Plateau borders in soap films
and Gauss’ capillarity theory

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 [WH99, Section 1.3], or in terms of the scale at which the soap film/foam is described [CCAE⁺13, Chapter 2, Section 3 and 4].

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. [WH99, Equilibrium rules A1, A2, page 24].

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

1.1 and [WH99, Fig. 1.8 and Fig. 1.9]. 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:

	the unit normal to a soap film/foam changes continuously		(1.1)
	$\displaystyle\mbox{{\it across the transition lines between wet and dry interfaces}}\,;$

see, e.g., [WH99, Equilibrium rule B, page 25] and [CCAE⁺13, Section 4.1.4]. 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 [LL65, JP92, BR97, KHS99, LC99, KHS00, GKJ05, SM15]. Postulate (1.1) 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 (1.1) 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 [MNR23a, MNR23b]; 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 [MSS19], where the idea is introduced of modeling soap films as regions $E$ of positive volume $|E|=v$ contained in the complement $\Omega=\mathbb{R}^{n+1}\setminus\mathbf{W}$ of a “wire frame” $\mathbf{W}$ ($n=2$ is the physical case, although the planar case $n=1$ is also quite interesting in applications). We associate to $E$ the surface tension energy $\mathcal{H}^{n}(\Omega\cap\partial E)$ (where $\mathcal{H}^{n}$ stands for $n$-dimensional (Hausdorff) measure, i.e., area when $n=2$ and length when $n=1$), and minimize $\mathcal{H}^{n}(\Omega\cap\partial E)$ under the constraints that $|E|=v$ (for some given $v>0$) and

$$\mbox{$\Omega\cap\partial E$ is spanning $\mathbf{W}$}\,.$$

(1.2)

From the mathematical viewpoint the meaning assigned to (1.2) 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 [KMS22a], this minimization process leads to the identification of generalized minimizers in the form of pairs $(K,E)$ with $E\subset\Omega$, $|E|=v$, and such that

$$\mbox{$\Omega\cap\partial E\subset K$ and $K$ is spanning $\mathbf{W}$}\,.$$

(1.3)

These pairs are minimizing in the sense that

$$\mathcal{H}^{n}(\Omega\cap\partial E)+2\,\mathcal{H}^{n}(K\setminus\partial E)\leq\mathcal{H}^{n}(\Omega\cap\partial E^{\prime})\,,$$

(1.4)

whenever $E^{\prime}\subset\Omega$, $|E^{\prime}|=v$ and $\Omega\cap\partial E^{\prime}$ is spanning $\mathbf{W}$.

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

Figure 1.2). We expect collapsing to occur whenever the Plateau problem for $\mathbf{W}$ 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 $\sup\{{\rm dist}(x,E_{v_{j}}):x\in\Sigma(S)\}\to 0$. 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 (1.4), and this is the content of [KMS22a, KMS21, KMS22b]. With reference to the cases when $n=1$ or $n=2$, one can deduce from (1.4) that if $\mathcal{H}^{n}(K\setminus\partial E)>0$, then $K\setminus\partial E$ is a smooth minimal surface (a union of segments if $n=1$) and that $\partial E$ contains a regular part $\partial^{*}E$ 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 (1.1) 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 (1.4), it turns out to be very difficult to say much about the “transition line”

$$\partial E\setminus\partial^{*}E$$

between the zero and the negative constant mean curvature interfaces in $K$, across which one should check the validity of (1.1). More precisely, all that descends from (1.4) and a direct application of Allard’s regularity theorem [All72] is that $\partial E\setminus\partial^{*}E$ 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 $\partial E\setminus\partial^{*}E$ could very well have positive $\mathcal{H}^{n}$-measure and be everywhere dense in $K$! With such poor understanding of $\partial E\setminus\partial^{*}E$, proving the validity of (1.1) – that is, the continuity of the unit normals to $K\setminus\partial E$ and $\partial^{*}E$ in passing across $\partial E\setminus\partial^{*}E$ – is of course out of question.

We overcome these difficulties by performing a major measure-theoretic overhaul of the Harrison–Pugh homotopic spanning condition [HP16, HP17] used in [MSS19, KMS22a, KMS21, KMS22b] to give a rigorous meaning to (1.2), 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 (1.4) of $(K,E)$ into the full minimality property

$$\mathcal{H}^{n}(\Omega\cap\partial E)+2\,\mathcal{H}^{n}(K\setminus\partial E)\leq\mathcal{H}^{n}(\Omega\cap\partial E^{\prime})+2\,\mathcal{H}^{n}(K^{\prime}\setminus\partial E^{\prime})$$

(1.5)

whenever $E^{\prime}\subset\Omega$, $|E^{\prime}|=v$, $\Omega\cap\partial E^{\prime}\subset K^{\prime}$ and $K^{\prime}$ is spanning $\mathbf{W}$. The crucial difference between (1.4) and (1.5) 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 $(K^{\prime},E^{\prime})$. Such comparisons are immediate when working with (1.5), but they are actually quite delicate to set up when we only have (1.4). In the latter case, given a competitor $(K^{\prime},E^{\prime})$, to set up the energy comparison with $(K,E)$ we first need to find a sequence of non-collapsed competitors $\{E^{\prime}_{j}\}_{j}$ (with $E^{\prime}_{j}\subset\Omega$, $|E^{\prime}_{j}|=v$, and $\Omega\cap\partial E^{\prime}_{j}$ spanning $\mathbf{W}$) such that $\mathcal{H}^{n}(\Omega\cap\partial E^{\prime}_{j})\to\mathcal{H}^{n}(\Omega\cap\partial E^{\prime})+2\,\mathcal{H}^{n}(K^{\prime}\setminus\partial E^{\prime})$. Intuitively, $E_{j}^{\prime}$ needs to be a $\delta_{j}$-neighborhood of $K^{\prime}\cup E^{\prime}$ for some $\delta_{j}\to 0^{+}$ and the energy approximation property has to be deduced from the theory of Minkowski content. But applying the theory of Minkowski content to $(K^{\prime},E^{\prime})$ (which is the approach followed, e.g., in [KMS22b]) requires $(K^{\prime},E^{\prime})$ to satisfy rectifiability and uniform density properties that substantially restrict the class of available competitors $(K^{\prime},E^{\prime})$.

In contrast, once the validity of (1.5) is established, a suitable generalization (Theorem 1.2) of the partition theorem of sets of finite perimeter into indecomposable components [ACMM01, Theorem 1] combined with a subtle variational argument (see Figure 1.7) allows us to show that, in any ball $B\subset\!\subset\Omega$ 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

$$\mathcal{H}^{n}(B\cap\partial U_{i})\leq\mathcal{H}^{n}(B\cap\partial V)+\Lambda\,|U_{i}\Delta V|\,,$$

(1.6)

with respect to completely arbitrary perturbations $V\subset B$, $V\Delta U_{i}\subset\!\subset B$. By a classical theorem of De Giorgi [DG60, Tam84], (1.6) implies (away from a closed singular set of codimension at least $8$, which is thus empty if $n\leq 6$) the $C^{1,\alpha}$-regularity of $B\cap\partial U_{i}$ for each $i$, and thus establishes the continuity of the normal stated in (1.1). In fact, locally at each $x$ on the transition line, $K$ is the union of the graphs of two $C^{1,\alpha}$-functions $u_{1}\leq u_{2}$ 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 [Sil05, FGS15] to deduce that the transition line $\partial E\setminus\partial^{*}E$ is indeed $(n-1)$-dimensional, and to improve the $C^{1,\alpha}$-regularity of $B\cap\partial U_{i}$ to $C^{1,1}$-regularity. Given the mean curvature jump across $\partial E\setminus\partial^{*}E$ 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 [MNR23a], 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 [MNR23b].

The rest of this introduction is devoted to a rigorous formulation of the results presented in this overview. We begin in Section 1.2 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 [Alm76, Tay76]. In Section 1.3 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 [CCDPM17, CCDPM14]. In Section 1.4 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 1.4). In Section 1.5 we prove the existence of generalized soap film minimizers (Theorem 1.5) and their convergence in energy to solutions to the Plateau problem. A sharp regularity theorem (Theorem 1.6) for these minimizers, which validates (1.1), is stated in Section 1.6. Finally, in Section 1.7 we reformulate the above results in the case of foams, see in particular Theorem 1.7.

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 $\mathbb{R}^{3}$ are carried by smooth minimal surfaces, and thus cannot model $Y$-type¹¹1Currents 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 [Alm76]: these are sets $S\subset\mathbb{R}^{n+1}$ that are relatively closed in a given open set $\Omega\subset\mathbb{R}^{n+1}$, and satisfy $\mathcal{H}^{n}(S)\leq\mathcal{H}^{n}(f(S))$ whenever $f:\Omega\to\Omega$ is a Lipschitz (not necessarily injective) map with $\{f\neq{\rm id}\,\}\subset\!\subset\Omega$. Taylor’s historical result [Tay76] validates the Plateau laws in this context, by showing that, when²²2Similar regularity assertions hold when $n=1$ (by elementary methods) and, in much more recent developments, when $n\geq 3$ [CES22]. $n=2$, Almgren minimal sets are locally $C^{1,\alpha}$-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 [Dav14]. In this direction, a major breakthrough has been obtained by Harrison and Pugh in [HP16] with the introduction of a new spanning condition, which, following the presentation in [DLGM17a], can be defined as follows:

Existence of minimizers of $\ell$ holds as soon as $\ell<\infty$, and minimizers $S$ of $\ell$ are Almgren minimal sets in $\Omega$ [HP16, DLGM17a] that are indeed going to exhibit Plateau-type singularities (this is easily seen in the plane, but see also [BM21] for a higher dimensional example). Moreover, given a same $\mathbf{W}$, different choices of $\mathcal{C}$ are possible and can lead to different minimizers, see

Figure 1.3. Finally, the approach is robust enough to provide the starting point for several important extensions [DPDRG16, DR18, HP17, FK18, DLDRG19, DPDRG20], 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 [MSS19, KMS22a], with the introduction of the model

$$\psi(v):=\inf\Big{\{}\mathcal{H}^{n}(\Omega\cap\partial E):|E|=v\,,\,\,\mbox{$\Omega\cap\partial E$ is $\mathcal{C}$-spanning $\mathbf{W}$}\Big{\}}\,,$$

(1.8)

where $E\subset\Omega$ is an open set with smooth boundary. Without the spanning condition, at small volumes, minimizers of $\mathcal{H}^{n}(\Omega\cap\partial E)$ would be small diffeomorphic images of half-balls [MM16]. However, the introduction of the $\mathcal{C}$-spanning constraint rules out small droplets, and forces the exploration of a different part of the energy landscape of $\mathcal{H}^{n}(\Omega\cap\partial E)$. As informally discussed in Section 1.1, this leads to the emergence of generalized minimizers $(K,E)$. More precisely, in [KMS22a] the existence is proved of $(K,E)$ in the class

	$\displaystyle\mathcal{K}=\Big{\{}(K,E):\mbox{$K$ is relatively closed and $\mathcal{H}^{n}$-rectifiable in $\Omega$, $E$ is open,}$		(1.9)
	$\displaystyle\hskip 76.82234pt\mbox{$E$ has finite perimeter in $\Omega$, and $\Omega\cap\mathrm{cl}\,(\partial^{*}E)=\Omega\cap\partial E\subset K$}\Big{\}}\,,$

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

$$\mathcal{H}^{n}(\Omega\cap\partial^{*}E)+2\,\mathcal{H}^{n}(\Omega\cap(K\setminus\partial^{*}E))\leq\mathcal{H}^{n}(\Omega\cap\partial E^{\prime})\,.$$

(1.10)

Starting from (1.10) one can apply Allard’s regularity theorem [All72] and various ad hoc comparison arguments [KMS21, KMS22b] to prove that $\Omega\cap\partial^{*}E$ is a smooth hypersurface with constant mean curvature (negative if $\mathcal{H}^{n}(K\setminus\partial^{*}E)>0$), $\Omega\cap(\partial E\setminus\partial^{*}E)$ has empty interior in $K$, and that $K\setminus(\Sigma\cup\partial E)$ is a smooth minimal hypersurface, where $\Sigma$ is a closed set with codimension at least $8$.

In a nutshell, the idea behind our measure theoretic revision the Harrison–Pugh homotopic spanning condition is the following. Rather than asking that $S\cap\gamma(\SS^{1})\neq\varnothing$ for every $\gamma\in\mathcal{C}$, as done in Definition A, we shall replace $\gamma$ with an open “tube” $T$ containing $\gamma(\SS^{1})$, 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

Figure 1.4. 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 [CCDPM17, CCDPM14] 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 [CPS20, Per22, Dom23]). 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 $T\subset\mathbb{R}^{n+1}$ (i.e., $x\in T^{{\scriptscriptstyle{(t)}}}$ if and only if $|T\cap B_{r}(x)|/\omega_{n+1}\,r^{n+1}\to t$ as $r\to 0^{+}$, where $\omega_{k}$ is the Lebesgue measure of the unit ball in $\mathbb{R}^{k}$), and by $\partial^{e}T=\mathbb{R}^{n+1}\setminus(T^{{\scriptscriptstyle{(0)}}}\cup T^{{\scriptscriptstyle{(1)}}})$ 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

$$\begin{split}&\mbox{$\{T_{1},T_{2}\}$ is a non-trivial Borel partition of $T$}\,,\\ &\mbox{and $T^{{\scriptscriptstyle{(1)}}}\cap\partial^{e}T_{1}\cap\partial^{e}T_{2}$ is $\mathcal{H}^{n}$-contained in $S$}\,.\end{split}$$

(1.11)

(For example, if $K$ is a set of full $\mathcal{L}^{1}$-measure in $[-1,1]$, then $S=K\times\{0\}$ essentially disconnects the unit disk in $\mathbb{R}^{2}$.) Moreover, we say that $T$ is essentially connected³³3Whenever $T$ 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 $\mathcal{H}^{n}$-contained in $F$” we mean that $\mathcal{H}^{n}(E\setminus F)=0$. We also notice that, in (1.11), we have $T^{{\scriptscriptstyle{(1)}}}\cap\partial^{e}T_{1}\cap\partial^{e}T_{2}=T^{{\scriptscriptstyle{(1)}}}\cap\partial^{e}T_{i}$ ($i=1,2$), a fact that is tacitly and repeatedly considered in the use of (1.11) in order to shorten formulas.

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

Before commenting on (1.12), we notice that the terminology of Definition B is coherent with that of Definition A thanks to the following theorem.

Theorem 1.1 is proved in Appendix A. There we also comment on the delicate reason why, in formulating (1.12), 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 1.4, but it is actually so when dealing with more complex situations; see Figure A.1.

Homotopic spanning according to Definition B is clearly stable under modifications of $S$ by $\mathcal{H}^{n}$-negligible sets, but there is more to it.

Indeed, even a notion like “$\mathcal{H}^{n}(S\cap T)>0$ for every $T\in\mathcal{T}(\mathcal{C})$” would be stable under modifications by $\mathcal{H}^{n}$-negligible sets, and would probably look more appealing in its simplicity. The catch, of course, is finding an extension of Definition A for which compactness theorems, like Theorem 1.4 below, hold true. This is evidently not the case, for example, if one tries to work with a notion like “$\mathcal{H}^{n}(S\cap T)>0$ for every $T\in\mathcal{T}(\mathcal{C})$”.

The first key insight on Definition B is that, if restricted to Borel sets $S$ that are locally $\mathcal{H}^{n}$-finite in $\Omega$, 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 [ACMM01, Theorem 1]. For an illustration of this result, see Figure 1.5.

Given $U$ and $S$ as in the statement of Theorem 1.2 we can define⁶⁶6Uniquely modulo $\mathcal{H}^{n}$-null sets thanks to Federer’s theorem recalled in (1.37) below. the union of the (reduced) boundaries (relative to $U$) of the essential partition induced by $S$ on $U$ by setting⁷⁷7Given a Borel set $E$, we denote by $\partial^{*}E$ its reduced boundary relative to the maximal open set $A$ wherein $E$ has locally finite perimeter.

$${\rm UBEP}(S;U)=U^{\scriptscriptstyle{(1)}}\cap\bigcup_{i}\partial^{*}U_{i}\,.$$

(1.13)

Two properties of ${\rm UBEP}$’s which well illustrate the concept are: first, if $\mathcal{R}(S)$ denotes the rectifiable part of $S$, then ${\rm UBEP}(S;U)$ is $\mathcal{H}^{n}$-equivalent to ${\rm UBEP}(\mathcal{R}(S);U)$; second, if $S^{*}$ is $\mathcal{H}^{n}$-contained in $S$, then ${\rm UBEP}(S;U)$ is $\mathcal{H}^{n}$-contained in ${\rm UBEP}(S;U)$; both properties are proved in Theorem 2.1 (an expanded restatement of Theorem 1.2).

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 1.6.

The most convenient setting for addressing the existence of minimizers in Gauss’ capillarity theory is of course that of sets of finite perimeter [Fin86, Mag12]. However, if the notion of homotopic spanning is limited to closed sets, as it is the case when working with Definition A, then one cannot directly use homotopic spanning on sets of finite perimeter, and this is the reason behind the specific formulation (1.8) of $\psi(v)$ used in [MSS19, KMS22a]. Equipped with Definition B 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

	$\displaystyle\psi_{\rm bk}(v)=\inf\Big{\{}\mathcal{H}^{n}(\Omega\cap\partial^{*}E):\mbox{$|E|=v$ and $\Omega\cap(\partial^{*}E\cup E^{\scriptscriptstyle{(1)}})$ is $\mathcal{C}$-spanning $\mathbf{W}$}\Big{\}}\,,$
	$\displaystyle\psi_{\rm bd}(v)=\inf\Big{\{}\mathcal{H}^{n}(\Omega\cap\partial^{*}E):\mbox{$|E|=v$ and $\Omega\cap\partial^{*}E$ is $\mathcal{C}$-spanning $\mathbf{W}$}\Big{\}}\,,$

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 $\Omega\cap(\partial^{*}E\cup E^{\scriptscriptstyle{(1)}})$ or via the (reduced) boundary of $E$. Inspired by the definition of the class $\mathcal{K}$ introduced in (1.9), we also introduce the class $\mathcal{K}_{\rm B}$ of generalized soap films defined by

	$\displaystyle\mathcal{K}_{\rm B}=\Big{\{}(K,E):\mbox{$K$ and $E$ are Borel subsets of $\Omega$,}$		(1.15)
	$\displaystyle\hskip 76.82234pt\mbox{$E$ has locally finite perimeter in $\Omega$ and $\partial^{*}E\cap\Omega\overset{\scriptscriptstyle{\mathcal{H}^{n}}}{\subset}K$}\Big{\}}\,.$

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

In the companion paper [Nov23] the following relaxation formulas for problems $\psi_{\rm bk}$ and $\psi_{\rm bd}$ are proved,

$$\psi_{\rm bk}(v)=\Psi_{\rm bk}(v)\,,\qquad\psi_{\rm bd}(v)=\Psi_{\rm bd}(v)\,,\qquad\forall v>0\,,$$

(1.16)

where the following minimization problems on $\mathcal{K}_{\rm B}$ are introduced

	$\displaystyle\Psi_{\rm bk}(v)=\inf\Big{\{}\mathcal{F}_{\rm bk}(K,E):(K,E)\in\mathcal{K}_{\rm B}\,,|E|=v\,,\mbox{$K\cup E^{{\scriptscriptstyle{(1)}}}$ is $\mathcal{C}$-spanning $\mathbf{W}$}\Big{\}}\,,\hskip 28.45274pt$		(1.17)
	$\displaystyle\Psi_{\rm bd}(v)=\inf\Big{\{}\mathcal{F}_{\rm bd}(K,E):(K,E)\in\mathcal{K}_{\rm B}\,,|E|=v\,,\mbox{$K$ is $\mathcal{C}$-spanning $\mathbf{W}$}\Big{\}}\,.$		(1.18)

Here $\mathcal{F}_{\rm bk}$ and $\mathcal{F}_{\rm bd}$ are the relaxed energies defined for $(K,E)\in\mathcal{K}_{\rm B}$ and $A\subset\Omega$ as

	$\displaystyle\mathcal{F}_{\rm bk}(K,E;A)=2\,\mathcal{H}^{n}(A\cap K\cap E^{{\scriptscriptstyle{(0)}}})+\mathcal{H}^{n}(A\cap\partial^{*}E)\,,$		(1.19)
	$\displaystyle\mathcal{F}_{\rm bd}(K,E;A)=2\,\mathcal{H}^{n}(A\cap K\setminus\partial^{*}E)+\mathcal{H}^{n}(A\cap\partial^{*}E)\,,$		(1.20)

(We also set, for brevity, $\mathcal{F}_{\rm bk}(K,E):=\mathcal{F}_{\rm bk}(K,E;\Omega)$ and $\mathcal{F}_{\rm bd}(K,E):=\mathcal{F}_{\rm bd}(K,E;\Omega)$.) 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 (1.16), 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 (1.7): more precisely, if we set

$$\ell_{\rm B}:=\frac{\Psi_{\rm bk}(0)}{2}=\frac{\Psi_{\rm bd}(0)}{2}=\inf\Big{\{}\mathcal{H}^{n}(S):\mbox{$S$ is a Borel set $\mathcal{C}$-spanning $\mathbf{W}$}\Big{\}}\,,$$

(1.21)

then, by Theorem 1.1, we evidently have $\ell_{\rm B}\leq\ell$; 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 $\Psi_{\rm bd}(v)$.

The delicate part of Theorem 1.4 is proving the closure statements. This will require first to extend the characterization of homotopic spanning from locally $\mathcal{H}^{n}$-finite sets to generalized soap films (Theorem 3.1), and then to discuss the behavior under weak-star convergence of the associated Radon measures of the objects appearing in conditions like (1.14) (Theorem 4.1).

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 [MNR23a]; (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 $\Psi_{\rm bk}(v)$.

The conclusions of Theorem 1.5 about $\Psi_{\rm bk}(v)$ can be read in parallel to the conclusions about $\psi(v)$ obtained in [KMS22a]. The crucial difference is that, in place of the “weak” minimality inequality (1.10), which in this context would be equivalent to $\mathcal{F}_{\rm bk}(K,E)\leq\mathcal{H}^{n}(\Omega\cap\partial^{*}E^{\prime})$ for every competitor $E^{\prime}$ in $\psi_{\rm bk}(v)$, we now have the proper minimality inequality

$$\mathcal{F}_{\rm bk}(K,E)\leq\mathcal{F}_{\rm bk}(K^{\prime},E^{\prime})$$

(1.25)

for every competitor $(K^{\prime},E^{\prime})$ in $\Psi_{\rm bk}(v)$. Not only the final conclusion is stronger, but the proof is also entirely different: whereas [KMS22a] 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.

We now formalize the validation of (1.1) 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 1.3/Theorem 3.1. 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 $\Omega$ induced by $K\cup E^{\scriptscriptstyle{(1)}}$ are actually $(\Lambda,r_{0})$-minimizers of the perimeter in $\Omega$, i.e., there exist $\Lambda$ and $r_{0}$ positive constants such that

$$P(U_{i};B_{r}(x))\leq P(V;B_{r}(x))+\Lambda\,|V\Delta U_{i}|\,,$$

whenever $V\Delta U_{i}\subset\!\subset\Omega$ and ${\rm diam}\,(V\Delta U_{i})<r_{0}$. 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 [DLGM17a], the slab competitors and exterior cup competitors of [KMS22a], etc.).

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

Figure 1.7, is the following. By Theorem 1.2, we can locally represent $\mathcal{F}_{\rm bk}(K,E;B_{r}(x))$ 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 $(K^{\prime},E^{\prime})$ so that the elements of the essential partition induced by $K^{\prime}\cup(E^{\prime})^{\scriptscriptstyle{(1)}}$ in $\Omega$, that can be used to represent $\mathcal{F}_{\rm bk}(K^{\prime},E^{\prime};B_{r}(x))$ as the sum $P(V_{i};B_{r}(x))+P(V_{j};B_{r}(x))+P(V_{k};B_{r}(x))$, are such that

$\displaystyle\mathcal{F}_{\rm bk}(K^{\prime},E^{\prime};B_{r}(x))-\mathcal{F}_{\rm bk}(K,E;B_{r}(x))=P(V;B_{r}(x))-P(U_{i};B_{r}(x))\,.$

(1.26)

The trick is that by suitably defining $K^{\prime}$ and $E^{\prime}$ we can recover the entirety of $B_{r}(x)\cap\partial^{*}U_{j}$ and $B_{r}(x)\cap\partial^{*}U_{k}$ by attributing different parts of these boundaries to different terms in the representation of $\mathcal{F}_{\rm bk}(K^{\prime},E^{\prime};B_{r}(x))$. In other words we are claiming that things can be arranged so that we still have

$$B_{r}(x)\cap\big{(}\partial^{*}U_{j}\cap\partial^{*}U_{k})\overset{\scriptscriptstyle{\mathcal{H}^{n}}}{\subset}K^{\prime}\cup(E^{\prime})^{\scriptscriptstyle{(1)}}\,.$$

(1.27)

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 $K^{\prime}\cup(E^{\prime})^{\scriptscriptstyle{(1)}}$ is still $\mathcal{C}$-spanning $\mathbf{W}$ by means of Theorem 1.3/Theorem 3.1.

By the regularity theory of $(\Lambda,r_{0})$-perimeter minimizers (see, e.g. [Mag12, Part III]) we can deduce the $C^{1,\alpha}$-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 $\Omega\cap(\partial E\setminus\partial^{*}E)$, 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.