in terms of unions of boundaries of induced essential partitions. We shall actually need this kind of characterization also for sets $S$ of the more general form $S=K\cup E^\one$, where $(K,E)\in\K_{\rm B}$. For an illustration of the proposed characterization of homotopic spanning on this type of sets, see

In panel (a) we have depicted a pair $(K,E)$ where $E$ is a tube inside $T$ and $K$ consists of the union of the boundary of $E$ and the non-spanning set $S$ of Figure <ref>-(a). Notice that $K$ is not $\C$-spanning, if we see things from the point of view of Definition <ref>, since it misses every loop $\g$ contained in the interior of $E$; while, of course, $K\cup E$ is $\C$-spanning because $E$ has been added. In panel (b) we have depicted the essential partition $\{U_i\}_{i=1}^5$ of $T$ induced by $K\cup T[s]$.  Notice that $E=U_1$, therefore no $\pa^*U_i\cap\pa^*U_j$ $\H^1$-containis $T[s]\cap E$. In particular, $T[s]\cap E$ (which $\H^1$-equivalent to $T[s]\setminus E^\zero$) is not $\H^1$-contained in ${\rm UBEP}(K\cup T[s];T)$, and we see again, this time from the point of view of Definition <ref> as reformulated in Theorem <ref>, that $K$ is not $\C$-spanning. As stated in Theorem <ref>, from the viewpoint of Definition <ref> it is only the $\H^1$-containment of $T[s]\cap E^\zero$ into ${\rm UBEP}(K\cup T[s];T)$ that establishes the $\C$-spanning property of $K\cup E$: and this $\H^1$-containment indeed holds, since $T[s]\cap E^\zero=T[s]\setminus\cl(E)$ is $\H^1$-contained in the union of $\pa^*U_2\cap\pa^*U_3$ and $\pa^*U_4\cap\pa^*U_5$.

Figure <ref>.

If $\wire\subset\R^{n+1}$ is a closed set in $\R^{n+1}$, $\C$ is a spanning class for $\wire$, $K$ is a Borel set locally $\H^n$-finite in $\Om$, and $E$ is of locally finite perimeter in $\Om$ such that
$\Om\cap\pa^*E$ is $\H^n$-contained in $K$, then the set

\begin{eqnarray}
\label{S equal to K union E}
S=\RR(K)\cup E^{\one}
\end{eqnarray}

is $\C$-spanning $\wire$ if and only if, for every $(\gamma,\Phi, T)\in \mathcal{T}(\C)$ and $\H^1$-a.e. $s\in\SS^1$,

\begin{eqnarray}\label{spanning and the S partition equation}
&&\mbox{$T[s]\cap E^\zero$ is $\H^n$-contained in ${\rm UBEP}(K\cup T[s];T)$}\,.
\end{eqnarray}

An immediate corollary of Theorem <ref> is that if $K$ is $\H^n$-finite and $(K,E)\in\KK_{\rm B}$ then $K\cup E^\one$ is $\C$-spanning $\wire$ if and only if $\RR(K)\cup E^\one$ is $\C$-spanning $\wire$. Indeed, $\RR(K\cup T[s])=\RR(K)\cup T[s]$, so that, by (<ref>), ${\rm UBEP}(K\cup T[s])={\rm UBEP}(\RR(K)\cup T[s])$.

This is Theorem <ref> with $E=\varnothing$.

Step one: We prove the following claim: If $S$ essentially disconnects $G$ into $\{G_1,G_2\}$ and $H\subset G$ satisfies

\begin{align}\label{nont parti}
\min\{|H \cap G_1|\,,\,|H \cap G_2|\} > 0\,,
\end{align}

then $S$ essentially disconnects $H$ into $H \cap G_1$ and $H \cap G_2$. Indeed, if $x\in H^{\one}$, then $x\in \partial^e(H \cap G_i)$ if and only if $x\in \partial^e G_i$ ($i=1,2$). Hence $H^{\one} \cap \partial^e (G_1 \cap H) \subset H^{\one} \cap \partial^e G_1 \subset G^{\one} \cap \partial^e G_1$, which, by (<ref>) and our assumption on $S$ and $G$, gives the desired conclusion.

Step two: Taking from now on $S$, $K$ and $E$ as in the statement we preliminary notice that if $(\g,\Phi,T)\in\T(\C)$, $s\in\SS^1$, and $\{U_i\}_i$ is the essential partition of $T$ induced by $(\RR(K)\cup T[s])$, then

\begin{align}\label{partial E containment}
T\cap\partial^* E  \shn T\cap\bigcup_{i} \partial^* U_i\,.
\end{align}

Indeed, since $\Om\cap\pa^*E$ is $\H^n$-contained in $\RR(K)$, if a Borel set $G$ is such that $|G\cap E|\,|G\setminus E|>0$ then, by step one, $\RR(K)$ essentially disconnects $G$. In particular, since, for each $i$, $\RR(K)\cup T[s]$ does not essentially disconnect $U_i$, we find that, for each $i$,

\begin{equation}\label{in or out of E prelude}
\mbox{either $U_i^{\one}\subset E^\zero$}\qquad\mbox{or $U_i^{\one}\subset E^{\one}$}\,.
\end{equation}

Clearly, (<ref>) immediately implies  (<ref>).

Step three: We prove the “only if” part of the statement, that is, given $(\g,\Phi,T)\in\T(\C)$ and $s\in \mathbb{S}^1$, we assume that

\begin{eqnarray}\label{spanning borel x}
&&\mbox{for $\H^n$-a.e. $x\in T[s]$}\,,
\\\nonumber
&&\mbox{$\exists$ a partition $\{T_1,T_2\}$ of $T$ with $x\in\partial^e T_1 \cap \partial^e T_2$}\,,
\\ \nonumber
&&\mbox{and s.t. $\RR(K)\cup E^{\one}\cup  T[s]$ essentially disconnects $T$ into $\{T_1,T_2\}$}\,,
\end{eqnarray}

and then prove that

\begin{eqnarray}\label{spanning and the S partition equation x}
&&\mbox{$T[s]\cap E^\zero$ is $\H^n$-contained in $\bigcup_i\pa^*U_i$}\,,
\end{eqnarray}

where $\{U_i\}_i$ is the essential partition of $T$ induced by $\RR(K)\cup T[s]$. To this end, arguing by contradiction, we suppose that for some $s\in \mathbb{S}^1$, there is $G \subset T[s] \cap E^\zero$ with $\H^n(G)>0$ and such that $G\cap\cup_i\pa^*U_i=\varnothing$. In particular, there is an index $i$ such that $\H^n(G \cap U_i^{\one})>0$, which, combined with (<ref>) and $G \subset E^\zero$, implies

\begin{equation}
\label{bibn}
U_i^{\one}\subset E^\zero\,.
\end{equation}

Now by (<ref>) and $\H^n(G \cap U^{\one}_i)>0$, we can choose $x\in G \cap U_i^{\one}$ such that $\RR(K)\cup E^{\one} \cup T[s]$ essentially disconnects $T$ into some $\{T_1,T_2\}$ such that $x\in \partial^e T_1 \cap \partial^e T_2$. Then, $\{U_i \cap T_1, U_i \cap T_2\}$ is a non-trivial partition of $U_i$, so that, by step one and (<ref>), $\RR(K) \cup T[s]$ essentially disconnects $U_i$ into $\{U_i \cap T_1,U_i \cap T_2\}$. This contradicts the defining property (<ref>) of essential partitions, and concludes the proof.

Step four: We prove the “if” part of the statement. More precisely, given $(\g,\Phi,T)\in\T(\C)$ and $s\in \mathbb{S}^1$, we assume that (<ref>) holds at $s$, and then proceed to prove that (<ref>) holds at $s$. We first notice that, since $\{E^{\one},E^\zero,\pa^*E\}$ is a partition of $\Om$ modulo $\H^n$, it is enough to prove (<ref>) for $\H^n$-a.e. $x\in T[s]\cap(E^{\one}\cup E^\zero\cup\pa^*E)$.

If $x\in T[s] \cap \partial^* E$, then by letting $T_1 = T \cap E$ and $T_2 = T \setminus E$ we obtain a partition of $T$ such that $x\in T\cap\pa^*E=T\cap\pa^*T_1\cap\pa^*T_2\subset\pa^eT_1\cap\pa^eT_2$, and such that $\pa^*E$ essentially disconnects $T$ into $\{T_1,T_2\}$. Since $\Om\cap\pa^*E$ is $\H^n$-contained in $\RR(K)$, we deduce (<ref>).

If $x\in T[s] \cap E^\zero$, then, thanks to (<ref>) and denoting by $\{U_i\}_i$ the essential partition of $T$ induced by $(\RR(K)\cup T[s])$, there is an index $i$ such that $x\in T \cap \partial^* U_i$. Setting $T_1=U_i$ and $T_2=T\setminus U_i$, we have that $T \cap \partial^* U_i$ (which contains $x$) is in turn contained into $\partial^e T_1 \cap \partial^e T_2 \cap T$. Since the latter set is non-empty, $\{T_1,T_2\}$ is a non-trivial partition of $T$. Moreover, by definition of essential partition,

\begin{align}\notag
T^{\one}\cap \partial^e T_1 \cap \partial^e T_2 = T \cap \partial^e U_i \shn \RR(K) \cup T[s]\,,
\end{align}

so that $\RR(K) \cup T[s]$ essentially disconnects $T$, and (<ref>) holds.

Finally, if $x\in T[s] \cap E^{\one}$, we let $s_1=s$, pick $s_2\neq s$, denote by $\{I_1,I_2\}$ the partition of $\SS^1$ defined by $\{s_1,s_2\}$, and set

\[
T_1=\Phi(I_1\times B_1^n)\cap E\,,\qquad T_2=\Phi(I_2\times B_1^n)\cup\,\Big(\Phi(I_1\times B_1^n)\setminus E\Big)\,.
\]

This is a Borel partition of $T$, and using the fact that $x\in E^{\one}$, we compute

\[
|T_1\cap B_r(x)|=|\Phi(I_1\times B_1^n)\cap E\cap B_r(x)|=|\Phi(I_1\times B_1^n)\cap B_r(x)|+{\rm o}(r^{n+1})=\frac{|B_r(x)|}2+{\rm o}(r^{n+1})\,.
\]

Therfore $x\in \partial^e T_1 \cap \partial^e T_2$, and by standard facts about reduced boundaries <cit.>,

\[
\partial^e T_1 \cap \partial^e T_2 \cap T^{\one} \shn \partial^* T_1  \cap T^{\one} \shn \big( \partial^* E \cup ((T[s_1]\cup T[s_2]) \cap E^{\one}) \big)\cap T^{\one}\,.
\]

Since $\Om\cap\pa^*E$ is $\H^n$-contained in $\RR(K)$, we have shown (<ref>).

§ THE FUNDAMENTAL CLOSURE THEOREM FOR HOMOTOPIC SPANNING CONDITIONS
In Theorem <ref> and Theorem <ref> we have presented two reformulations of the homotopic spanning condition in terms of $\H^n$-containment into union of boundaries of essential partitions. The goal of this section is discussing the closure of such reformulations, and provide a statement (Theorem <ref> below) which will lie at the heart of the closure theorems proved in Section <ref>.

Let $\wire\subset\R^{n+1}$ be closed and let $\C$ be a spanning class for $\wire$. Let us assume that:

(a): $K_j$ are $\H^n$-finite Borel subsets of $\Om$ with $\H^n\mres K_j\weakstar\mu$ as Radon measures in $\Om$;

(b): $(\g,\Phi,T)\in\T(\C)$, $\{s_j\}_j$ is a sequence in $\SS^1$ with $s_j\to s_0$ as $j\to\infty$;

(c): if $\{U_i^j\}_i$ denotes the essential partition  of $T$ induced by $K_j\cup T[s_j]$, then there is a limit partition $\{U_i\}_i$ of $\{U_i^j\}_i$ in the sense of (<ref>) in Lemma <ref>;

Under these assumptions, if $\mu(T[s_0])=0$, $F_j,F\subset\Om$ are sets of finite perimeter with $F_j\to F$ as $j\to\infty$ and such that, for every $j$, $\Om\cap\pa^*F_j$ is $\H^n$-contained in $K_j$ and

\begin{equation}\label{Tsj Fj in Kjstar}
\mbox{$T[s_j]\cap F_j^\zero$ is $\H^n$-contained in $K^*_j$}\,,
\end{equation}

\begin{equation}\label{Ts0 F in Kstar}
\mbox{$T[s_0]\cap F^\zero$ is $\H^n$-contained in $K^*$}\,,
\end{equation}

where we have set

\begin{equation}
\label{def of Kjstar and Kstar}
K_j^*={\rm UBEP}(K_j\cup T[s_j];T)=T\cap\bigcup_i\pa^*U_i^j\,,\qquad K^*=T\cap\bigcup_i\pa^*U_i\,.
\end{equation}

Notice that $\{U_i\}_i$ may fail to be the essential partition of $T$ induced by $K^*$ (which is the “optimal” choice of a Borel set potentially inducing $\{U_i\}_i$ on $T$): indeed, some of the sets $U_i$ may fail to be essentially connected, even though $U_i^j\to U_i$ as $j\to\infty$ and every $U_i^j$, as an element of an essential partition, is necessarily essentially connected; see

The situation in the proof of Theorem <ref> in the basic case when $K_j=\Om\cap\pa^*F_j$. The essential partition of $T$ induced by $K_j\cup T[s_j]$ is denoted by $\{U_i^j\}_i$. The limit partition $\{U_i\}_i$ of $\{U_i^j\}_i$ may fail to be the essential partition of $T$ induced by $K^*=T\cap\cup_i\pa^*U_i$, since some of the $U_i$ may be essentially disconnected. In the picture, denoting by $\{V_k\}_k$ the essential partition of $T$ induced by $K^*$, we have  $U_5=V_5\cup V_6=T\cap F$. We also notice, in reference to the notation set in (<ref>), that $X_1^j=\{5\}$ and $X_0^j=\{1,2,3,4\}$.
Figure <ref>.

Step one:  We start by showing that, for each $j$ and $i$ such that $|U_i^j|>0$, we have

\begin{eqnarray}
\label{doesnt cross partial Ej}
\mbox{either}\quad (U_i^j)^{\one} \subset F_j^{\one}\,,\qquad\mbox{or}\quad(U_i^j)^{\one} \subset F_j^\zero\,,
\end{eqnarray}

and for each $i$ such that $|U_i|>0$,

\begin{equation}
\label{doesnt cross partial E}
\mbox{either}\quad U_i^{\one} \subset F^{\one}\,,\qquad\mbox{or}\quad U_i^{\one} \subset F^\zero\,.
\end{equation}

Postponing for the moment the proof of (<ref>) and (<ref>), let us record several consequences of these inclusions. First, if we set

\begin{align}\label{def of Xj}
&X^j_1=\big\{i:|U_i^j|>0\,,\, (U_i^j)^{\one} \subset F_j^{\one}\big\}\,,\qquad X^j_0=\big\{i:|U_i^j|>0\,,\,(U_i^j)^{\one} \subset F_j^\zero\big\}\,,
\\
\label{def of X}
&X_1=\big\{i:|U_i|>0\,,\,U_i^{\one} \subset F^{\one}\big\}\,,\qquad\hspace{0.5cm} X_0=\big\{i:|U_i|>0\,,\,U_i^{\one} \subset F^\zero\big\}\,,
\end{align}

then, thanks to (<ref>) and (<ref>), we have

\begin{eqnarray}
\label{Xj deco}
X^j:=\{i:|U_i^j|>0\}=X_0^j\cup X_1^j\,, \qquad X:=\{i:|U_i|>0\}=X_0\cup X_1\,.
\end{eqnarray}

Combining (<ref>) and (<ref>) with $F_j\to F$ and $U_i^j\to U_i$, we find that for every $i\in X$, there is $J_i\in \mathbb{N}$ such that, for every $m\in\{0,1\}$,

\begin{align}\label{index agreement}
\mbox{if $i\in X_m$, then $i \in X_m^j$ for all $j\geq J_i$.}
\end{align}

Lastly, $\{U_i^j\}_{i\in X_1^j}$ is a Lebesgue partition of $T \cap F_j$, and thus, by Federer's theorem (<ref>),

\begin{eqnarray}\label{partition of Ej}
T \cap  F_j^\one\,\, \shn \,\,\bigcup_{i\in X_1^j} (U_i^j)^\one\cup \pa^*U_i^j\,,
\qquad  T \cap \partial^* F_j \,\,\shn \,\,T \cap \bigcup_{i\in X_1^j} \partial^* U_i^j \,\,\subset\,\, T \cap K_j^*\,.
\end{eqnarray}

To prove (<ref>) and (<ref>): Since $\{U_i^j\}_i$ is the essential partition of $T$ induced by $K_j \cup T[s_j]$ and $K_j^*={\rm UBEP}(K_j\cup T[s_j];T)$, we have

\begin{eqnarray}
\label{Uij ess part union}
&&\mbox{$K^*_j$ is $\H^n$-contained in  $K_j \cup T[s_j]$}\,,\qquad\forall j\,,
\\
\label{Uij ess part disco}
\mbox{$K_j\cup T[s_j]$ does not essentially disconnect $U_i^j$}\,,\qquad\forall i,j\,.
\end{eqnarray}

Since $\Om\cap\pa^*F_j$ is $\H^n$-contained in $K_j\cup T[s_j]$, the combination of (<ref>) with Federer's theorem (<ref>) gives (<ref>). The combination of $|U_i^j\Delta U_i|\to 0$ as $j\to\infty$ with (<ref>) gives (<ref>).

Step two: We reduce the proof of (<ref>) to that of

\begin{eqnarray}\label{luk}
\H^n(U_i^\one\cap T[s_0])=0\,,\qquad\forall i\in X_0\,.
\end{eqnarray}

Indeed, $\{U_i^\one:i\in X_0\}\cup\{F^\zero\cap\pa^* U_i:i\in X_0\}$ is an $\H^n$-partition of $T\cap F^\zero$. In particular, $T\cap F^\zero$ is $\H^n$-contained in $\cup_{i\in X_0} U_i^{\one}\cup \partial^* U_i$, so that, should (<ref>) hold,  then $T[s_0]\cap F^\zero$ would be $\H^n$-contained in $\cup_{i\in X_0}\pa^*U_i$, and thus in $K^*$, thus proving (<ref>).

Step three: We change variables from $T$ to[Here we identify $\SS^1$ with $\R/(2\pi\Z)$ and, with a slight abuse of notation, denote by $\L^{n+1}$ the “Lebesgue measure on $\SS^1\times B_1^n$”, which we use to define sets of finite perimeter and points of density in $\SS^1\times B_1^n$.] $Y=\Phi^{-1}(T)=\SS^1\times B_1^n$. We set $Y[s]=\Phi^{-1}(T[s])=\{s\}\times B_1^n$ for the $s$-slice of $Y$, and

\begin{equation}
\label{def of Yi interface}
Y_i=\Phi^{-1}(U_i)\,,\qquad Y_i^j=\Phi^{-1}(U_i^j)\,,\qquad W_i=Y\setminus Y_i\,,\qquad W_i^j=Y\setminus Y_i^j\,,
\end{equation}

Since $\Phi$ is a diffeomorphism, by <cit.> and the area formula we have that

\begin{equation}
\label{diffeo boundaries}
\pa^*\Phi^{-1}(H)=\Phi^{-1}(\pa^*H)\,,\qquad (\Phi^{-1}(H))^{\scriptscriptstyle{(m)}}=\Phi^{-1}(H^{\scriptscriptstyle{(m)}})\,, m\in\{0,1\}\,,
\end{equation}

for every set of finite perimeter $H\subset T$; in particular, setting

\[
M_j=\Phi^{-1}(F_j\cap T)\,,\qquad M=\Phi^{-1}(F\cap T)\,,
\]

by Federer's theorem (<ref>), we see that (<ref>) is equivalent

\begin{equation}
\label{basic closure assumption Y}
\mbox{$Y[s_j]$ is $\H^n$-contained in $\bigcup_i\pa^*Y_i^j \cup M_j^\one \cup \partial^* M_j $}\,,
\end{equation}

By (<ref>) and (<ref>), we may rewrite (<ref>) as

\begin{equation}\label{alternative basic closure assumption Y}
\mbox{$Y[s_j]$ is $\H^n$-contained in $\bigcup_{i\in \mathbb{N}}\pa^*Y_i^j \cup \bigcup_{i\in X_1^j} (Y_i^j)^\one $}\,.
\end{equation}

Similarly, $Y_i^\one=\Phi^{-1}(U_i^\one)$ for every $i$, and thus (<ref>) is equivalent to

\begin{equation}
\label{basic closure conclusion Y}
\H^n(Y_i^{\one} \cap Y[s_0])=0\,,\qquad\forall i\in X_0\,.
\end{equation}

We are thus left to prove that (<ref>) implies (<ref>).

To this end, let us denote by $\pp$ the projection of $Y=\SS^1\times B_1^n$ onto $B_1^n$, and consider the sets

\[
G_i= \pp\big( Y_i^{\one} \cap Y[s_0]\big)\,,\qquad G_i^*=G^*\cap G_i\,,
\]

corresponding to the set $G^*\subset B_1^n$ with $\H^n(B_1^n\setminus G^*)=0$ defined as follows:

(i) denoting by $H_y=\{s\in \mathbb{S}^1 : (s,y)\in H\}$ the “circular slice of $H\subset Y$ above $y$”, if $y\in G^*$, $j\in\N$, $k$ is an index for the partitions $\{Y_k\}_k$ and $\{Y_k^j\}$, and $H\in\{Y_k,W_k,Y_k^j,W_k^j\}$, then $H_y$ is a set of finite perimeter in $\mathbb{S}^1$ with

\begin{equation}
\label{equivalence of reduced boundaries}
H_y \overset{\H^1}{=} (H_y)^{{\one}_{\mathbb{S}^1}}\,, \qquad
\partial^*_{\mathbb{S}^1}(H_y) \overset{\H^0}{=} (\partial^* H)_y\,,
\end{equation}

(and thus with $\partial^*_{\mathbb{S}^1}(H_y)=(\partial^* H)_y$); this is a standard consequence of the slicing theory for sets of finite perimeter, see, e.g., <cit.> or <cit.>;

(ii) for every $y\in G^*$ and $j\in \N$,

\begin{equation}
\label{ii}
(s_j,y)\in\bigcup_{k\in \mathbb{N}}\pa^*Y_k^j \cup \bigcup_{k\in X_1^j} (Y_k^j)^\one\,;
\end{equation}

this is immediate from (<ref>);

(iii) for every $y\in G^*$, and $k$ an index for the partitions $\{Y_k\}_k$ and $\{Y_k^j\}$,

\begin{equation}
\label{immediate}
\lim_{j\to\infty}\H^1((Y_k)_y\Delta (Y_k^j)_y)=0\,;
\end{equation}

this is immediate from Fubini's theorem and $Y_k^j\to Y_k$ as $j\to\infty$;

(iv) for every $y\in G^*$,

\begin{align}\label{property iv of Gstar}
\sum_k \H^0((\partial^* Y_k^j)_y ) < \infty;
\end{align}

indeed, by applying in the order the coarea formula, the area formula and (<ref>) we find

\begin{eqnarray*}
\sum_k \int_{B_1^n}\H^0((\partial^* Y_k^j)_y )\,d\H^n &\leq& \sum_k P(Y_k^j;Y) \leq (\Lip\Phi^{-1})^n\,\sum_k P(U_k^j;T)
\\
&\leq&2\,(\Lip\Phi^{-1})^n\,\H^n(K_j \cup T[s_j])\,.
\end{eqnarray*}

Now, let us pick $y\in G_i^*$. Since $y\in G_i$ implies $(s_0,y)\in Y_i^\one$, and $Y_i^\one\cap\pa^*Y_i=\varnothing$, we find $(s_0,y)\not\in\pa^*Y_i$, i.e. $s_0\not\in(\pa^*Y_i)_y$. By $y\in G^*$, we have $(\pa^*Y_i)_y=\partial^*_{\mathbb{S}^1}(Y_i)_y$, so that

\begin{equation}
\label{heyhey}
s_0\not \in \partial^*_{\mathbb{S}^1}(Y_i)_y\,.
\end{equation}

Since $(Y_i)_y$ has finite perimeter, $\partial^*_{\mathbb{S}^1}(Y_i)_y$ is a finite set, and so (<ref>) implies the existence of an open interval $\mathcal{A}_y\subset \mathbb{S}^1$, containing $s_0$, $\H^1$-contained either in $(Y_i)_y$ or in $(W_i)_y$, and such that

\begin{equation}
\label{heyhey2}
\partial_{\mathbb{S}^1}\mathcal{A}_y \subset (\partial^* Y_i)_y=\pa^*_{\SS^1}(W_i)_y\,.
\end{equation}

We claim that there is $G_i^{**}\subset G_i^*$, with full $\H^n$-measure in $G_i^*$ (and thus in $G_i$), such that

\begin{equation}
\label{bellali}
\mbox{$\mathcal{A}_y$ is $\H^1$-contained in $(Y_i)_y$}\,,\qquad\forall y\in G_i^{**}\,.
\end{equation}

Indeed, let us consider the countable decomposition $\{G_{i,m}^*\}_{m=1}^\infty$ of $G_i^*$ given by

\[
G_{i,m}^*=\Big\{y\in G_i^*:\dist\big(\{s_0\},\pa_{\SS^1}\mathcal{A}_y\big)\in\big[1\big/(m+1),1\big/m\big)\Big\}\subset B_1^n\,,
\]

and let

\[
Z_{i,m}=\big\{y\in G_{i,m}^*: \mbox{$\mathcal{A}_y$ is $\H^1$-contained in $(W_i)_y$}\big\}\,.
\]

If $\H^n(Z_{i,m})>0$, then there is $y^*\in Z_{i,m}^\one$, so that $\H^n(Z_{i,m}\cap B_r^n(y^*))=\om_n\,r^n+{\rm o}(r^n)$. Therefore, if $r<1/(m+1)$ and $B_r^1(s_0)$ denotes the open interval of center $s_0$ and radius $r$ inside $\SS^1$, then

\begin{eqnarray*}
&&\L^{n+1}\big(Y_i\cap\big(B_r^1(s_0)\times B_r^n(y^*)\big)\big)=\int_{B_r^n(y^*)}\H^1(B_r^1(s_0)\cap (Y_i)_y)\,d\H^n_y
\\
&=&\int_{Z_{i,m}\cap B_r^n(y^*)}\H^1(B_r^1(s_0)\cap (Y_i)_y)\,d\H^n_y+{\rm o}(r^{n+1}) ={\rm o}(r^{n+1})
\end{eqnarray*}

where in the last identity we have used the facts that $y\in Z_{i,m}\cap B_r^n(y^*)$, $s_0\in\mathcal{A}_y$, and $r<1/(m+1)$ to conclude that $B_r^1(s_0)$ is $\H^1$-contained in $(W_i)_y$; in particular, $(s_0,y^*)\in Y_i^\zero$, against the fact that $Z_{i,m}\subset G_i(=\pp(Y[s_0]\cap Y_i^\one))$. We have thus proved that each $Z_{i,m}$ is $\H^n$-negligible, and therefore that there is $G_i^{**}\subset G_i^*$ and $\H^n$-equivalent to $G_i^*$, such that (<ref>) holds true.

Having proved (<ref>), we now notice that, by (<ref>), $y\in G_i^*$ implies

\begin{equation}
\label{combined with sj}
s_j\in\bigcup_{k\in \mathbb{N}}(\pa^*Y_k^j)_y \cup \bigcup_{k\in X_1^j} \big((Y_k^j)^\one\big)_y=\bigcup_{k}\pa^*_{\SS^1}(Y_k^j)_y \cup \bigcup_{k\in X_1^j} \big((Y_k^j)_y \big)^{\one_{\mathbb{S}^1}}\,.%=\bigcup_k\pa^*_{\SS^1}(W_k^j)_y
\end{equation}

If (<ref>) holds because $s_j\in\pa^*_{\SS^1}(Y_k^j)_y$ for some $k$, then, thanks to (<ref>) there must $k'\ne k$ such that $s_j\in\pa^*_{\SS^1}(Y_{k'}^j)_y$ too; since either $k$ or $k'$ must be different from $i$, we conclude that $s_i\in\pa^*_{\SS^1}(Y_{k(i)}^j)_y$ for some $k(i)\ne i$; if, instead, (<ref>) holds because $s_j\in\big((Y_k^j)_y \big)^{\one_{\mathbb{S}^1}}$ for some $k\in X_1^j$, then we can recall that, thanks to (<ref>), $i\in X_0^j$ for every $j\ge J_i$, and thus $i\ne k$; in summary, for each $y\in G_i^*$,

\begin{equation}
\label{spanning yay}
\mbox{if $j\ge J_i$, then $\exists k(j)\ne i$ s.t.}\,\,s_j\in \partial^*_{\mathbb{S}^1} (Y_{k(j)}^j)_y \cup \big((Y_{k(j)}^j)_y\big)^{\one_{\mathbb{S}^1}}\,.
\end{equation}

With the goal of obtaining a lower bound on the relative perimeters of the sets $Y_i^j$ in a neighborhood of $G_i$ (see (<ref>) below), we now consider $y\in G_i^{**}$, and pick $r>0$ such that $\cl B_r^1(s_0) \subset \mathcal{A}_y$. Correspondingly, since $s_j\to s_0$ and (<ref>) holds, we can find $J^*=J^*(i,y,r)\ge J_i$ such that, for $j\ge J^*$,

\begin{equation}
\label{bellali2}
s_j\in B_r^1(s_0) \cap   \big[\partial^*_{\mathbb{S}^1} (Y_{k(j)}^j)_y\cup \big((Y_{k(j)}^j)_y\big)^{\one_{\mathbb{S}^1}}\big]\subset\mathcal{A}_y\cap   \big[\partial^*_{\mathbb{S}^1} (Y_{k(j)}^j)_y\cup \big((Y_{k(j)}^j)_y\big)^{\one_{\mathbb{S}^1}}\big]\,.
\end{equation}

Now, by (<ref>), $k(j)\neq i$, and $\mathcal{A}_y \overset{\H^1}{\subset} (Y_i)_y$, we have

\begin{equation}
\label{bellali3}
\lim_{j\to\infty}\H^1(\mathcal{A}_y \cap (Y_{k(j)}^j)_y ) =0\,.
\end{equation}

Since, by (<ref>), $(Y_{k(j)}^j)_y$ is $\H^1$-equivalent to a finite union of intervals, (<ref>) implies the existence of an open interval $\mathcal{I}_y^j$ such that

\begin{eqnarray}\label{last}
s_j\in \cl_{\SS^1}\mathcal{I}_y^j\,,\qquad \mathcal{I}_y^j\overset{\H^1}{\subset}  (Y_{k(j)}^j)_y\,,
\qquad
\pa_{\SS^1}\mathcal{I}_y^j\subset(\pa^*Y_{k(j)}^j)_y\subset(\pa^*W_i^j)_y\,,
\end{eqnarray}

which, due to (<ref>) and (<ref>), must satisfy

\[
\lim_{j\to\infty}\diam \big(\mathcal{I}_y^j \big)=0\,.
\]

In particular,

\[
\pa_{\SS^1}\,\mathcal{I}_y^j\subset B_r^1(s_0)\,,\qquad\forall j\ge J^*\,,
\]

and thus, by the last inclusion in (<ref>),

\[
\H^0\big(B_r^1(s_0)\cap\partial^*_{\SS^1}(W_i^j)_y\big)
\ge
\H^0(B_r^1(s_0)\cap \pa_{\SS^1}\mathcal{I}_j^y)\ge 2\,,
\]

whenever $j\ge J^*$. Since $y\in G_i^{**}$ and $r>0$ were arbitrary, by the coarea formula and Fatou's lemma,

\begin{eqnarray}
\nonumber
\liminf_{j\to\infty}P(W_i^j;B_r^1(s_0)\times G_i^{**})&\ge&\liminf_{j\to\infty}\int_{G_i^{**}}\H^0\big(B_r^1(s_0)\cap\partial^*_{\SS^1}(W_i^j)_y\big)\,d\H^n_y
\\
\label{bad multiplicity take 2}
\end{eqnarray}

Now, since $\pa^*W_i^j=\pa^* Y_i^j=\Phi^{-1}(\pa^*U_i^j)$, by (<ref>) we have

\[
\mbox{$Y\cap\bigcup_i\pa^*W_i^j$ is $\H^n$-contained in $Y[s_j]\cup\Phi^{-1}\big(T\cap K_j\big)$}\,,
\]

which implies, for every $j$ large enough to have $s_j\in B_r^1(s_0)$,

\begin{eqnarray}
\nonumber
&&P(W_i^j;B_r^1(s_0)\times G_i^{**})
\\\nonumber
\H^n\big(Y[s_j]\cap( B_r^1(s_0)\times G_1^{**})\big)+\H^n\big( \Phi^{-1}(T\cap K_j)\cap (B_r^1(s_0)\times B_1^n)\big)
\\\nonumber
\H^n(G_i^{**})+\H^n\big( \Phi^{-1}(T\cap K_j)\cap (B_r^1(s_0)\times B_1^n)\big)
\\\label{gelato}
\H^n(G_i)+\Lip(\Phi^{-1})^n\,\H^n\big(K_j\cap \Phi(B_r^1(s_0)\times B_1^n)\big)\,.
\end{eqnarray}

By combining (<ref>) with (<ref>) we conclude that for every $r>0$

\begin{equation}
\label{in}
\H^n(G_i)\le\Lip(\Phi^{-1})^n\,\mu\big(\Phi(\cl(B_r^1(s_0))\times B_1^n)\big)\,,
\end{equation}

By $\mu(T[s_0])=0$, if we let $r\to 0^+$ in (<ref>), we conclude that $\H^n(G_i)=0$. Now, since $G_i= \pp\big( Y_i^{\one} \cap Y[s_0]\big)$, we have

\begin{equation}
\label{soft}
\H^n\big( Y_i^{\one} \cap Y[s_0]\big)=\H^n(G_i)\,,
\end{equation}

thus proving (<ref>), and hence the theorem.

§ DIRECT METHOD ON GENERALIZED SOAP FILMS (THEOREM <REF>)
In Section <ref> we prove Theorem <ref>, while in Section <ref> we notice the changes to that argument that are needed to prove a different closure theorem that will be crucial in the companion papers [42, 43]. In particular, Section <ref> will not be needed for the other main results of this paper (although it is included here since it is definitely easier to understand in this context).

§.§ Proof of Theorem <ref>
Let us first of all recall the setting of the theorem. We are given a closed set $\wire$ in $\mathbb{R}^{n+1}$, a spanning class $\C$ for $\wire$, and a sequence $\{(K_j,E_j)\}_j$ in $\K_{\rm B}$ such that

\begin{equation}
\label{energy bound capillarity 2}
\sup_j\,\H^n(K_j)<\infty\,,
\end{equation}

and, for some Borel set $E$ and Radon measures $\mu_{\rm bk}$ and $\mu_{\rm bd}$ in $\Om$, it holds that $E_j\toloc E$ and

\begin{eqnarray}
\label{def of mub}
&&\H^n\mres (\Om\cap\pa^*E_j) + 2\,\H^n \mres (\mathcal{R}(K_j) \cap E_j^\zero) \weakstar \mu_{\rm bk}\,,
\\
\label{def of mu}
&&\H^n\mres (\Om\cap\pa^*E_j) + 2\,\H^n \mres (\mathcal{R}(K_j) \setminus \pa^* E_j) \weakstar \mu_{\rm bd}\,,
\end{eqnarray}

as $j\to\infty$. In this setting we want to prove that the sets

\begin{eqnarray}
\label{def of Kbk}
K_{\rm bk}\!\! &:=&\!\!\big(\Om\cap\partial^* E\big) \cup \Big\{x\in \Omega \cap E^\zero : \theta^n_*(\mu_{\rm bk})(x)\geq 2 \Big\}\,,
\\
\label{def of Kbd}
K_{\rm bd}\!\!&:=&\!\!\big(\Om\cap\partial^* E\big) \cup \Big\{x\in \Omega \setminus\pa^*E : \theta^n_*(\mu_{\rm bd})(x)\geq 2 \Big\}\,,
\end{eqnarray}

are such that $(K_{\rm bk},E),(K_{\rm bd},E)\in\K_{\rm B}$ and

\begin{eqnarray}
\label{mub limit lb}
\mu_{\rm bk}\!\!&\ge&\!\! \H^n\mres (\Om\cap\pa^*E) + 2\,\H^n \mres (K_{\rm bk} \cap E^\zero)\,,
\\
\label{mu limit lb}
\mu_{\rm bd}\!\!&\ge&\!\!\H^n\mres (\Om\cap\pa^*E) + 2\,\H^n \mres (K_{\rm bd} \setminus\pa^*E)\,,
\end{eqnarray}

\begin{equation}
\label{lsc Fb}
\liminf_{j\to\infty}\F_{\rm bk}(K_j,E_j)\ge\F_{\rm bk}(K_{\rm bk},E)\,,\qquad
\liminf_{j\to\infty}\F_{\rm bd}(K_j,E_j)\ge\F_{\rm bd}(K_{\rm bd},E)\,;
\end{equation}

and that the closure statements

\begin{eqnarray}\label{hp bulk spanning}
&&\mbox{if $K_j\cup E_j^\one$ is $\C$-spanning $\wire$ for every $j$,}
\\
\label{limit bulk spanning}
&&\mbox{then $K_{\rm bk}\cup E^\one$ is $\C$-spanning $\wire$}\,,
\end{eqnarray}

\begin{eqnarray}
\label{hp interface spanning}
&&\mbox{if $K_j$ is $\C$-spanning $\wire$ for every $j$,}
\\
\label{limit interface spanning}
&&\mbox{then $K_{\rm bd}$ is $\C$-spanning $\wire$}\,,
\end{eqnarray}

hold true.

By $\Om\cap\pa^*E\subset K_{\rm bk}\cap K_{\rm bd}$ we have $(K_{\rm bk},E),(K_{\rm bd},E)\in\K_{\rm B}$. By <cit.>, $\tnl(\mu_{\rm bk}) \geq 2$ on $K_{\rm bk}\cap E^\zero$ implies $\mu_{\rm bk} \mres  (K_{\rm bk} \cap E^\zero) \geq 2\H^n \mres (K_{\rm bk} \cap E^\zero)$, and, similarly, we have $\mu_{\rm bd}\mres (K_{\rm bd}\setminus\pa^*E)\ge 2\,\H^n\mres(K_{\rm bd}\setminus\pa^*E)$. Since,  by the lower semicontinuity of distributional perimeter, we have $\min\{\mu_{\rm bk},\mu_{\rm bd}\}\geq \H^n \mres (\partial^* E\cap \Omega)$,  (<ref>), (<ref>) and (<ref>) follow. We are thus left to prove that if either (<ref>) or (<ref>) holds, then (<ref>) or (<ref>) holds respectively. We divide the proof into three parts, numbered by Roman numerals.

I. Set up of the proof: Fixing from now on a choice of $(\gamma,\Phi,T)\in \mathcal{T}(\C)$ against which we want to test the $\C$-spanning properties (<ref>) and (<ref>), we introducing several key objects related to $(\gamma,\Phi,T)$.

Introducing $s_0$: Up to extracting subsequences, let $\mu$ be the weak-star limit of $\H^n\mres\,K_j$, and set

\begin{align}\label{def of J with mustar}
J=\{s\in \SS^1:\mu(T[s])=0\}\,,
\end{align}

so that $\H^1(\SS^1\setminus J)=0$. We fix $s_0\in J$.

Introducing $s_j$, $\{U_i^j\}_i$, and $K_j^*$: For $\H^1$-a.e. $s\in\SS^1$ it holds that $\H^n(K_j\cap T[s])=0$ for every $j$ and (thanks to Theorem <ref>/Theorem <ref>) the essential partition $\{U_i^j[s]\}_i$ induced on $T$ by $K_j\cup T[s]$ is such that

\begin{eqnarray*}
\mbox{$T[s]\cap E_j^\zero$ is $\H^n$-contained in ${\rm UBEP}(K_j\cup T[s];T)$}\,,&&\qquad\mbox{(if \eqref{hp bulk spanning} holds)}\,,
\\
\mbox{$T[s]$ is $\H^n$-contained in ${\rm UBEP}(K_j\cup T[s];T)$}\,,&&\qquad\mbox{(if \eqref{hp interface spanning} holds)}\,.
\end{eqnarray*}

Therefore we can find a sequence $s_j\to s_0$ as $j\to\infty$ such that

\begin{equation}
\label{def of sj}
\H^n(K_j\cap T[s_j])=0\qquad\forall j\,,
\end{equation}

and, denoting by $\{U_i^j\}_i$ the essential partition of $T$ induced by $K_j\cup T[s_j]$ (i.e. $U_i^j=U_i^j[s_j]$), and setting for brevity

\begin{equation}
\label{def of Kj star}
K_j^*={\rm UBEP}(K_j\cup T[s_j];T)=T\cap\bigcup_i\pa^*U_i^j\,,
\end{equation}

we have

\begin{eqnarray}
\label{whatabout Kj 2}
\mbox{$T[s_j]\cap E_j^\zero$ is $\H^n$-contained in $K_j^*$}\,,&&\qquad\mbox{(if \eqref{hp bulk spanning} holds)}\,,
\\
\label{whatabout Kj 2 interface}
\mbox{$T[s_j]$ is $\H^n$-contained in $K_j^*$}\,,&&\qquad\mbox{(if \eqref{hp interface spanning} holds)}\,.
\end{eqnarray}

Introducing $\{U_i\}_i$ and $K^*$: By (<ref>), Lemma <ref>, and up to extract a subsequence we can find a Lebesgue partition $\{U_i\}_i$ of $T$ such that,

\begin{equation}
\label{def of Ui}
\mbox{$\{U_i\}_i$ is the limit of $\{\{U_i^j\}_i\}_j$ in the sense specified by \eqref{partition convergence}}\,.
\end{equation}

Correspondingly we set

\begin{equation}
\label{def of Kstar}
\end{equation}

Having introduced $s_0$, $s_j$, $\{U_i^j\}_i$, $K_j^*$, $\{U_i\}_i$, and $K^*$,  we notice that if (<ref>) holds, then we can apply Theorem <ref> with $F_j=E_j$ and find that

\begin{eqnarray}
\label{whatabout K^*}
\mbox{$T[s_0]\cap E^\zero$ is $\H^n$-contained in $K^*$}\,,&&\mbox{(if  \eqref{hp bulk spanning} holds)}\,;
\end{eqnarray}

if, instead, (<ref>) holds, then Theorem <ref> can be applied with $F_j=F=\varnothing$ to deduce

\begin{eqnarray}
\label{whatabout K^* int}
\mbox{$T[s_0]$ is $\H^n$-contained in $K^*$}\,,&&\mbox{(if \eqref{hp interface spanning}  holds)}\,.
\end{eqnarray}

We now make the following claim:

Claim: We have

\begin{eqnarray}\label{Kstar inclusion}
&&\mbox{$K^*\setminus (T[s_0]\cup E^\one)$ is $\H^n$-contained in $K_{\rm bk}$}\,,
\\
\label{Kstar inclusion int}
&&\mbox{$K^*\setminus T[s_0]$ is $\H^n$-contained in $K_{\rm bd}$}\,.
\end{eqnarray}

The rest of the proof of the theorem is then divided in two parts: the conclusion follows from the claim, and the proof of the claim.

II. Conclusion of the proof from the claim: Proof that (<ref>) implies (<ref>): By $\H^1(\SS^1\setminus J)=0$, the arbitrariness of $s_0\in J$, and that of $(\g,\Phi,T)\in\T(\C)$, thanks to Theorem <ref> we can conclude that $K_{\rm bd}$ is $\C$-spanning $\wire$ by showing that

\begin{equation}
\label{chiaro1}
\mbox{$T[s_0]$ is $\H^n$-contained in ${\rm UBEP}(K_{\rm bd}\cup T[s_0];T)$.}
\end{equation}

Now, since $\{U_i\}_i$ is a Lebesgue partition of $T$ induced by $K^*$ (in the very tautological sense that $K^*$ is defined as $T\cap\cup_i\pa^*U_i$!) and, by (<ref>) in claim one, $K^*$ is $\H^n$-contained in $K_{\rm bd}\cup T[s_0]$, by Theorem <ref>-(a) we have that if $\{Z_i\}_i$ is the essential partition of $T$ induced by $K_{\rm bd}\cup T[s_0]$, then $\cup_i\pa^*U_i$ is $\H^n$-contained in $\cup_i\pa^*Z_i$: therefore, by definition of $K^*$ and by definition of ${\rm UBEP}$, we have that

\begin{equation}
\label{f1}
\mbox{$K^*$ is $\H^n$-contained in ${\rm UBEP}\big(K_{\rm bd}\cup T[s_0];T\big)$}\,.
\end{equation}

By combining (<ref>) with (<ref>) we immediately deduce (<ref>) and conclude.

Proof that (<ref>) implies (<ref>): Thanks to Theorem <ref> it suffices to prove that

\begin{equation}
\label{chiaro2}
\mbox{$T[s_0]\cap E^\zero$ is $\H^n$-contained in ${\rm UBEP}(K_{\rm bk}\cup T[s_0];T)$}\,.
\end{equation}

By (<ref>), the proof of (<ref>) can be reduced to that of

\begin{equation}
\label{chiaro3}
\mbox{$K^*\cap E^\zero$ is $\H^n$-contained in ${\rm UBEP}(K_{\rm bk}\cup T[s_0];T)$}\,.
\end{equation}

Now, let us consider the Lebesgue partition of $T$ defined by $\{V_k\}_k=\{U_i\setminus E\}_i\cup\{T\cap E\}$. By <cit.> we easily see that for each $i$

\begin{equation}
\label{chiaro5}
E^\zero\cap\pa^*U_i\,\shn\, \pa^*(U_i\setminus E)\,\shn \,\big(E^\zero\cap\pa^*U_i\big)\cup\pa^*E\,,
\end{equation}

which combined with $T\cap\pa^*(T\cap E)=T\cap\pa^*E\subset K_{\rm bk}$ and with (<ref>) in claim one, gives

\begin{eqnarray}
\nonumber
T\cap\bigcup_k\pa^*V_k&=&(T\cap\pa^*E)\cup \Big\{T\cap\bigcup_i\pa^*(U_i\setminus E)\Big\}
\,\shn\,(T\cap\pa^*E)\cup\,\big(E^\zero\cap K^*\big)
\\
\label{chiaro4}
&\shn&(T\cap\pa^*E)\cup\,\big(K^*\setminus E^\one)\,\shn\,K_{\rm bk}\cup T[s_0]\,.
\end{eqnarray}

By (<ref>) we can exploit Theorem <ref>-(a) to conclude that

\begin{equation}
\label{chiaro6}
\mbox{$T\cap\bigcup_k\pa^*V_k$ is $\H^n$-contained in ${\rm UBEP}(K_{\rm bk}\cup T[s_0];T)$}\,.
\end{equation}

By the first inclusion in (<ref>), $E^\zero\cap K^*$ is $\H^n$-contained in $T\cap\bigcup_k\pa^*V_k$, therefore (<ref>) implies (<ref>), as required. We are thus left to prove the two claims.

III. Proof of the claim: We finally prove that $K^*\setminus (T[s_0]\cup E^\one)$ is $\H^n$-contained in $K_{\rm bk}$ (that is (<ref>)), and that $K^*\setminus T[s_0]$ is $\H^n$-contained in $K_{\rm bd}$ (that is (<ref>)).

To this end, repeating the argument in the proof of Theorem <ref> with $F_j=E_j$ and $F=E$ we see that, if we set $X^j_m=\{i:(U_i^j)^\one\subset E_j^{\scriptscriptstyle{(m)}}\}$ and $X_m=\{i:U_i^\one\subset E^{\scriptscriptstyle{(m)}}\}$ for $m\in\{0,1\}$ (see (<ref>) and (<ref>)), then

\begin{eqnarray}
\label{Xj deco next}
X^j:=\{i:|U_i^j|>0\}=X_0^j\cup X_1^j\,, \qquad X:=\{i:|U_i|>0\}=X_0\cup X_1\,;
\end{eqnarray}

and, moreover, for every $i$ there is $j(i)$ such that $i\in X_m$ implies $i\in X_m^j$ for every $j\ge j(i)$. Thanks to (<ref>) we easily see that $K_j^*=T\cap\cup_i\pa^*U_i^j$ can be decomposed as

\begin{equation}
\label{manca 0}
K_j^*\ehn\bigcup_{(i,k)\in X_0^j\times X_0^j\,, i\ne j}M_{ik}^j
\cup\bigcup_{(i,k)\in X_1^j\times X_1^j\,, i\ne j}M_{ik}^j
\cup\bigcup_{(i,k)\in X_0^j\times X_1^j}M_{ik}^j\,,
\end{equation}

where $M_{ik}^j=T\cap\pa^*U_i^j\cap\pa^*U_k^j$ (an analogous decomposition of $K^*$ holds as well, and will be used in the following, but is not explicitly written for the sake of brevity). We now prove that

\begin{eqnarray}
\label{Mikj zero}
&&M_{ik}^j \subset E_j^\zero\,,\qquad\forall i,k\in X^j_0\,, i\ne k\,,
\\
\label{Mikj bordo}
&&M_{ik}^j\subset\pa^e E_j\,,\qquad\forall i\in X^j_0\,,k\in X^j_1\,,
\\
\label{Mikj one}
&&M_{ik}^j \subset E_j^{\one}\,,\qquad\forall i,k\in X^j_1\,, i\ne k\,.
\end{eqnarray}

To prove (<ref>) and (<ref>): if $i\ne k$, $i,k\in X^j_0$, and $x\in M_{ik}^j$, then (by $|U_i^j\cap U_k^j|=0$) $U_i^j$ and $U_k^j$ blow-up two complementary half-spaces at $x$, an information that combined with the $\L^{n+1}$-inclusion of $U_i^j\cup U_k^j$ in $\R^{n+1}\setminus E_j$ implies

\begin{eqnarray*}
|B_r(x)|+{\rm o}(r^{n+1})=|B_r(x)\cap U_i^j|+|B_r(x)\cap U_k^j|\le|B_r(x)\setminus E_j|\,,
\end{eqnarray*}

that is, $x\in E_j^\zero$, thus proving (<ref>); the proof of (<ref>) is analogous.

To prove (<ref>): if $i\in X^j_0$, $k\in X^j_1$, and $x\in M_{ik}^j$, then

\begin{eqnarray*}
&&|B_r(x)\cap E_j|\ge|B_r(x)\cap U_k^j|=\frac{|B_r(x)|}2+{\rm o}(r^{n+1})\,,
\\
&&|B_r(x)\setminus E_j|\ge|B_r(x)\cap U_i^j|=\frac{|B_r(x)|}2+{\rm o}(r^{n+1})\,,
\end{eqnarray*}

so that $x\not\in E_j^\zero$ and $x\not\in E_j^{\one}$, i.e. $x\in\pa^eE_j$, that is (<ref>).

With (<ref>)–(<ref>) at hand, we now prove that

\begin{eqnarray}
\label{Mikj bordo decompo}
T\cap\pa^*E_j\ehn \bigcup_{(i,k)\in X_0^j\times X_1^j}M_{ik}^j\,,
\\
\label{Mikj zero decompo}
K_j^*\cap E_j^\zero\ehn \bigcup_{(i,k)\in X_0^j\times X_0^j\,,k\ne i}M_{ik}^j\,.
\end{eqnarray}

(Analogous relations hold with $K^*$ and $E$ in place of $K^*_j$ and $E_j$.)

To prove (<ref>): By $\pa^*E_j\subset\pa^eE_j$ and (<ref>) we find $\pa^*E_j\cap (U_i^j)^{\one}=\varnothing$ for every $i,j$; hence, since $\{(U_i^j)^{\one}\}_i\cup\{\pa^*U_i^j\}_i$ is an $\H^n$-partition of $T$, and by repeatedly applying (<ref>), (<ref>) and (<ref>), we find

\begin{eqnarray*}
\bigcup_{(i,k)\in X_0^j\times X_1^j}M_{ik}^j&\shn& T\cap\pa^*E_j\ehn\bigcup_i (T\cap\pa^*E_j\cap\pa^*U_i^j)\ehn\bigcup_{i,k} M_{ik}^j\cap\pa^*E_j
\\
&\ehn&\bigcup_{(i,k)\in X_0^j\times X_1^j} M_{ik}^j\cap\pa^*E_j\,,
\end{eqnarray*}

which gives (<ref>).

To prove (<ref>): By (<ref>), (<ref>), and (<ref>), $M_{ik}^j$ has empty intersection with $E_j^\zero$ unless $i,k\in X_0^j$, in which case $M_{ik}^j$ is $\H^n$-contained in $E_j^\zero$: hence,

\[
\bigcup_{(i,k)\in X_0^j\times X_0^j\,,k\ne i}M_{ik}^j\shn K_j^*\cap E_j^\zero=\bigcup_{(i,k)\in X_0^j\times X_0^j\,, k\ne i}E_j^\zero\cap M_{ik}^j\,,
\]

that is  (<ref>).

With (<ref>) and (<ref>) at hand, we now prove the following perimeter formulas: for every open set $A\subset T$ and every $j$,

\begin{eqnarray}
\label{perimeter Uij on Xj0}
\sum_{i\in X^j_0}P(U_i^j;A)=\H^n\big(A\cap\pa^* E_j\big)+2\,\H^n\big( A\cap K^*_j\cap E_j^\zero\big)\,,
\\
\label{perimeter Uij on Xj1}
\sum_{i\in X^j_1}P(U_i^j;A)=\H^n\big(A\cap\pa^* E_j\big)+2\,\H^n\big( A\cap K^*_j\cap E_j^\one\big)\,.
\end{eqnarray}

Analogously, for $\a=0,1$,

\begin{equation}\label{perimeter Ui on Xalfa}
\sum_{i\in X_\a}P(U_i;A)=\H^n\big(A\cap\pa^* E\big)+2\,\H^n\big( A\cap K^*\cap E^{\scriptscriptstyle{(\a)}}\big)\,.
\end{equation}

To prove (<ref>) and (<ref>): Indeed, by (<ref>) and (<ref>),

\begin{eqnarray*}
\sum_{i\in X_0^j}P(U_i^j;A)
&=&\sum_{(i,k)\in X_0^j\times X_1^j}\H^n(A\cap  M_{ik}^j)
+\sum_{i  \in X_0^j}\sum_{k\in X_0^j\setminus\{i\}}\H^n(A\cap M_{ik}^j)
\\
&=&\H^n\Big(\bigcup_{(i,k)\in X_0^j\times X_1^j} A\cap M_{ik}^j\Big)+2\,\H^n\Big(\bigcup_{(i,k)\in X_0^j\times X_0^j\,,i\ne k} A\cap M_{ik}^j\Big)
\\
&=&\H^n(A\cap\pa^*E)+2\,\H^n\big(A\cap K_j^*\cap E_j^\zero\big)\,,
\end{eqnarray*}

that is (<ref>). The proof of (<ref>) is analogous (since (<ref>) is (<ref>) applied to the complements of the $E_j$'s – recall indeed that $\Om\cap\pa^*E_j=\Om\cap\pa^*(\Om\setminus E_j)$).

Conclusion of the proof of (<ref>) in the claim: We want to prove that $K^*\setminus (T[s_0]\cup E^\one)$ is $\H^n$-contained in $K_{\rm bk}$. Since $\{E^\zero,E^\one,\pa^*E\}$ is an $\H^n$-partition of $\Om$, and $\Om\cap\pa^*E$ is contained in $K_{\rm bk}$, looking back at the definition (<ref>) of $K_{\rm bk}$ it is enough to show that

\begin{equation}
\label{Kstar inclusion proof}
\mbox{$\theta^n_*(\mu_{\rm bk})(x)\ge 2$ for $\H^n$-a.e. $x\in (K^*\cap E^\zero)\setminus T[s_0]$}\,.
\end{equation}

To this end, we begin noticing that, if $Y_0$ is an arbitrary finite subset of $X_0$, then there is $j(Y_0)$ such that $Y_0\subset X_0^j$ for every $j\ge j(Y_0)$; correspondingly,

\[
\sum_{i\in Y_0}P(U_i;A)\le\liminf_{j\to\infty}\sum_{i\in Y_0}P(U_i^j;A)\le
\liminf_{j\to\infty}\sum_{i\in X_0^j}P(U_i^j;A)\,.
\]

By arbitrariness of $Y_0$, (<ref>) with $\a=0$, (<ref>), and (<ref>) (notice that the $\H^n$-containment of the $\H^n$-rectifiable set $K^*_j$ into $K_j\cup T[s_0]$ is equivalent to its $\H^n$-containment in $\RR(K_j\cup T[s_j])=\RR(K_j)\cup T[s_j]$) we conclude that, if $A\subset T$ is open and such that $\cl(A)\cap T[s_0]=\varnothing$, so that $A\cap T[s_j]=\varnothing$ for $j$ large enough, then

\begin{eqnarray}\nonumber
&&\H^n\big(A\cap\pa^* E\big)+2\,\H^n\big( A\cap K^*\cap E^\zero\big)
\\\nonumber
&&=\sum_{i\in X_0}P(U_i;A)\le\liminf_{j\to\infty}\sum_{i\in X_0^j}P(U_i^j;A)
\\\nonumber
&&=\liminf_{j\to\infty}\H^n\big(A\cap\pa^* E_j\big)+2\,\H^n\big( A\cap K^*_j\cap E_j^\zero\big)
\\
\nonumber
&&\le\liminf_{j\to\infty}\H^n\big(A\cap\pa^* E_j\big)+2\,\H^n\big( A\cap \big(\RR(K_j)\cup T[s_j]\big)\cap E_j^\zero\big)
\\
\label{nm}
&&=\liminf_{j\to\infty}\H^n\big(A\cap\pa^* E_j\big)+2\,\H^n\big( A\cap \RR(K_j)\cap E_j^\zero\big)\le\mu_{\rm bk}(\cl(A))\,,
\end{eqnarray}

where we have used the definition (<ref>) of $\mu_{\rm bk}$. Now, if $x\in (K^*\cap E^\zero)\setminus T[s_0]$, then we we can apply (<ref>) with $A=B_s(x)$ and $s>0$ such that $\cl(B_s(x))\cap T[s_0]=\varnothing$, together with the fact that $x\in E^\zero$ implies $\H^n(B_s(x)\cap\pa^*E)={\rm o}(s^n)$ as $s\to 0^+$, to conclude that

\begin{align}\label{nm applied}
\mu_{\rm bk}(\cl(B_s(x)))&\ge 2\,\H^n\big(B_s(x)\cap K^*\cap E^\zero\big)+{\rm o}(s^n)\,,\qquad\mbox{as $s\to0^+$}\,.
\end{align}

Since $K^*\cap E^\zero$ is an $\H^n$-rectifiable set, and thus $\H^n\big(B_s(x)\cap K^*\cap E^\zero\big)=\om_n\,s^n+{\rm o}(s^n)$ for $\H^n$-a.e. $x\in K^*\cap E^\zero$, we deduce (<ref>) from (<ref>).

Conclusion of the proof of (<ref>) in the claim: We want to prove the $\H^n$-containment of $K^*\setminus T[s_0]$ in $K_{\rm bd}$. As in the proof of (<ref>), combining Federer's theorem (<ref>) with the definition (<ref>) of $K_{\rm bd}$, we are left to prove that

\begin{equation}
\label{Kstar inclusion int proof}
\mbox{$\theta_*^n(\mu_{\rm bd})(x)\ge 2$ for $\H^n$-a.e. $x\in K^*\setminus(T[s_0]\cup\pa^*E)$}\,.
\end{equation}

As proved in (<ref>), if $A\subset T$ is open and such that $\cl(A)\cap T[s_0]=\varnothing$, then by exploiting (<ref>) and (<ref>) with $\a=0$ we have

\begin{eqnarray}\label{nm again}
&&\H^n\big(A\cap\pa^* E\big)+2\,\H^n\big( A\cap K^*\cap E^\zero\big)
\\\nonumber
&&\le\liminf_{j\to\infty}\H^n\big(A\cap\pa^* E_j\big)+2\,\H^n\big( A\cap \RR(K_j)\cap E_j^\zero\big)\,;
\end{eqnarray}

the same argument, this time based on (<ref>) and (<ref>) with $\a=1$, also gives

\begin{eqnarray}\label{nm again one}
&&\H^n\big(A\cap\pa^* E\big)+2\,\H^n\big( A\cap K^*\cap E^\one\big)
\\\nonumber
&&\le\liminf_{j\to\infty}\H^n\big(A\cap\pa^* E_j\big)+2\,\H^n\big( A\cap \RR(K_j)\cap E_j^\one\big)\,;
\end{eqnarray}

and, finally, since $\Om\setminus\pa^*E$ is $\H^n$-equivalent to $\Om\cap(E^\zero\cup E^\one)$, the combination of (<ref>) and (<ref>) gives

\begin{eqnarray}\label{nm again bordo}
&&\H^n\big(A\cap\pa^* E\big)+2\,\H^n\big( A\cap K^*\setminus\pa^* E\big)
\\\nonumber
&&\le\liminf_{j\to\infty}\H^n\big(A\cap\pa^* E_j\big)+2\,\H^n\big( A\cap \RR(K_j)\setminus\pa^* E_j\big)\le\mu_{\rm bd}(\cl(A))\,,
\end{eqnarray}

where we have used the definition (<ref>) of $\mu_{\rm bd}$. Now, for $\H^n$-a.e. $x\in K^*\setminus(T[s_0]\cup\pa^*E)$ we have $\H^n(B_r(x)\cap\pa^*E)={\rm o}(r^n)$ and $\H^n(B_r(x)\cap K^*\setminus \pa^*E)=\om_n\,r^n+{\rm o}(r^n)$ as $r\to 0^+$, as well as  $\cl(B_r(x))\cap T[s_0]=\varnothing$ for $r$ small enough, so that (<ref>) with $A=B_r(x)$ readily implies (<ref>). The proof of the claim, and thus of the theorem, is now complete.

§.§ A second closure theorem
We now present a variant of the main arguments presented in this section and alternative closure theorem to Theorem <ref>. As already noticed, this second closure theorem, Theorem <ref> below, will play a role only in the companion paper [42], where Plateau's laws will be studied in the relation to the Allen–Cahn equation, so that this section can be omitted on a first reading focused on Gauss' capillarity theory alone.

To introduce Theorem <ref>, let us consider the following question: given an $\H^n$-finite set $S$ which is $\C$-spanning $\wire$, what parts of $S$ are essential to its $\C$-spanning property? We already know from Lemma <ref> that the unrectifiable part of $S$ is not necessary, since $\RR(S)$ is also $\C$-spanning. However, some parts of $\RR(S)$ could be discarded too – indeed rectifiable sets can be “porous at every scale”, and thus completely useless from the point of view of achieving $\C$-spanning. To make an example, consider the rectifiable set $P\subset\R^2$ obtained by removing from $[0,1]$ all the intervals $(q_i-\e_i,q_i+\e_i)$ where $\{q_i\}_i$ are the rational numbers in $[0,1]$ and $2\,\sum_i\e_i=\e$ for some given $\e\in(0,1)$: it is easily seen that $P$ is a rectifiable set with positive $\H^1$-measure in $\R^2$, contained in $\R\times\{0\}$, which fails to essentially disconnect any stripe of the form $(a,b)\times\R$ with $(a,b)\cc(0,1)$. Intuitively, if a set like $P$ stands as an isolated portion of $S$, then $\RR(S)\setminus P$ should still be $\C$-spanning.

We can formalize this idea as follows. Denoting as usual $\Om=\R^{n+1}\setminus\wire$, we consider the open covering $\{\Om_k\}_k$ of $\Om$ defined by

\begin{equation}
\label{def of Omega i}
\{\Om_k\}_k=\{B_{r_{mh}}(x_m)\}_{m,h}\,,
\end{equation}

where $\{x_m\}_m=\Q^{n+1}\cap\Om$ and $\{r_{mh}\}_h=\Q\cap(0,\dist(x_m,\pa\Om))$. For every $\H^n$-finite set $S$ we define the essential spanning part of $S$ in $\Om$ as the Borel set

\[
{\rm ESP}(S)=\bigcup_k\,{\rm UBEP}(S;\Om_k)=\bigcup_k\,\Big\{\Om_k\cap\bigcup_i\pa^*U_i[\Om_k]\Big\}\,,
\]

if $\{U_i[\Om_k]\}_i$ denotes the essential partition of $\Om_k$ induced by $S$. Since each ${\rm UBEP}(S;\Om_k)$ is a countable union of reduced boundaries and is $\H^n$-contained in the $\H^n$-finite set $S$, we see that ${\rm ESP}(S)$ is always $\H^n$-rectifiable. The idea is that by following the unions of boundaries of essential partitions induced by $S$ over smaller and smaller balls we are capturing all the parts of $S$ that may potentially contribute to achieve a spanning condition with respect to $\wire$. Thinking about Figure <ref>: the tendrils of $S$ appearing in panel (a) and not captured by ${\rm UBEP}(S;U)$, will eventually be included into ${\rm ESP}(S)$ by considering ${\rm UBEP}$'s of $S$ relative to suitable subsets of $U$. Another way to visualize the construction of ${\rm ESP}(S)$ is noticing that if $B_r(x)\subset B_s(x)\subset\Om$, then

\[
B_r(x)\cap {\rm UBEP}(S;B_s(x))\subset {\rm UBEP}(S;B_r(x))\,,
\]

which points to the monotonicity property behind the construction of ${\rm ESP}(S)$. Intuitively, we expect that

\begin{equation}
\label{intro S 1}
\mbox{if $S$ is $\C$-spanning $\wire$, then ${\rm ESP}(S)$ is $\C$-spanning $\wire$}
\end{equation}

(where $\C$ is an arbitrary spanning class for $\wire$). This fact will proved in a moment as a particular case of Theorem <ref> below.

Next, we introduce the notion of convergence behind our second closure theorem. Consider a sequence $\{S_j\}_j$ of Borel subsets of $\Om$ such that $\sup_j\H^n(S_j)<\infty$. If we denote by $\{U_i^j[\Om_k]\}_i$ the essential partition induced on $\Om_k$ by $S_j$, then a diagonal argument based on Lemma <ref> shows the existence of a (not relabeled) subsequence in $j$, and, for each $k$, of a Borel partition $\{U_i[\Om_k]\}_i$ of $\Om_k$ such that  $\{U_i^j[\Om_k]\}_i$ converges to $\{U_i[\Om_k]\}_i$ as $j\to\infty$ in the sense specified by (<ref>). Since ${\rm UBEP}(S_j;\Om_k)=\Om_k\cap\bigcup_i\pa^*U_i^j[\Om_k]$, we call any set $S$ of the form[The limit partition $\{U_i[\Om_k]\}_i$ appearing in (<ref>) may not be the essential partition induced by $S$ on $\Om_k$ since the individual $U_i[\Om_k]$, arising as $L^1$-limits, may fail to be essentially connected. This said, $\{U_i[\Om_k]\}_i$ is automatically a partition of $\Om_k$ induced by $S_0$.]

\begin{equation}
\label{spl def}
\end{equation}

a subsequential partition limit of $\{S_j\}_j$ in $\Om$. Having in mind (<ref>), it is natural to ask if the following property holds:

\begin{eqnarray}\nonumber
&&\mbox{if $S_j$ is $\C$-spanning $\wire$ for each $j$}\,,
\\
\nonumber
&&\mbox{and $S$ is a subsequential partition limit of $\{S_j\}_j$ in $\Om$}\,,
\\\label{intro S 2}
&&\mbox{then $S$ is $\C$-spanning $\wire$}\,.
\end{eqnarray}

Our next theorem implies both (<ref>) and (<ref>) as particular cases (corresponding to be taking $E_j=\varnothing$ and, respectively, $K_j=S$ and $K_j=S_j$ for every $j$).

Let $\wire$ be a closed set in $\mathbb{R}^{n+1}$, $\C$ a spanning class for $\wire$, and $\{(K_j,E_j)\}_j$ a sequence in $\K_{\rm B}$ such that $\sup_j\H^n(K_j)<\infty$ and $K_j\cup E_j^\one$ is $\C$-spanning $\wire$ for every $j$.

If $S_0$ and $E_0$ are, respectively, a subsequential partition limit of $\{K_j\}_j$ in $\Om$ and an $L^1$-subsequential limit of $\{E_j\}_j$ (corresponding to a same not relabeled subsequence in $j$), and we set

\[
K_0=(\Om\cap\pa^*E_0)\cup S_0\,,
\]

then $(K_0,E_0)\in\KK_{\rm B}$ and $K_0\cup E_0^\one$ is $\C$-spanning $\wire$.

Since $\Om\cap\pa^*E_0\subset K_0$ by definition of $K_0$ we trivially have $(K_0,E_0)\in\KK_{\rm B}$. Aiming to prove that $K_0\cup E_0^\one$ is $\C$-spanning $\wire$, we fix $(\g,\Phi,T)\in\T(\C)$, and define $s_0$, $s_j$, $\{U_i^j\}_i$ and $\{U_i\}_i$ exactly as in part I of the proof of Theorem <ref>. Thanks to Theorem <ref> and by arguing as in part II of the proof of Theorem <ref>, we have reduced to prove that

\begin{equation}
\label{upon}
\mbox{$K^*\setminus (T[s_0]\cup E^\one)$ is $\H^n$-contained in $K_0$}\,.
\end{equation}

By Federer's theorem (<ref>) and since $\Om\cap\pa^*E\subset K_0$ it is enough to prove

\[
\mbox{$(K^*\cap E^\zero)\setminus T[s_0]$ is $\H^n$-contained in $S_0$}\,,
\]

and, thanks to the construction of $S_0$, we shall actually be able to prove

\begin{equation}
\label{upon3}
\mbox{$K^* \setminus T[s_0]$ is $\H^n$-contained in $S_0$}\,.
\end{equation}

To this end let us pick $k$ such that $\Om_k\cc T$ and $\Om_k\cap T[s_0]=\emptyset$. Then, for $j\ge j(k)$, we have $\Om_k\cap T[s_j]=\varnothing$, so that

\[
\Om_k\cap {\rm UBEP}\big(K_j\cup T[s_j];T\big)\subset {\rm UBEP}\big(K_j\cup T[s_j];\Om_k\big)={\rm UBEP}\big(K_j;\Om_k\big)\,.
\]

Since $\{U_i^j\}_i$ is the essential partition of $T$ induced by $K_j\cup T[s_j]$, if $\{U_m^j[\Om_k]\}_m$ is the essential partition of $\Om_k$ induced by $K_j$, we have just claimed that, for every $i$ and $j\ge j(k)$,

\begin{equation}
\label{localize it}
\Om_k\cap\pa^*U_i^j\subset\Om_k\cap\bigcup_m \pa^*U_m^j[\Om_k]\,.
\end{equation}

Since $\{U_m^j[\Om_k]\}_m$ is a Lebesgue partition of $\Om_k$ into essentially connected sets, by (<ref>) the indecomposable components of $\Om_k\cap U_i^j$ must belong to $\{U_m^j[\Om_k]\}_m$. In other words, for each $i$ and each $j\ge j(k)$ there is $M(k,i,j)$ such that

\[
\Om_k\cap U_i^j=\bigcup_{m\in M(k,i,j)}U_m^j[\Om_k]\,.
\]

As a consequence of $U_i^j\to U_i$ and of $U_m^j[\Om_k]\to U_m[\Om_k]$ as $j\to\infty$ we find that, for a set of indexes $M(k,i)$, it must be

\[
\Om_k\cap U_i=\bigcup_{m\in M(k,i)}U_m[\Om_k]\,,
\]

and therefore

\[
\Om_k\cap\pa^*U_i\,\,\shn\,\,\bigcup_{m\in M(k,i)}\pa^*U_m[\Om_k]\subset S_0\,.
\]

Since we have proved this inclusion for every $i$ and for every $k$ such that $\Om_k\cc T$ with $\Om_k\cap T[s_0]=\emptyset$, it follows that $K^*\setminus T[s_0]$ is $\H^n$-contained in $S_0$, that is (<ref>).

§ EXISTENCE OF MINIMIZERS AND CONVERGENCE TO PLATEAU'S PROBLEM (THEOREM <REF>)
In this section we prove two main results: the first one (Theorem <ref>) concerns the equivalence of Harrison–Pugh Plateau's problem $\ell$ with its measure theoretic reformulation $\ell_{\rm B}$ (see (<ref>)); the second (Theorem <ref>) is a very refined version of Theorem <ref>.

If $\wire\subset\R^{n+1}$ is closed, $\C$ is a spanning class for $\wire$, and the Harrison–Pugh formulation of the Plateau problem

\begin{align}\notag
\ell= \inf\big\{\H^n(S) : \mbox{$S$ is a closed subset $\Om$, $S$ is $\C$-spanning $\wire$}\big\}
\end{align}

is finite, then the problem

\begin{align}\notag
\ell_{\rm B}= \inf\big\{\H^n(S) : \mbox{$S$ is a Borel subset $\Om$, $S$ is $\C$-spanning $\wire$}\big\}
\end{align}

admits minimizers, and given any minimizer $S$ for $\ell_{\rm B}$, there exists relatively closed $S^*$ which is $\H^n$-equivalent to $S$ and a minimizer for $\ell$. In particular, $\ell=\ell_{\rm B}$.

If $\wire$ is a compact set in $\mathbb{R}^{n+1}$ and $\C$ is a spanning class for $\wire$ such that $\ell<\infty$, then for every $v>0$ there exist minimizers $(K,E)$ of $\Psi_{\rm bk}(v)$. Moreover,

(i): if $(K_*,E_*)$ is a minimizer of $\Psi_{\rm bk}(v)$, then there is $(K,E)\in\KK$ such that $K$ is $\H^n$-equivalent to $K^*$, $E$ is Lebesgue equivalent to $E_*$, $(K,E)$ is a minimizer of $\Psi_{\rm bk}(v)$, both $E$ and $K$ are bounded, $K\cup E$ is $\C$-spanning $\wire$, $K\cap E^\one=\varnothing$, and there is $\lambda\in \mathbb{R}$ such that

\begin{eqnarray}
\label{first var cap theorem section}
\lambda \int_{\Om\cap\partial^* E} X \cdot \nu_{E} \,d\H^n = \int_{\Om\cap\partial^* E} \mathrm{div}^{K}\,X\,d\H^n + 2\int_{K \cap E^\zero} \mathrm{div}^{K}\, X\,d\H^n\,,&&
\\\nonumber
\hspace{2.3cm}\forall X\in C^1_c(\mathbb{R}^{n+1};\mathbb{R}^{n+1})\quad\mbox{with $X\cdot \nu_\Omega =0$ on $\partial \Omega$}\,,&&
\end{eqnarray}

and there are positive constants $c=c(n)$ and $r_1=r_1(K,E)$ such that

\begin{equation}
\label{upper density estimates proof}
%\H^n(K\cap B_r(x))\ge c\,r^n\,,\qquad
|E\cap B_\rho(y)|\le (1-c)\,\om_{n+1}\,\rho^{n+1}\,,
\end{equation}

for every $y\in\Om\cap\pa E$ and $\rho<\min\{r_1,\dist(y,\wire)\}$; under the further assumption that $\partial \wire$ is $C^2$, then there is positive $r_0=r_0(n,\wire,|\l|)$ such that

\begin{equation}
\label{lower density estimates proof}
\H^n(K\cap B_r(x))\ge c\,r^n%\,,\qquad
%|E\cap B_\rho(y)|\le (1-c)\,\om_{n+1}\,\rho^{n+1}\,,
\end{equation}

for every $x\in\cl(K)$ and $r<r_0$;

(ii): if $(K_j,E_j)$ is a sequence of minimizers for $\Psi_{\rm bk}(v_j)$ with $v_j\to 0^+$, then there exists a minimizer $S$ of $\ell$ such  that, up to extracting subsequences, as Radon measures in $\Om$,

\begin{align}
\label{convergence as measures section}
\H^n\mres(\Om\cap\partial^* E_j) + 2\,\H^n\mres (K_j \cap E_j^{\zero}) \weakstar 2\H^n\mres S\,,
\end{align}

as $j\to\infty$. In particular, $\Psi_{\rm bk}(v)\to 2\,\ell=\Psi_{\rm bk}(0)$ as $v\to 0^+$.

By Theorem <ref>, if  $\ell<\infty$, then $\ell_{\rm B}<\infty$. Let now $\{S_j\}_j$ be a minimizing sequence for $\ell_{\rm B}$, then $\{(S_j,\varnothing)\}_j$ is a sequence in $\KK_{\rm B}$ satisfying (<ref>). By Theorem <ref>, we find a Borel set $S$ which is $\C$-spanning $\wire$ and is such that

\[
2\,\liminf_{j\to\infty}\H^n(S_j)=\liminf_{j\to\infty}\F_{\rm bk}(S_j,\varnothing)\ge\F_{\rm bk}(S,\varnothing)=2\,\H^n(S)\,.
\]

This shows that $S$ is a minimizer of $\ell_{\rm B}$. By Lemma <ref>, $S$ is $\H^n$-rectifiable, for, otherwise, $\mathcal{R}(S)$ would be admissible for $\ell_{\rm B}$ and have strictly less area than $S$. We conclude the proof by showing that up to modifications on a $\H^n$-null set, $S$ is relatively closed in $\Omega$ (and thus is a minimizer of $\ell$ too). Indeed the property of being $\C$-spanning $\wire$ is preserved under diffeomorphism $f$ with $\{f\ne\id\}\cc\Omega$. In particular, $\H^n(S) \leq \H^n(f(S))$ for every such $f$, so that the multiplicity one rectifiable varifold $V_S=\var(S,1)$ associated to $S$ is stationary. By a standard application of the monotonicity formula, we can find $S^*$ $\H^n$-equivalent to $S$ such that $S^*$ is relative closed in $\Om$. Since $\H^n(S)=\H^n(S^*)$ and $\C$-spanning is preserved under $\H^n$-null modifications, we conclude the proof.