# Partial Plateau’s Problem with $H$-mass

Enrique Alvarado<EMAIL_ADDRESS>Qinglan Xia
<EMAIL_ADDRESS>
###### Abstract

Classically, Plateau’s problem asks to find a surface of least area with a
given boundary $B$. In this article, we investigate a version of Plateau’s
problem, where the boundary of an admissible surface is only required to
partially span $B$. Our boundary data is given by a flat $(m-1)$-chain $B$ and
a smooth compactly supported differential $(m-1)$-form $\Phi$. We are
interested in minimizing $\mathbf{M}(T)-\int_{\partial T}\Phi$ over all
$m$-dimensional rectifiable currents $T$ in $\mathbb{R}^{n}$ such that
$\partial T$ is a subcurrent of the given boundary $B$. The existence of a
rectifiable minimizer is proven with Federer and Fleming’s compactness
theorem. We generalize this problem by replacing the mass $\mathbf{M}$ with
the $H$-mass of rectifiable currents. By minimizing over a larger class of
objects, called scans with boundary, and by defining their $H$-mass as a type
of lower-semicontinuous envelope over the $H$-mass of rectifiable currents, we
prove an existence result for this problem by using Hardt and De Pauw’s BV
compactness theorem.

##  1 Introduction

#### A brief history of Plateau’s problem.

Loosely speaking, Plateau’s problem asks to find a surface of least area with
a given boundary. The various answers to this question depend on the precise
notion of a surface, of its area, and of its boundary. Although this problem
was classically studied for surfaces in $\mathbb{R}^{3}$ with the use of
mappings of surfaces [14], geometric measure theory has now provided us with
precise formulations and definitions of these italicized terms [7, 17, 13, 2,
16]. Motivated by De Rham’s notion of currents, in the seminal work [8],
Federer and Fleming solved a version of Plateau’s problem with a class of
generalized oriented submanifolds called integral currents, which come
equipped with a natural notion of a boundary and surface area. We refer the
reader to [11] for a historical overview of Plateau’s problem and geometric
measure theory.

#### The $H$-mass and scans.

The mass $\mathbf{M}(T)$ of a current $T$ is used as a way to measure its
area, and for a rectifiable current $T=\mathcal{H}^{m}\,\hbox{\vrule
height=7.0pt,depth=0.0pt\vrule
height=0.4pt,depth=0.0pt,width=6.0pt}\,M\wedge\theta\xi$, its mass is given by
$\int_{M}\theta\,d\mathcal{H}^{m}$ where $\theta:M\to\mathbb{N}$ is its
multiplicity function (see §2). For surfaces in $\mathbb{R}^{3}$, a mass
minimizing current $T$ provides a good model for some but not all soap films.
For example, in [10], Fleming showed that 2-dimensional mass minimizing
currents in $\mathbb{R}^{3}$ are smooth embedded surfaces away from their
boundary, whereas general soap films have interior singularities. To get a
better model for soap films, Almgren [1] introduced the notion of size for
rectifiable currents and investigated size minimizing currents. However, in
general, the existence of size minimizing currents is difficult to obtain. In
part, this is due to the fact that size minimizing sequences may have
unbounded mass, as can be seen in examples provided by F. Morgan in [12]. The
lack of bounded mass prevents one from applying Federer and Fleming’s
compactness theorem [8] to size minimizing sequences.

Another important measurement of surface area is the $\alpha$-mass,

$\mathbf{M}_{\alpha}(T)=\int_{M}\theta^{\alpha}\,d\mathcal{H}^{m}$ (1)

of a real rectifiable current $T=\mathcal{H}^{m}\,\hbox{\vrule
height=7.0pt,depth=0.0pt\vrule
height=0.4pt,depth=0.0pt,width=6.0pt}\,M\wedge\theta\xi$, where the parameter
$\alpha\in[0,1]$. The $\alpha$-mass acts as a bridge between the mass and the
size of a real rectifiable current. When $\alpha=0$, we get the size of $T$,
and when $\alpha=1$ we get its mass. Given two probability measures $\mu_{+}$
and $\mu_{-}$, any real rectifiable $1$-dimensional current $T$ with $\partial
T=\mu_{+}-\mu_{-}$ is called a transport path from $\mu_{+}$ to $\mu_{-}$ in
the study of branched (ramified) optimal transport [18, 3, 19]. The
$\alpha$-mass of $T$ is called the $\alpha$-cost of the transport path $T$. A
more general version of the $\alpha$-mass is the $H$-mass. For a concave
integrand (Definition 4.1) $H:[0,\infty)\to[0,\infty)$, the $H$-mass of a real
rectifiable current $T=\mathcal{H}^{m}\,\hbox{\vrule
height=7.0pt,depth=0.0pt\vrule
height=0.4pt,depth=0.0pt,width=6.0pt}\,M\wedge\theta\xi$ is given by
$\int_{M}H\circ\theta\,d\mathcal{H}^{m}$.

In article [5], Hardt and De Pauw investigated a Plateau-type problem:
minimizing the $H$-mass over all rectifiable currents with a fixed boundary.
Although $\mathbf{M}_{H}$ minimizing sequences of rectifiable currents may not
necessarily have bounded mass, they showed that for any fixed $\epsilon>0$,
the minimizing sequences for the approximate problem:
$\mathbf{M}_{H}+\epsilon\mathbf{M}$ do. They then considered a sequence of
rectifiable currents, each of which is a minimizer to the approximate problem
for a sequence $\epsilon_{i}\to 0$. Using their general BV-compactness theorem
on this sequence, they obtained an $\mathbf{M}_{H}$ minimizing subsequence of
rectifiable currents, that when viewed as scans, converges to a rectifiable
scan. An $m$-dimensional scan in $\mathbb{R}^{n}$ is a type of measurable
function $\mathcal{T}$ from the space of $(n-m)$ planes, to the space of $0$
dimensional currents in $\mathbb{R}^{n}$ endowed with the $H$-flat distance
(Definition 4.3). For example, an $m$-dimensional rectifiable current in
$\mathbb{R}^{n}$ can be viewed as a scan by slicing it with $n-m$ planes (see
§4.1).

#### Partial Plateau’s problem.

Figure 1: The solution to the standard Plateau’s problem for the closed curve
shown in $(A)$ is the surface shown in $(B)$. When one of the outer arcs is
removed as shown in $(C)$, the soap film shown in $(D)$ may still form.
Although it is no longer a solution to the standard Plateau’s problem, it is a
solution to partial Plateau’s problem studied in the examples of §3.

In the above Plateau-type problems, the boundaries of the surfaces are
prescribed. In this article, we propose a new Plateau-type problem, where the
boundary is not required to completely fulfill the prescribed one, but rather,
is only required to satisfy a portion of it. We call this partial Plateau’s
problem. To motivate it, let us consider the following scenario. Suppose that
we have an oriented wire that has the form of a sunflower as in Figure 1($A$).
The solution to the standard oriented Plateau’s problem with this boundary
will be the union of all petals as in Figure 1($B$). Now, assume that one
petal accidentally falls off the flower and the boundary wire becomes the one
shown in Figure 1($C$). In this case, there is no solution to the
corresponding Plateau’s problem anymore since the resulting wire is no longer
a closed curve. Nevertheless, the remaining wire still generates a soap film
in the form of the remaining pedals as shown in Figure 1($D$). This scenario
(where the soap film only spans a portion of the given wire) motivates us to
relax the boundary condition. In Example 3.6, we investigate this example in
detail, and show that the solution shown in Figure 1($D$) is given by a
solution to our partial Plateau’s problem.

Another motivation for this problem comes from the study of ramified optimal
transportation as given in [20]. More precisely, let $\mu$ and $\nu$ be two
Radon measures on a convex compact subset $X$ of the Euclidean space
$\mathbb{R}^{n}$, let $\mathbf{M}_{\alpha}$ be the $\alpha$-mass as defined in
(1) for some $\alpha\in[0,1)$, and let $h$ be a continuous function on the
support of the signed measure $\nu-\mu$. Then, [20] considered the following
resource allocation problem: Minimize

$\mathbf{E}_{\alpha}^{h}(T):=\mathbf{M}_{\alpha}(T)-\int_{X}h\,d(\partial T)$
(2)

among all 1 dimensional real rectifiable currents $T$ with $\partial
T\preceq\nu-\mu$, in the sense that its Jordan decomposition $\partial
T=\tilde{\nu}-\tilde{\mu}$ satisfies $\tilde{\mu}\leq\mu$ and
$\tilde{\nu}\leq\nu$. In [20], this problem was motivated by considering the
following example: Consider a firm that produces and sells a product in
various regions. The measures $\mu$ and $\nu$ represent, respectively, the
distributions of production capacities and market sizes. The function $h$
represents the sale price on the support of $\nu$, and the production cost on
the support of $\mu$. The firm aims to maximize its profit (the negative of
$\mathbf{E}_{\alpha}^{h}(T)$) defined as sale revenue
$\int_{X}h\,d\tilde{\nu}$ minus the cost involved in transportation
$\mathbf{M}_{\alpha}(T)$ and production $\int_{X}h\,d\tilde{\mu}$. Note that
the boundary $\partial T$ is not required to equal $\nu-\mu$, but rather to be
a portion of it. Similar kinds of optimal partial mass transportation problems
have been studied for instance by Caffarelli and McCann [4] and also Figalli
[9] for the scenario of Monge-Kantorovich problems with particular attention
to the quadratic cost.

#### Partial Plateau’s problem with $H$-mass.

The above Plateau-type problems either look to minimize functionals involving

  * •

the mass, or more generally, the $H$-mass over $m$-dimensional rectifiable
currents with a fixed boundary condition, or

  * •

the $\alpha$-mass over 1 dimensional real rectifiable currents with a partial
boundary condition.

In this article, we investigate a Plateau type problem that minimizes a
functional involving the $H$-mass over $m$-dimensional rectifiable currents
with a partial boundary condition. Precisely, let $H$ be a concave integrand,
let $B$ be an $(m-1)$-dimensional rectifiable current with finite $H$-mass,
and let $\Phi$ be a smooth, compactly supported $(m-1)$-differential form in
$\mathbb{R}^{n}$. We are interested in minimizing

$\mathbf{E}_{H,\Phi}(T):=\mathbf{M}_{H}(T)-\int_{\partial T}\Phi$

over all $m$-dimensional rectifiable currents $T$ in $\mathbb{R}^{n}$ with
$\partial T\preceq B$. By definition, $\partial T\preceq B$ means that
$\partial T$ is a subcurrent [15] of $B$, in the sense that
$\mathbf{M}(B)=\mathbf{M}(B-\partial T)+\mathbf{M}(\partial T)$. For the sake
of intuition, it is worth mentioning that whenever $B$ is a real rectifiable
current with finite mass, $\partial T\preceq B$ means that $\partial T$ is
supported within the support of $B$, has the same orientation as $B$, but only
has a fraction of its density [15, Lemma 3.7].

We have thus far formulated our Plateau type problems as minimization problems
over rectifiable currents. However, to obtain our main existence results
(Theorem 4.18, Theorem 5.1, and Corollary 5.5), we perform our minimization
over a larger class of objects, called $m$-dimensional scans with boundary
(see Definitions 4.7 and 4.10). Loosely speaking, these are scans (introduced
in [5]) with well-defined boundaries that are obtained as pointwise limits of
rectifiable currents (when viewed as scans). The reason that we minimize over
scans with boundary is that when viewed as scans, the class of $m$-dimensional
rectifiable currents is not closed under pointwise almost everywhere
convergence.

#### Organization of the article.

In §2, we go over relevant definitions and notations that are by now standard
in geometric measure theory literature. Then in §3, we look at partial
Plateau’s problem with mass: Given an $(m-1)$-dimensional flat chain $B$ in
$\mathbb{R}^{n}$ with finite mass, and a smooth, compactly supported
$(m-1)$-differential form $\Phi$ in $\mathbb{R}^{n}$, consider

$\mathcal{P}_{(\mathbf{E}_{\Phi},\,\preceq
B,\,\mathbb{R}^{n})}\begin{cases}\text{minimize
}\mathbf{E}_{\Phi}(T):=\mathbf{M}(T)-\partial T(\Phi)\\\ \text{among
}T\in\mathscr{R}_{m}(\mathbb{R}^{n})\text{ such that }\partial T\preceq
B.\end{cases}$ (3)

In Theorem 3.5, we prove the existence of this problem by using the standard
compactness arguments of rectifiable currents. Here, for any
$\mathbf{E}_{\Phi}$ minimizing sequence $\\{T_{i}\\}$, we have uniform bounds
for $\\{\mathbf{M}(T_{i})+\mathbf{M}(\partial T_{i})\\}$. Although this is a
special case of partial Plateau’s problem with $H$-mass (studied in §5) when
$H(\theta)=\theta$, we do not have to minimize over $m$-dimensional scans in
$\mathbb{R}^{n}$ as in the following problems that involve the general
$H$-mass.

In §4, we re-investigate Plateau’s problem with $H$-mass, which was first
investigated by De Pauw and Hardt in their seminal work [5]. Formulated over
rectifiable currents, the problem is stated as follows: Given a rectifiable
current $T_{0}\in\mathscr{R}_{m}(\mathbb{R}^{n})$,

$\begin{cases}\text{minimize }\mathbf{M}_{H}(T)\\\ \text{among
}T\in\mathscr{R}_{m}(\mathbb{R}^{n})\text{ such that }\partial T=\partial
T_{0}.\end{cases}$ (4)

To get an existence-type result [5, Theorem 3.5.2.] for problem (4), De Pauw
and Hardt innovatively relax the notion of a rectifiable current to a
rectifiable scan. In their proof of the theorem, they construct a sequence of
$\mathbf{M}_{H}$ minimizing rectifiable currents that pointwise converge (as
scans) to a rectifiable scan $\mathcal{T}^{\ast}$. By Fateau’s lemma, it is
clear that $\mathbf{M}_{H}(\mathcal{T}^{\ast})$ is less than or equal to the
infimum value of problem (4). Nevertheless, since $\mathcal{T}^{\ast}$ is not
necessarily a rectifiable current, one cannot conclude that
$\mathbf{M}_{H}(\mathcal{T}^{\ast})$ achieves the infimum value of problem (4)
as claimed in [5, Theorem 3.5.2.].

To overcome this issue, in §4.2 we amend their existence result [5, Theorem
3.5.2.] by reformulating problem (4) with a modified version of the $H$-mass,
denoted by $\underline{\mathbf{M}}_{H}$, and by minimizing
$\underline{\mathbf{M}}_{H}$ over $Scan_{m}(\mathbb{R}^{n})$, the class of
$m$-dimensional scans with boundary (Definitions 4.7 and 4.10). In particular,
we provide the following reformulation of Plateau’s problem with $H$-mass:
Given a scan $\mathcal{T}_{0}\in{\it Scan}_{m}(\mathbb{R}^{n})$ with boundary,

$\mathcal{P}_{(\underline{\mathbf{M}}_{H},\,\partial\mathcal{T}_{0},\,\mathbb{R}^{n})}\begin{cases}\text{minimize
}\underline{\mathbf{M}}_{H}(\mathcal{T})\\\ \text{among }\mathcal{T}\in
Scan_{m}(\mathbb{R}^{n})\text{ such that
}\partial\mathcal{T}=\partial\mathcal{T}_{0}.\end{cases}$ (5)

It is worth mentioning the following two facts about
$Scan_{m}(\mathbb{R}^{n})$. The class of $m$-dimensional scans with boundary
is larger than the class of $m$-dimensional rectifiable currents, and any
element in $Scan_{m}(\mathbb{R}^{n})$ has a well-defined boundary, which is
not necessarily the case for the general $m$-dimensional scans studied in [5].
Under such modifications, in Theorem 4.18 we are able to get the existence of
an $\underline{\mathbf{M}}_{H}$ minimizing $m$-dimensional scan with fixed
boundary.

Given $B$, $\Phi$, and $H$ as above, and the additional constraint that $B$ is
rectifiable and has finite $H$-mass, in §5 we consider partial Plateau’s
problem with $H$-mass over rectifiable currents:

$\begin{cases}\text{minimize
}\mathbf{E}_{\Phi,H}(T):=\mathbf{M}_{H}(T)-\partial T(\Phi)\\\ \text{among
}T\in\mathscr{R}_{m}(\mathbb{R}^{n})\text{ such that }\partial T\preceq
B.\end{cases}$ (6)

Unfortunately, as in Plateau’s problem (4) with $H$-mass, this problem does
not always have a solution among rectifiable currents. To resolve this issue,
we reformulate the problem by using our modified version
$\underline{\mathbf{M}}_{H}$ of the $H$-mass on the class
$Scan_{m}(\mathbb{R}^{n})$, and investigate our main problem of interest,
called partial Plateau’s problem with $H$-mass:

$\mathcal{P}_{(\mathbf{E}_{\Phi,H},\,\preceq
B,\,\mathbb{R}^{n})}\begin{cases}\text{minimize
}\mathbf{E}_{\Phi,H}(\mathcal{T}):=\underline{\mathbf{M}}_{H}(\mathcal{T})-\partial\mathcal{T}(\Phi)\\\
\text{among }\mathcal{T}\in Scan_{m}(\mathbb{R}^{n})\text{ such that
}\partial\mathcal{T}\preceq\mathcal{S}(B).\end{cases}$ (7)

Our main existence result, Theorem 5.1, is proven within a slightly more
general setting, and the existence of minimizer $\mathcal{T}^{\ast}$ to
problem (7) is then given in Corollary 5.5. Moreover, Corollary 5.5 also shows
us that

$\mathbf{E}_{\Phi,H}(\mathcal{T}^{*})=\inf\\{\mathbf{E}_{\Phi,H}(T):T\in\mathscr{R}_{m}(\mathbb{R}^{n}),\partial
T\preceq B\\}.$

In other words, despite the fact that one can not expect to find an
$\mathbf{E}_{H,\Phi}$ minimizer to problem (6) among rectifiable currents, its
infimum value is obtained by the corresponding $\mathbf{E}_{H,\Phi}$ minimizer
of problem (7), where we minimize over scans with boundary under the same
partial boundary condition.

##  2 Notation and preliminary definitions

Table 1: Notation $\mathbb{N}$ | $:=$ |  {1, 2, 3, …}, the set of all natural numbers
---|---|---
$H$ |  |  A concave integrand (Def. 4.1)
$\mathbf{O}^{\ast}(n,k)$ |  |  the space of all orthogonal projections of $\mathbb{R}^{n}$ onto $\mathbb{R}^{k}$
$\boldsymbol{\theta}_{n,k}^{\ast}$ |  |  the $\mathbf{O}(n)$ invariant measure on $\mathbf{O}^{\ast}(n,k)$
$\mathcal{L}^{k}$ |  |  the Lebesgue measure on $\mathbb{R}^{k}$
$\mathcal{H}^{k}$ |  |  the $k$ dimensional Hausdorff measure
Currents
$\mathscr{D}_{m}(\mathbb{R}^{n})$ |  |  $m$-dimensional currents in $\mathbb{R}^{n}$
$\mathscr{E}_{m}(\mathbb{R}^{n})$ |  |  compactly supported $m$-dimensional currents in $\mathbb{R}^{n}$
$\mathbf{M}(T)$ |  |  the mass of $T\in\mathscr{D}_{m}(\mathbb{R}^{n})$ (8)
$\mathscr{R}_{m}(\mathbb{R}^{n})$ |  |  $m$-dimensional rectifiable currents in $\mathbb{R}^{n}$
$\mathbf{F}_{K}(T)$ |  |  flat norm of $T\in\mathscr{D}_{m}(\mathbb{R}^{n})$ (11)
$\mathbf{I}_{m}(\mathbb{R}^{n})$ |  |  $m$-dimensional integral currents in $\mathbb{R}^{n}$ (12)
$\mathscr{F}_{m}(\mathbb{R}^{n})$ |  |  $m$-dimensional integral flat chains in $\mathbb{R}^{n}$ (12)
$\mathbf{N}_{m}(\mathbb{R}^{n})$ |  |  $m$-dimensional normal currents in $\mathbb{R}^{n}$ (13)
$\mathbf{F}_{m}(\mathbb{R}^{n})$ |  |  $m$-dimensional flat chains in $\mathbb{R}^{n}$ (13)
$\mathbf{M}_{H}(T)$ |  |  $H$-mass of $T\in\mathscr{D}_{m}(\mathbb{R}^{n})$ (Def. 4.2)
$\mathscr{F}_{K}^{H}(T)$ |  |  $H$-flat distance of $T\in\mathscr{D}_{m}(\mathbb{R}^{n})$ (Def. 4.3)
$A\preceq B$ |  |  $A$ is a subcurrent of $B$ for $A,B\in\mathscr{D}_{m}(\mathbb{R}^{n})$ (15)
Scans
$\mathcal{S}(T)$ |  |  the measurable map $(p,y)\mapsto\langle T,p,y\rangle$ associated to a flat chain $T$ (22)
$\mathbf{M}_{H}(f)$ |  |  the $H$-mass of a measurable map $f:\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}\to\mathbf{I}_{0,K}(\mathbb{R}^{n})$ (24)
$\mathcal{S}_{m}(\mathbb{R}^{n})$ |  |  scans obtained from rectifiable currents in $\mathbb{R}^{n}$ (27)
${\it Scan}_{m}(\mathbb{R}^{n})$ |  |  $m$-dimensional scans in $\mathbb{R}^{n}$ with boundary (Def. 4.7 and 4.10)
$\underline{\mathbf{M}}_{H}(\mathcal{T})$ |  |  the lower-semicontinuous envelope of $\mathbf{M}_{H}$ over a.e. pointwise convergence of $\\{\mathcal{S}(R):R\in\mathscr{R}_{m}(\mathbb{R}^{n}),\partial\mathcal{T}=\partial R\\}$ (Def. 4.13)

In this article, most of our notations and definitions are consistent with
that of Federer’s geometric measure theory book [7] along with De Pauw and
Hardt’s article [5]. For the convenience of the reader, we will now give a
brief review of the standard concepts and results from geometric measure
theory that will be important for us in this article.

Let $0\leq m\leq n$. We let $\mathscr{D}^{m}(\mathbb{R}^{n})$ denote the real
vector space of smooth differential $m$-forms with compact support. The class
$\mathscr{D}_{m}(\mathbb{R}^{n})$ is the vector space of all continuous real-
valued linear functions on $\mathscr{D}^{m}(\mathbb{R}^{n})$. Elements of
$\mathscr{D}_{m}(\mathbb{R}^{n})$ are called $m$-dimensional currents in
$\mathbb{R}^{n}$, and the subspace $\mathscr{E}_{m}(\mathbb{R}^{n})$ of
$\mathscr{D}_{m}(\mathbb{R}^{n})$ contains all $m$-dimensional currents with
compact support. The mass of a current $T\in\mathscr{D}_{m}(\mathbb{R}^{n})$
is defined as

$\mathbf{M}(T):=\sup\\{T(\phi):\phi\in\mathscr{D}^{m}(\mathbb{R}^{n}),\text{
with }\langle e_{1}\wedge\dots\wedge e_{m},\phi(x)\rangle\leq 1,\ \forall
x\in\mathbb{R}^{n}\text{ and }e_{i}\in\mathbb{S}^{n-1}\\}.$ (8)

For $m\geq 1$, the boundary of a current $T\in\mathscr{D}_{m}(\mathbb{R}^{n})$
is the current $\partial T\in\mathscr{D}_{m-1}(\mathbb{R}^{n})$ defined by
$\partial T(\phi):=T(d\phi)$ for $\phi\in\mathscr{D}^{m-1}(\mathbb{R}^{n})$,
and $\partial T$ is defined to be the zero-current whenever
$T\in\mathscr{D}_{0}(\mathbb{R}^{n})$. For a sequence
$\\{T_{i}\\}_{i=1}^{\infty}\subseteq\mathscr{D}_{m}(\mathbb{R}^{n})$ of
$m$-dimensional currents and a current $T\in\mathscr{D}_{m}(\mathbb{R}^{n})$,
we say $T_{i}$ converges (weakly) to $T$, written as $T_{i}\rightharpoonup T$,
if $T_{i}(\phi)\to T(\phi)$ for all $\phi\in\mathscr{D}^{m}(\mathbb{R}^{n})$.

The rectifiable currents of Federer and Fleming play an important role in this
article. With $\mathcal{H}^{m}$ denoting the $m$-dimensional Hausdorff measure
on $\mathbb{R}^{n}$, a subset $M$ of $\mathbb{R}^{n}$ is called
$(\mathcal{H}^{m},m)$-rectifiable if $\mathcal{H}^{m}(M)<\infty$ and
$\mathcal{H}^{m}(M\setminus\cup_{i=1}^{\infty}N_{i})=0$ for some finite or
countable family $\\{N_{i}\\}_{i\in I}$ of $m$-dimensional
$C^{1}$-submanifolds of $\mathbb{R}^{n}$. The following is an important
characterization of the class $\mathscr{R}_{m}(\mathbb{R}^{n})$ of
$m$-dimensional rectifiable currents.

Let $0\leq m\leq n$. An $m$-dimensional rectifiable current
$T\in\mathscr{R}_{m}(\mathbb{R}^{n})$ is given by the following three things:

  1. 1.

an $\mathcal{H}^{m}$-measurable and bounded $(\mathcal{H}^{m},m)$-rectifiable
subset $M$ of $\mathbb{R}^{n}$,

  2. 2.

an $\mathcal{H}^{m}$-measurable $m$-vectorfield
$\xi:M\to\wedge_{m}\mathbb{R}^{n}$ such that for $\mathcal{H}^{m}$-a.e. $x\in
M$, $\xi(x)=e_{1}\wedge\dots\wedge e_{m}$ for some orthonormal basis
$\\{e_{i}\\}_{i=1}^{m}$ of the approximate tangent space $\mathrm{Tan}(M,x)$,

  3. 3.

and an $(\mathcal{H}^{m}\,\hbox{\vrule height=7.0pt,depth=0.0pt\vrule
height=0.4pt,depth=0.0pt,width=6.0pt}\,M)$-summable function
$\theta:M\to\mathbb{N}$ called the multiplicity function.

We will be denoting such a rectifiable current
$T\in\mathscr{R}_{m}(\mathbb{R}^{n})$ by $\mathcal{H}^{m}\,\hbox{\vrule
height=7.0pt,depth=0.0pt\vrule
height=0.4pt,depth=0.0pt,width=6.0pt}\,M\wedge\theta\xi$ as in [7]. Other
common notation is also given by $\underline{\underline{\tau}}(M,\theta,\xi)$
as in [17], or by $\llbracket M,\xi,\theta\rrbracket$. Its action on
$\phi\in\mathscr{D}^{m}(U)$ is given by

$(\mathcal{H}^{m}\,\hbox{\vrule height=7.0pt,depth=0.0pt\vrule
height=0.4pt,depth=0.0pt,width=6.0pt}\,M\wedge\theta\xi)(\phi)=\int_{M}\langle\phi,\xi\rangle\theta\,d\mathcal{H}^{m},$

and its mass has the simple form

$\mathbf{M}(\mathcal{H}^{m}\,\hbox{\vrule height=7.0pt,depth=0.0pt\vrule
height=0.4pt,depth=0.0pt,width=6.0pt}\,M\wedge\theta\xi)=\int_{M}\theta\,d\mathcal{H}^{m}.$
(9)

######  Remark 2.1.

Notice that the $m$-dimensional rectifiable currents we are considering all
have finite mass and compact support.

We will now give a brief overview of some special classes of currents.
Whenever $K\subseteq\mathbb{R}^{n}$ is a compact Lipschitz neighborhood
retract (CLNR) [7, 4.1.29.], (e.g., compact convex subsets of
$\mathbb{R}^{n}$), we have the identity

$\mathscr{R}_{m,K}(\mathbb{R}^{n})=\mathscr{R}_{m}(\mathbb{R}^{n})\cap\\{T:\mathrm{spt}(T)\subseteq
K\\}.$ (10)

When a subset $K$ is simply a compact subset of $\mathbb{R}^{n}$,
$\mathscr{R}_{m,K}(\mathbb{R}^{n})$ has a more technical definition. However,
since we will mainly be dealing with CLNRs of $\mathbb{R}^{n}$, we will forgo
stating its definition here, and instead refer the reader to [7, 4.1.24.].

Let $K$ be a compact subset of $\mathbb{R}^{n}$. We define the following
classes of currents

  * •

$\mathbf{I}_{0,K}(\mathbb{R}^{n}):=\mathscr{R}_{0,K}(\mathbb{R}^{n})$, and
$\mathbf{I}_{m,K}(\mathbb{R}^{n}):=\\{T:T\in\mathscr{R}_{m,K}(\mathbb{R}^{n}),\
\partial T\in\mathscr{R}_{m-1,K}(\mathbb{R}^{n})\\}$ for $m>0$;

  * •

$\mathscr{F}_{m,K}(\mathbb{R}^{n}):=\\{R+\partial
S:R\in\mathscr{R}_{m,K}(\mathbb{R}^{n}),\
S\in\mathscr{R}_{m+1,K}(\mathbb{R}^{n})\\}$;

  * •

$\mathbf{N}_{m,K}(\mathbb{R}^{n}):=\\{T\in\mathscr{D}_{m}(\mathbb{R}^{n}):\mathbf{M}(T)+\mathbf{M}(\partial
T)<\infty\text{ with }\mathrm{spt}(T)\subseteq K\\}$;

  * •

$\mathbf{F}_{m,K}(\mathbb{R}^{n}):=$ the $\mathbf{F}_{K}$ closure of
$\mathbf{N}_{m,K}(\mathbb{R}^{n})$ in $\mathscr{D}_{m}(\mathbb{R}^{n})$, where
$\mathbf{F}_{K}:\mathscr{D}_{m}(\mathbb{R}^{n})\to[0,\infty]$ is the flat norm
defined as

$\mathbf{F}_{K}(T):=\inf\\{\mathbf{M}(T-\partial
S)+\mathbf{M}(S):S\in\mathscr{D}_{m+1}(\mathbb{R}^{n})\text{ with
}\mathrm{spt}(S)\subseteq K\\},\forall T\in\mathscr{D}_{m}(\mathbb{R}^{n}).$
(11)

######  Remark 2.2.

Any $0$ dimensional integral current $T\in\mathbf{I}_{0,K}(\mathbb{R}^{n})$
has a representation

$T=\sum_{i=1}^{k}a_{i}\boldsymbol{\delta}_{x_{i}}$

where $a_{1},\dots,a_{k}$ are positive integers, and $x_{1},\dots,x_{k}\in K$.

The members of the abelian groups

$\displaystyle\mathbf{I}_{m}(\mathbb{R}^{n})$
$\displaystyle:=\bigcup_{K\subseteq\mathbb{R}^{n}\text{
compact}}\mathbf{I}_{m,K}(\mathbb{R}^{n}),$
$\displaystyle\mathscr{F}_{m}(\mathbb{R}^{n}):=\bigcup_{K\subseteq\mathbb{R}^{n}\text{
compact}}\mathscr{F}_{m,K}(\mathbb{R}^{n}),$ (12)
$\displaystyle\mathbf{N}_{m}(\mathbb{R}^{n})$
$\displaystyle:=\bigcup_{K\subseteq\mathbb{R}^{n}\text{
compact}}\mathbf{N}_{m,K}(\mathbb{R}^{n}),$
$\displaystyle\mathbf{F}_{m}(\mathbb{R}^{n}):=\bigcup_{K\subseteq\mathbb{R}^{n}\text{
compact}}\mathbf{F}_{m,K}(\mathbb{R}^{n})$ (13)

are called $m$-dimensional integral currents, integral flat chains, normal
currents, and flat chains in $\mathbb{R}^{n}$, respectively. One can show the
following set-theoretic relations between the classes of currents that we have
defined so far:

${\mathbf{I}_{m}(\mathbb{R}^{n})}$${\mathscr{R}_{m}(\mathbb{R}^{n})}$${\mathscr{F}_{m}(\mathbb{R}^{n})}$${\mathbf{N}_{m}(\mathbb{R}^{n})}$${\\{T\in\mathbf{F}_{m}(\mathbb{R}^{n}):\mathbf{M}(T)<\infty\\}}$${\mathbf{F}_{m}(\mathbb{R}^{n})}$${\mathscr{D}_{m}(\mathbb{R}^{n}).}$${\subseteq}$${\subseteq}$${\subseteq}$${\subseteq}$${\subseteq}$${\subseteq}$${\subseteq}$${\subseteq}$

######  Remark 2.3.

The flat norm endows $\mathscr{D}_{m}(\mathbb{R}^{n})$ with a metric defined
by $(T,T^{\prime})\mapsto\mathbf{F}_{K}(T-T^{\prime})$. By [7, 4.2.18.], one
can metrize integral flat chains $\mathscr{F}_{m,K}(\mathbb{R}^{n})$ with

$\mathscr{F}_{K}(T):=\min\\{\mathbf{M}(R)+\mathbf{M}(S):R\in\mathscr{R}_{m,K}(\mathbb{R}^{n}),\text{
and }\ S\in\mathscr{R}_{m+1,K}(\mathbb{R}^{n})\text{ with }T=R+\partial S\\}.$
(14)

######  Remark 2.4.

For any sequence
$\\{T_{i}\\}_{i=1}^{\infty}\subseteq\mathscr{D}_{m}(\mathbb{R}^{n})$ of
currents, and a current $T\in\mathscr{D}_{m}(\mathbb{R}^{n})$, if
$\lim_{i\to\infty}\mathbf{F}_{K}(T_{i}-T)=0$ for some compact subset $K$ of
$\mathbb{R}^{n}$, then $T_{i}\rightharpoonup T$. Conversely, if
$\\{T_{i}\\}_{i=1}^{\infty}\subseteq\mathbf{I}_{m}(\mathbb{R}^{n})$ is a
sequence of integral currents with
$\sup_{i}\mathbf{M}(T_{i})+\mathbf{M}(\partial T_{i})<\infty$, then
$T_{i}\rightharpoonup T$ implies $\lim_{i\to\infty}\mathbf{F}_{K}(T_{i}-T)=0$
for all compact subsets $K$ of $\mathbb{R}^{n}$. We refer the reader to Leon
Simon’s book on geometric measure theory [17, Theorem 31.2] for a proof of
this fact.

##  3 Partial Plateau’s problem with mass

As in [15], for any two $k$ dimensional currents $A$ and $B$ of finite mass,
we say that $A$ is a subcurrent of $B$, denoted by $A\preceq B$, if

$\mathbf{M}(B)=\mathbf{M}(B-A)+\mathbf{M}(A).$ (15)

Notice that the zero-current $\boldsymbol{0}$ is always a subcurrent of $B$
and that $\mathbf{M}(A)\leq\mathbf{M}(B)$ whenever $A\preceq B$. In
particular, $A\preceq{\bf 0}$ if and only if $A={\bf 0}$.

######  Remark 3.1.

Let $B=\mathscr{H}^{m}\,\hbox{\vrule height=7.0pt,depth=0.0pt\vrule
height=0.4pt,depth=0.0pt,width=6.0pt}\,M\wedge\theta\xi$ be a real-rectifiable
current. If $A$ is any $m$-dimensional subcurrent of $B$, Paolini and Stepanov
[15, Lemma 3.7] showed that $A=\mathscr{H}^{m}\,\hbox{\vrule
height=7.0pt,depth=0.0pt\vrule
height=0.4pt,depth=0.0pt,width=6.0pt}\,M\wedge\lambda\theta\xi$ for some Borel
function $\lambda:\mathbb{R}^{n}\to[0,1]$. In particular, $A$ is a real-
rectifiable current with $\mathrm{spt}(A)\subseteq\mathrm{spt}(B)$.

######  Lemma 3.2.

Let $B$ be a $k$ dimensional current and $\\{A_{i}\\}_{i=1}^{\infty}$ be a
sequence of $k$ dimensional currents such that $A_{i}\preceq B$ for all
$i\in\mathbb{N}$. If $A$ is an $k$ dimensional current such that
$A_{i}\rightharpoonup A$, then $A\preceq B$.

###### Proof.

Since $\mathbf{M}$ satisfies the triangle inequality, it is sufficient to show
that $\mathbf{M}(B-A)+\mathbf{M}(A)\leq\mathbf{M}(B)$. By lower-semicontinuity
of $\mathbf{M}$ with respect to weak convergence,

$\mathbf{M}(B-A)\leq\liminf_{i\to\infty}\mathbf{M}(B-A_{i})\quad\text{ and
}\quad\mathbf{M}(A)\leq\liminf_{i\to\infty}\mathbf{M}(A_{i}).$

Therefore since $A_{i}\preceq B$ for all $i\in\mathbb{N}$,

$\displaystyle\mathbf{M}(B-A)+\mathbf{M}(A)$
$\displaystyle\leq\liminf_{i\to\infty}\mathbf{M}(B-A_{i})+\liminf_{i\to\infty}\mathbf{M}(A_{i})$
$\displaystyle\leq\liminf_{i\to\infty}\left(\mathbf{M}(B-A_{i})+\mathbf{M}(A_{i})\right)\leq\mathbf{M}(B).$

∎

######  Remark 3.3.

We mention that Lemma 3.2 holds true if we replace $A_{i}\rightharpoonup A$ by
$\mathbf{F}_{K}(A_{i}-A)\rightarrow 0$ for some compact subset $K$ of
$\mathbb{R}^{n}$. This is because by Remark 2.4,
$\mathbf{F}_{K}(A_{i}-A)\rightarrow 0$ implies $A_{i}\rightharpoonup A$.

We are interested in what we call, partial Plateau’s problem: Given a flat
chain $B\in\mathbf{F}_{m-1}(\mathbb{R}^{n})$ with mass $\mathbf{M}(B)<\infty$,
a smooth compactly supported form $\Phi\in\mathscr{D}^{m-1}(\mathbb{R}^{n})$,
and a rectifiable current $T_{0}\in\mathscr{R}_{m}(\mathbb{R}^{n})$, consider

$\mathcal{P}_{(\mathbf{E}_{\Phi},\preceq
B,T_{0},\mathbb{R}^{n})}\begin{cases}\text{minimize }\mathbf{E}_{\Phi}(T)\\\
\text{among }T\in\mathscr{R}_{m}(\mathbb{R}^{n})\text{ such that
}\partial(T-T_{0})\preceq B,\end{cases}$

where

$\mathbf{E}_{\Phi}(T):=\mathbf{M}(T)-\partial T(\Phi).$ (16)

######  Remark 3.4.

Note the following:

  * •

When $B={\bf 0}$, $\partial(T-T_{0})\preceq\boldsymbol{0}$ implies $\partial
T=\partial T_{0}$. Thus, up to a constant, this problem corresponds to the
standard Plateau’s problem of rectifiable currents: Minimize
$\mathbf{E}_{\Phi}(T):=\mathbf{M}(T)-\partial T_{0}(\Phi)$ among all
$T\in\mathscr{R}_{m}(\mathbb{R}^{n})$ with $\partial T=\partial T_{0}$.

  * •

The introduction of $T_{0}$ makes this a slightly more general problem than
$\mathcal{P}_{(\mathbf{E}_{\Phi},\,\preceq B,\,\mathbb{R}^{n})}$, given in
(3). When $T_{0}={\bf 0}$, the boundary of $T$ is a subcurrent of $B$.

  * •

We require $\mathbf{M}(B)<\infty$ but do not require $\mathbf{M}(\partial
T_{0})<\infty$. Thus for any admissible $T$, the mass of its boundary
$\mathbf{M}(\partial T)$ may be unbounded.

We now state the existence result of the partial Plateau’s problem.

######  Theorem 3.5.

Let $\Phi\in\mathscr{D}^{m-1}(\mathbb{R}^{n})$ and
$B\in\mathbf{F}_{m-1}(\mathbb{R}^{n})$ with $\mathbf{M}(B)<\infty$. Given
$T_{0}\in\mathscr{R}_{m}(\mathbb{R}^{n})$, there exists a rectifiable current
$T^{\ast}\in\mathscr{R}_{m}(\mathbb{R}^{n})$ with
$\partial(T^{\ast}-T_{0})\preceq B$ such that

$\mathbf{E}_{\Phi}(T^{\ast})=\min\\{\mathbf{E}_{\Phi}(T):T\in\mathscr{R}_{m}(\mathbb{R}^{n}),\partial(T-T_{0})\preceq
B\\}.$

###### Proof.

Notice that, since the zero-current ${\bf 0}\preceq B$, this class contains
$T_{0}$. By definition, both $T_{0}$ and $B$ have compact support. Let $K$ be
a compact convex subset of $\mathbb{R}^{n}$ containing
$\mathrm{spt}(T_{0})\cup\mathrm{spt}(B)$ and let
$\pi_{K}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ denote the nearest point projection
onto $K$ [7, 4.1.15].

For any $T\in\mathscr{R}_{m}(\mathbb{R}^{n})$ with $\partial(T-T_{0})\preceq
B$, it holds that
$\mathbf{E}_{\Phi}((\pi_{K})_{\\#}T)\leq\mathbf{E}_{\Phi}(T)$. Indeed, observe
that $\mathrm{spt}(\partial T)\subseteq K$ because $\partial
T=\partial(T-T_{0})+\partial T_{0}$,
$\mathrm{spt}(\partial(T-T_{0}))\subseteq\mathrm{spt}(B)$ (see Remark 3.1),
and $\mathrm{spt}(T_{0})\cup\mathrm{spt}(B)\subseteq K$. Thus,
$\partial\left((\pi_{K})_{\\#}T\right)=(\pi_{K})_{\\#}\partial T=\partial T$.
Since also $\mathbf{M}((\pi_{K})_{\\#}T)\leq\mathbf{M}(T)$, it follows that

$\mathbf{E}_{\Phi}((\pi_{K})_{\\#}T)=\mathbf{M}((\pi_{K})_{\\#}T)-\partial((\pi_{K})_{\\#}T)(\Phi)\leq\mathbf{M}(T)-\partial
T(\Phi)=\mathbf{E}_{\Phi}(T).$ (17)

Let $\\{T_{i}\\}_{i=1}^{\infty}\subseteq\mathscr{R}_{m}(\mathbb{R}^{n})$ be an
$\mathbf{E}_{\Phi}$-minimizing sequence with $\partial(T_{i}-T_{0})\preceq B$.
Thus, for each $i$,

$\mathbf{M}(\partial(T_{i}-T_{0}))\leq\mathbf{M}(B)<\infty.$

By (17), without loss of generality we may assume
$\mathrm{spt}(T_{i})\subseteq K$ and
$\mathbf{E}_{\Phi}(T_{i})\leq\mathbf{E}_{\Phi}(T_{0})$. Then,

$\mathbf{M}(T_{i})=\mathbf{E}_{\Phi}(T_{i})+\partial
T_{i}(\Phi)\leq\mathbf{E}_{\Phi}(T_{0})+\partial
T_{i}(\Phi)=\mathbf{M}(T_{0})+\partial(T_{i}-T_{0})(\Phi)\leq\mathbf{M}(T_{0})+\mathbf{M}(B)||\Phi||,$

where $||\Phi||$ is the comass norm of $\Phi$, and

$\mathbf{M}(T_{i}-T_{0})\leq\mathbf{M}(T_{i})+\mathbf{M}(T_{0})\leq
2\mathbf{M}(T_{0})+\mathbf{M}(B)||\Phi||<\infty.$

Thus,
$\\{T_{i}-T_{0}\\}\subseteq\mathbf{I}_{m,K}(\mathbb{R}^{n})\cap\\{T:\mathbf{N}(T)<c\\}$
for some large enough $c>0$. By the Federer and Fleming’s compactness theorem
for integral currents [7, 4.2.17(b)], a subsequence of $\\{T_{i}-T_{0}\\}$,
still denoted by $\\{T_{i}-T_{0}\\}$, $\mathscr{F}_{K}$ converges to some
$R\in\mathbf{I}_{m,K}(\mathbb{R}^{n})$. Thus, $T_{i}-T_{0}\rightharpoonup R$
and $T_{i}\rightharpoonup
T^{\ast}:=R+T_{0}\in\mathscr{R}_{m}(\mathbb{R}^{n})$. Since
$\partial(T_{i}-T_{0})\preceq B$, and
$\partial(T_{i}-T_{0})\rightharpoonup\partial R=\partial(T^{\ast}-T_{0})$, by
Lemma 3.2, $\partial(T^{\ast}-T_{0})\preceq B$ as well. Also, by the lower-
semicontinuity of $\mathbf{M}$,

$\mathbf{E}_{\Phi}(T^{\ast})=\mathbf{M}(T^{\ast})-\partial
T^{\ast}(\Phi)\leq\liminf_{i\rightarrow\infty}\mathbf{M}(T_{i})-\partial
T_{i}(\Phi)=\liminf_{i\rightarrow\infty}\mathbf{E}_{\Phi}(T_{i}).$

Since $\\{T_{i}\\}$ is an $\mathbf{E}_{\Phi}$-minimizing sequence, we have
$T^{\ast}$ is the desired $\mathbf{E}_{\Phi}$-minimizer. ∎

######  Example 3.6.

We now investigate the sunflower example as shown in Figure 1 with more
detail. Let $B$ be the 1 dimensional rectifiable current as shown in Figure
2(A) with density 2 on the inner circle, and density 1 on the outer circular
arcs. Let $\llbracket P_{i}\rrbracket$ (for $i=1,\dots,8$) and $\llbracket
D\rrbracket$ denote the 2 dimensional rectifiable currents with clockwise
orientation which respectively represent the petals and the central disk of
the sunflower, as shown in Figure 2(B). Then any
$T\in\mathscr{R}_{2}(\mathbb{R}^{2})$ with $\partial T\preceq B$ has the form

$T=a\llbracket D\rrbracket+\sum_{i=1}^{8}c_{i}\llbracket P_{i}\rrbracket$ (18)

where $a$ and $c_{i}$ are integers such that

$0\leq c_{i}\leq 1\quad\text{ and }\quad 0\leq c_{i}-a\leq 2\quad\text{ for
each }i=1,2,\dots,8.$ (19)

Figure 2: The boundary $B$ is shown in $(A)$ along with its orientation and
density. The petals $\\{P_{i}\\}_{i=1}^{8}$ and central disk $D$ shown in
$(B)$ have a clockwise orientation. When $\Phi$ is given, the petals are
partitioned into the three families shown in (C): negative petals (shown in
green), neutral petals (shown in pink), and positive petals (shown in gray).

If $\Phi=0$, then clearly we get that the $0$-current $\boldsymbol{0}$ is the
unique minimizer. For any smooth, compactly supported $1$-form
$\Phi\in\mathscr{D}^{1}(\mathbb{R}^{2})$,

$\displaystyle\mathbf{E}_{\Phi}(T)$ $\displaystyle=\mathbf{M}(T)-\partial
T(\Phi)$ $\displaystyle=|a|\mathbf{M}(\llbracket
D\rrbracket)+\sum_{i=1}^{8}|c_{i}|\mathbf{M}(\llbracket
P_{i}\rrbracket)-\left(a\partial\llbracket
D\rrbracket(\Phi)+\sum_{i=1}^{8}c_{i}\partial\llbracket
P_{i}\rrbracket(\Phi)\right)$ $\displaystyle=|a|\mathbf{M}(\llbracket
D\rrbracket)-a\partial\llbracket
D\rrbracket(\Phi)+\sum_{i=1}^{8}c_{i}(\mathbf{M}(\llbracket
P_{i}\rrbracket)-\partial\llbracket P_{i}\rrbracket(\Phi)).$

We now investigate the possible solutions for
$\mathcal{P}_{(\mathbf{E}_{\Phi},\preceq B,\boldsymbol{0},\mathbb{R}^{2})}$.
First, partition the collection $\\{\llbracket P_{i}\rrbracket\\}$ of petals
into the following sets

$\displaystyle\mathcal{P}_{-}$ $\displaystyle=\\{\llbracket
P_{i}\rrbracket:\mathbf{M}(\llbracket P_{i}\rrbracket)-\partial\llbracket
P_{i}\rrbracket(\Phi)<0\\}\text{ of negative petals,}$
$\displaystyle\mathcal{P}_{0}$ $\displaystyle=\\{\llbracket
P_{i}\rrbracket:\mathbf{M}(\llbracket P_{i}\rrbracket)-\partial\llbracket
P_{i}\rrbracket(\Phi)=0\\}\text{ of neutral petals, and}$
$\displaystyle\mathcal{P}_{+}$ $\displaystyle=\\{\llbracket
P_{i}\rrbracket:\mathbf{M}(\llbracket P_{i}\rrbracket)-\partial\llbracket
P_{i}\rrbracket(\Phi)>0\\}\text{ of positive petals}.$

In Figure 2(C), the set of negative petals $\\{\llbracket
P_{1}\rrbracket,\llbracket P_{2}\rrbracket,\llbracket
P_{3}\rrbracket,\llbracket P_{4}\rrbracket\\}$ are colored green, the neutral
petals $\\{\llbracket P_{5}\rrbracket,\llbracket P_{6}\rrbracket\\}$ are
colored pink, and the positive petals $\\{\llbracket
P_{7}\rrbracket,\llbracket P_{8}\rrbracket\\}$ are colored gray.

Any possible solution $T$ for $\mathcal{P}_{(\mathbf{E}_{\Phi},\preceq
B,\boldsymbol{0},\mathbb{R}^{2})}$ must have the form described in (18). From
inequalities (19), we get that $a\in\\{-2,-1,0,1\\}$. We describe the possible
solutions:

  * •

When $a=-2$, all the $c_{i}$’s must be $0$. In this case, the only possible
solution is

$T_{-2}=-2\llbracket D\rrbracket$

as illustrated in Figure 3(A).

  * •

When $a=-1$ all $c_{i}$ are allowed to be either $0$ or $1$ for any admissible
candidate. However, for a solution $T_{-1}$ with $a=-1$, the coefficients
$c_{i}$ must be $1$ on the negative petals and $0$ on the positive petals. On
the neutral petals, $c_{i}$ may be chosen to be either $0$ or $1$. This is
because removing any positive petal or adding any negative petal from a
candidate minimizer can only decrease $\mathbf{E}_{\Phi}$, while removing or
adding a neutral petal does not change the value of $\mathbf{E}_{\Phi}$. That
is, any possible solution must be in the form

$T_{-1}=-\llbracket D\rrbracket+\sum_{P_{i}\in\mathcal{P}_{-}}\llbracket
P_{i}\rrbracket+\sum_{P_{i}\in\mathcal{P}_{0}^{\ast}}\llbracket
P_{i}\rrbracket\quad\text{ for any
}\mathcal{P}_{0}^{\ast}\subseteq\mathcal{P}_{0},$

as illustrated in Figure 3(B).

  * •

When $a=0$, for the same reason as in the case of $a=-1$, any possible
solution must be in the form

$T_{0}=\sum_{P_{i}\in\mathcal{P}_{-}}\llbracket
P_{i}\rrbracket+\sum_{P_{i}\in\tilde{\mathcal{P}_{0}}}\llbracket
P_{i}\rrbracket\quad\text{ for any
}\tilde{\mathcal{P}_{0}}\subseteq\mathcal{P}_{0},$

as illustrated in Figure 3(C).

  * •

When $a=1$, $c_{i}=1$ for all $i=1,\dots,8$. Hence, the only possible solution
is

$T_{1}=\llbracket D\rrbracket+\sum_{i=1}^{8}\llbracket P_{i}\rrbracket,$

as illustrated in Figure 3(D).

Figure 3: From left to right, we show the four possible solutions
$T_{-2},T_{-1},T_{0},T_{1}$ for $\mathcal{P}_{(\mathbf{E}_{\Phi},\preceq
B,\boldsymbol{0},\mathbb{R}^{2})}$ with boundary $B$ and the 1-form $\Phi$ as
shown in Figure 2.

So far, we have described the possible solutions of
$\mathcal{P}_{(\mathbf{E}_{\Phi},\preceq B,\boldsymbol{0},\mathbb{R}^{2})}$ by
only taking into account the subcurrent restrictions on their boundaries and
the way that the negative, neutral, and positive petals change the value of
$\mathbf{E}_{\Phi}$. We will now characterize the solutions for
$\mathcal{P}_{(\mathbf{E}_{\Phi},\preceq B,\boldsymbol{0},\mathbb{R}^{2})}$ in
terms of how $\partial\llbracket D\rrbracket$ acts on $\Phi$. First note the
following values of $\mathbf{E}_{\Phi}(T_{i})$ for $i=-2,-1,0,1$:

$\displaystyle\mathbf{E}_{\Phi}(T_{-2})=2\mathbf{M}(\llbracket
D\rrbracket)+2\partial\llbracket D\rrbracket(\Phi)$
$\displaystyle\mathbf{E}_{\Phi}(T_{-1})=\mathbf{M}(\llbracket
D\rrbracket)+\partial\llbracket
D\rrbracket(\Phi)+\sum_{P_{i}\in\mathcal{P}_{-}}(\mathbf{M}(\llbracket
P_{i}\rrbracket)-\partial\llbracket P_{i}\rrbracket(\Phi))$
$\displaystyle\mathbf{E}_{\Phi}(T_{0})=\sum_{P_{i}\in\mathcal{P}_{-}}(\mathbf{M}(\llbracket
P_{i}\rrbracket)-\partial\llbracket P_{i}\rrbracket(\Phi))$
$\displaystyle\mathbf{E}_{\Phi}(T_{1})=\mathbf{M}(\llbracket
D\rrbracket)-\partial\llbracket
D\rrbracket(\Phi)+\sum_{P_{i}\in\mathcal{P}_{-}}(\mathbf{M}(\llbracket
P_{i}\rrbracket)-\partial\llbracket
P_{i}\rrbracket(\Phi))+\sum_{P_{i}\in\mathcal{P}_{+}}(\mathbf{M}(\llbracket
P_{i}\rrbracket)-\partial\llbracket P_{i}\rrbracket(\Phi)).$

Thus,

  * •

$\mathbf{E}_{\Phi}(T_{-2})\leq\mathbf{E}_{\Phi}(T_{-1})$ whenever

$\partial\llbracket D\rrbracket(\Phi)\leq-\mathbf{M}(\llbracket
D\rrbracket)+\sum_{P_{i}\in\mathcal{P}_{-}}(\mathbf{M}(\llbracket
P_{i}\rrbracket)-\partial\llbracket P_{i}\rrbracket(\Phi))=:\lambda_{-2},$

  * •

$\mathbf{E}_{\Phi}(T_{-1})\leq\mathbf{E}_{\Phi}(T_{0})$ whenever

$\partial\llbracket D\rrbracket(\Phi)\leq-\mathbf{M}(\llbracket
D\rrbracket)=:\lambda_{-1},\text{ and }$

  * •

$\mathbf{E}_{\Phi}(T_{0})\leq\mathbf{E}_{\Phi}(T_{1})$ whenever

$\partial\llbracket D\rrbracket(\Phi)\leq\mathbf{M}(\llbracket
D\rrbracket)+\sum_{P_{i}\in\mathcal{P}_{+}}\left(\mathbf{M}(\llbracket
P_{i}\rrbracket)-\partial\llbracket P_{i}\rrbracket(\Phi)\right)=:\lambda_{0}$

Notice also that we have the following inequalities

$\lambda_{-2}\leq\lambda_{-1}\leq\lambda_{0}.$

As a result,

  * •

$T_{-2}$ is a solution when $\partial\llbracket
D\rrbracket(\Phi)\leq\lambda_{-2}$,

  * •

$T_{-1}$ is a solution when $\partial\llbracket
D\rrbracket(\Phi)\in[\lambda_{-2},\lambda_{-1}]$,

  * •

$T_{0}$ is a solution when $\partial\llbracket
D\rrbracket(\Phi)\in[\lambda_{-1},\lambda_{0}]$, and

  * •

$T_{1}$ is a solution when $\partial\llbracket
D\rrbracket(\Phi)\geq\lambda_{0}$.

In addition, the solution is unique whenever $\partial\llbracket
D\rrbracket(\Phi)\notin\\{\lambda_{-2},\lambda_{-1},\lambda_{0},\lambda_{1}\\}$.

Figure 4: Given the boundary $\tilde{B}$ as shown in $(A)$, the three possible
solutions $T_{-2},T_{-1},T_{0}$ for
$\mathcal{P}_{(\mathbf{E}_{\Phi},\preceq\tilde{B},\boldsymbol{0},\mathbb{R}^{2})}$
are shown in $(B)$, $(C)$, and $(D)$, respectively.

######  Example 3.7.

Let $\tilde{B}$ be the $1$ dimensional rectifiable current shown in Figure
4(A). As in Example 3.6, $T_{-2}$ is a solution when $\partial\llbracket
D\rrbracket(\Phi)\leq\lambda_{-2}$, $T_{-1}$ is a solution whenever
$\partial\llbracket D\rrbracket(\Phi)\in[\lambda_{-2},\lambda_{-1}]$, and
$T_{0}$ is a solution when $\partial\llbracket
D\rrbracket(\Phi)\geq\lambda_{-1}$. However, $T_{1}$ cannot be a solution for
any $\Phi$ since its boundary $\partial T_{1}$ is no longer a subcurrent of
$\tilde{B}$.

##  4 Plateau’s problem with $H$-mass for scans with boundary

Building upon the work of De Pauw and Hardt [5], in this section, we further
investigate Plateau’s problem with $H$-mass. We define a subfamily of
measurable maps from the space of all $n-m$ planes in $\mathbb{R}^{n}$ to the
space of 0 dimensional integral currents, called scans with boundary and
modify the definition of the $H$-mass to prove the existence of a solution of
Plateau’s problem with $H$-mass for scans with boundary in Theorem 4.18. The
definitions and techniques from this section will also be used when we study
partial Plateau’s problem in §5.

###  4.1 The $H$-mass and the $H$-flat distance

We now list a few concepts and results from [5] that are particularly useful
for the analysis in this article. We start by defining the $H$-mass of real
rectifiable currents with finite size and by defining the $H$-flat distance on
integral flat chains.

######  Definition 4.1 ([5, Definition 3.2.1.]).

A function $H:[0,\infty)\to[0,\infty)$ is called a concave integrand if it
satisfies the following conditions:

  1. 1.

$H(0)=0$;

  2. 2.

$H(1)=1$;

  3. 3.

$H(\theta_{1})<H(\theta_{2})$ whenever $0\leq\theta_{1}<\theta_{2}$;

  4. 4.

$H(\theta_{1}+\theta_{2})\leq H(\theta_{1})+H(\theta_{2})$ whenever
$\theta_{1},\theta_{2}\geq 0$;

  5. 5.

$\lim_{\theta\to\infty}H(\theta)=\infty$.

######  Definition 4.2 ([5, Definition 3.2.2.]).

For a concave integrand $H$ and a real rectifiable current
$T\in\mathscr{D}_{m}(\mathbb{R}^{n})$ with $\mathbf{S}(T)<\infty$, the
$H$-mass of $T$ is defined as

$\mathbf{M}_{H}(T):=\int_{\mathbb{R}^{n}}H(\Theta^{m}(\|T\|,x))\,d\mathcal{H}^{m}(x).$
(20)

######  Definition 4.3 ([5, Definition 3.2.8.]).

For a compact subset $K$ of $\mathbb{R}^{n}$, a concave integrand $H$, and an
integral flat chain $T\in\mathscr{F}_{m,K}(\mathbb{R}^{n})$ we define

$\mathscr{F}^{H}_{K}(T):=\inf\\{\mathbf{M}_{H}(T-\partial
S)+\mathbf{M}_{H}(S):S\in\mathscr{R}_{m+1,K}(\mathbb{R}^{n}),\ T-\partial
S\in\mathscr{R}_{m,K}(\mathbb{R}^{n})\\}.$ (21)

As in [7, 1.7.4, 2.7.16], let $\mathbf{O}(n)$ be the orthogonal group of
$\mathbb{R}^{n}$ and $\mathbf{O}^{\ast}(n,k)$ be the space of all orthogonal
projections of $\mathbb{R}^{n}$ to $\mathbb{R}^{k}$ with its $\mathbf{O}(n)$
invariant measure $\theta^{\ast}_{n,k}$. In [5, Proposition 3.1.3. and
3.1.5.], it shows that any flat chain $T\in\mathbf{F}_{m,K}(\mathbb{R}^{n})$
and $k\in\\{1,\dots,m\\}$ can be represented by the
$\boldsymbol{\theta}^{\ast}_{n,k}\times\mathcal{L}^{k}$ measurable map

$\mathcal{S}(T):\mathbf{O}^{\ast}(n,k)\times\mathbb{R}^{k}\to\mathbf{F}_{m-k,K}(\mathbb{R}^{n}),\quad\mathcal{S}(T)(p,y):=\langle
T,p,y\rangle,$ (22)

defined for every $p\in\mathbf{O}^{\ast}(n,k)$ and for $\mathcal{L}^{k}$
almost every $y\in\mathbb{R}^{k}$. The $k$ dimensional flat chain $\langle
T,p,y\rangle$ is obtained by slicing $T$ with $p^{-1}(y)$ under the orthogonal
projection map $p$ for $y\in\mathbb{R}^{m}$. For the remainder of the paper,
the map $\mathcal{S}(T)$ is to be defined as in (22) for $k=m$.

Let $T\in\mathscr{R}_{m,K}(\mathbb{R}^{n})$. For $\mathcal{L}^{k}$ almost
every $y\in\mathbb{R}^{m}$, $\langle
T,p,y\rangle\in\mathbf{I}_{0,K}(\mathbb{R}^{n})$ [7, 4.3.6.], and hence
$\mathcal{S}(T):\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}\to\mathbf{I}_{0,K}(\mathbb{R}^{n})$.
In addition, as in [5, (17)], the integral-geometric identity says that its
mass can be given by integrating the mass of its $0$ dimensional slices:

$\mathbf{M}(T)=\frac{1}{\boldsymbol{\beta}_{1}(m,n)}\int_{\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}}\mathbf{M}(\mathcal{S}(T)(p,y))\,d(\mathbf{\theta}_{n,m}^{\ast}\times\mathcal{L}^{m})(p,y),$

where $\boldsymbol{\beta}_{1}(m,n)$ is the constant given in [7, 2.7.16]. By
[5, (21)], we also have a similar equality for the $H$-mass of a rectifiable
current $T\in\mathscr{R}_{m}(\mathbb{R}^{n})$:

$\mathbf{M}_{H}(T)=\frac{1}{\boldsymbol{\beta}_{1}(m,n)}\int_{\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}}\mathbf{M}_{H}(\mathcal{S}(T)(p,y))\,d(\mathbf{\theta}_{n,m}^{\ast}\times\mathcal{L}^{m})(p,y).$
(23)

Motivated by (23), De Pauw and Hardt [5] defined the $H$ mass of a rectifiable
scan. They also introduced the concept of a scan cycle from the observation
that a flat chain $T\in\mathbf{F}_{m}(\mathbb{R}^{n})$ having boundary zero is
equivalent to a corresponding condition on its zero dimensional slices [5,
Proposition 3.1.6]. Here, we extend their definitions to measurable maps from
$\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}$ to
$\mathbf{I}_{0,K}(\mathbb{R}^{n})$ in the following definition.

######  Definition 4.4.

For any $\mathbf{\theta}_{n,m}^{\ast}\times\mathcal{L}^{m}$ measurable map
$f:\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}\to\mathbf{I}_{0,K}(\mathbb{R}^{n})$,
we define the $H$-mass of $f$ to be

$\mathbf{M}_{H}(f):=\frac{1}{\boldsymbol{\beta}_{1}(m,n)}\int_{\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}}\mathbf{M}_{H}(f(p,y))\,d(\mathbf{\theta}_{n,m}^{\ast}\times\mathcal{L}^{m})(p,y)$
(24)

and define the variation of $f$ as

$\mathbf{D}(f):=\sup\\{\int_{\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}}|D(\phi\circ
f)|:\ \phi:\mathbf{I}_{0,K}(\mathbb{R}^{n})\to\mathbb{R}\text{ with
}\mathrm{Lip}(\phi)\leq 1\\}.$ (25)

Moreover we say that $f$ is a cycle, and write $\boldsymbol{\partial}f=0$, if
for $\mathbf{\theta}_{n,m}^{\ast}\times\mathcal{L}^{m}$ almost every
$(p,y)\in\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}$,

$f(p,y)(\boldsymbol{1})=0,$

where $\boldsymbol{1}$ denotes the constant function $\boldsymbol{1}(x)=1$ on
$\mathbb{R}^{n}$.

From equations (23) and (24), we can see that for any rectifiable current
$T\in\mathscr{R}_{m}(\mathbb{R}^{n})$,

$\mathbf{M}_{H}(T)=\mathbf{M}_{H}(\mathcal{S}(T)).$ (26)

For any compact $K\subseteq\mathbb{R}^{n}$, we will denote the class of
measurable maps obtained by rectifiable currents in
$\mathcal{R}_{m,K}(\mathbb{R}^{n})$ as
$\mathcal{S}_{m,K}(\mathbb{R}^{n}):=\\{\mathcal{S}(T):T\in\mathcal{R}_{m,K}(\mathbb{R}^{n})\\}$
and let

$\mathcal{S}_{m}(\mathbb{R}^{n}):=\bigcup_{K\subseteq\mathbb{R}^{n}\text{
compact}}\mathcal{S}_{m,K}(\mathbb{R}^{n}).$ (27)

We now recall one of the most important theorems from [5], and then state a
direct corollary that we will use in the proofs of Theorems 4.18 and 5.1.

######  Theorem 4.5 ([5, Theorem 3.4.1]).

Suppose $X$ is a $k$ dimensional Riemannian manifold, $Y$ is a weakly
separable metric space, $M:Y\to\mathbb{R}^{+}$ is lower semicontinuous, and
$M^{-1}([0,R])$ is sequentially compact in $Y$ for all $R>0$. If $f_{j}:X\to
Y$ is measurable, and

$\int_{X}M(f_{j}(x))\,d\mathcal{H}^{k}x+\int_{X}|D(\phi\circ
f_{j})|\leq\Lambda<\infty,$

for all $j=1,2,\dots$ and maps $\phi:Y\to\mathbb{R}$ with
$\mathrm{Lip}(\phi)\leq 1$, then some subsequence $f_{j^{\ast}}$ converges
pointwise $\mathcal{H}^{k}$ a.e. to a function $f:X\to Y$ with

$\int_{X}M(f(x))\,d\mathcal{H}^{k}x+\int_{X}|D(\phi\circ f)|\leq\Lambda$

for all such $\phi$.

######  Corollary 4.6.

Let
$f_{j}:\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}\to\mathbf{I}_{0,K}(\mathbb{R}^{n})$
be a sequence of $\boldsymbol{\theta}^{\ast}_{n,m}\times\mathcal{L}^{m}$
measurable maps with

$\mathbf{M}_{H}(f_{j})+\mathbf{D}(f_{j})\leq\Lambda<\infty,$

for all $j=1,2,\dots$. Then there exists a subsequence $(f_{j_{k}})$ of
$(f_{j})$ and an $\boldsymbol{\theta}^{\ast}_{n,m}\times\mathcal{L}^{m}$
measurable map
$f:\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}\to\mathbf{I}_{0,K}(\mathbb{R}^{n})$
such that

$\mathbf{M}_{H}(f)+\mathbf{D}(f)\leq\Lambda,$

and $(f_{j_{k}})$ converges pointwise a.e. to $f$ in the sense that for
$\boldsymbol{\theta}^{\ast}_{n,m}\times\mathcal{L}^{m}$ almost every
$(p,y)\in\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}$,

$\mathscr{F}_{K}^{H}[f_{j_{k}}(p,y)-f(p,y)]\to 0\text{ as }k\to\infty.$

###### Proof.

This is a special case of [5, Theorem 3.4.1] with

$\displaystyle X$ $\displaystyle=\mathrm{the\ Riemannian\ manifold}\
\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}$ $\displaystyle Y$
$\displaystyle=\mathbf{I}_{0,K}(\mathbb{R}^{n})\ \mathrm{with}\
\mathrm{dist}_{Y}(T,\tilde{T})=\mathscr{F}_{K}^{H}(T-\tilde{T})$
$\displaystyle M$ $\displaystyle=\mathbf{M}_{H}.$

∎

###  4.2 Existence of $H$-mass minimizing scans

In this subsection, we amend the seminal work [5, Theorem 3.5.2.] of De Pauw
and Hardt, who consider the following Plateau-type problem: Given
$T_{0}\in\mathscr{R}_{m}(\mathbb{R}^{n})$,

$\begin{cases}\text{minimize }\mathbf{M}_{H}(T)\\\ \text{among
}T\in\mathscr{R}_{m}(\mathbb{R}^{n})\text{ such that }\partial T=\partial
T_{0},\end{cases}$

where $\mathbf{M}_{H}(T)$ denotes the $H$-mass of
$T\in\mathscr{R}_{m}(\mathbb{R}^{n})$ as given in (20). Let
$\Gamma(\mathbf{M}_{H},T_{0},\mathbb{R}^{n})$ be the infimum of that problem:

$\Gamma(\mathbf{M}_{H},T_{0},\mathbb{R}^{n}):=\inf\\{\mathbf{M}_{H}(T):T\in\mathscr{R}_{m}(\mathbb{R}^{n})\text{
and }\partial T=\partial T_{0}\\}.$ (28)

For the convenience of the reader, we now state [5, Theorem 3.5.2.] of De Pauw
and Hardt.

######  Theorem ([5, Theorem 3.5.2.]).

Let $T_{0}\in\mathscr{R}_{m}(\mathbb{R}^{n})$ with
$\mathcal{I}_{1}^{m}(\mathrm{spt}(\partial T_{0}))=0$ and let $H$ be a concave
integrand. Then there exists an $m$-dimensional rectifiable scan $\mathcal{T}$
in $\mathbb{R}^{n}$ such that
$\boldsymbol{\partial}(\mathcal{T}-\mathcal{S}(T_{0}))=0$ and
$\mathbf{M}_{H}(\mathcal{T})=\Gamma(\mathbf{M}_{H},T_{0},\mathbb{R}^{n})$.
Moreover, if $\mathrm{spt}(\partial T_{0})$ is an $m-1$ dimensional compact
propertly embedded $C^{1,1}$ submanifold then there exists
$T\in\mathbf{I}_{m}(\mathbb{R}^{n})$ with $\partial T=\partial T_{0}$ and
$\mathbf{M}_{H}(T)=\Gamma(\mathbf{M}_{H},T_{0},\mathbb{R}^{n})$.

Before we discuss their proof, with the notation in (27) and by equation (26),
observe that (28) has the following equivalent formulation:

$\Gamma(\mathbf{M}_{H},T_{0},\mathbb{R}^{n})=\inf\\{\mathbf{M}_{H}(\mathcal{R}):\mathcal{R}\in\mathcal{S}_{m}(\mathbb{R}^{n})\text{
and }\boldsymbol{\partial}(\mathcal{R}-\mathcal{S}(T_{0}))=0\\}.$

In other words, $\mathcal{S}_{m}(\mathbb{R}^{n})$ is the class of objects that
[5] minimizes $\mathbf{M}_{H}$ over.

Now, in the proof, De Pauw and Hardt construct a sequence of $\mathbf{M}_{H}$
minimizing rectifiable currents, and use their BV compactness theorem [5,
Theorem 3.4.1.] to obtain a subsequence $\\{T_{j}\\}$ such that
$\mathcal{S}(T_{j})$ converges pointwise almost everywhere to a rectifiable
scan $\mathcal{T}$. By the lower-semicontinuity of $\mathbf{M}_{H}$ for $0$
dimensional integral currents (under the $H$-flat distance), and by Fatou’s
Lemma, it follows that

$\mathbf{M}_{H}(\mathcal{T})\leq\Gamma(\mathbf{M}_{H},T_{0},\mathbb{R}^{n}).$

However, since the rectifiable scan $\mathcal{T}$ is not necessarily a
rectifiable current, we cannot conclude that
$\mathcal{T}\in\mathcal{S}_{m}(\mathbb{R}^{n})$, and in turn, can neither
conclude that

$\mathbf{M}_{H}(\mathcal{T})=\Gamma(\mathbf{M}_{H},T_{0},\mathbb{R}^{n}).$

To overcome this issue (the lack of closedness of
$\mathcal{S}_{m}(\mathbb{R}^{n})$ under pointwise almost everywhere
convergence), we introduce a modified class of objects that we will minimize
over, denoted by ${\it Scan}_{m}(\mathbb{R}^{n})$, as well as a modified
version of the $H$-mass.

######  Definition 4.7.

Whenever $K$ is a compact subset of $\mathbb{R}^{n}$, we say that an
$\boldsymbol{\theta}^{\ast}_{n,m}\times\mathcal{L}^{m}$ measurable map
$\mathcal{T}:\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}\to\mathscr{F}_{0,K}(\mathbb{R}^{n})$
is an $m$-dimensional scan in $K$ if it is in the closure of
$\mathcal{S}_{m,K}(\mathbb{R}^{n})$. That is, if there exists a sequence of
rectifiable currents $T_{1},T_{2},\dots\in\mathscr{R}_{m,K}(\mathbb{R}^{n})$
such that $\mathcal{S}(T_{i})\to\mathcal{T}$ pointwise almost everywhere, in
the sense that

$\lim_{i\to\infty}\mathscr{F}_{K}^{H}[\langle
T_{i},p,y\rangle-\mathcal{T}(p,y)]=0\text{ for
}\boldsymbol{\theta}^{\ast}_{n,m}\times\mathcal{L}^{m}\text{ almost every
}(p,y)\in\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}.$ (29)

We define

${\it Scan}_{m,K}(\mathbb{R}^{n})$

to be the collection of all $m$-dimensional scans in $K$ such that there
exists a rectifiable current $T\in\mathscr{R}_{m,K}(\mathbb{R}^{n})$ with
$\boldsymbol{\partial}(\mathcal{T}-\mathcal{S}(T))=0$.111By Definition 4.4,
$\boldsymbol{\partial}(\mathcal{T}-\mathcal{S}(T))=0$ means that
$(\mathcal{T}-\mathcal{S}(T))(p,y)(\boldsymbol{1})=0$ for
$\theta^{\ast}_{n,m}\times\mathcal{L}^{m}$ almost every
$(p,y)\in\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}$. Members of ${\it
Scan}_{m,K}(\mathbb{R}^{n})$ are called $m$-dimensional scans in $K$ with
boundary.

Let us now investigate a couple of facts concerning the classes ${\it
Scan}_{m,K}(\mathbb{R}^{n})$ and its members.

######  Proposition 4.8.

Let $K_{1},K_{2}$ be two compact subsets of $\mathbb{R}^{n}$. If
$K_{1}\subseteq K_{2}$, then ${\it
Scan}_{m,K_{1}}(\mathbb{R}^{n})\subseteq{\it Scan}_{m,K_{2}}(\mathbb{R}^{n})$.

###### Proof.

For any $\mathcal{T}\in{\it Scan}_{m,K_{1}}(\mathbb{R}^{n})$, by definition,
$\mathcal{T}:\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}\to\mathscr{F}_{0,K_{1}}(\mathbb{R}^{n})$
is an $m$-dimensional scan in $K_{1}$ and
$\boldsymbol{\partial}(\mathcal{T}-\mathcal{S}(T))=0$ for some
$T\in\mathscr{R}_{m,K_{1}}(\mathbb{R}^{n})$. Since $K_{1}\subseteq K_{2}$,
$\mathcal{T}$ is also an $m$-dimensional scan in $K_{2}$ and
$T\in\mathscr{R}_{m,K_{2}}(\mathbb{R}^{n})$. So $\mathcal{T}\in{\it
Scan}_{m,K_{2}}(\mathbb{R}^{n})$ as well. ∎

######  Lemma 4.9.

Let $K$ be a compact subset of $\mathbb{R}^{n}$. For any integral flat chain
$T\in\mathscr{F}_{m,K}(\mathbb{R}^{n})$, the
$\boldsymbol{\theta}^{\ast}_{n,m}\times\mathcal{L}^{m}$ measurable map

$\mathcal{S}(T):\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}\to\mathscr{F}_{0,K}(\mathbb{R}^{n}),\quad\mathcal{S}(T)(p,y):=\langle
T,p,y\rangle,$

is an $m$-dimensional scan in $K$ with boundary.

###### Proof.

We first show that $\mathcal{S}(T)$ is an $m$-dimensional scan in $K$. Indeed,
since $T\in\mathscr{F}_{m,K}$ is an integral flat chain, there exists a
sequence of rectifiable currents
$T_{1},T_{2},\cdots\in\mathscr{R}_{m,K}(\mathbb{R}^{n})$ such that
$\lim_{i\to\infty}\mathcal{F}_{K}(T-T_{i})=0$. By [5, Remark 3.2.9.],
$\mathcal{F}_{K}^{H}(T-T_{i})\leq H(2)\mathcal{F}_{K}(T-T_{i})$, and hence
$\lim_{i\to\infty}\mathcal{F}_{K}^{H}(T-T_{i})=0$. By [5, Remark 3.2.11.],
$\int_{\mathbb{R}^{m}}\mathcal{F}_{K}^{H}(\langle
T-T_{i},p,y\rangle)\,d\mathcal{L}^{m}y\leq\mathcal{F}_{K}^{H}(T-T_{i})$ for
any $p\in\mathbf{O}^{\ast}(n,m)$. Thus,

$\int_{\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}}\mathcal{F}_{K}^{H}(\langle
T-T_{i},p,y\rangle)\,d(\boldsymbol{\theta}^{\ast}_{n,m}\times\mathcal{L}^{m})(p,y)\leq\boldsymbol{\beta}_{1}(n,m)\mathcal{F}_{K}^{H}(T-T_{i})\to
0\quad\text{ as }i\to\infty.$

By [6, Theorem 1.21], there exists a subsequence $(T_{ij})$ of $(T_{i})$ such
that

$\lim_{j\to\infty}\mathcal{F}_{K}^{H}(\langle T-T_{ij},p,y\rangle)=0$

for $\boldsymbol{\theta}^{\ast}_{n,m}\times\mathcal{L}^{m}$ almost every
$(p,y)\in\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}$. As a result,
$\mathcal{S}(T)$ is an $m$-dimensional scan in $K$.

We now show that $\mathcal{S}(T)$ is also in ${\it
Scan}_{m,K}(\mathbb{R}^{n})$. Since $T\in\mathscr{F}_{m,K}(\mathbb{R}^{n})$,
there exists $R\in\mathscr{R}_{m,K}(\mathbb{R}^{n})$ and
$Q\in\mathscr{R}_{m+1,K}(\mathbb{R}^{n})$ such that $T=R+\partial Q$. Thus, by
[5, Proposition 3.1.6.],

$\boldsymbol{\partial}(\mathcal{S}(T)-\mathcal{S}(R))=\boldsymbol{\partial}(\mathcal{S}(T-R))=\boldsymbol{\partial}(\mathcal{S}(\partial
Q))=0.$ (30)

Therefore, $\mathcal{S}(T)\in{\it Scan}_{m,K}(\mathbb{R}^{n})$. ∎

######  Definition 4.10.

We define the class

${\it Scan}_{m}(\mathbb{R}^{n}):=\bigcup_{K\subseteq\mathbb{R}^{n}\text{
compact}}{\it Scan}_{m,K}(\mathbb{R}^{n}),$

whose members are called $m$-dimensional scans in $\mathbb{R}^{n}$ with
boundary. For each $\mathcal{T}\in{\it Scan}_{m}(\mathbb{R}^{n})$, the
boundary of $\mathcal{T}$ is defined by

$\partial\mathcal{T}:=\mathcal{S}(\partial T)$ (31)

for any $T\in\mathscr{R}_{m}(\mathbb{R}^{n})$ satisfying
$\boldsymbol{\partial}(\mathcal{T}-\mathcal{S}(T))=0$.

With this definition we can now say that by (30),
$\partial(\mathcal{S}(T))=\mathcal{S}(\partial T)$ for any
$T\in\mathscr{F}_{m}(\mathbb{R}^{n})$.

######  Proposition 4.11.

Equation (31) defines a boundary operator $\partial:{\it
Scan}_{m}(\mathbb{R}^{n})\rightarrow{\it Scan}_{m-1}(\mathbb{R}^{n})$ with
$\partial^{2}=0$, and $\partial:{\it
Scan}_{m,K}(\mathbb{R}^{n})\rightarrow{\it Scan}_{m-1,K}(\mathbb{R}^{n})$ for
$K$ compact.

###### Proof.

For any $\mathcal{T}\in{\it Scan}_{m}(\mathbb{R}^{n})$, we first show that
$\partial\mathcal{T}:=\mathcal{S}(\partial T)$ is well-defined in the sense
that $\mathcal{S}(\partial T)$ is independent of the choices of
$T\in\mathscr{R}_{m,K}(\mathbb{R}^{n})$ that satisfies
$\boldsymbol{\partial}(\mathcal{T}-\mathcal{S}(T))=0$ and the choice of the
compact set $K\subseteq\mathbb{R}^{n}$. Indeed, suppose there are two compact
sets $K_{1},K_{2}$ and an associated rectifiable current
$T_{j}\in\mathscr{R}_{m,K_{j}}(\mathbb{R}^{n})\subseteq\mathscr{R}_{m,K_{1}\cup
K_{2}}(\mathbb{R}^{n})$ with
$\boldsymbol{\partial}(\mathcal{T}-\mathcal{S}(T_{j}))=0$ for $j=1,2$. Here,
$\mathcal{T}-\mathcal{S}(T_{j}):\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}\to\mathscr{F}_{0,K_{j}}(\mathbb{R}^{n})\subseteq\mathscr{F}_{0,K_{1}\cup
K_{2}}(\mathbb{R}^{n})$. So,
$\boldsymbol{\partial}(\mathcal{S}(T_{1}-T_{2}))=\boldsymbol{\partial}(\mathcal{S}(T_{1})-\mathcal{S}(T_{2}))=0$.
By [5, Proposition 3.1.6.], $\partial(T_{1}-T_{2})=0$ for
$T_{1}-T_{2}\in\mathscr{R}_{m,K_{1}\cup K_{2}}(\mathbb{R}^{n})$. Therefore,
$\partial T_{1}=\partial T_{2}$ and hence $\mathcal{S}(\partial
T_{1})=\mathcal{S}(\partial T_{2})$. This shows that the definition
$\partial\mathcal{T}:=\mathcal{S}(\partial T)$ is well-defined. By definition,
$\partial T\in\mathscr{F}_{m-1,K}(\mathbb{R}^{n})$. Thus by Lemma 4.9,
$\partial\mathcal{T}=\mathcal{S}(\partial T)\in{\it
Scan}_{m-1,K}(\mathbb{R}^{n})$ and
$\partial^{2}\mathcal{T}=\partial(\mathcal{S}(\partial
T))=\mathcal{S}(\partial^{2}T)=\mathcal{S}(0)=0$. ∎

######  Remark 4.12.

Recall that the symbol $\boldsymbol{\partial}$ is only used in the form of
$\boldsymbol{\partial}\mathcal{T}=0$, denoting that $\mathcal{T}$ is a cycle
and that we cannot apply this symbol to other maps, while $\partial$ is a
well-defined boundary operator on ${\it Scan}_{m}(\mathbb{R}^{n})$. In ${\it
Scan}_{m}(\mathbb{R}^{n})$ the boundary operator $\partial$ is equivalent to
$\boldsymbol{\partial}$, in the sense that for any $\mathcal{T}\in{\it
Scan}_{m}(\mathbb{R}^{n})$, $\partial\mathcal{T}=0$ if and only if
$\boldsymbol{\partial}\mathcal{T}=0$.

For each compact set $K\subseteq\mathbb{R}^{n}$, the following functional,
denoted by $\underline{\mathbf{M}}_{H,K}$, serves as a modification of the
$H$-mass. Lemma 4.15 shows that it agrees with $\mathbf{M}_{H}$ on the class
of scans of rectifiable currents.

######  Definition 4.13.

For any compact $K\subseteq\mathbb{R}^{n}$, we define the functional
$\underline{\mathbf{M}}_{H,K}:{\it Scan}_{m,K}(\mathbb{R}^{n})\to\mathbb{R}$
by

$\underline{\mathbf{M}}_{H,K}(\mathcal{T}):=\inf\\{\liminf_{i\to\infty}\mathbf{M}_{H}(T_{i}):T_{i}\in\mathscr{R}_{m,K}(\mathbb{R}^{n}),\mathcal{S}(T_{i})\to\mathcal{T},\text{
and }\partial(\mathcal{S}(T_{i}))=\partial\mathcal{T}\\}.$ (32)

We will simply write $\underline{\mathbf{M}}_{H}(\mathcal{T})$ instead of
$\underline{\mathbf{M}}_{H,K}(\mathcal{T})$ if $K$ is clear from context.

######  Remark 4.14.

In general, the value of $\underline{\mathbf{M}}_{H,K}(\mathcal{T})$ depends
on the compact set $K$. Nevertheless, if $K_{1}\subseteq K_{2}$ are nonempty
compact subsets of $\mathbb{R}^{n}$ and $\mathcal{T}\in{\it
Scan}_{m,K_{1}}(\mathbb{R}^{n})\cap{\it Scan}_{m,K_{2}}(\mathbb{R}^{n})$,
since
$\mathscr{R}_{m,K_{1}}(\mathbb{R}^{n})\subseteq\mathscr{R}_{m,K_{2}}(\mathbb{R}^{n})$
and $\mathscr{F}_{K_{2}}^{H}\leq\mathscr{F}_{K_{1}}^{H}$, it follows that
$\underline{\mathbf{M}}_{H,K_{2}}(\mathcal{T})\leq\underline{\mathbf{M}}_{H,K_{1}}(\mathcal{T})$.

######  Lemma 4.15.

For any $T\in\mathscr{R}_{m,K}(\mathbb{R}^{n})$,
$\underline{\mathbf{M}}_{H}(\mathcal{S}(T))=\mathbf{M}_{H}(T)=\mathbf{M}_{H}(\mathcal{S}(T))$.

###### Proof.

Taking the constant sequence $T_{i}=T$ in (32) gives us that
$\underline{\mathbf{M}}_{H}(\mathcal{S}(T))\leq\mathbf{M}_{H}(T)$. We will now
show that $\mathbf{M}_{H}(T)\leq\underline{\mathbf{M}}_{H}(\mathcal{S}(T))$.
Let $T_{i}\in\mathscr{R}_{m,K}(\mathbb{R}^{n})$ with
$\mathcal{S}(T_{i})\to\mathcal{S}(T)$. By definition, for
$\boldsymbol{\theta}^{\ast}_{n,m}\times\mathcal{L}^{m}$ almost every
$(p,y)\in\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}$,
$\mathscr{F}_{K}^{H}[\langle T_{i},p,y\rangle-\langle T,p,y\rangle]\to 0$ as
$i\to\infty$. By lower semi-continuity of $\mathbf{M}_{H}$ under
$\mathscr{F}_{K}^{H}$ convergence [5, Lemma 3.2.14.],

$\mathbf{M}_{H}(\langle
T,p,y\rangle)\leq\liminf_{i\to\infty}\mathbf{M}_{H}(\langle
T_{i},p,y\rangle).$

Integrating both sides and using Fatou’s lemma, we get

$\displaystyle\mathbf{M}_{H}(\mathcal{S}(T))$ $\displaystyle=$
$\displaystyle\frac{1}{\boldsymbol{\beta}_{1}(m,n)}\int_{\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}}\mathbf{M}_{H}(\langle
T,p,y\rangle)\,d(\mathbf{\theta}_{n,m}^{\ast}\times\mathcal{L}^{m})(p,y)$
$\displaystyle\leq$
$\displaystyle\frac{1}{\boldsymbol{\beta}_{1}(m,n)}\int_{\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}}\liminf_{i\to\infty}\mathbf{M}_{H}(\langle
T_{i},p,y\rangle)\,d(\mathbf{\theta}_{n,m}^{\ast}\times\mathcal{L}^{m})(p,y)$
$\displaystyle\leq$
$\displaystyle\liminf_{i\to\infty}\frac{1}{\boldsymbol{\beta}_{1}(m,n)}\int_{\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}}\mathbf{M}_{H}(\langle
T_{i},p,y\rangle)\,d(\mathbf{\theta}_{n,m}^{\ast}\times\mathcal{L}^{m})(p,y)$
$\displaystyle=$
$\displaystyle\liminf_{i\to\infty}\mathbf{M}_{H}(\mathcal{S}(T_{i})).$

Thus, $\mathbf{M}_{H}(\mathcal{S}(T))\leq\underline{\mathbf{M}}_{H}(T)$ and
hence the result follows from
$\mathbf{M}_{H}(\mathcal{S}(T))=\mathbf{M}_{H}(T)$. ∎

The following proposition says that $\underline{\mathbf{M}}_{H}$ is
subadditive on ${\it Scan}_{m}(\mathbb{R}^{n})$.

######  Proposition 4.16.

For any nonempty compact subset $K$ of $\mathbb{R}^{n}$,
$\underline{\mathbf{M}}_{H}(\mathcal{T}_{1}+\mathcal{T}_{2})\leq\underline{\mathbf{M}}_{H}(\mathcal{T}_{1})+\underline{\mathbf{M}}_{H}(\mathcal{T}_{2})$
for any $\mathcal{T}_{1},\mathcal{T}_{2}\in{\it Scan}_{m,K}(\mathbb{R}^{n})$.

###### Proof.

Let $K$ be a nonempty compact subset of $\mathbb{R}^{n}$ such that both
$\mathcal{T}_{1},\mathcal{T}_{2}\in{\it Scan}_{m,K}(\mathbb{R}^{n})$. For each
$k=1,2$, let $\\{T^{k}_{i}\\}_{i=1}^{\infty}$ be a sequence in
$\mathscr{R}_{m,K}(\mathbb{R}^{n})$ such that
$\mathcal{S}(T_{i}^{k})\to\mathcal{T}_{k}$ with
$\partial(\mathcal{S}(T_{i}^{k}))=\partial\mathcal{T}_{k}$ for all
$i\in\mathbb{N}$. By picking a subsequence if necessary, we may assume that
for each $k=1,2$, $\\{\mathbf{M}_{H}(T^{k}_{i})\\}_{i=1}^{\infty}$ converges.
Now consider the sequence $\\{T_{i}^{1}+T_{i}^{2}\\}_{j=1}^{\infty}$ and
notice that $\mathcal{T}_{1}+\mathcal{T}_{2}\in{\it
Scan}_{m,K}(\mathbb{R}^{n})$,
$\mathcal{S}(T_{i}^{1}+T_{i}^{2})\to(\mathcal{T}_{1}+\mathcal{T}_{2})$, and
$\partial(\mathcal{S}(T_{i}^{1}+T_{i}^{2}))=\partial(\mathcal{T}_{1}+\mathcal{T}_{2})$.
Indeed, for $\boldsymbol{\theta}^{\ast}_{n,m}\times\mathcal{L}^{m}$ almost
every $(p,y)\in\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}$,

$\mathscr{F}_{K}^{H}[(\mathcal{S}(T_{i}^{1}+T_{i}^{2})(p,y)-(\mathcal{T}_{1}+\mathcal{T}_{2})(p,y)]\leq\mathscr{F}_{K}^{H}[(\mathcal{S}(T_{i}^{1})-\mathcal{T}_{1})(p,y)]+\mathscr{F}_{K}^{H}[(\mathcal{S}(T_{i}^{2})-\mathcal{T}_{2})(p,y)]\to
0$

as $i\to\infty$, and

$((\mathcal{T}_{1}+\mathcal{T}_{2})-\mathcal{S}(T_{i}^{1}+T_{i}^{2}))(p,y)(\boldsymbol{1})=(\mathcal{T}_{1}-\mathcal{S}(T_{i}^{1})(p,y)(\boldsymbol{1})+(\mathcal{T}_{2}-\mathcal{S}(T_{i}^{2})(p,y)(\boldsymbol{1})=0$

for all $i\in\mathbb{N}$. By definition, this implies
$\boldsymbol{\partial}((\mathcal{T}_{1}+\mathcal{T}_{2})-\mathcal{S}(T_{i}^{1}+T_{i}^{2}))=0$
for all $i\in\mathbb{N}$. Thus,

$\displaystyle\underline{\mathbf{M}}_{H}(\mathcal{T}_{1}+\mathcal{T}_{2})$
$\displaystyle\leq\liminf_{i\to\infty}\mathbf{M}_{H}(T^{1}_{i}+T^{2}_{i})\leq\liminf_{i\to\infty}\mathbf{M}_{H}(T_{i}^{1})+\mathbf{M}_{H}(T_{i}^{2})$
$\displaystyle=\lim_{i\to\infty}\mathbf{M}_{H}(T_{i}^{1})+\lim_{i\to\infty}\mathbf{M}_{H}(T_{i}^{2})=\liminf_{i\to\infty}\mathbf{M}_{H}(T_{i}^{1})+\liminf_{i\to\infty}\mathbf{M}_{H}(T_{i}^{2}).$

Since the sequences $\\{T^{1}_{i}\\}_{i=1}^{\infty}$ and
$\\{T_{i}^{2}\\}_{i=1}^{\infty}$ were arbitrary, we have
$\underline{\mathbf{M}}_{H}(\mathcal{T}_{1}+\mathcal{T}_{2})\leq\underline{\mathbf{M}}_{H}(\mathcal{T}_{1})+\underline{\mathbf{M}}_{H}(\mathcal{T}_{2})$.
∎

######  Lemma 4.17.

Let $K$ be a compact subset of $\mathbb{R}^{n}$ and let
$T_{i}\in\mathbf{I}_{0}(\mathbb{R}^{n})$ be a sequence of $0$ dimensional
integral currents. If $\lim_{i\to\infty}\mathscr{F}_{K}^{H}(T_{i})=0$, then
$\lim_{i\to\infty}T_{i}(\boldsymbol{1})=0$.

###### Proof.

By definition of $\mathscr{F}_{K}^{H}$, there exists a sequence
$\\{S_{i}\\}_{i=1}^{\infty}$ in $\mathscr{R}_{1,K}(\mathbb{R}^{n})$ such that
$\lim_{i\to\infty}\mathbf{M}_{H}(T_{i}-\partial S_{i})=0$. It is sufficient to
prove that $\lim_{i\to\infty}\mathscr{F}_{K}(T_{i}-\partial S_{i})=0$ since
this implies that

$\lim_{i\to\infty}T_{i}(\boldsymbol{1})=\lim_{i\to\infty}T_{i}(\boldsymbol{1})-S_{i}(d(\boldsymbol{1}))=\lim_{i\to\infty}(T_{i}-\partial
S_{i})(\boldsymbol{1})=0.$

To prove that $\lim_{i\to\infty}\mathscr{F}_{K}(T_{i}-\partial S_{i})=0$ we
will apply [5, Lemma 3.2.13.] which says that any sequence of integral flat
chains $\\{Q_{i}\\}_{i=1}^{\infty}$ with
$\sup_{i}\\{\mathbf{M}(Q_{i})+\mathbf{M}(\partial Q_{i})\\}<\infty$ and
$\lim_{i\to\infty}\mathscr{F}_{K}^{H}(Q_{i})=0$ will also have that
$\lim_{i\to\infty}\mathscr{F}_{K}(Q_{i})=0$. To apply this lemma to our
sequence $\\{T_{i}-\partial S_{i}\\}_{i=1}^{\infty}$ of $0$ dimensional
integral currents, we only need to show that
$\sup_{i}\\{\mathbf{M}(T_{i}-\partial S_{i})\\}<\infty$ and
$\lim_{i\to\infty}\mathscr{F}_{K}^{H}(T_{i}-\partial S_{i})=0$. Our last claim
is shown by $\mathscr{F}_{K}^{H}(T_{i}-\partial
S_{i})\leq\mathbf{M}_{H}(T_{i}-\partial S_{i})\to 0.$ We may also assume that
$\sup_{i}\\{\mathbf{M}_{H}(T_{i}-\partial S_{i})\\}<\infty$. Remark 3.2.7. in
[5] says that $H(\mathbf{M}(T))\leq\mathbf{M}_{H}(T)$ for all
$T\in\mathbf{I}_{0}(\mathbb{R}^{n})$ and hence
$\sup_{i}\\{H(\mathbf{M}(T_{i}-\partial S_{i}))\\}<\infty$. Since $H$ is
strictly increasing, we lastly get that $\sup_{i}\\{\mathbf{M}(T_{i}-\partial
S_{i})\\}<\infty$. ∎

Using this modified version of the $H$-mass $\underline{\mathbf{M}}_{H}$ on
the class ${\it Scan}_{m,K}(\mathbb{R}^{n})$ of $m$-dimensional scans in $K$
with boundary, we now show that the Plateau’s problem with $H$-mass (5) has a
solution for each $K$.

######  Theorem 4.18.

Let $K$ be a nonempty compact subset of $\mathbb{R}^{n}$. For any
$\mathcal{T}_{0}\in{\it Scan}_{m,K}(\mathbb{R}^{n})$, there exists a scan
$\mathcal{T}^{\ast}\in{\it Scan}_{m,K}(\mathbb{R}^{n})$ with
$\partial\mathcal{T}^{\ast}=\partial\mathcal{T}_{0}$ such that

$\underline{\mathbf{M}}_{H}(\mathcal{T}^{\ast})=\min\\{\underline{\mathbf{M}}_{H}(\mathcal{T}):\mathcal{T}\in{\it
Scan}_{m,K}(\mathbb{R}^{n}),\ \partial\mathcal{T}=\partial\mathcal{T}_{0}\\}.$
(33)

Moreover, $\mathcal{T}^{\ast}(p,y)\in\mathbf{I}_{0,K}(\mathbb{R}^{n})$ for
$\boldsymbol{\theta}^{\ast}_{n,m}\times\mathcal{L}^{m}$ almost every
$(p,y)\in\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}$.

###### Proof.

By assumption $\\{\mathcal{T}\in{\it
Scan}_{m,K}(\mathbb{R}^{n}):\partial\mathcal{T}=\partial\mathcal{T}_{0}\\}$
contains $\mathcal{T}_{0}$, and hence is non-empty. We may assume that
$\inf\\{\underline{\mathbf{M}}_{H}(\mathcal{T}):\mathcal{T}\in{\it
Scan}_{m,K}(\mathbb{R}^{n}),\
\partial\mathcal{T}=\partial\mathcal{T}_{0}\\}<\infty$ as otherwise
$\underline{\mathbf{M}}_{H}(\mathcal{T}_{0})=\infty$ and we may simply pick
$\mathcal{T}^{\ast}=\mathcal{T}_{0}$. Let
$\\{\mathcal{T}^{i}\\}_{i=1}^{\infty}$ be an $\underline{\mathbf{M}}_{H}$
minimizing sequence in this collection. That is, $\mathcal{T}^{i}\in{\it
Scan}_{m,K}(\mathbb{R}^{n})$,
$\partial\mathcal{T}^{i}=\partial\mathcal{T}_{0}$, and

$\lim_{i\to\infty}\underline{\mathbf{M}}_{H}(\mathcal{T}^{i})=\inf\\{\underline{\mathbf{M}}_{H}(\mathcal{T}):\mathcal{T}\in{\it
Scan}_{m,K}(\mathbb{R}^{n}),\ \partial\mathcal{T}=\partial\mathcal{T}_{0}\\}.$
(34)

Since $\\{\mathcal{T}^{i}\\}_{i=1}^{\infty}$ is an
$\underline{\mathbf{M}}_{H}$ minimizing sequence, without loss of generality,
we may assume that

$\sup_{i}\underline{\mathbf{M}}_{H}(\mathcal{T}^{i})\leq C<\infty$ (35)

for some $C>0$. By definition of $\underline{\mathbf{M}}_{H}$ in (32), for
each $i\in\mathbb{N}$, there exists a sequence
$\\{T_{j}^{i}\\}_{j=1}^{\infty}$ in $\mathscr{R}_{m,K}(\mathbb{R}^{n})$ such
that

$\lim_{j\to\infty}\mathbf{M}_{H}(T_{j}^{i})=\underline{\mathbf{M}}_{H}(\mathcal{T}^{i}),\quad\text{
and
}\quad\partial(\mathcal{S}(T_{j}^{i}))=\partial\mathcal{T}^{i}=\partial\mathcal{T}_{0}\text{
for all }j\in\mathbb{N}.$ (36)

Since $\mathcal{T}_{0}\in{\it Scan}_{m,K}(\mathbb{R}^{n})$, there exists a
$T_{0}\in\mathscr{R}_{m,K}(\mathbb{R}^{n})$ such that
$\partial\mathcal{T}_{0}=\partial(\mathcal{S}(T_{0}))$. By the proof of
Proposition 4.11, the condition
$\partial(\mathcal{S}(T_{j}^{i}))=\partial\mathcal{T}_{0}$ imply that
$\partial T_{j}^{i}=\partial T_{0}$ for all $i,j\in\mathbb{N}$. By (36), for
each $i\in\mathbb{N}$ there exists an $N_{i}\in\mathbb{N}$ such that

$|\mathbf{M}_{H}(T^{i}_{N_{i}})-\underline{\mathbf{M}}_{H}(\mathcal{T}^{i})|\leq\frac{1}{i}.$
(37)

Hence by (35), $\sup_{i}\mathbf{M}_{H}(T^{i}_{N_{i}})\leq C+1<\infty$. Thus,

$\mathbf{M}_{H}(\mathcal{S}(T^{i}_{N_{i}}-T_{0}))=\mathbf{M}_{H}(T^{i}_{N_{i}}-T_{0})\leq
C+1+\mathbf{M}_{H}(T_{0})\quad\text{ for all }i\in\mathbb{N}.$

On the other hand, by [5, Remark 3.3.3], [5, Lemma 3.1.2], and the fact that
$\partial T^{i}_{N_{i}}=\partial T_{0}$,

$\displaystyle\mathbf{D}(\mathcal{S}(T^{i}_{N_{i}}-T_{0}))$
$\displaystyle\leq\mathbf{c}_{1}[\mathbf{M}_{H}((T^{i}_{N_{i}}-T_{0})\times\llbracket\mathbf{O}^{\ast}(n,m)\rrbracket)+\mathbf{M}_{H}(\partial((T^{i}_{N_{i}}-T_{0})\times\llbracket\mathbf{O}^{\ast}(n,m)\rrbracket))]$
$\displaystyle=\mathbf{c}_{1}\mathbf{M}_{H}((T^{i}_{N_{i}}-T_{0})\times\llbracket\mathbf{O}^{\ast}(n,m)\rrbracket)$
$\displaystyle\leq\mathbf{c}_{2}\mathbf{M}_{H}(T^{i}_{N_{i}}-T_{0})$
$\displaystyle\leq\mathbf{c}_{2}[C+1+\mathbf{M}_{H}(T_{0})],$

where the constants $\mathbf{c}_{1}$ and $\mathbf{c}_{2}$ depend only on $n$
and $K$. By Corollary 4.6 of the BV compactness theorem [5, Theorem 3.4.1] of
De Pauw and Hardt, and replacing $\\{T_{N_{i}}^{i}\\}$ by a subsequence if
necessary, there exists a
$\boldsymbol{\theta}^{\ast}_{n,m}\times\mathcal{L}^{m}$ measurable map
$\mathcal{R}:\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}\to\mathbf{I}_{0,K}(\mathbb{R}^{n})$
such that

$\lim_{i\rightarrow\infty}\mathscr{F}_{K}^{H}[\mathcal{S}(T_{N_{i}}^{i}-T_{0})(p,y)-\mathcal{R}(p,y)]=0$
(38)

for $\boldsymbol{\theta}^{\ast}_{n,m}\times\mathcal{L}^{m}$ almost every
$(p,y)\in\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}$.

Now, let $\mathcal{T}^{\ast}=\mathcal{R}+\mathcal{S}(T_{0})$. By definition,
(38) implies
$\mathcal{S}(T_{N_{i}}^{i})\to\mathcal{R}+\mathcal{S}(T_{0})=\mathcal{T}^{\ast}$.
By Lemma 4.17, this implies

$(\mathcal{T}^{\ast}-\mathcal{S}(T_{0}))(p,y)(\boldsymbol{1})=\lim_{i\to\infty}\langle
T_{N_{i}}^{i}-T_{0},p,y\rangle(\boldsymbol{1})=0$

$\boldsymbol{\theta}^{\ast}_{n,m}\times\mathcal{L}^{m}$ almost every $(p,y)$,
where the second equality is given by [5, Proposition 3.1.6.] since $\partial
T_{N_{i}}^{i}=\partial T_{0}$ for all $i\in\mathbb{N}$. This gives us our
boundary condition,
$\boldsymbol{\partial}(\mathcal{T}^{\ast}-\mathcal{S}(T_{0}))=0$, i.e.,
$\partial\mathcal{T}^{\ast}=\partial(\mathcal{S}(T_{0}))=\partial\mathcal{T}_{0}$.
As a result, $\mathcal{T}^{\ast}\in{\it Scan}_{m,K}(\mathbb{R}^{n})$ with
$\partial\mathcal{T}=\partial\mathcal{T}_{0}$. Hence

$\underline{\mathbf{M}}_{H}(\mathcal{T}^{\ast})\geq\inf\\{\underline{\mathbf{M}}_{H}(\mathcal{T}):\mathcal{T}\in{\it
Scan}_{m,K}(\mathbb{R}^{n}),\ \partial\mathcal{T}=\partial\mathcal{T}_{0}\\}.$

Additionally, by (32), (37), and (34),

$\displaystyle\underline{\mathbf{M}}_{H}(\mathcal{T}^{\ast})\leq\liminf_{i\to\infty}\mathbf{M}_{H}(T_{N_{i}}^{i})=\liminf_{i\to\infty}\underline{\mathbf{M}}_{H}(\mathcal{T}^{i})=\inf\\{\underline{\mathbf{M}}_{H}(\mathcal{T}):\mathcal{T}\in{\it
Scan}_{m,K}(\mathbb{R}^{n}),\ \partial\mathcal{T}=\partial\mathcal{T}_{0}\\}.$

Therefore, $\mathcal{T}^{\ast}$ satisfies (33) as desired. ∎

######  Corollary 4.19.

Let $K$ be a nonempty compact and convex subset of $\mathbb{R}^{n}$, and $H$
be a concave integrand. For any $T_{0}\in\mathscr{R}_{m,K}(\mathbb{R}^{n})$
and

$\mathcal{T}^{\ast}\in\mathrm{argmin}\\{\underline{\mathbf{M}}_{H}(\mathcal{T}):\mathcal{T}\in{\it
Scan}_{m,K}(\mathbb{R}^{n}),\ \partial\mathcal{T}=\partial\mathcal{T}_{0}\\}$
(39)

with $\mathcal{T}_{0}:=\mathcal{S}(T_{0})$, we have that
$\underline{\mathbf{M}}_{H}(\mathcal{T}^{\ast})=\Gamma(\mathbf{M}_{H},T_{0},\mathbb{R}^{n})$.

###### Proof.

First, recall that
$\Gamma(\mathbf{M}_{H},T_{0},\mathbb{R}^{n})=\inf\\{\mathbf{M}_{H}(T):T\in\mathscr{R}_{m,K}(\mathbb{R}^{n}),\
\partial T=\partial T_{0}\\}$ since
$\mathbf{M}_{H}({\pi_{K}}_{\\#}(T))\leq\mathbf{M}_{H}(T)$ for any
$T\in\mathscr{R}_{m}(\mathbb{R}^{n})$ under the nearest point projection map
$\pi_{K}$.

Since $\mathcal{S}_{m,K}(\mathbb{R}^{n})\subseteq{\it
Scan}_{m,K}(\mathbb{R}^{n})$, by Lemma 4.15, we have
$\underline{\mathbf{M}}_{H}(\mathcal{T}^{\ast})\leq\inf\\{\mathbf{M}_{H}(T):T\in\mathscr{R}_{m,K}(\mathbb{R}^{n}),\
\partial T=\partial T_{0}\\}=\Gamma(\mathbf{M}_{H},T_{0},\mathbb{R}^{n})$. On
the other hand, since each
$\mathbf{M}_{H}(T^{i}_{j})\geq\inf\\{\mathbf{M}_{H}(T):T\in\mathscr{R}_{m,K}(\mathbb{R}^{n}),\
\partial T=\partial T_{0}\\}$, by (36), we have
$\underline{\mathbf{M}}_{H}(\mathcal{T}^{\ast})\geq\Gamma(\mathbf{M}_{H},T_{0},\mathbb{R}^{n})$
as well. Therefore,
$\underline{\mathbf{M}}_{H}(\mathcal{T}^{\ast})=\Gamma(\mathbf{M}_{H},T_{0},\mathbb{R}^{n})$
as desired. ∎

##  5 Partial Plateau’s problem with $H$-mass

In this section, we assume that $\Phi\in\mathscr{D}^{m-1}(\mathbb{R}^{n})$,
$H$ is a concave integrand, and that $B\in\mathscr{R}_{m-1}(\mathbb{R}^{n})$
with $\mathbf{M}_{H}(B)<\infty$.

We are interested in what we call, partial Plateau’s problem with $H$-mass:

$\inf\\{\mathbf{E}_{H,\Phi}(T):T\in\mathscr{R}_{m}(\mathbb{R}^{n}),\partial
T\preceq
B\\}\quad\mathrm{where}\quad\mathbf{E}_{H,\Phi}(T):=\mathbf{M}_{H}(T)-\partial
T(\Phi).$ (40)

To consider this problem, as in the previous section, we will consider an
extension of $\mathbf{E}_{H,\Phi}$ to elements in ${\it
Scan}_{m,K}(\mathbb{R}^{n})$ for a fixed compact set
$K\subseteq\mathbb{R}^{n}$.

For any $\mathcal{T}\in{\it Scan}_{m,K}(\mathbb{R}^{n})$, by definition, there
exists a $T\in\mathscr{R}_{m,K}(\mathbb{R}^{n})$ such that
$\partial\mathcal{T}=\mathcal{S}(\partial T)$. We simply write
$\partial\mathcal{T}\preceq\mathcal{S}(B)$ whenever $\partial T\preceq B$.
Also, we define $\partial\mathcal{T}(\Phi):=\partial T(\Phi)$ and

$\mathbf{E}_{H,\Phi}(\mathcal{T}):=\underline{\mathbf{M}}_{H}(\mathcal{T})-\partial\mathcal{T}(\Phi).$

It extends the definition of $\mathbf{E}_{H,\Phi}$ in (40) since by Lemma
4.15, it holds that
$\mathbf{E}_{H,\Phi}(\mathcal{S}(T))=\mathbf{E}_{H,\Phi}(T)$ for any
$T\in\mathscr{R}_{m,K}(\mathbb{R}^{n})$.

######  Theorem 5.1.

Let $K$ be a nonempty, compact, and convex subset of $\mathbb{R}^{n}$,
$\Phi\in\mathscr{D}^{m-1}(\mathbb{R}^{n})$, $H$ a concave integrand, and let
$B\in\mathscr{R}_{m-1}(\mathbb{R}^{n})$ with $\mathbf{M}_{H}(B)<\infty$. For
any $\mathcal{T}_{0}\in{\it Scan}_{m,K}(\mathbb{R}^{n})$, there exists a scan
$\mathcal{T}^{\ast}:\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}\to\mathbf{I}_{0,K}(\mathbb{R}^{n})$
such that $\mathcal{T}^{\ast}\in{\it Scan}_{m,K}(\mathbb{R}^{n})$,
$\partial(\mathcal{T}^{\ast}-\mathcal{T}_{0})\preceq\mathcal{S}(B)$, and

$\mathbf{E}_{H,\Phi}(\mathcal{T}^{\ast})=\min\\{\mathbf{E}_{H,\Phi}(\mathcal{T}):\mathcal{T}\in{\it
Scan}_{m,K}(\mathbb{R}^{n}),\
\partial(\mathcal{T}-\mathcal{T}_{0})\preceq\mathcal{S}(B)\\}.$ (41)

Moreover, if $\mathcal{T}_{0}=\mathcal{S}(T_{0})$ for some
$T_{0}\in\mathscr{R}_{m,K}(\mathbb{R}^{n})$, then

$\mathbf{E}_{H,\Phi}(\mathcal{T}^{\ast})=\inf\\{\mathbf{E}_{H,\Phi}(T):T\in\mathscr{R}_{m}(\mathbb{R}^{n}),\partial(T-T_{0})\preceq
B\\}.$ (42)

The proof of this theorem will make use of the following two lemmas.

######  Lemma 5.2.

Suppose that $B\in\mathscr{R}_{k}(\mathbb{R}^{n})$ for $k\geq 0$. If $A\preceq
B$, then $A$ is a $k$ dimensional real rectifiable current with
$\mathbf{M}_{H}(A)\leq\mathbf{M}_{H}(B)$.

###### Proof.

Since $B\in\mathscr{R}_{k}(\mathbb{R}^{n})$,

$B=\mathcal{H}^{k}\,\hbox{\vrule height=7.0pt,depth=0.0pt\vrule
height=0.4pt,depth=0.0pt,width=6.0pt}\,M\wedge\theta\xi$

such that $M$ is an $(\mathcal{H}^{k},k)$-rectifiable subset of
$\mathbb{R}^{n}$, $\theta:M\to\mathbb{Z}$ is summable, and
$\xi:M\to\wedge_{k}(\mathbb{R}^{n})$ is a simple $k$-vectorfield on $M$ that
coincides with the approximate tangent space of $M$. By [15, Lemma 3.7],
$A\preceq B$ implies that

$A=\mathcal{H}^{m}\,\hbox{\vrule height=7.0pt,depth=0.0pt\vrule
height=0.4pt,depth=0.0pt,width=6.0pt}\,M\wedge\sigma\theta\xi\text{ for some
Borel function }\sigma:\mathbb{R}^{n}\to[0,1].$

Thus,

$\mathbf{M}_{H}(A)=\int_{M}H(\sigma(x)\theta(x))\,d\mathcal{H}^{m}(x)\leq\int_{M}H(\theta(x))\,d\mathcal{H}^{m}(x)=\mathbf{M}_{H}(B)$
(43)

since $0\leq\sigma\leq 1$ and $H$ is monotonically non-decreasing. ∎

######  Lemma 5.3.

Let $m\in\mathbb{N}$, $K$ be a nonempty, convex, compact subset of
$\mathbb{R}^{n}$, $Z\in\mathscr{R}_{m-1,K}(\mathbb{R}^{n})$ with $\partial
Z=0$, and let
$\\{Z_{j}\\}_{j=1}^{\infty}\subseteq\mathscr{R}_{m-1,K}(\mathbb{R}^{n})$ with
$\partial Z_{j}=0$ for all $j\in\mathbb{N}$. If
$\lim_{j\to\infty}\mathscr{F}_{K}(Z_{j}-Z)=0$, then there exists
$C,C_{1},C_{2},\dots$ in $\mathscr{R}_{m,K}(\mathbb{R}^{n})$ such that
$\partial C=Z$, $\partial C_{j}=Z_{j}$ for all $j\in\mathbb{N}$, and
$\lim_{j\to\infty}\mathscr{F}_{K}(C_{j}-C)=0$.

###### Proof.

For each $j\in\mathbb{N}$, let
$C_{j}=\boldsymbol{\delta}_{v}\hbox{{$\times$}\kern-6.49994pt{$\times$}}Z_{j}$
and $C=\boldsymbol{\delta}_{v}\hbox{{$\times$}\kern-6.49994pt{$\times$}}Z$ be
the cones over $Z_{j}$ and $Z$ respectively with the fixed vertex $v\in K$ [8,
4.1.11]. Note that since $K$ is convex, the cones $C_{j}$ and $C$ are in
$\mathscr{R}_{m,K}(\mathbb{R}^{n})$. By the homotopy formula,

$\partial
C_{j}=\partial(\boldsymbol{\delta}_{v}\hbox{{$\times$}\kern-6.49994pt{$\times$}}Z_{j})=Z_{j}-\boldsymbol{\delta}_{v}\hbox{{$\times$}\kern-6.49994pt{$\times$}}\partial
Z_{j}=Z_{j}\text{ and }\partial
C=\partial(\boldsymbol{\delta}_{v}\hbox{{$\times$}\kern-6.49994pt{$\times$}}Z)=Z-\boldsymbol{\delta}_{v}\hbox{{$\times$}\kern-6.49994pt{$\times$}}\partial
Z=Z.$

For any $\epsilon>0$, since $\lim_{j\to\infty}\mathscr{F}_{K}(Z_{j}-Z)=0$,
there exists an $N\in\mathbb{N}$ such that whenever $j\geq N$, it holds that
$\mathscr{F}_{K}(Z_{j}-Z)<\epsilon$. Thus, by the definition of the flat norm
$\mathscr{F}_{K}$, there exists $R_{j}$ and $S_{j}$ such that

$Z_{j}-Z=R_{j}+\partial S_{j},\text{ and
}\mathbf{M}(R_{j})+\mathbf{M}(S_{j})<\epsilon.$

Now,

$\displaystyle C_{j}-C$
$\displaystyle=\boldsymbol{\delta}_{v}\hbox{{$\times$}\kern-6.49994pt{$\times$}}Z_{j}-\boldsymbol{\delta}_{v}\hbox{{$\times$}\kern-6.49994pt{$\times$}}Z=\boldsymbol{\delta}_{v}\hbox{{$\times$}\kern-6.49994pt{$\times$}}(Z_{j}-Z)=\boldsymbol{\delta}_{v}\hbox{{$\times$}\kern-6.49994pt{$\times$}}(R_{j}+\partial
S_{j})$
$\displaystyle=\boldsymbol{\delta}_{v}\hbox{{$\times$}\kern-6.49994pt{$\times$}}R_{j}+\boldsymbol{\delta}_{v}\hbox{{$\times$}\kern-6.49994pt{$\times$}}\partial
S_{j}=\boldsymbol{\delta}_{v}\hbox{{$\times$}\kern-6.49994pt{$\times$}}R_{j}+S_{j}-\partial(\boldsymbol{\delta}_{v}\hbox{{$\times$}\kern-6.49994pt{$\times$}}S_{j}),$

where we used the homotopy formula in the last equality. Assume that the
compact set $K$ is contained in the ball $B(0,\rho)$ for some $\rho>0$. Then,

$\mathbf{M}(\boldsymbol{\delta}_{v}\hbox{{$\times$}\kern-6.49994pt{$\times$}}R_{j})\leq\rho\mathbf{M}(R_{j})\text{
and
}\mathbf{M}(\boldsymbol{\delta}_{v}\hbox{{$\times$}\kern-6.49994pt{$\times$}}S_{j})\leq\rho\mathbf{M}(S_{j}),$

and thus

$\mathscr{F}_{K}(C_{j}-C)\leq\mathbf{M}(\boldsymbol{\delta}_{v}\hbox{{$\times$}\kern-6.49994pt{$\times$}}R_{j}+S_{j})+\mathbf{M}(\boldsymbol{\delta}_{v}\hbox{{$\times$}\kern-6.49994pt{$\times$}}S_{j})\leq\rho\mathbf{M}(R_{j})+\mathbf{M}(S_{j})+\rho\mathbf{M}(S_{j})\leq(1+\rho)\epsilon.$

This shows that $\lim_{j\to\infty}\mathscr{F}_{K}(C_{j}-C)=0$ as desired. ∎

######  Remark 5.4.

The following proof of Theorem 5.1 is similar to the one of Theorem 4.18.
Nevertheless, as the boundary of the minimizing sequence is not fixed, one can
not apply directly Theorem 4.18 here.

###### Proof of Theorem 5.1.

Since $\mathcal{T}_{0}\in{\it Scan}_{m,K}(\mathbb{R}^{n})$, by definition,
there exists a $T_{0}\in\mathscr{R}_{m,K}(\mathbb{R}^{n})$ with
$\partial\mathcal{T}_{0}=\partial(\mathcal{S}(T_{0}))$. Since the zero-current
${\bf 0}\preceq B$, both $\mathcal{T}_{0}$ and $\mathcal{S}(T_{0})$ are
contained in the set $\\{\mathcal{T}\in{\it
Scan}_{m,K}(\mathbb{R}^{n}):\partial(\mathcal{T}-\mathcal{T}_{0})\preceq\mathcal{S}(B)\\}$.
Let $\\{\mathcal{T}_{i}\\}_{i=1}^{\infty}$ be an $\mathbf{E}_{H,\Phi}$
minimizing sequence in this collection. That is, $\mathcal{T}_{i}\in{\it
Scan}_{m,K}(\mathbb{R}^{n})$ with
$\partial(\mathcal{T}_{i}-\mathcal{T}_{0})\preceq\mathcal{S}(B)$ for each
$i=1,2,\dots$, and

$\lim_{i\to\infty}\mathbf{E}_{H,\Phi}(\mathcal{T}_{i})=\inf\\{\mathbf{E}_{H,\Phi}(\mathcal{T}):\mathcal{T}\in{\it
Scan}_{m,K}(\mathbb{R}^{n}),\
\partial(\mathcal{T}_{i}-\mathcal{T}_{0})\preceq\mathcal{S}(B)\\}.$ (44)

Without loss of generality, we may assume that
$\mathbf{E}_{H,\Phi}(\mathcal{T}_{i})\leq\mathbf{E}_{H,\Phi}(\mathcal{S}(T_{0}))=\mathbf{E}_{H,\Phi}(T_{0})$
for all $i\in\mathbb{N}$. Note that by definition, our boundary condition
$\partial(\mathcal{T}_{i}-\mathcal{T}_{0})\preceq\mathcal{S}(B)$ implies that
for each $i\in\mathbb{N}$, there exists a
$T_{i}\in\mathscr{R}_{m,K}(\mathbb{R}^{n})$ such that
$\partial\mathcal{T}_{i}=\partial(\mathcal{S}(T_{i}))$ and
$\partial(T_{i}-T_{0})\preceq B$. Note also that by the boundary
rectifiability theorem [17],
$\partial(T_{i}-T_{0})\in\mathscr{R}_{m-1}(\mathbb{R}^{n})$ and hence
$T_{i}-T_{0}$ are integral currents. Now,

$\displaystyle\underline{\mathbf{M}}_{H}(\mathcal{T}_{i})$
$\displaystyle=\mathbf{E}_{H,\Phi}(\mathcal{T}_{i})+\partial
T_{i}(\Phi)\leq\mathbf{E}_{H,\Phi}(T_{0})+\partial
T_{i}(\Phi)=\mathbf{M}_{H}(T_{0})+\partial(T_{i}-T_{0})(\Phi)$ (45)
$\displaystyle\leq\mathbf{M}_{H}(T_{0})+\mathbf{M}(\partial(T_{i}-T_{0}))||\Phi||\leq\mathbf{M}_{H}(T_{0})+\mathbf{M}(B)||\Phi||<\infty,$
(46)

since $\mathbf{M}_{H}(T_{0})\leq H(2)\mathbf{M}(T_{0})<\infty$ by using the
fact that $H(\theta)\leq H(2)\theta$ for all $\theta\geq 1$ in equation (20).
By definition of $\underline{\mathbf{M}}_{H}$ in (32), for each
$i\in\mathbb{N}$, there exists a sequence $\\{T_{j}^{i}\\}_{j=1}^{\infty}$ in
$\mathscr{R}_{m,K}(\mathbb{R}^{n})$ such that

$\lim_{j\to\infty}\mathbf{M}_{H}(T_{j}^{i})=\underline{\mathbf{M}}_{H}(\mathcal{T}_{i}),\quad\text{
and }\quad\partial(\mathcal{S}(T_{j}^{i}))=\partial\mathcal{T}_{i}\text{ for
all }j\in\mathbb{N}.$ (47)

Notice that since
$\partial(\mathcal{S}(T_{j}^{i}))=\partial\mathcal{T}_{i}=\partial(\mathcal{S}(T_{i}))$
it follows that $\partial T^{i}_{j}=\partial T_{i}$ for all
$i,j\in\mathbb{N}$. By (47), for each $i\in\mathbb{N}$ there exists an
$N_{i}\in\mathbb{N}$ such that

$|\mathbf{M}_{H}(T^{i}_{N_{i}})-\underline{\mathbf{M}}_{H}(\mathcal{T}_{i})|\leq\frac{1}{i}.$
(48)

Hence by (46) and (48), for all $i\in\mathbb{N}$,

$\displaystyle\mathbf{M}_{H}(\mathcal{S}(T^{i}_{N_{i}}-T_{0}))$
$\displaystyle=\mathbf{M}_{H}(T^{i}_{N_{i}}-T_{0})\leq\mathbf{M}_{H}(T^{i}_{N_{i}})+\mathbf{M}_{H}(T_{0})$
$\displaystyle\leq\underline{\mathbf{M}}_{H}(\mathcal{T}_{i})+1+\mathbf{M}_{H}(T_{0})\leq
2\mathbf{M}_{H}(T_{0})+\mathbf{M}(B)||\Phi||+1.$

Furthermore, by [5, Remark 3.3.3], [5, Lemma 3.1.2], Lemma 5.2 and the fact
that $\partial T^{i}_{N_{i}}=\partial T_{i}$, it holds that

$\displaystyle\mathbf{D}(\mathcal{S}(T^{i}_{N_{i}}-T_{0}))$
$\displaystyle\leq\mathbf{c}_{1}[\mathbf{M}_{H}((T^{i}_{N_{i}}-T_{0})\times\llbracket\mathbf{O}^{\ast}(n,m)\rrbracket)+\mathbf{M}_{H}(\partial((T^{i}_{N_{i}}-T_{0})\times\llbracket\mathbf{O}^{\ast}(n,m)\rrbracket))]$
$\displaystyle=\mathbf{c}_{1}[\mathbf{M}_{H}((T^{i}_{N_{i}}-T_{0})\times\llbracket\mathbf{O}^{\ast}(n,m)\rrbracket)+\mathbf{M}_{H}(\partial((T_{i}-T_{0})\times\llbracket\mathbf{O}^{\ast}(n,m)\rrbracket))]$
$\displaystyle\leq\mathbf{c}_{2}[\mathbf{M}_{H}(T^{i}_{N_{i}}-T_{0})+\mathbf{M}_{H}(\partial(T_{i}-T_{0}))]$
$\displaystyle\leq\mathbf{c}_{2}[2\mathbf{M}_{H}(T_{0})+\mathbf{M}(B)||\Phi||+1+\mathbf{M}_{H}(B)],$

where the constants $\mathbf{c}_{1}$ and $\mathbf{c}_{2}$ depend only on $n$
and $K$. Thus, for each $i$,

$\displaystyle\mathbf{M}_{H}(\mathcal{S}(T^{i}_{N_{i}}-T_{0}))+\mathbf{D}(\mathcal{S}(T^{i}_{N_{i}}-T_{0}))$
$\displaystyle\leq(1+\mathbf{c}_{2})(2\mathbf{M}_{H}(T_{0})+\mathbf{M}(B)||\Phi||+1)+\mathbf{c}_{2}\mathbf{M}_{H}(B)<\infty.$

Now, by Corollary 4.6 and replacing $\\{T_{N_{i}}^{i}\\}$ by a subsequence if
necessary, we may assume that

$\lim_{i\rightarrow\infty}\mathscr{F}_{K}^{H}[\mathcal{S}(T_{N_{i}}^{i}-T_{0})(p,y)-\mathcal{R}(p,y)]=0$
(49)

for $\boldsymbol{\theta}^{\ast}_{n,m}\times\mathcal{L}^{m}$ almost every
$(p,y)\in\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}$, where
$\mathcal{R}:\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}\to\mathbf{I}_{0,K}(\mathbb{R}^{n})$
is a measurable map. Let $\mathcal{T}^{\ast}=\mathcal{R}+\mathcal{S}(T_{0})$.
By definition, (49) implies $\mathcal{S}(T_{N_{i}}^{i})\to\mathcal{T}^{\ast}$.
By Lemma 4.17, this implies for a.e. $(p,y)$,

$\lim_{i\to\infty}(\mathcal{T}^{\ast}-\mathcal{S}(T_{i}))(p,y)(\boldsymbol{1})=\lim_{i\to\infty}(\mathcal{T}^{\ast}-\mathcal{S}(T_{N_{i}}^{i}))(p,y)(\boldsymbol{1})+\lim_{i\to\infty}\langle
T_{N_{i}}^{i}-T_{i},p,y\rangle(\boldsymbol{1})=0,$ (50)

where the second equality is given by [5, Proposition 3.1.6.] since $\partial
T_{N_{i}}^{i}=\partial T_{i}$ for all $i\in\mathbb{N}$.

Our next goal is to show that $\mathcal{T}^{\ast}\in{\it
Scan}_{m,K}(\mathbb{R}^{n})$ and satisfy the boundary condition
$\partial(\mathcal{T}^{\ast}-\mathcal{T}_{0})\preceq\mathcal{S}(B)$. Since
$\partial(\mathcal{T}^{\ast}-\mathcal{T}_{0})=\partial(\mathcal{T}^{\ast}-\mathcal{S}(T_{0}))=\partial\mathcal{R}$,
it is sufficient to show that $\partial\mathcal{R}=\partial(\mathcal{S}(C))$
for some $C\in\mathscr{R}_{m,K}(\mathbb{R}^{n})$ with $\partial C\preceq B$.
Since $\partial(T_{i}-T_{0})\preceq B$, we have
$\mathbf{M}(\partial(T_{i}-T_{0}))\leq\mathbf{M}(B)<\infty$ for each $i$. By
the compactness theorem of rectifiable currents and replacing $(T_{i}-T_{0})$
by a subsequence if necessary, we may assume that
$\lim_{i\rightarrow\infty}\mathscr{F}_{K}(\partial(T_{i}-T_{0})-S)=0$ for some
$S\in\mathscr{R}_{m-1}(\mathbb{R}^{n})$. Since $\partial(T_{i}-T_{0})\preceq
B$, by Lemma 3.2, $S\preceq B$.

Let
$C_{i}=\boldsymbol{\delta}_{v}\hbox{{$\times$}\kern-6.49994pt{$\times$}}\partial(T_{i}-T_{0})$
and $C=\boldsymbol{\delta}_{v}\hbox{{$\times$}\kern-6.49994pt{$\times$}}S$ be
the cones with a fixed cone vertex $v\in K$. Note that both $C_{i}$ and $C$
are in $\mathscr{R}_{m,K}(\mathbb{R}^{n})$ since $K$ is convex. Then,
$\partial C_{i}=\partial(T_{i}-T_{0})$ and by Lemma 5.3,
$\displaystyle\lim_{i\to\infty}\mathscr{F}_{K}(C_{i}-C)=0$. By the integral-
geometric equality,

$\int_{\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}}\mathscr{F}_{K}(\langle
C_{i}-C,p,y\rangle)\,d\left(\boldsymbol{\theta}^{\ast}_{n,m}\times\mathcal{L}^{m}\right)(p,y)\leq\boldsymbol{\beta}_{1}(n,m)\mathscr{F}_{K}(C_{i}-C)\to
0.$

Thus, for $\boldsymbol{\theta}^{\ast}_{n,m}\times\mathcal{L}^{m}$-a.e.
$(p,y)\in\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}$,
$\lim_{i\to\infty}\mathscr{F}_{K}(\langle C_{i}-C,p,y\rangle)=0$. As a result,
for $\boldsymbol{\theta}^{\ast}_{n,m}\times\mathcal{L}^{m}$-a.e.
$(p,y)\in\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}$, by (50),

$\displaystyle(\mathcal{R}-\mathcal{S}(C))(p,y)(\boldsymbol{1})$
$\displaystyle=(\mathcal{T}^{\ast}-\mathcal{S}(T_{0})-\mathcal{S}(C))(p,y)(\boldsymbol{1})$
$\displaystyle=\lim_{i\to\infty}(\mathcal{T}^{\ast}-\mathcal{S}(T_{i}))(p,y)(\boldsymbol{1})+\lim_{i\to\infty}\langle
T_{i}-T_{0}-C_{i},p,y\rangle(\boldsymbol{1})=0,$

because $\partial C_{i}=\partial(T_{i}-T_{0})$. This shows that
$\partial(\mathcal{R}-\mathcal{S}(C))=0$. Since $\partial C=S\preceq B$, we
have $\partial\mathcal{R}=\mathcal{S}(\partial C)\preceq S(B)$. This gives us
our boundary condition,
$\partial(\mathcal{T}^{\ast}-\mathcal{S}(T_{0}+C))=\partial(\mathcal{R}-\mathcal{S}(C))=0$.
Since $\mathcal{S}(T_{N_{i}}^{i})\to\mathcal{T}^{\ast}$, it follows that
$\mathcal{T}^{\ast}\in{\it Scan}_{m,K}(\mathbb{R}^{n})$ with
$\partial(\mathcal{T}^{\ast}-\mathcal{S}(T_{0}))=\partial\mathcal{R}\preceq
B$. Hence $\mathcal{T}^{\ast}$ belongs to the collection
$\\{\mathcal{T}\in{\it
Scan}_{m,K}(\mathbb{R}^{n}):\partial(\mathcal{T}-\mathcal{S}(T_{0}))\preceq\mathcal{S}(B)\\}$.
Additionally, by (32), (48), and (44),

$\displaystyle\mathbf{E}_{H,\Phi}(\mathcal{T}^{*})$
$\displaystyle=\underline{\mathbf{M}}_{H}(\mathcal{T}^{\ast})-\partial(T_{0}+C)(\Phi)$
$\displaystyle\leq\liminf_{i\to\infty}\mathbf{M}_{H}(T_{N_{i}}^{i})-\lim_{i\to\infty}\partial(T_{0}+C_{i})(\Phi)$
$\displaystyle=\liminf_{i\to\infty}\underline{\mathbf{M}}_{H}(\mathcal{T}_{i})-\lim_{i\to\infty}\partial
T_{i}(\Phi)$
$\displaystyle=\liminf_{i\to\infty}\mathbf{E}_{H,\Phi}(\mathcal{T}_{i})$
$\displaystyle=\inf\\{\mathbf{E}_{H,\Phi}(\mathcal{T}):\mathcal{T}\in{\it
Scan}_{m,K}(\mathbb{R}^{n}),\
\partial(\mathcal{T}-\mathcal{S}(T_{0}))\preceq\mathcal{S}(B)\\}.$

Therefore, $\mathcal{T}^{\ast}$ is a desired $\mathbf{E}_{H,\Phi}$ minimizer
satisfying (41).

Now, suppose that $\mathcal{T}_{0}=\mathcal{S}(T_{0})$ for some
$T_{0}\in\mathscr{R}_{m,K}(\mathbb{R}^{n})$. We prove (42) as follows. Since
$\\{\mathcal{S}(T):T\in\mathscr{R}_{m,K}(\mathbb{R}^{n})\\}\subseteq{\it
Scan}_{m,K}(\mathbb{R}^{n})$, by Lemma 4.15, we have

$\displaystyle\mathbf{E}_{H,\Phi}(\mathcal{T}^{*})$
$\displaystyle\leq\inf\\{\mathbf{E}_{H,\Phi}(T):T\in\mathscr{R}_{m,K}(\mathbb{R}^{n}),\partial(T-T_{0})\preceq
B\\}$
$\displaystyle=\inf\\{\mathbf{E}_{H,\Phi}(T):T\in\mathscr{R}_{m}(\mathbb{R}^{n}),\partial(T-T_{0})\preceq
B\\}\leq\mathbf{E}_{H,\Phi}(T^{i}_{N_{i}})$

for each $i$. Taking $i\rightarrow\infty$, by (47), we have the desired
equality (42). ∎

In particular, when $T_{0}=0$, we have the following corollary.

######  Corollary 5.5.

Let $K$ be a nonempty, compact, and convex subset of $\mathbb{R}^{n}$,
$\Phi\in\mathscr{D}^{m-1}(\mathbb{R}^{n})$, $H$ a concave integrand, and let
$B\in\mathscr{R}_{m-1}(\mathbb{R}^{n})$ with $\mathbf{M}_{H}(B)<\infty$. Then
there exists
$\mathcal{T}^{\ast}:\mathbf{O}^{\ast}(n,m)\times\mathbb{R}^{m}\to\mathbf{I}_{0,K}(\mathbb{R}^{n})$
such that $\mathcal{T}^{\ast}\in{\it Scan}_{m,K}(\mathbb{R}^{n})$,
$\partial\mathcal{T}^{\ast}\preceq\mathcal{S}(B)$, and

$\displaystyle\mathbf{E}_{H,\Phi}(\mathcal{T}^{*})$
$\displaystyle=\min\\{\mathbf{E}_{H,\Phi}(\mathcal{T}):\mathcal{T}\in{\it
Scan}_{m,K}(\mathbb{R}^{n}),\partial\mathcal{T}\preceq\mathcal{S}(B)\\}$
$\displaystyle=\inf\\{\mathbf{E}_{H,\Phi}(T):T\in\mathscr{R}_{m}(\mathbb{R}^{n}),\partial
T\preceq B\\}.$

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