# A non-parametric Plateau problem with partial free boundary

Giovanni Bellettini 111 Dipartimento di Ingegneria dell’Informazione e Scienze
Matematiche, Università di Siena, 53100 Siena, Italy, and International Centre
for Theoretical Physics ICTP, Mathematics Section, 34151 Trieste, Italy.
E-mail<EMAIL_ADDRESS>Roberta Marziani222 Angewandte Mathematik,
WWU Münster, Germany. E-mail<EMAIL_ADDRESS>Riccardo
Scala333 Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche,
Università di Siena, 53100 Siena, Italy. E-mail<EMAIL_ADDRESS>
###### Abstract

We consider a Plateau problem in codimension $1$ in the non-parametric
setting. A Dirichlet boundary datum is given only on part of the boundary
$\partial\Omega$ of a bounded convex domain $\Omega\subset\mathbb{R}^{2}$.
Where the Dirichlet datum is not prescribed, we allow a free contact with the
horizontal plane. We show existence of a solution, and prove regularity for
the corresponding minimal surface. Finally we compare these solutions with the
classical minimal surfaces of Meeks and Yau, and show that they are equivalent
when the Dirichlet boundary datum is assigned in at most $2$ disjoint arcs of
$\partial\Omega$.

Key words: Plateau problem, relaxation, Cartesian currents, area functional,
minimal surfaces.

AMS (MOS) 2020 Subject Classification: 49J45, 49Q05, 49Q15, 28A75.

##  1 Introduction

Let $\Omega\subset\mathbb{R}^{2}$ be a bounded open convex set; in this paper
we look for an area-minimizing surface which can be written as a graph over a
subset of $\Omega$, and spanning a Jordan curve
$\Gamma_{\sigma}=\gamma\cup\sigma\subset\mathbb{R}^{2}\times[0,+\infty)$. Here
$\gamma$ is fixed (Dirichlet condition) and is given by a family
$\\{\gamma_{i}\\}_{i=1}^{n}\subset\partial\Omega\times[0,+\infty)$ of
$n\in\mathbb{N}$ curves each joining pairs of points
$\\{(p_{i},q_{i})\\}_{i=1}^{n}$ of $\partial\Omega$. Whereas $\sigma$, which
represents the free boundary, consists of (the image of) $n$ curves
$(\sigma_{1},\dots,\sigma_{n})$ sitting in the plane containing $\Omega$ (also
called free boundary plane), and joining the endpoints of $\gamma$ in order
that $\gamma\cup\sigma$ forms a Jordan curve $\Gamma_{\sigma}$ in
$\mathbb{R}^{3}$. We assume that each $\gamma_{i}$ is Cartesian, i.e., it can
be expressed as the graph of a given nonnegative function $\varphi$ defined on
a corresponding portion of $\partial\Omega$. This allows to restrict ourselves
to the Cartesian setting, and to assume that the competitors for the Plateau
problem are expressed by graphs of functions $\psi$ defined on a suitable
subdomain of $\Omega$ depending on $\sigma$.

Our prototypical example is given by the catenoid. Consider a cylinder in
$\mathbb{R}^{3}$ with a circle of radius $r$ as basis, and height $l$. Choose
a system of Cartesian coordinates in which the $x_{1}x_{2}$-plane contains the
cylinder axis, and restrict attention to the half-space $\\{x_{3}\geq 0\\}$ as
in Figure 1, where $\Omega=(0,l)\times(-r,r)$ and $n=2$. Write

$\partial\Omega=\partial^{0}_{1}\Omega\cup\overline{\partial^{D}_{1}\Omega}\cup\partial^{0}_{2}\Omega\cup\overline{\partial^{D}_{2}\Omega},$

where $\partial^{0}_{1}\Omega=(0,l)\times\\{r\\}$,
$\partial^{0}_{2}\Omega=(0,l)\times\\{-r\\}$,
$\partial^{D}_{1}\Omega=\\{0\\}\times(-r,r)$, and
$\partial^{D}_{2}\Omega=\\{l\\}\times(-r,r)$. On the Dirichlet boundary
$\partial^{D}\Omega=\partial^{D}_{1}\Omega\cup\partial^{D}_{2}\Omega$ we
prescribe a continuous function $\varphi$ whose graph consists of the two
half-circles $\gamma_{1}$ and $\gamma_{2}$. The endpoints of $\gamma_{1}$ and
$\gamma_{2}$ live on the free boundary plane (the horizontal plane) and are
$p_{1}=(0,-r)$, $q_{1}=(0,r)$, and $p_{2}=(l,r)$, $q_{2}=(l,-r)$ respectively.
The free boundary $\sigma$ consists of two curves $\sigma_{1}$ and
$\sigma_{2}$ with endpoints $q_{1},p_{2}$, and $q_{2},p_{1}$, respectively,
constrained to stay in $\overline{\Omega}$. The concatenation of
$\gamma=\gamma_{1}\cup\gamma_{2}$ and $\sigma$ forms a Jordan curve in
$\mathbb{R}^{3}$

$\Gamma_{\sigma}=\gamma_{1}\cup\sigma_{1}\cup\gamma_{2}\cup\sigma_{2}.$

Therefore we proceed to look for an area-minimizer among all Cartesian
surfaces $S$ with boundary $\Gamma_{\sigma}$ keeping $\sigma$ free, i.e. we
minimize the area among all pairs $(\sigma,S)$. In this particular case of the
catenoid, a minimizing sequence $((\sigma_{k},S_{k}))$ tends (in a suitable
way specified in the sequel) to a minimizer $(\sigma,S)$ which allows for two
different possibilities. If $l$ is small, $\sigma_{1}$ and $\sigma_{2}$ remain
disjoint and the classical catenoid (half of it, namely the intersection
between the catenoid and the half-space $\\{x_{3}\geq 0\\}$) is the surface
$S$, in turn coinciding with the graph of a function $\psi$ defined on the
region of $\Omega$ “enclosed” by $\sigma$. If instead $l$ is large, the two
curves $\sigma_{1}$ and $\sigma_{2}$ merge and the region of $\Omega$ enclosed
by $\sigma$ tends to become empty (it reduces to the two segments
$\partial_{1}^{D}\Omega\cup\partial_{2}^{D}\Omega$). This describes the
solution given by two (half) discs.

Figure 1: The catenoid: when $l$ is large enough the two dotted curves
$\sigma_{1}$ and $\sigma_{2}$ merge and the (generalized) graph of $\psi$
reduces to two vertical half-circles on
$\partial^{D}\Omega=\partial_{1}^{D}\Omega\cup\partial_{2}^{D}\Omega$. In this
case $\partial^{D}\Omega\subset\partial E(\sigma_{1})\cup\partial
E(\sigma_{2})$.

A peculiarity of our problem is the presence of a free boundary. The problem
of Plateau with partial free boundary has been exhaustively studied (see for
instance [10]) but never investigated, to our best knowledge, with the non-
parametric approach.

Referring to Section 2 for the precise description of the mathematical
framework, here we just describe it with few details. We fix some distinct
points $p_{1},q_{1},p_{2},q_{2},\dots,p_{n},q_{n}\in\partial\Omega$ taken in
clockwise order. The part of $\partial\Omega$ between the points $p_{i}$ and
$q_{i}$ is noted by $\partial_{i}^{D}\Omega$, and the part between $q_{i}$ and
$p_{i+1}$ by $\partial_{i}^{0}\Omega$. We fix a nonnegative continuous
function $\varphi\colon\partial\Omega\to[0,+\infty)$ which is positive on
$\partial^{D}\Omega=\cup_{i=1}^{n}\partial^{D}_{i}\Omega$ and vanishes on
$\\{p_{i},q_{i}\\}_{i=1}^{n}\cup\partial^{0}\Omega$ with
$\partial^{0}\Omega=\cup_{i=1}^{n}\partial^{0}_{i}\Omega$ , and we consider
Lipschitz injective and mutually disjoint curves $\sigma_{i}$ in
$\overline{\Omega}$, $i=1,\dots,n$, joining $p_{i}$ to $q_{i+1}$. We suppose
the graph of $\varphi$ on $\partial^{D}\Omega$ to be a Lipschitz curve in
$\mathbb{R}^{3}$. We define $E(\sigma):=\cup_{i=1}^{n}E(\sigma_{i})$, with
$E(\sigma_{i})$ the planar closed region enclosed between
$\partial^{0}_{i}\Omega$ and $\sigma_{i}$.

We define the two classes

$\displaystyle\widehat{\Sigma}:=\\{\sigma=(\sigma_{1},\sigma_{2},\dots,\sigma_{n}):[0,1]\rightarrow\overline{\Omega}^{n}\text{
be curves as above}\\},$ (1.1)
$\displaystyle\mathcal{X}_{\varphi}:=\\{(\sigma,\psi)\in\widehat{\Sigma}\times
W^{1,1}(\Omega):\psi=0\text{ a.e. in }E(\sigma)\text{ and }\psi=\varphi\text{
on }\partial^{D}\Omega\\}.$ (1.2)

We want to find a solution to the following minimum problem:

$\displaystyle\inf_{(\sigma,\psi)\in\mathcal{X}_{\varphi}}\mathbb{A}(\psi;\Omega\setminus
E(\sigma)),$ (1.3)

where $\mathbb{A}$ denotes the classical area integral, i.e.,

$\mathbb{A}(\psi;\Omega\setminus E(\sigma)):=\int_{\Omega\setminus
E(\sigma)}\sqrt{1+|\nabla\psi|^{2}}dx.$ (1.4)

Since, in general, existence of minimizers is not guaranteed in the class
$\mathcal{X}_{\varphi}$, we need to formulate this problem to a more suited
space of admissible pairs. Specifically, a standard relaxation procedure leads
one to analyse the problem above for pairs $(\sigma,\psi)$ belonging to
$\Sigma\times BV(\Omega)$, where $\Sigma$ is a suitable class containing
$\widehat{\Sigma}$ but which also allows for partial overlapping of the curves
$\sigma_{i}$ (a precise definition is given in Section 2.3). Therefore we
shall be concerned with the study of the functional $\mathcal{F}_{\varphi}$
defined as

$\displaystyle\mathcal{F}_{\varphi}(\sigma,\psi):=\mathcal{A}(\psi;\Omega)-|E(\sigma)|+\int_{\partial\Omega}|\psi-\varphi|d\mathcal{H}^{1},$
(1.5)

where $(\sigma,\psi)\in\mathcal{W}\subset\Sigma\times BV(\Omega)$,
$\mathcal{W}$ is the space of pairs $(\sigma,\psi)\in\Sigma\times BV(\Omega)$
such that $\psi=0$ a.e. on $E(\sigma)$, and $\mathcal{A}(\psi;\Omega)$ is the
relaxed area functional defined as in (2.1), which accounts for the area of
the generalized graph of the map $\psi$ on $\Omega$. The functional
$\mathcal{F}_{\varphi}$ extends the area integral $\mathbb{A}$ to the larger
class $\mathcal{W}$.

Figure 2: An example of the setting (in 3D), when $n=3$. On the boundary of
the convex set $\Omega$ we have fixed the points $p_{i}$, $q_{i}$; the arc of
$\partial\Omega$ joining $p_{i}$ to $q_{i}$ is $\partial_{i}^{D}\Omega$, while
the arc joining $q_{i}$ to $p_{i+1}$ is $\partial^{0}_{i}\Omega$
($p_{4}:=p_{1}$). On $\partial^{D}\Omega$ the Dirichlet boundary datum
$\varphi$ is imposed, whose graph has been depicted. The dotted arcs are the
free planar curves $\sigma_{i}$ joining the pairs $(q_{i},p_{i+1})$.

We then prove the following result, accounting for existence and regularity of
minimizers of $\mathcal{F}_{\varphi}$.

######  Theorem 1.1.

There exists a minimizer of $\mathcal{F}_{\varphi}$ on $\mathcal{W}$.
Moreover, any minimizer $(\sigma,\psi)\in\mathcal{W}$ of
$\mathcal{F}_{\varphi}$ satisfies the following regularity properties:

  * $(1)$

The region $E(\sigma)$ consists of a family of closed convex sets. The
boundary $\partial E(\sigma)$ is given by the union of the arcs
$\partial^{0}\Omega$ and a family of disjoint Lipschitz curves in
$\overline{\Omega}$ (joining the points $p_{i}$ and $q_{j}$, in some order).
Moreover, if $\partial^{D}_{i}\Omega$ is not a straight segment, then
$\partial^{D}_{i}\Omega\cap\partial E(\sigma)=\mathchar 31\relax$. If instead
$\partial^{D}_{i}\Omega$ is a straight segment, then either
$\partial^{D}_{i}\Omega\cap\partial E(\sigma)=\mathchar 31\relax$ or
$\partial^{D}_{i}\Omega\cap\partial E(\sigma)=\partial^{D}_{i}\Omega$.

  * $(2)$

The function $\psi$ is real analytic in $\Omega\setminus E(\sigma)$, and is
continuous on $\partial^{D}\Omega\setminus\partial E(\sigma)$ where it attains
the boundary value $\psi=\varphi$.

  * $(3)$

If $\Omega\cap\partial E(\sigma)\neq\mathchar 31\relax$, there is at least a
minimizer $(\sigma,\psi)$ such that $\psi$ is continuous and null on
$\Omega\cap\partial E(\sigma)$, and moreover $\Omega\cap\partial E(\sigma)$
consists of a family of mutually disjoint smooth curves (joining $p_{i}$ and
$q_{j}$ in some order).

A comparison with classical solutions of the Plateau problem in parametric
form is in order. Denoting by $\gamma_{i}$ the graph of the map $\varphi$ on
$\overline{\partial_{i}^{D}\Omega}$, we consider also ${\rm sym}(\gamma_{i})$,
namely the graph of $-\varphi$ on $\overline{\partial_{i}^{D}\Omega}$, which
is symmetric to $\gamma_{i}$ with respect to the plane containing $\Omega$.
Setting $\Gamma_{i}:=\gamma_{i}\cup{\rm sym}(\gamma_{i})$, this turns out to
be a simple Jordan curve in $\mathbb{R}^{3}$, for all $i=1,\dots,n$. Hence we
can consider the classical Plateau problem for the curves $\Gamma_{i}$. In the
case $n=1$ it is intuitive that a disc-type minimal surface $S$ spanning
$\Gamma=\Gamma_{1}$ will be symmetric with respect to the plane containing
$\Omega$, and that $S^{+}:=S\cap\\{x_{3}\geq 0\\}$ will be a minimal disc with
partial free boundary on $\Omega$. It is interesting to compare such a minimal
disc with the graph of $\psi$, where $(\sigma,\psi)\in\mathcal{W}$ is a
minimizer as in Theorem 1.1. Actually, in this simple case $n=1$, it is not
difficult to see that $S^{+}$ is Cartesian, and it is the graph of a function
$\psi$ which is positive outside the convex region $E(\sigma)$ enclosed by
$\sigma$ and $\partial^{0}\Omega$, and further $(\sigma,\psi)$ is a minimizer
as provided by Theorem 1.1. Also the converse is true: Any minimizer
$(\sigma,\psi)$ that satisfies (1)-(3) of Theorem 1.1 has as graph of $\psi$ a
disc-type surface $S^{+}$ whose double $S=S^{+}\cup S^{-}$ is a classical
solution to the Plateau problem for the curve $\Gamma$.

This result is rigorously stated in Theorem 6.1 of Section 6.1. In Section 6.2
we instead analyse the case $n=2$. In this case one might look for minimal
surfaces obtained as union of two discs spanning $\Gamma_{1}$ and
$\Gamma_{2}$, or else for a catenoid-type surface spanning
$\Gamma=\Gamma_{1}\cup\Gamma_{2}$ together. Appealing to an existence result
due to Meeks and Yau [18], we are able to show the counterpart of Theorem 6.1:
Theorem 6.5, that essentially states that any minimizer
$(\sigma,\psi)\in\mathcal{W}$ of $\mathcal{F}_{\varphi}$ satisfying properties
(1)–(3) of Theorem 1.1 is (the nonnegative half of) a Meeks-Yau solution, and
vice-versa. In order to prove Theorem 6.5 we will strongly use the convexity
of the domain $\Omega$, which implies that the cylinder
$\Omega\times\mathbb{R}$, which contains $\Gamma$ on its boundary, is convex,
and so the results of Meeks and Yau are applicable.

Due to the highly nontrivial arguments used to prove this result, we restrict
our analysis to the case $n=2$, since a generalization to the case $n>2$
probably requires heavy modifications. Indeed, some of the lemmas needed to
prove Theorem 6.5 employ crucially the fact that $\partial^{0}\Omega$ consists
of only two connected components. For this reason we leave the case $n>2$ for
future investigations.

Figure 3:  A possible configuration of the sets $E(\sigma_{i})$ (in the
relaxed problem). Also in this example $n=3$. The (clockwise oriented) arcs
$\wideparen{p_{1}q_{1}}=\partial^{D}_{1}\Omega$,
$\wideparen{p_{2}q_{2}}=\partial^{D}_{2}\Omega$, and
$\wideparen{p_{3}q_{3}}=\partial^{D}_{3}\Omega$ are the set where $\varphi$ is
prescribed and positive. In the set
$\partial^{0}\Omega=\wideparen{q_{1}p_{2}}\cup\wideparen{q_{2}p_{3}}\cup\wideparen{q_{3}p_{1}}$
and on $E(\sigma)=E(\sigma_{1})\cup E(\sigma_{2})\cup E(\sigma_{3})$ we
prescribe $\psi=0$. The curves $\sigma_{i}$ joining $q_{i}$ to $p_{i+1}$ (with
the corresponding set $E(\sigma_{i})$) are indicated. On the dotted segment
$\sigma_{1}$ and $\sigma_{2}$ overlaps with opposite orientations. The
emphasized region $\Omega\setminus E(\sigma)$ is the one where $\psi$ is not
necessarily null.

Let us now come to the reasons for our study. One motivation is the
description of a cluster of soap films which are constrained to wet a given
system of wires $\gamma$ emanating from a given free boundary plane. The soap
films are expected to arrange in such a way to form a free boundary on the
plane. Therefore, the questions of existence of a minimal configuration and
its regularity naturally arise. A second motivation is related to the
description of the singular part of the $L^{1}$-relaxation of the Cartesian
2-codimensional area functional

$\int_{U}\sqrt{1+|\nabla u_{1}|^{2}+|\nabla u_{2}|^{2}+({\rm det}\nabla
u)^{2}}~{}dx,\qquad u=(u_{1},u_{2})\in C^{1}(U;\mathbb{R}^{2}),$

computed on nonsmooth maps. The $L^{1}$-relaxed area functional [1, 14],
denoted by $\mathcal{A}(\cdot;U)$, is mostly unknown, up to a few exceptions,
see [1, 7, 5, 20, 6]. One of the remarkable exceptions is the case of the
vortex map
$u_{V}:B_{l}(0)\setminus\\{0\\}\subset\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$,
defined by $u_{V}(x)=\frac{x}{|x|}$: in this case it can be proved that

$\displaystyle\mathcal{A}(u_{V};B_{l}(0))=\int_{B_{l}(0)}\sqrt{1+|\nabla
u_{V}|^{2}}dx+\inf\mathcal{F}_{\varphi}(\sigma,\psi),$ (1.6)

where the infimum is taken over all pairs $(\sigma,\psi)\in\Sigma\times
BV(R_{2l})$ with $\psi=0$ a.e. on $E(\sigma)$. Here the setting is the
following: $n=1$, $R_{2l}=(0,2l)\times(-1,1)$,
$\partial^{0}R_{2l}=(0,2l)\times\\{1\\}$,
$\partial^{D}R_{2l}=(\\{0\\}\times(-1,1))\cup([0,2l]\times\\{-1\\})\cup(\\{2l\\}\times(-1,1))$,
$p=(0,1)$, $q=(2l,1)$, and $\sigma$ is a unique curve in $\overline{R}_{2l}$
joining $p$ to $q$. The Dirichlet datum
$\varphi:\partial^{D}R_{2l}\rightarrow[0,\infty)$ is the function
$\varphi(w_{1},w_{2})=\sqrt{1-w_{2}^{2}}$. This setting is similar to the
catenoid case, with the difference that the Dirichlet boundary is here
extended to include the basis $(0,2l)\times\\{-1\\}$ and the free curve
$\sigma$ is just one simple curve (see Figure 4).

Figure 4: The domain $R_{2l}$ of the vortex map. The graph of $\varphi$ on
$\partial^{D}R_{2l}$ is emphasized (in particular $\varphi=0$ on the lower
horizontal side), together with an admissible curve $\sigma$, which in this
specific case partially overlaps the Dirichlet boundary. In this example
$n=1$.

In order to construct a recovery sequence for the relaxed area (1.6) of the
vortex map, it is essential to analyse the existence and regularity of
minimizers of $\mathcal{F}_{\varphi}$. In particular, it is necessary to show
that there is at least one sufficiently regular444Conditions provided by
Theorem 1.1 are sufficient. minimizer $(\sigma,\psi)$. The shape of the curve
$\sigma$ and the graph of $\psi$ are related to the vertical part of a
Cartesian current in $B_{l}(0)\times\mathbb{R}^{2}$ which arises as limit of
(the graphs of) a recovery sequence $(v_{k})\subset
C^{1}(B_{l}(0);\mathbb{R}^{2})$ for $\mathcal{A}(u_{V};B_{l}(0))$.

According to what happens for the catenoid, also in this case we have a
dichotomy for the behaviour of minimizers $(\sigma,\psi)$. When $l$ is small,
the solution $(\sigma,\psi)$ consists of a curve $\sigma$ joining $p$ and $q$
whose interior is contained in $R_{2l}$, and its shape is so that $E(\sigma)$
is convex; at the same time the graph of $\psi$ on $R_{2l}\setminus E(\sigma)$
is a sort of half-catenoid, so that if we double it considering also its
symmetric with respect to the plane containing $R_{2l}$, it becomes a sort of
catenoid spanning two radius one circles, and constrained to contain the
segment $(0,2l)\times\\{-1\\}$. When instead $l$ is larger than a certain
threshold, then the solution reduces to two circles spanning the two radius
one and parallel circles.

The structure of the paper is as follows. In Section 2 we introduce the
setting of the problem in detail. In order to prove existence of minimizers of
$\mathcal{F}_{\varphi}$ we first restrict ourselves to prove the result in a
smaller class $\mathcal{W}_{\textrm{conv}}\subset\mathcal{W}$ of admissible
pairs $(\sigma,\psi)$, where compactness is easier and allows to make use of
the direct method. Roughly speaking, the class $\mathcal{W}_{\textrm{conv}}$
accounts only for specific geometries of the free boundary $\sigma$, namely,
it considers configurations for which each set $E(\sigma_{i})$ is convex. In
Section 3 we prove the existence of minimizers of $\mathcal{F}_{\varphi}$ in
$\mathcal{W}_{\textrm{conv}}$. Next, in Section 4, we show the existence of
minimizers in the wider class $\mathcal{W}$ where, essentially, $\sigma$ is
not constrained to the previous geometric features; this result is contained
in Corollary 4.3. To show this we consider a minimizing sequence in
$\mathcal{W}$ and we modify it, by a cut and paste procedure, in order to
construct a minimizing sequence in $\mathcal{W}_{\textrm{conv}}$. In Section 5
we study the regularity properties of minimizers. Specifically, we state and
prove Theorem 5.1, which rephrases in a more precise way the results contained
in Theorem 1.1. Theorem 1.1 follows from Theorem 3.1, Corollary 4.3, and
Theorem 5.1. Eventually in Section 6 we compare the solutions we found with
the classical minimal surfaces spanning $\Gamma$. Here, as anticipated, we
restrict our analysis to the case $n=1,2$, the case $n=2$ essentially giving
rise to either a catenoid-type minimal surface, or two disc-type surfaces
spanning $\Gamma_{1}$ and $\Gamma_{2}$. The main theorems here are Theorems
6.1 and 6.5. The proof of the former, for the case $n=1$, is quite simple,
whereas Theorem 6.5, for the case $n=2$, requires a series of lemmas. In
particular, if $S$ is a Meeks-Yau catenoid-type minimal surface, at one step,
we need to employ a Steiner symmetrization of the $3$-dimensional finite
perimeter set in $\Omega\times\mathbb{R}$ enclosed by $S$. In turn, using
standard results on the condition of equality for the perimeters of a set and
its symmetrization, we are able to show that the starting surface $S$ were
already symmetric with respect to the plane containing $\Omega$, and already
Cartesian, and the conclusion of the proof of Theorem 6.5 is achieved.

##  2 Preliminaries

###  2.1 Area of the graph of a $BV$ function

Let $U\subset\mathbb{R}^{2}$ be a bounded open set. For any $\psi\in BV(U)$ we
denote by $D\psi$ its distributional gradient, so that

$D\psi=\nabla\psi\mathcal{L}^{2}+D^{s}\psi,$

where $\nabla\psi$ is the approximate gradient of $\psi$ and $D^{s}\psi$
denotes the singular part of $D\psi$. We recall that the $L^{1}$-relaxed area
functional reads as [15]

$\displaystyle\mathcal{A}(\psi;U):=\int_{U}\sqrt{1+|\nabla\psi|^{2}}dx+|D^{s}\psi|(U).$
(2.1)

In what follows we denote by $\partial^{*}A$ the reduced boundary of a set of
finite perimeter $A\subset\mathbb{R}^{3}$ (see [2]). For any $\psi\in BV(U)$
we denote by $R_{\psi}\subset U$ the set of regular points of $\psi$, namely
the set of points $x\in U$ which are Lebesgue points for $\psi$, $\psi(x)$
coincides with the Lebesgue value of $\psi$ at $x$ and $\psi$ is approximately
differentiable at $x$. We define the subgraph $SG_{\psi}$ of $\psi$ as

$SG_{\psi}:=\\{(x,y)\in R_{\psi}\times\mathbb{R}\colon y<\psi(x)\\}.$

This turns out to be a finite perimeter set in $U\times\mathbb{R}$. Its
reduced boundary in $U\times\mathbb{R}$ is the generalised graph
$\mathcal{G}_{\psi}:=\\{(x,\psi(x))\colon x\in R_{\psi}\\}$ of $\psi$, which
turns out to be a $2$-rectifiable set. If
$\text{{\rm\textlbrackdbl}}{SG_{\psi}}\text{{\rm\textrbrackdbl}}\in\mathcal{D}_{3}(\mathbb{R}^{3})$
denotes the integral current given by integration over $SG_{\psi}$ and
$\partial\text{{\rm\textlbrackdbl}}{SG_{\psi}}\text{{\rm\textrbrackdbl}}\in\mathcal{D}_{2}(\mathbb{R}^{3})$
is its boundary in the sense of currents, then

$\text{{\rm\textlbrackdbl}}{\mathcal{G}_{\psi}}\text{{\rm\textrbrackdbl}}=\partial\text{{\rm\textlbrackdbl}}{SG_{\psi}}\text{{\rm\textrbrackdbl}}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits(U\times\mathbb{R}),$

with
$\text{{\rm\textlbrackdbl}}{\mathcal{G}_{\psi}}\text{{\rm\textrbrackdbl}}$
denoting the integer multiplicity $2$-current given by integration over
$\mathcal{G}_{\psi}$ (suitably oriented; see [13] for more details).

###  2.2 Hausdorff distance

If $A,B\subset\mathbb{R}^{2}$ are nonempty, the symbol $d_{H}(A,B)$ stands for
the Hausdorff distance between $A$ and $B$, that is

$d_{H}(A,B):=\max\left\\{\sup_{a\in A}\,{\rm d}_{B}(a)\,,\,\sup_{b\in B}\,{\rm
d}_{A}(b)\right\\},$

where $\,{\rm d}_{F}(\cdot)$ is the distance from the nonempty set
$F\subseteq\mathbb{R}^{2}$. If we restrict $d_{H}$ to the class of closed
sets, then $d_{H}$ defines a metric. Moreover:

  1. (H1)

$\,{\rm d}_{A}(x)\leq\,{\rm d}_{B}(x)+d_{H}(A,B)$ for every
$x\in\mathbb{R}^{2}$;

  2. (H2)

$(\mathcal{K},d_{H})$ with
$\mathcal{K}:=\\{K\subset\mathbb{R}^{2}\,\text{nonempty and compact}\\}$ is a
complete metric space;

  3. (H3)

If $A,B\subset\mathbb{R}^{2}$ are bounded, closed, nonempty and convex sets,
then $d_{H}(A,B)=d_{H}(\partial A,\partial B)$;

  4. (H4)

If $A\in\mathcal{K}$ is convex, then there exists a sequence
$(A_{n})_{n}\subset\mathcal{K}$ of convex sets with boundary of class
$C^{\infty}$ such that $d_{H}(A_{n},A)\to 0$ as $n\to\infty$;

  5. (H5)

Let $(A_{n})_{n}$ be a sequence of closed convex sets in $\mathbb{R}^{2}$,
$A\subset\mathbb{R}^{2}$ and $d_{H}(A_{n},A)\to 0$ as $n\to+\infty$. Then $A$
is convex as well;

  6. (H6)

Let $(A_{n})_{n}$ and $A$ be compact convex subsets of $\mathbb{R}^{2}$ such
that $d_{H}(A_{n},A)\to 0$ and let $x\in{\rm int}(A)$; then $x\in A_{n}$
definitely in $n$;

  7. (H7)

Let $A$ and $B$ be closed subsets of $\mathbb{R}^{2}$ with
$d_{H}(A,B)=\varepsilon$. Then $A\subset B^{+}_{\varepsilon}$ and $B\subset
A^{+}_{\varepsilon}$ where, for all $E\subset\mathbb{R}^{2}$, we have set
$E^{+}_{\varepsilon}:=\\{x\in\mathbb{R}^{2}\colon\,{\rm
d}_{E}(x)\leq\varepsilon\\}$.

######  Remark 2.1.

Property (H1) is straightforward, while (H2) is well-known. Also property (H3)
is easily obtained (see, e.g. [21]). Concerning property (H4) we refer to,
e.g., [4, Corollary 2]. To see (H5), from (H1) we have that $\,{\rm
d}_{A_{n}}\to\,{\rm d}_{A}$ pointwise, and therefore since $\,{\rm d}_{A_{n}}$
is convex, also $\,{\rm d}_{A}$ is convex, which implies $A$ convex555Since
$A$ is closed, it coincides with the sublevel $\\{x:d(x,A)\leq 0\\}$, which is
convex.. Let us now prove (H6) by contradiction; assume that there exists a
subsequence $(n_{k})$ such that $\,{\rm d}_{A_{n_{k}}}(x)>0$ for all
$k\in\mathbb{N}$; then $x\in\mathbb{R}^{2}\setminus A_{n_{k}}$, $\,{\rm
d}_{A_{n_{k}}}(x)=\,{\rm d}_{\partial A_{n_{k}}}(x)$, and using (H1) twice,

$\displaystyle\,{\rm d}_{\partial A}(x)$ $\displaystyle\leq\,{\rm d}_{\partial
A_{n_{k}}}(x)+d_{H}(\partial A_{n_{k}},\partial A)=\,{\rm
d}_{A_{n_{k}}}(x)+d_{H}(A_{n_{k}},A)$ $\displaystyle\leq\,{\rm
d}_{A}(x)+2d_{H}(A,A_{n_{k}})=2d_{H}(A,A_{n_{k}})\rightarrow 0,$

the first equality following from (H3). This implies $x\in\partial A$, a
contradiction.

We begin with the following standard result that will be useful later:

######  Lemma 2.2.

Let $K\subset\mathbb{R}^{2}$ be a convex compact set with nonempty interior.
Then there exists a $1$-periodic curve
$\widehat{\sigma}\in{\rm{Lip}}(\mathbb{R};\mathbb{R}^{2})$, injective on
$[0,1)$, such that $\widehat{\sigma}([0,1])=\partial K$ and

$\widehat{\sigma}(t)=\widehat{\sigma}(0)+\ell(\widehat{\sigma})\int_{0}^{t}\widehat{\gamma}(s)\,ds,\quad\widehat{\gamma}(t)=(\cos(\widehat{\theta}(t))\,,\,\sin(\widehat{\theta}(t)))\quad\text{for
all }\,t\in[0,1],$

with $\widehat{\theta}$ a non-decreasing function satisfying
$\widehat{\theta}(t+1)-\widehat{\theta}(t)=2\pi$ for all $t\in\mathbb{R}$.

Notice that $\widehat{\sigma}$ is differentiable a.e. in $\mathbb{R}$ and
$\widehat{\sigma}^{\prime}(t)=\ell(\widehat{\sigma})\widehat{\gamma}(t)$, so
that the speed modulus of the curve
$|\hat{\sigma}^{\prime}(t)|=\ell(\widehat{\sigma})$ is constant and coincides
with the length of the curve
$\ell(\widehat{\sigma})=\int_{0}^{1}|\widehat{\sigma}^{\prime}(s)|ds$.

###### Proof.

We start by approximating $K$ by convex sets with $C^{\infty}$ boundary. By
(H4) for all $n\in\mathbb{N}$ there is a convex compact set
$K_{n}\subset\mathbb{R}^{2}$ with boundary of class $C^{\infty}$ and such that
$d_{H}(K_{n},K)\to 0$ as $n\to\infty$. For any $n\in\mathbb{N}$ we let
$\widehat{\sigma}_{n}\in C^{\infty}(\mathbb{R};\mathbb{R}^{2})$ be a
$1$-periodic function injectively parametrizing $\partial K_{n}$ on $[0,1)$;
therefore $\widehat{\sigma}_{n}([0,1])=\partial K_{n}$, and

$\widehat{\sigma}_{n}(t)=\widehat{\sigma}_{n}(0)+\ell(\widehat{\sigma}_{n})\int_{0}^{t}\widehat{\gamma}_{n}(s)\,ds,\quad\widehat{\gamma}_{n}(t)=(\cos(\widehat{\theta}_{n}(t))\,,\,\sin(\widehat{\theta}_{n}(t)))\quad\forall
t\in[0,1],$

where $\widehat{\theta}_{n}\in C^{\infty}(\mathbb{R})$ is a non-decreasing
function with $\widehat{\theta}_{n}(t+1)-\widehat{\theta}_{n}(t)=2\pi$, for
all $t\in\mathbb{R}$. In view of (H2), by construction we can find $x_{0}\in
K$, $R>r>0$ such that $B_{r}(x_{0})\subset K_{n}\subset B_{R}(x_{0})$ for all
$n\in\mathbb{N}$, and therefore $\mathcal{H}^{1}(\partial
B_{r}(x_{0}))\leq\ell(\widehat{\sigma}_{n})=\mathcal{H}^{1}(\mathcal{\partial}K_{n})\leq\mathcal{H}^{1}(\partial
B_{R}(x_{0}))$; thus, up to subsequence,
$\ell(\widehat{\sigma}_{n})\to\widehat{m}\in\mathbb{R}^{+}$ as $n\to\infty$.
Moreover, up to subsequence, we might assume
$\widehat{\sigma}_{n}(0)\rightarrow p\in\partial K$. On the other hand
observing that

$\int_{t}^{t+1}|\widehat{\theta}^{\prime}_{n}(s)|ds=\int_{t}^{t+1}\widehat{\theta}_{n}^{\prime}(s)ds=2\pi,\text{
for all }t\in\mathbb{R},$

we have that, again up to subsequence,
$\widehat{\theta}_{n}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\widehat{\theta}\in
BV_{{\rm loc}}(\mathbb{R})$ and pointwise (by Helly selection principle), with
$\widehat{\theta}$ a non-decreasing function with
$\widehat{\theta}(t+1)-\widehat{\theta}(t)=2\pi$ for all $t\in\mathbb{R}$. We
also have
$\widehat{\gamma}_{n}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\widehat{\gamma}$
in $BV_{{\rm loc}}(\mathbb{R};\mathbb{R}^{2})$ where
$\widehat{\gamma}(t)=(\cos(\widehat{\theta}(t))\,,\,\sin(\widehat{\theta}(t)))$.
We let $\widehat{\sigma}\in{\rm{Lip}}(\mathbb{R};\mathbb{R}^{2})$ be the
($1$-periodic) simple closed curve defined as

$\widehat{\sigma}(t):=p+\widehat{m}\int_{0}^{t}\widehat{\gamma}(s)\,ds\quad\forall
t\in\mathbb{R}.$ (2.2)

Note that $\widehat{m}=\ell(\widehat{\sigma})$. Then clearly
$\widehat{\sigma}_{n}\to\widehat{\sigma}$ in $W^{1,1}([0,1];\mathbb{R}^{2})$,
since

$\displaystyle\|\widehat{\sigma}^{\prime}_{n}-\widehat{\sigma}^{\prime}\|_{L^{1}([0,1];\mathbb{R}^{2})}$
$\displaystyle=\int_{0}^{1}|\ell(\widehat{\sigma}_{n})\widehat{\gamma}_{n}(t)-\ell(\widehat{\sigma})\widehat{\gamma}(t)|dt$
$\displaystyle\leq|\ell(\widehat{\sigma}_{n})-\ell(\widehat{\sigma})|+\ell(\widehat{\sigma})\int_{0}^{1}|\widehat{\gamma}_{n}(t)-\widehat{\gamma}(t)|dt\to
0.$

By the continuous embedding $W^{1,1}([0,1];\mathbb{R}^{2})\subset
C^{0}([0,1];\mathbb{R}^{2})$ (and by $1$-periodicity, on $\mathbb{R}$) we also
get $\widehat{\sigma}_{n}\to\widehat{\sigma}$ uniformly on $[0,1]$. This,
together with property (H3) gives

$d_{H}(\partial K,\widehat{\sigma}([0,1]))\leq d_{H}(\partial K,\partial
K_{n})+d_{H}(\widehat{\sigma}_{n}([0,1]),\widehat{\sigma}([0,1]))\to 0,$

which in turn implies $\widehat{\sigma}([0,1])=\partial K$. The injectivity of
$\widehat{\sigma}$ on $[0,1)$ follows from expression (2.2), the fact that
$\widehat{m}>0$, and the fact that $K$ is convex with nonempty interior.

∎

######  Corollary 2.3.

Let $K\subset\mathbb{R}^{2}$ be a convex compact set with nonempty interior.
Let $q,p$ be two distinct points on $\partial K$, and let $\wideparen{pq}$ be
the relatively open, connected curve contained in $\partial K$ with endpoints
$q$ and $p$ clockwise ordered. Then there exists an injective curve
$\sigma\in{\rm{Lip}}([0,1];\mathbb{R}^{2})$ such that
$\sigma((0,1))=\wideparen{pq}$, $\sigma(0)=q$, $\sigma(1)=p$, and

$\sigma(t)=q+\ell(\sigma)\int_{0}^{t}\gamma(s)\,ds,\quad\gamma(t)=(\cos(\theta(t))\,,\,\sin(\theta(t))\quad\text{for
all }\,t\in[0,1],$

with $\theta$ a non-decreasing function satisfying $\theta(1)-\theta(0)\leq
2\pi$.

###### Proof.

Lemma 2.2 provides $\widehat{\sigma}\in{\rm Lip}([0,1];\mathbb{R}^{2})$
parametrizing $\partial K$. Then there are two values $t_{1},t_{2}\in[0,1]$,
$t_{1}<t_{2}$, with $q=\sigma(t_{1})$ and $p=\sigma(t_{2})$. Then the
existence of $\sigma$ follows by reparametrization of the interval
$[t_{1},t_{2}]$, and all the properties follows from the corresponding
properties of $\widehat{\sigma}$. ∎

###  2.3 Setting of the problem

We fix $\Omega\subset\mathbb{R}^{2}$ to be an open bounded convex set (strict
convexity is not required) which will be our reference domain. Given two
points $p,q\in\partial\Omega$ in clockwise order, $\wideparen{pq}$ stands for
the relatively open arc on $\partial\Omega$ joining $p$ and $q$.

Let $n\in\mathbb{N}$, $n\geq 1$, and let $\\{p_{i}\\}_{i=1}^{n}$ be distinct
points on $\partial\Omega$ chosen in clockwise order; we set $p_{n+1}:=p_{1}$.
For all $i=1,\dots,n$ let $q_{i}$ be a point in
$\wideparen{p_{i}p_{i+1}}\subset\partial\Omega$. We set

$\displaystyle\partial_{i}^{D}\Omega:=\wideparen{p_{i}q_{i}},\qquad\partial^{0}_{i}\Omega:=\wideparen{q_{i}p_{i+1}}\qquad\text{
for }i=1,\dots,n,$ (2.3)

and

$\partial^{D}\Omega:=\bigcup_{i=1}^{n}\partial_{i}^{D}\Omega,\qquad\partial^{0}\Omega:=\bigcup_{i=1}^{n}\partial^{0}_{i}\Omega.$
(2.4)

Since $\partial_{i}^{D}\Omega$ and $\partial^{0}_{i}\Omega$ are relatively
open in $\partial\Omega$, so are $\partial^{D}\Omega$ and
$\partial^{0}\Omega$. It follows that $\partial\Omega$ is the disjoint union

$\partial\Omega=\cup_{i=1}^{n}\\{p_{i},q_{i}\\}\cup\partial^{D}\Omega\cup\partial^{0}\Omega.$

We fix a continuous function $\varphi:\partial\Omega\rightarrow[0,+\infty)$
such that

$\varphi=0\ {\rm on}~{}\partial^{0}\Omega\qquad{\rm and}\qquad\varphi>0\ {\rm
on}~{}\partial^{D}\Omega,$

see Figures 1, 2. We will make a further regularity assumption on $\varphi$:
we require that the graph $\mathcal{G}_{\varphi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\partial^{D}_{i}\Omega}=\\{(x,\varphi(x)):x\in\partial^{D}_{i}\Omega\\}$
of $\varphi$ on $\partial^{D}_{i}\Omega$ is a Lipschitz curve in
$\mathbb{R}^{3}$, for all $i=1,\dots,n$.

######  Remark 2.4.

The hypothesis $\varphi>0$ on $\partial^{D}\Omega$ excludes from our analysis
the example in Figure 4 of the introduction. We will further comment on this
later on (see Section 5.1); the presence of pieces of $\partial^{D}\Omega$
where $\varphi=0$ will bring to some additional technical difficulties that we
prefer to avoid here. However, the setting in Figure 4 can be easily achieved
by an approximation argument. Namely, one considers a suitable regularization
$\varphi_{\varepsilon}$ of $\varphi$ on $\partial^{D}\Omega$ such that
$\varphi_{\varepsilon}>0$, and then letting $\varepsilon\rightarrow 0$ one
obtains a solution to the problem with Dirichlet datum $\varphi$.

We will analyse the functional $\mathcal{F}=\mathcal{F}_{\varphi}$ defined in
(1.5), namely

$\displaystyle\mathcal{F}(\sigma,\psi):=\mathcal{A}(\psi;\Omega)-|E(\sigma)|+\int_{\partial\Omega}|\psi-\varphi|d\mathcal{H}^{1},$
(2.5)

where the pair $(\sigma,\psi)$ belongs to the admissible class $\mathcal{W}$,
defined as follows:

$\begin{split}&\mathcal{W}:=\Big{\\{}(\sigma,\psi)\in\Sigma\times
BV(\Omega):\psi=0\text{ a.e. in }E(\sigma)\Big{\\}},\\\
&\Sigma:=\Big{\\{}\sigma=(\sigma_{1},\dots,\sigma_{n})\in({\rm
Lip}([0,1];\overline{\Omega}))^{n}\text{ satisfies
(i')-(ii')}\Big{\\}},\end{split}$ (2.6)

where

  1. (i’)

$\sigma=(\sigma_{1},\dots,\sigma_{n})$ with $\sigma_{i}$ injective,
$\sigma_{i}(0)=q_{i}$ and $\sigma_{i}(1)=p_{i+1}$, for all $i=1,\dots,n$;

  2. (ii’)

For $i=1,\dots,n$, denoting by $E(\sigma_{i})\subset\overline{\Omega}$ the
closed region enclosed between $\partial^{0}_{i}\Omega$ and
$\sigma_{i}([0,1])$, we assume
$\textrm{int}(E(\sigma_{i}))\cap\textrm{int}(E(\sigma_{j}))=\mathchar
31\relax$ for $i\neq j$ where int denotes the interior part; we also set

$E(\sigma):=\bigcup_{i=1}^{n}E(\sigma_{i}).$ (2.7)

######  Remark 2.5.

The injectivity property in (i’) guarantees that the sets $E(\sigma_{i})$ are
simply connected (not necessarily connected). The assumption that the interior
${\rm int}(E(\sigma_{i}))$ of the sets $E(\sigma_{i})$ are mutually disjoint
is an hypothesis on the curves $\sigma_{i}$, which essentially translates into
the fact that these curves cannot cross transversally each other, but might
overlap. Notice that ${\rm int}(E(\sigma_{i}))$ might be empty, as the case
$\partial^{0}_{i}\Omega=\sigma_{i}([0,1])$ is not excluded.

The strategy to show existence and regularity of minimizers of the functional
(2.5) (see (3.1)) is to reduce to study the same functional on a restricted
class of competitors, more precisely to reduce our analysis to the case where
the sets $E(\sigma_{i})$ are convex. Specifically, we define:

$\begin{split}&{\mathcal{W}_{\rm conv}}:=\Big{\\{}(\sigma,\psi)\in\Sigma_{\rm
conv}\times BV(\Omega):\ \psi=0\text{ a.e. in }E(\sigma)\Big{\\}},\\\
&\Sigma_{\rm
conv}:=\Big{\\{}\sigma=(\sigma_{1},\dots,\sigma_{n})\in\Sigma:\sigma\text{
satisfies (i)}\Big{\\}},\end{split}$ (2.8)

and:

  1. (i)

For all $i=1,\dots,n$ the set $E(\sigma_{i})$ is convex.

As we have already said, the sets $\textrm{int}(E(\sigma_{i}))$ might also be
empty, since from assumption (i’) we cannot exclude that $\sigma_{i}$ overlaps
$\partial^{0}_{i}\Omega$: Recalling that $\Omega$ is convex, this can happen,
by (ii’) and (i), only if $\wideparen{q_{i}p_{i+1}}$ is a straight
segment666We shall prove that, for a minimizer, $\sigma_{i}([0,1])$ cannot
intersect $\partial^{D}\Omega$ unless $\partial^{D}\Omega$ is locally a
segment, see Theorem 5.1.. Clearly,

${\mathcal{W}_{\rm conv}}\subset\mathcal{W}.$ (2.9)

######  Remark 2.6.

Exploiting the characterization of the boundaries of convex sets given in
Corollary 2.3, we see that conditions (i’),(ii’) and (i) for the curves in
$\Sigma_{\rm conv}$ imply the following:

  1. (P)

For all $i=1,\dots,n$ there is a nondecreasing function
$\theta_{i}\colon[0,1]\to\mathbb{R}$ with $\theta_{i}(1)-\theta_{i}(0)\leq
2\pi$, and such that, setting
$\gamma_{i}(t):=(\cos(\theta_{i}(t))\,,\,\sin(\theta_{i}(t)))$ for all
$t\in[0,1]$, we have

$\sigma_{i}(t)=q_{i}+\ell(\sigma_{i})\int_{0}^{t}\gamma_{i}(s)\,ds\quad\quad\forall
t\in[0,1].$

Here we have denoted the length of $\sigma_{i}$ by $\ell(\sigma_{i})$.

##  3 Existence of minimizers of $\mathcal{F}$ in ${\mathcal{W}_{\rm conv}}$

The main result of this section reads as follows.

######  Theorem 3.1 (Existence of a minimizer of $\mathcal{F}$ in
${\mathcal{W}_{\rm conv}}$).

Let $\mathcal{F}$ and ${\mathcal{W}_{\rm conv}}$ be as in (2.5) and (2.8)
respectively. Then there is a solution to

$\min_{(\sigma,\psi)\in{\mathcal{W}_{\rm conv}}}\mathcal{F}(\sigma,\psi).$
(3.1)

We prove Theorem 3.1 using the direct method. To this aim we need to introduce
a notion of convergence in ${\mathcal{W}_{\rm conv}}$.

######  Definition 3.2 (Convergence in ${\mathcal{W}_{\rm conv}}$).

We say that the sequence
$(((\sigma)_{k},\psi_{k}))_{k}\subset{\mathcal{W}_{\rm conv}}$, with
$(\sigma)_{k}=((\sigma_{1})_{k},...,(\sigma_{n})_{k})$, converges to
$(\sigma,\psi)\in{\mathcal{W}_{\rm conv}}$ if:

  1. $(a)$

$(\sigma_{i})_{k}$ converges to $\sigma_{i}$ uniformly in $[0,1]$ for all
$i=1,\dots,n$;

  2. $(b)$

$(\psi_{k})_{k}$ converges to $\psi$ weakly star in $BV(\Omega)$, i.e.,
$\psi_{k}\to\psi$ in $L^{1}(\Omega)$ and $D\psi_{k}\rightharpoonup D\psi$
weakly star in $\Omega$ in the sense of measures as $k\to+\infty$.

######  Remark 3.3.

For any $i=1,\dots,n$ we have
$\lim_{k\to+\infty}d_{H}(E((\sigma_{i})_{k}),E(\sigma_{i}))=0$, since by
property (H3)

$d_{H}(E((\sigma_{i})_{k}),E(\sigma_{i}))=d_{H}(\partial
E((\sigma_{i})_{k}),\partial
E(\sigma_{i}))=d_{H}((\sigma_{i})_{k}([0,1]),\sigma_{i}([0,1]))\to 0.$

######  Lemma 3.4 (Compactness of ${\mathcal{W}_{\rm conv}}$).

Let $\big{(}((\sigma)_{k},\psi_{k})\big{)}_{k}\subset{\mathcal{W}_{\rm conv}}$
be a sequence with
$\sup_{k}\mathcal{F}((\sigma)_{k},\psi_{k})<+\infty$. Then
$\big{(}((\sigma)_{k},\psi_{k})\big{)}_{k}$ admits a subsequence converging to
an element of ${\mathcal{W}_{\rm conv}}$.

###### Proof.

We divide the proof in two steps.

Step 1: Compactness of $(\sigma)_{k}$. For simplicity we use the notation
$\sigma_{ik}=(\sigma_{i})_{k}$ for every $k\in\mathbb{N}$ and
$i\in\\{1,\dots,n\\}$. By condition (P) in Remark 2.6, for every
$k\in\mathbb{N}$ and $i\in\\{1,\dots,n\\}$ there exists a non-decreasing
function $\theta_{ik}\colon[0,1]\to\mathbb{R}$,
$\theta_{ik}(1)-\theta_{ik}(0)\leq 2\pi$, such that

$\sigma_{ik}(t)=q_{i}+\ell(\sigma_{ik})\int_{0}^{t}\gamma_{ik}(s)ds,\quad\gamma_{ik}(t):=(\cos\theta_{ik}(t)\,,\,\sin\theta_{ik}(t))\quad\forall
t\in[0,1],$

and with $\sigma_{ik}(1)=p_{i+1}$. We observe that

$\ell(\sigma_{ik})=\int_{0}^{1}|\sigma_{ik}^{\prime}(t)|dt\leq\mathcal{H}^{1}(\partial\Omega),$

since the orthogonal projection
$\Pi_{ki}\colon\partial\Omega\setminus\partial^{0}_{i}\Omega\to
E(\sigma_{ik})$ is a contraction and
$\mathcal{H}^{1}(\partial\Omega\setminus\partial^{0}_{i}\Omega)\leq\mathcal{H}^{1}(\partial\Omega)$.
Hence, up to a (not relabelled) subsequence, $\ell(\sigma_{ik})\to
m_{i}\in\mathbb{R}^{+}$ as $k\to+\infty$. The number $m_{i}$ is positive
since, for all $k$ and $i$, we have $\ell(\sigma_{ik})\geq|q_{i}-p_{i+1}|>0$.
Moreover

$\int_{0}^{1}|\theta^{\prime}_{ik}(t)|dt=\int_{0}^{1}\theta_{ik}^{\prime}(t)dt\leq
2\pi;$

hence, up to a subsequence,
$\theta_{ki}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\theta_{i}$ in
$BV([0,1])$ and $\theta_{i}$ is non-decreasing with
$\theta_{i}(1)-\theta_{i}(0)\leq 2\pi$. Furthermore
$\gamma_{ik}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\gamma_{i}$ in
$BV([0,1];\mathbb{R}^{2})$ with
$\gamma_{i}(t)=(\cos(\theta_{i}(t))\,,\,\sin(\theta_{i}(t)))$.

As a consequence $\sigma_{ik}\to\sigma_{i}$ in
$W^{1,1}([0,1];\mathbb{R}^{2})$, where

$\sigma_{i}(t):=q_{i}+m_{i}\int_{0}^{t}\gamma_{i}(s)ds=q_{i}+\ell(\sigma_{i})\int_{0}^{t}\gamma_{i}(s)ds.$

Indeed we have

$\displaystyle\|\sigma_{ik}^{\prime}-\sigma^{\prime}_{i}\|_{L^{1}([0,1];\mathbb{R}^{2})}$
$\displaystyle=\int_{0}^{1}|\ell(\sigma_{ik})\gamma_{ik}(t)-\ell(\sigma_{i})\gamma_{i}(t)|dt$
$\displaystyle\leq\mathcal{H}^{1}(\partial\Omega)\int_{0}^{1}|\gamma_{ik}(t)-\gamma_{i}(t)|dt+|\ell(\sigma_{ik})-\ell(\sigma_{i})|.$
(3.2)

Now taking the limit as $k\to+\infty$ in (3.2) we conclude. Thus
$\lim_{k\to+\infty}\sigma_{ik}=\sigma_{i}$ uniformly, hence we also conclude
that $\sigma_{i}$ takes values in $\overline{\Omega}$.
It remains to show that $E(\sigma_{i})$ is convex for any
$i\in\\{1,\dots,n\\}$. The uniform convergence of $(\sigma_{ik})$ yields

$\lim_{k\to+\infty}d_{H}(\partial E(\sigma_{ik}),\partial E(\sigma_{i}))=0.$

This, together with property (H3), gives for $h\geq k$,

$\displaystyle d_{H}(E(\sigma_{ik}),E(\sigma_{ih}))=d_{H}(\partial
E(\sigma_{ik}),\partial E(\sigma_{ih}))$ $\displaystyle\leq$ $\displaystyle
d_{H}(\partial E(\sigma_{ik}),\partial E(\sigma_{i}))+d_{H}(\partial
E(\sigma_{ih}),\partial E(\sigma_{i}))\to 0\quad\text{as}\quad k\to+\infty,$

and so $(E(\sigma_{ik}))_{k\in\mathbb{N}}$ is a Cauchy sequence in the space
of compact subsets of $\mathbb{R}^{2}$ endowed with the Hausdorff distance
(see (H2)). We find $K\subset\mathbb{R}^{2}$ convex compact such that
$d_{H}(E(\sigma_{ik}),K)\to 0$. Eventually from (H3) we get

$\displaystyle d_{H}(\partial K,\partial E(\sigma_{i}))\leq d_{H}(\partial
E(\sigma_{ik}),\partial K)+d_{H}(\partial E(\sigma_{ik}),\partial
E(\sigma_{i}))$ $\displaystyle=$ $\displaystyle
d_{H}(E(\sigma_{ik}),K)+d_{H}(\partial E(\sigma_{ik}),\partial
E(\sigma_{i}))\to 0\quad\text{as}\quad k\to+\infty.$

Therefore we conclude that $\partial K=\partial E(\sigma_{i})$, so
$E(\sigma_{ik})\rightarrow E(\sigma_{i})$ in the Hausdorff distance, and
$E(\sigma_{i})$ is convex by property (H5).

Step 2: Compactness of $(\psi_{k})$. Setting
$F_{k}=\cup_{i=1}^{n}E(\sigma_{ik})$ we have

$|D\psi_{k}|(\Omega)\leq\mathcal{A}(\psi_{k},\Omega)\leq\mathcal{F}((\sigma)_{k},\psi_{k})+|F_{k}|\leq
C<+\infty\qquad\forall k>0,$

where we used that $|F_{k}|\leq|\Omega|$. Therefore, up to a subsequence,
$\psi_{k}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\psi$ in $BV(\Omega)$ as
$k\to+\infty$. To conclude it remains to show that $\psi=0$ a.e. in
$E(\sigma)=\cup_{i}E(\sigma_{i})$. By
$\lim_{k\to+\infty}d_{H}(F_{k},E(\sigma))=0$, property (H6) yields

$\text{if}\quad x\in\text{int}(E(\sigma))\quad\text{then}\quad x\in
F_{k}\quad\text{definitely in }k,$

and hence since $\lim_{k\to+\infty}\psi_{k}=\psi$ a.e. in $\Omega$, we infer
$\psi=0$ a.e. in $E(\sigma)$.

∎

######  Lemma 3.5 (Lower semicontinuity of $\mathcal{F}$ in ${\mathcal{W}_{\rm
conv}}$).

Let $\big{(}((\sigma)_{k},\psi_{k})\big{)}_{k}\subset{\mathcal{W}_{\rm conv}}$
be a sequence converging to $(\sigma,\psi)\in{\mathcal{W}_{\rm conv}}$. Then

$\mathcal{F}(\sigma,\psi)\leq\liminf_{k\to+\infty}\mathcal{F}((\sigma)_{k},\psi_{k}).$

###### Proof.

By a standard argument [15], the functional

$\psi\in
BV(\Omega)\mapsto\mathcal{A}(\psi;\Omega)+\int_{\partial\Omega}|\psi-\varphi|d\mathcal{H}^{1}$

is $L^{1}(\Omega)$-lower semicontinuous. We now show that the map
$\sigma\in\Sigma_{\rm conv}\mapsto|E(\sigma)|$ is continuous. Let
$(\sigma)_{k}\subset\Sigma_{\rm conv}$, $\sigma\in\Sigma_{\rm conv}$, and
suppose that $(\sigma_{i})_{k}$ uniformly converges to $\sigma_{i}$ for all
$i=1,\dots,n$ as $k\to+\infty$. Set $F_{k}:=\cup_{i=1}^{n}E((\sigma_{i})_{k})$
and recall that $E(\sigma)=\cup_{i=1}^{n}E(\sigma_{i})$. By Remark 3.3
$\lim_{h\to+\infty}d_{H}(E((\sigma_{i})_{k}),E(\sigma_{i}))=0$ for all
$i=1,\dots,n$ and therefore $d_{H}(F_{k},E(\sigma))=:\varepsilon_{k}\to
0^{+}$.

By invoking (H7) we have $E(\sigma)\subset(F_{k})^{+}_{\varepsilon_{k}}$.
Moreover, since $d_{H}((F_{k})_{\varepsilon_{k}}^{+},E(\sigma))\leq
2\varepsilon_{k}$, we get
$(F_{k})^{+}_{\varepsilon_{k}}\subseteq(E(\sigma))_{2\varepsilon_{k}}^{+}$,
and so

$|E(\sigma)|\leq|(F_{k})^{+}_{\varepsilon_{k}}|\leq|(E(\sigma))_{2\varepsilon_{k}}^{+}|.$

This implies

$\limsup_{k\rightarrow+\infty}|F_{k}|\leq\limsup_{k\rightarrow+\infty}|(F_{k})^{+}_{\varepsilon_{k}}|\leq\mathcal{|}E(\sigma)|.$

The converse inequality is a consequence of Fatou’s Lemma and (H6), indeed

$|E(\sigma)|\leq\int_{\Omega}\liminf_{k\rightarrow+\infty}{\chi_{F_{k}}(x)}dx\leq\liminf_{k\rightarrow+\infty}\int_{\Omega}{\chi_{F_{k}}(x)}dx=\liminf_{k\rightarrow+\infty}|F_{k}|.$

The assertion of the lemma follows. ∎

###### Proof of Theorem 3.1.

By Lemma 3.4 and Lemma 3.5 we can apply the direct method and conclude. ∎

##  4 Existence of a minimizer of $\mathcal{F}$ in $\mathcal{W}$

In this section we extend the previous results to the minimization of
${\mathcal{F}}$ in the larger class $\mathcal{W}$ of competitors.

One issue we find in minimizing the functional ${\mathcal{F}}$ on
$\mathcal{W}$, is that the class $\Sigma$ in (2.6) is not closed under uniform
convergence, since a uniform limit of elements in $\Sigma$ needs not be formed
by injective curves. To overcome this difficulty, in Theorem 4.1 we prove that
the infimum of ${\mathcal{F}}$ over $\mathcal{W}$ coincides with the infimum
of ${\mathcal{F}}$ over ${\mathcal{W}_{\rm conv}}$. Thus in particular, by
Theorem 3.1, we derive the existence of a minimizer for ${\mathcal{F}}$ in
$\mathcal{W}$ (Corollary 4.3).

######  Theorem 4.1 (Existence of a minimizer of $\mathcal{F}$ in
$\mathcal{W}$).

There exists $(\sigma,\psi)\in{\mathcal{W}_{\rm conv}}$ such that

${\mathcal{F}}(\sigma,\psi)=\inf_{(s,\zeta)\in\mathcal{W}}{\mathcal{F}}(s,\zeta).$

Moreover every connected component of $E(\sigma)$ is convex.

######  Remark 4.2.

Since the $\sigma_{i}$’s may overlap, the assumption that every
$E(\sigma_{i})$ is convex does not imply in general that every connected
component of $E(\sigma)=\cup_{i=1}^{n}E(\sigma_{i})$ is convex.

As a direct consequence of Theorem 4.1 we have:

######  Corollary 4.3.

Let $(\sigma,\psi)\in{\mathcal{W}_{\rm conv}}$ be a minimizer as in Theorem
3.1. Then $(\sigma,\psi)$ is also a minimizer of ${\mathcal{F}}$ in the class
$\mathcal{W}$.

For the reader convenience we split the proof of Theorem 4.1 into a sequence
of intermediate results: Lemmas 4.4, 4.5, 4.6, and the conclusion. First we
need to introduce some notation.

Let $(\sigma,\psi)\in\mathcal{W}$. We fix an extension $\widehat{\varphi}\in
W^{1,1}(B)$ of $\varphi$ on an open ball $B\supset\overline{\Omega}$.
Extending $\psi$ in $B\setminus\overline{\Omega}$ as $\widehat{\varphi}$
(still denoting by $\psi$ such an extension), we can rewrite
$\mathcal{F}(\sigma,\psi)$ as

${\mathcal{F}}(\sigma,\psi)=\mathcal{A}(\psi;B)-|E(\sigma)|-\mathcal{A}(\psi;B\setminus\overline{\Omega}).$
(4.1)

######  Lemma 4.4.

Let $u\in BV(\mathbb{R}\times(0,+\infty))$ be a nonnegative function with
compact support in an open ball $B_{r}$. Then

$\int_{(\mathbb{R}\times\\{0\\})\cap
B_{r}}u(s)\;d\mathcal{H}^{1}(s)\leq\mathcal{A}(u;B_{r}\cap(\mathbb{R}\times(0,+\infty)))-|E_{B_{r}}|,$
(4.2)

where

$E_{B_{r}}:=\\{x\in B_{r}\cap(\mathbb{R}\times(0,+\infty)):u(x)=0\\}.$

Notice that the function $u$ is defined only on the half-plane
$\mathbb{R}\times(0,+\infty)$, and in (4.2) the symbol $u(s)$ denotes its
trace on the line $\mathbb{R}\times\\{0\\}$.

###### Proof.

We denote by $x=(x_{1},x_{2})\in\mathbb{R}^{2}$ the coordinates in
$\mathbb{R}^{2}$. Set $H^{+}:=\mathbb{R}\times(0,+\infty)$, $Z:=(B_{r}\cap
H^{+})\times\mathbb{R}$. Let

$L:=\\{(x,y)\in Z\colon x\in R_{u},\
y\in(-u(x),u(x))\\}\subset\mathbb{R}^{3},$

where $R_{u}$ is the set of regular points of $u$. We have, recalling the
notation in Section 2.1,

$\begin{split}2\mathcal{A}(u;B_{r}\cap H^{+})&=\mathcal{A}(u;B_{r}\cap
H^{+})+\mathcal{A}(-u;B_{r}\cap H^{+})\\\ &=\mathcal{H}^{2}(\partial^{*}(Z\cap
SG_{u}))+\mathcal{H}^{2}(\partial^{*}(Z\cap SG_{-u}))\\\
&=\mathcal{H}^{2}(Z\cap\partial^{*}L)+2|E_{B_{r}}|.\end{split}$ (4.3)

Suppose $B_{r}\cap(\mathbb{R}\times\\{0\\})=(a,b)\times\\{0\\}$. Then, looking
at $\mathcal{G}_{u}$ as an integral current, a slicing argument yields

$\begin{split}\mathcal{H}^{2}(Z\cap\partial^{*}L)&\geq\int_{a}^{b}\mathcal{H}^{1}(Z\cap\\{x_{1}=t\\}\cap\partial^{*}L)dt\\\
&=\int_{a}^{b}\mathcal{H}^{1}(Z\cap\\{x_{1}=t\\}\cap(\textrm{spt}(\mathcal{G}_{u}-\mathcal{G}_{-u})))dt\\\
&\geq\int_{a}^{b}2u(t,0)dt=2\int_{(\mathbb{R}\times\\{0\\})\cap
B_{r}}u(s)\;d\mathcal{H}^{1}(s)\,,\end{split}$ (4.4)

where the last inequality follows from the following fact: If we denote by
$\text{{\rm\textlbrackdbl}}{\mathcal{G}_{u}}\text{{\rm\textrbrackdbl}}_{t}$
the slice of the current
$\text{{\rm\textlbrackdbl}}{\mathcal{G}_{u}}\text{{\rm\textrbrackdbl}}$ on the
line $\\{x_{1}=t\\}$, then

$\partial\text{{\rm\textlbrackdbl}}{\mathcal{G}_{u}}\text{{\rm\textrbrackdbl}}_{t}=\delta_{(t,0,u(t,0))}-\delta_{(t,s_{t},0)}\qquad{\rm
for~{}a.e.~{}}t\in(a,b),$

where $s_{t}\geq 0$ is such that
$(t,s_{t})=B_{r}\cap(\\{t\\}\times\mathbb{R}^{+})$, and in writing
$\delta_{(t,s_{t},0)}$ we are using that $u$ has compact support in $B_{r}$.
This can be seen, for instance, by approximation of $u$ by smooth maps777With
respect to the strict convergence of $BV(B_{r}\cap(\mathbb{R}\times\\{0\\}))$,
which guarantees the approximation also of the trace of $u$ on
$\partial\big{(}B_{r}\cap(\mathbb{R}\times\\{0\\})\big{)}$.. Therefore

$\partial(\text{{\rm\textlbrackdbl}}{\mathcal{G}_{u}}\text{{\rm\textrbrackdbl}}_{t}-\text{{\rm\textlbrackdbl}}{\mathcal{G}_{-u}}\text{{\rm\textrbrackdbl}}_{t})=\delta_{(t,0,u(t,0))}-\delta_{(t,0,-u(t,0))}\qquad{\rm
for~{}a.e.~{}}t\in(a,b).$

This justifies the last inequality in (4.4), and the proof is achieved. ∎

We now turn to two technical lemmas which are necessary to prove Theorem 4.1.
We need to introduce a class of sets whose boundaries are regular enough so
that the trace of a $BV$ function on them is well-defined. Precisely we say
that an open subset of $\mathbb{R}^{2}$ is piecewise Lipschitz if it can be
written as the union of a finite family of (not necessarily disjoint)
Lipschitz open sets. Notice that, by (2.1) if $V\subset\subset U$ is a
piecewise Lipschitz subset of an open and bounded $U\subset\mathbb{R}^{2}$,
then

$\mathcal{A}(\psi,\overline{V})=\mathcal{A}(\psi,V)+\int_{\partial
V}|\psi^{+}-\psi^{-}|d\mathcal{H}^{1},$ (4.5)

where $\psi^{+}$ (respectively $\psi^{-}$) denotes the trace of
$\psi\mathop{\hbox{\vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits V$ (respectively
$\psi\mathop{\hbox{\vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits(U\setminus\overline{V})$) on
$\partial V$.

######  Lemma 4.5 (Reduction of energy, I).

For $N\geq 1$ let $F_{1},\dots,F_{N}$ be nonempty connected subsets of
$\overline{\Omega}$, each $F_{i}$ being the closure of a piecewise Lipschitz
set, with $F_{i}\cap F_{j}=\mathchar 31\relax$ for $i,j\in\\{1,\dots,N\\}$,
$i\neq j$. Let $\psi\in BV(B)$ satisfy

$\psi=0\quad\text{a.e. in}\quad
G:=\bigcup_{i=1}^{N}F_{i}\quad\text{and}\quad\psi=\widehat{\varphi}\quad\text{in}\quad
B\setminus{\Omega}\,.$ (4.6)

Then, for any $i\in\\{1,\dots,N\\}$,

$\displaystyle\mathcal{A}(\psi_{i}^{\star};B)-|G_{i}^{\star}|-\mathcal{A}(\psi_{i}^{\star};B\setminus\overline{\Omega})\leq\mathcal{A}(\psi;B)-|G|-\mathcal{A}(\psi;B\setminus\overline{\Omega}),$

where

$G_{i}^{\star}:=\bigcup_{j\neq i}F_{j}\cup{\rm
conv}(F_{i})\quad\text{and}\quad\psi_{i}^{\star}:=\begin{cases}0&\text{{\rm
in} }{\rm conv}(F_{i})\\\ \psi&\text{{\rm otherwise}.}\end{cases}$ (4.7)

###### Proof.

Fix $i\in\\{1,\dots,N\\}$. By the convexity of $\Omega$, we have
$\psi=\psi_{i}^{\star}$ in $B\setminus\overline{\Omega}$, hence it suffices to
show that

$\mathcal{A}(\psi_{i}^{\star};B)-|G_{i}^{\star}|\leq\mathcal{A}(\psi;B)-|G|.$

We start by observing that we may assume $F_{i}$ to be simply connected.
Indeed, if not, we can replace it with the set obtained by filling the holes
of $F_{i}$, and by setting $\psi$ equal to zero in the holes. This procedure
reduces the energy. Indeed, since $F_{i}$ is piecewise Lipschitz, any hole $H$
of it satisfies $\partial H\subset\cup_{j=1}^{n}\partial A_{j}$ where
$A_{j}$’s are the Lipschitz sets whose union is $F_{i}$. Hence the trace of
$\psi\mathop{\hbox{\vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits H$ on $\partial H$ is well-
defined, and the external trace $\psi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits(B\setminus H)$ vanishes.

We have that $(\partial{\rm conv}(F_{i}))\setminus\partial F_{i}$ is a
countable union of segments. We will next modify $\psi$ by iterating at most
countably many operations, setting $\psi=0$ in the region between each of
these segments and $\partial F_{i}$.

Step 1: Base case. Let $l$ be one of such segments, and $U$ be the open region
enclosed between $\partial F_{i}$ and $l$. We define $\psi^{\prime}\in
BV(\Omega)$ as

$\psi^{\prime}:=\begin{cases}0&\text{in}\ \ U\\\
\psi&\text{otherwise}\,.\end{cases}$

We claim that

$\mathcal{A}(\psi^{\prime};B)-|G^{\prime}|\leq\mathcal{A}(\psi;B)-|G|\,,$
(4.8)

where $G^{\prime}:=G\cup\overline{U}$. To prove the claim we introduce the
sets

$H:={\rm int}(F_{i}\cup U),\qquad V:=U\cap(\cup_{j\neq i}F_{j}).$

Note that $H$ is a piecewise Lipschitz set. By construction

$|G^{\prime}|=|{H}|+|\cup_{j\neq i}F_{j}|-|V|\,,$

and (4.8) will follow if we show that

$\begin{split}\mathcal{A}(\psi^{\prime};B)-|{H}|\leq\mathcal{A}(\psi;B)-|\cup_{j}F_{j}|+|\cup_{j\neq
i}F_{j}|-|V|=\mathcal{A}(\psi;B)-|F_{i}\cup V|\,.\end{split}$

Since $|H|=|F_{i}\cup V|+|U\setminus V|$, this can also be written as

$\mathcal{A}(\psi^{\prime};B)\leq\mathcal{A}(\psi;B)+|U\setminus V|\,.$

In turn
$\mathcal{A}(\psi^{\prime};B)=\mathcal{A}(\psi^{\prime};\overline{U})+\mathcal{A}(\psi^{\prime};B\setminus\overline{U})$
(and similarly for $\psi$), so we have reduced ourselves with proving

$\mathcal{A}(\psi^{\prime};\overline{U})\leq\mathcal{A}(\psi;\overline{U})+|U\setminus
V|\,.$ (4.9)

In view of the definition of $\psi^{\prime}$ which is zero in $U$, we
have888Notice that we use the precise integral formula (4.5) thanks to the
boundary regularity of $U$. More precisely we have $\partial U\setminus
l\subset\partial F_{i}\subset\cup_{j=1}^{n}\partial A_{j}$, where $A_{j}$’s
are the Lispchitz sets whose union is $F_{i}$.
$\mathcal{A}(\psi^{\prime};\overline{U})=\int_{l}|{\psi}^{+}|d\mathcal{H}^{1}+|U|$
($\psi^{+}$ denoting the trace of $\psi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits(B\setminus U)$ on the segment
$l$) implying that (4.9) is equivalent to

$\int_{l}|{\psi}^{+}|d\mathcal{H}^{1}\leq\mathcal{A}(\psi;\overline{U})-|V|\,.$

Finally, if $\psi_{U}$ denotes the trace of $\psi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits U$ on $l$, we write
$\mathcal{A}(\psi;\overline{U})=\mathcal{A}(\psi;\overline{U}\setminus
l)+\int_{l}|\psi^{+}-\psi_{U}|d\mathcal{H}^{1}$, and the expression above is
equivalent to

$\int_{l}|{\psi}^{+}|d\mathcal{H}^{1}\leq\int_{l}|\psi^{+}-\psi_{U}|d\mathcal{H}^{1}+\mathcal{A}(\psi;\overline{U}\setminus
l)-|V|\,.$ (4.10)

We now prove (4.10). Fix a Cartesian coordinate system $(x_{1},x_{2})$ so that
$l$ belongs to the $x_{1}$-axis and $U$ belongs to the half-plane
$\\{x_{2}>0\\}$. Let $u$ be an extension of $\psi$ in
$\mathbb{R}\times(0,+\infty)$ which vanishes outside $U$. Lemma 4.4, applied
to $u$ with the ball $B_{r}=B$, implies

$\int_{l}|{\psi}_{U}|d\mathcal{H}^{1}=\int_{\\{x_{2}=0\\}\cap
B}u\;d\mathcal{H}^{1}\leq\mathcal{A}(u;B\cap(\mathbb{R}\times(0,+\infty)))-|E_{B}|\leq\mathcal{A}(\psi;\overline{U}\setminus
l)-|V|.$

Here the last inequality follows by recalling that $\psi$ (and thus $u$)
vanishes on $V$. From this and the inequality
$\int_{l}|{\psi}^{+}|d\mathcal{H}^{1}\leq\int_{l}|\psi^{+}-\psi_{U}|d\mathcal{H}^{1}+\int_{l}|{\psi}_{U}|d\mathcal{H}^{1}$
the proof of (4.10) is achieved, so that (4.8) follows.

Step 2: Iterative case. We set $\partial({\rm conv}(F_{i}))\setminus\partial
F_{i}=\cup_{j=1}^{\infty}l_{j}$ with $l_{j}$ mutually disjoint segments. For
every $h\geq 1$ we define the pair $(\psi_{h},G_{h})$ as follows:

  * •

if $h=1$

$\psi_{1}:=\begin{cases}0&\text{in}\ U_{1}\\\
\psi&\text{otherwise,}\end{cases}\qquad\text{and}\quad
G_{1}:=G\cup\overline{U}_{1}\,,$

where $U_{1}$ is the open region enclosed between $\partial F_{i}$ and
$l_{1}$. We also define $H_{1}:={\rm int}(\overline{F_{i}\cup U_{1}})$.

  * •

if $h\geq 2$

$\psi_{h}:=\begin{cases}0&\text{in}\ U_{h}\\\
\psi_{h-1}&\text{otherwise,}\end{cases}\quad\text{and}\quad
G_{h}:=G_{h-1}\cup\overline{U}_{h}\,,$

where $U_{h}$ is the open region enclosed between $\partial H_{h-1}$ and
$l_{h}$ and $H_{h}:={\rm int}(\overline{H_{h-1}\cup U_{h}})$.

By construction each $H_{h}$ is simply connected and piecewise Lipschitz,
$H_{h}\subset H_{h+1}$, $G_{h}\subset G_{h+1}\subset\overline{\Omega}$ for
every $h\geq 1$, and moreover

$\lim_{h\to+\infty}|H_{h}|=|{\rm
conv}(F_{i})|\,,\qquad\lim_{h\to+\infty}|G_{h}|=|G^{\star}_{i}|\,,$ (4.11)

where $G^{\star}_{i}:=\cup_{h=1}^{\infty}G_{h}=\cup_{j\neq i}F_{j}\cup{\rm
conv}(F_{i})$. For any $h\geq 2$ we apply step 1, and after $h$ iterations we
get

$\mathcal{A}(\psi_{h};B)-|G_{h}|\leq\mathcal{A}(\psi_{h-1};B)-|G_{h-1}|\leq\cdot\cdot\cdot\leq\mathcal{A}(\psi_{1};B)-|G_{1}|\leq\mathcal{A}(\psi;B)-|G|\,.$
(4.12)

In particular,

$|D\psi_{h}|(B)\leq\mathcal{A}(\psi_{h};B)\leq\mathcal{A}(\psi;B)+|G_{h}\setminus
G|\leq\mathcal{A}(\psi;B)+|\Omega\setminus G|\,,$

for all $h\geq 1$, and then we easily see that, up to a subsequence,
$\psi_{h}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\psi^{\star}_{i}$ in
$BV(B)$, where $\psi_{i}^{*}$ is defined as in (4.7). Now the lower
semicontinuity of $\mathcal{A}(\cdot;B)$ yields

$\liminf_{h\to+\infty}\mathcal{A}(\psi_{h},B)\geq\mathcal{A}(\psi_{i}^{\star};B)\,.$
(4.13)

Finally, gathering together (4.11)-(4.13) we infer

$\mathcal{A}(\psi_{i}^{\star};B)-|G^{\star}_{i}|\leq\liminf_{h\to+\infty}\mathcal{A}(\psi_{h};B)-\lim_{h\to+\infty}|G_{h}|\leq\mathcal{A}(\psi;B)-|G|\,.$

This concludes the proof. ∎

######  Lemma 4.6 (Reduction of energy, II).

Let $N\geq 1$, $F_{1},\dots,F_{N},G$ and $\psi$ be as in Lemma 4.5. Then there
exist $\tilde{n}\in\\{1,\dots,N\\}$ and mutually disjoint closed convex sets
$\widetilde{F}_{1},\dots,\widetilde{F}_{\tilde{n}}\subset\overline{\Omega}$
such that

$G\subset\bigcup_{i=1}^{\tilde{n}}\widetilde{F}_{i}=:G^{\star}\,,$ (4.14)

and

$\mathcal{A}(\psi^{\star};B)-|G^{\star}|-\mathcal{A}(\psi^{\star};B\setminus\overline{\Omega})\leq\mathcal{A}(\psi;B)-|G|-\mathcal{A}(\psi;B\setminus\overline{\Omega})\,,$
(4.15)

where

$\psi^{\star}:=\begin{cases}0&\text{in}\ G^{\star}\\\
\psi&\text{otherwise}\,.\end{cases}$ (4.16)

###### Proof.

Base case: $(h=1)$. We take the sets

${\rm conv}(F_{1})\,,\ F_{2}\,,\ \dots\,,\ F_{N}\quad\text{and}\quad
G_{1}^{\star}:=\cup_{i=2}^{N}F_{i}\cup{\rm conv}(F_{1})\,,$ (4.17)

and let

$\psi_{1}^{\star}:=\begin{cases}0&\text{in}\ G_{1}^{\star}\\\
\psi&\text{otherwise}\,.\end{cases}$

Then by Lemma 4.5,

$\displaystyle\mathcal{A}(\psi_{1}^{\star};B)-|G_{1}^{\star}|-\mathcal{A}(\psi_{1}^{\star};B\setminus\overline{\Omega})\leq\mathcal{A}(\psi;B)-|G|-\mathcal{A}(\psi;B\setminus\overline{\Omega})\,.$
(4.18)

The next step is not necessary if $N=1$.

Iterative step: $(h>1)$. Suppose $N>1$. Let $1<h\leq m\leq N$ be natural
numbers, and let $F_{1,h},\dots,F_{m,h}$ be connected closed subsets of
$\overline{\Omega}$ with nonempty interior that satisfy the following
property: There exists $1\leq k<h$ such that:

  1. ($1$)

$F_{1,h},\dots,F_{k,h}$ are convex;

  2. ($2$)

$F_{i,h}\cap F_{j,h}=\mathchar 31\relax$ for all $i,j\neq k$, $i\neq j$.

Notice that, if $m>1$, for $h=2$ the sets

$F_{1,2}:={\rm conv}(F_{1})\,,\ F_{2,2}:=F_{2}\,,\ \dots\,,\
F_{N,2}:=F_{N}\,,$

in the base case satisfy ($1$), ($2$) with $m=N$ and $k=1$. We then set
$I_{k}:=\\{1\leq i\leq m,\,i\neq k\colon F_{i,h}\cap F_{k,h}\neq\mathchar
31\relax\\}$ and construct a new family of sets using the following algorithm,
distinguishing the two cases (a) and (b):

  * (a)

If $I_{k}=\mathchar 31\relax$ we define the sets

$F_{i,h+1}:=\begin{cases}F_{i,h}&\text{for}\ i\neq{k+1}\\\ {\rm
conv}(F_{k+1,h})&\text{for}\ i=k+1\,,\end{cases}$

and

$G_{h+1}^{\star}:=\cup_{i=1}^{m}F_{i,h+1}\,;$

  * (b)

if $I_{k}\neq\mathchar 31\relax$, up to relabelling the indices, we may assume
that

$\displaystyle I_{k}=\\{k_{1},k_{1}+1,\dots k_{2}\\}\setminus\\{k\\},$

for some $k_{1}\neq k_{2}$ with $1\leq k_{1}\leq k\leq k_{2}\leq m$, so that

$\\{1,\dots,m\\}\setminus\\{k\\}\setminus
I_{k}=\\{1,\dots,k_{1}-1\\}\cup\\{k_{2}+1,\dots,m\\}.$

Then we set

$F_{i,h+1}:=\begin{cases}F_{i,h}&\text{for}\ i=1,\dots,k_{1}-1\\\ {\rm
conv}(F_{k,h}\cup(\cup_{j\in I_{k}}F_{j,h}))&\text{for}\ i=k_{1}\\\
F_{i+k_{2}-k_{1},h}&\text{for}\ i=k_{1}+1,\dots,m-k_{2}+k_{1}\,,\end{cases}$

and

$G_{h+1}^{\star}:=\cup_{i=1}^{m-k_{2}+k_{1}}F_{i,h+1}\,.$

In both cases (a) and (b) a direct check shows that the produced sets satisfy
properties (1) and (2).

We define also the function

$\psi_{h+1}^{\star}:=\begin{cases}0&\text{in}\ G_{h+1}^{\star}\\\
\psi_{h}^{\star}&\text{otherwise}\,.\end{cases}$

Then, by induction, for all $h$ we use Lemma 4.5, and in view of (4.18) we
infer

$\displaystyle\mathcal{A}(\psi_{h+1}^{\star};B)-|G_{h+1}^{\star}|-\mathcal{A}(\psi_{h+1}^{\star};B\setminus\overline{\Omega})\leq$
$\displaystyle\mathcal{A}(\psi_{h}^{\star};B)-|G_{h}^{\star}|-\mathcal{A}(\psi_{h}^{\star};B\setminus\overline{\Omega})$
$\displaystyle\leq\mathcal{A}(\psi;B)-|G|-\mathcal{A}(\psi;B\setminus\overline{\Omega})\,.$

Conclusion. If $N=1$ it is sufficient to apply only the base case. If instead
$N>1$ after a finite number $h^{\star}\leq N$ of iterations we obtain a
collections of mutually disjoint and closed convex sets
$F_{1}:=F_{1,h^{\star}},\dots,F_{\tilde{n}}:=F_{\tilde{n},h^{\star}}$ with
$1\leq\tilde{n}\leq n$ such that

$G\subset\cup_{i=1}^{\tilde{n}}F_{i}=:G^{\star}\,,$

and

$\mathcal{A}(\psi^{\star};B)-|G^{\star}|-\mathcal{A}(\psi^{\star};B\setminus\overline{\Omega})\leq\mathcal{A}(\psi;B)-|G|-\mathcal{A}(\psi;B\setminus\overline{\Omega})\,,$

with

$\psi^{\star}:=\psi_{h^{\star}}^{\star}=\begin{cases}0&\text{in}\ G^{\star}\\\
\psi&\text{otherwise}\,.\end{cases}$

∎

###### Proof of Theorem 4.1.

By Theorem 3.1 it is enough to show that

$\inf_{(\sigma,\psi)\in\mathcal{W}}{\mathcal{F}}(\sigma,\psi)=\inf_{(\sigma,\psi)\in{\mathcal{W}_{\rm
conv}}}\mathcal{F}(\sigma,\psi)\,.$

Since from (2.9) it follows

$\inf_{(\sigma,\psi)\in\mathcal{W}}{\mathcal{F}}(\sigma,\psi)\leq\inf_{(\sigma,\psi)\in{\mathcal{W}_{\rm
conv}}}\mathcal{F}(\sigma,\psi),$

we only need to show the converse inequality. Take a pair
$(\bar{\sigma},\bar{\psi})\in\mathcal{W}$; we suitably modify
$(\bar{\sigma},\bar{\psi})$ into a new pair $(\sigma,\psi)\in{\mathcal{W}_{\rm
conv}}$ satisfying

$\mathcal{F}(\sigma,\psi)\leq{\mathcal{F}}(\bar{\sigma},\bar{\psi}),$

and this will conclude the proof.

Let $E(\bar{\sigma}_{1}),\dots,E(\bar{\sigma}_{n})$ be the closed sets with
mutually disjoint interiors corresponding to $\bar{\sigma}$ (as in (ii’) of
Section 2.3) and let $G:=\cup_{i=1}^{n}E(\bar{\sigma}_{i})$. Consider the
(closure of the) connected components $F_{1},\dots,F_{N}$ of $G$, $N\leq n$ .
Then by Lemma 4.6 there exist $1\leq\tilde{n}\leq N$ and
$\widetilde{F}_{1},\dots,\widetilde{F}_{\tilde{n}}\subset\overline{\Omega}$
mutually disjoint closed and convex satisfying (4.14), (4.15) and (4.16).
Therefore, by construction, for every $i=1,\dots,n$, $q_{i}$ and $p_{i+1}$
belong to $\widetilde{F}_{j}$ for a unique $j\in\\{1,\dots,\tilde{n}\\}$. For
every $j=1,\dots,\tilde{n}$ we denote by

$q_{j_{1}},p_{j_{1}+1},\dots,q_{j_{n_{j}}},p_{j_{n_{j}}+1},$

the ones that belong to $F_{j}$. Then we conclude by taking
$(\sigma,\psi)\in{\mathcal{W}_{\rm conv}}$ with
$\sigma:=(\sigma_{1},\dots,\sigma_{n})$ and

$\sigma_{j_{k}}([0,1])=\begin{cases}\overline{q_{j_{k}}p_{j_{k}+1}}&\text{for}\quad
k=1,\dots,n_{j}-1\\\ \partial
F_{j}\setminus\Bigl{(}\cup_{h=1}^{n_{j}}\partial^{0}_{j_{h}}\Omega\Bigr{)}\cup\Bigl{(}\cup_{h=1}^{n_{j}-1}\overline{q_{j_{h}}p_{j_{h}+1}}\Bigr{)}&\text{for}\quad
k=n_{j},\end{cases}$

for every $j=1,\dots,\tilde{n}$ and $\psi:=\psi^{\star}$. ∎

##  5 Regularity of minimizers

In this section we investigate regularity properties of minimizers of
$\mathcal{F}$. The main result reads as follows.

######  Theorem 5.1 (Structure of minimizers).

Every minimizer $(\sigma,\psi)\in{\mathcal{W}_{\rm conv}}$ of $\mathcal{F}$ in
$\mathcal{W}$, namely

${\mathcal{F}}(\sigma,\psi)=\min_{(s,\zeta)\in\mathcal{W}}{\mathcal{F}}(s,\zeta)\,,$

satisfies the following properties:

  1. 1

Each connected component of $E(\sigma)$ is convex;

  2. 2

$\psi$ is positive and real analytic in $\Omega\setminus E(\sigma)$;

  3. 3

If $\partial^{D}_{i}\Omega$ is not a segment for some $i=1,\dots,n$, then
$\partial E(\sigma)\cap\partial_{i}^{D}\Omega=\mathchar 31\relax$, $\psi$ is
continuous up to $\partial_{i}^{D}\Omega$, and $\psi=\varphi$ on
$\partial_{i}^{D}\Omega$;

  4. 4

If $\partial^{D}_{i}\Omega$ is a segment for some $i=1,\dots,n$, then either
$\partial E(\sigma)\cap\partial_{i}^{D}\Omega=\mathchar 31\relax$ or $\partial
E(\sigma)\cap\partial_{i}^{D}\Omega=\partial_{i}^{D}\Omega$. In the first case
$\psi$ is continuous up to $\partial_{i}^{D}\Omega$ and $\psi=\varphi$ on
$\partial_{i}^{D}\Omega$.

Moreover, there is a minimizer $(\sigma,\psi)\in{\mathcal{W}_{\rm conv}}$ such
that

  1. 5

$\Omega\cap\partial E(\sigma)$ consists of a finite number of disjoint curves
of class $C^{\infty}$, and $\psi$ is continuous and null on $\partial
E(\sigma)\setminus\partial^{D}\Omega$.

######  Remark 5.2.

If $\partial_{i}^{D}\Omega$ is a straight segment nothing ensures that
$\partial E(\sigma)\cap\partial_{i}^{D}\Omega=\mathchar 31\relax$. However, if
this intersection is nonempty, then it must be
$\partial_{i}^{D}\Omega\subset\partial E(\sigma)$. The prototypical example is
given by the classical catenoid, as explained in the introduction (see also
Figure 1) where, if the basis of the rectangle $\Omega=R_{2\ell}$ is large
enough, a solution $\psi$ is identically zero, and
$\partial^{D}\Omega\subset\partial E(\sigma)$.

This also explains why in point 5 of Theorem 5.1 we write $\partial
E(\sigma)\setminus\partial^{D}\Omega$, since $\partial^{D}\Omega$ might be
partially included in $\partial E(\sigma)$ if $\partial_{i}^{D}\Omega$ is a
segment (for some $i=1,\dots,n$).

For the reader convenience we divide the proof in a number of steps.

######  Lemma 5.3.

Every minimizer $(\sigma,\psi)\in{\mathcal{W}_{\rm conv}}$ of $\mathcal{F}$ in
$\mathcal{W}$ satisfies 1, 2 and $\psi=\varphi$ on
$\partial^{D}\Omega\setminus\partial E(\sigma)$.

###### Proof.

Item 1 follows by Theorem 4.1. By [15, Theorem 14.13] we also have that $\psi$
is real analytic in $\Omega\setminus E(\sigma)$. Together with the strong
maximum principle [15, Theorem C.4], this implies that, in $\Omega\setminus
E(\sigma)$, either $\psi>0$ or $\psi\equiv 0$. On the other hand, since
$\Omega$ is convex we can apply [15, Theorem 15.9] and get that $\psi$ is
continuous up to $\partial^{D}\Omega\setminus\partial E(\sigma)$; in
particular

$\psi=\varphi>0\quad\text{on}\ \partial^{D}\Omega\setminus\partial
E(\sigma)\,,$ (5.1)

which in turn implies $\psi>0$ in $\Omega\setminus E(\sigma)$ .

∎

######  Lemma 5.4.

Let $\Gamma\subset\mathbb{R}^{3}$ be a rectifiable, simple, closed and non-
planar curve satisfying the following properties:

  1. $(1)$

$\Gamma\subset\partial(F\times\mathbb{R})$ for some closed bounded convex set
$F\subset\mathbb{R}^{2}$ with nonempty interior;

  2. $(2)$

$\Gamma$ is symmetric with respect to the horizontal plane
$\mathbb{R}^{2}\times\\{0\\}$;

  3. $(3)$

There are an arc $\wideparen{pq}\subset\partial F$, with endpoints $p$ and
$q$, and $f\in C^{0}(\wideparen{pq}\cup\\{p,q\\};[0,+\infty))$ such that $f$
is positive in $\wideparen{pq}$ and

$\Gamma\cap\\{x_{3}\geq
0\\}=\mathcal{G}_{f}\cup(\\{p\\}\times[0,f(p)])\cup(\\{q\\}\times[0,f(q)]).$
(5.2)

Let $S$ be a solution to the classical Plateau problem for $\Gamma$, i.e., a
disc-type area-minimizing surface among all disc-type surfaces spanning
$\Gamma$. Then:

  1. $(1^{\prime})$

$\beta_{p,q}:=S\cap(\mathbb{R}^{2}\times\\{0\\})\subset F$ is a simple curve
of class $C^{\infty}$ joining $p$ and $q$ such that $\beta_{p,q}\cap\partial
F=\\{p,q\\}$;

  2. $(2^{\prime})$

$S$ is symmetric with respect to $\mathbb{R}^{2}\times\\{0\\}$;

  3. $(3^{\prime})$

The surface $S^{+}:=S\cap\\{x_{3}\geq 0\\}$ is the graph of a function
$\widetilde{\psi}\in W^{1,1}(U_{p,q})\cap C^{0}(\overline{U}_{p,q})$, where
$U_{p,q}\subset{\rm int}(F)$ is the open region enclosed between
$\wideparen{pq}$ and $\beta_{p,q}$. Moreover $\widetilde{\psi}$ is analytic in
$U_{p,q}$;

  4. $(4^{\prime})$

The curve $\beta_{p,q}$ is contained in the closed convex hull of $\Gamma$,
and $F\setminus U_{p,q}$ is convex.

######  Remark 5.5.

If the function $f$ in $(3)$ is such that $f(p)=f(q)=0$ then (5.2) becomes
$\Gamma\cap\\{x_{3}\geq 0\\}=\mathcal{G}_{f}$. For later convenience we prove
Lemma 5.4 under the more general assumption $(3)$.

###### Proof.

Even though several arguments are standard, we give the proof for
completeness.

Step 1: $\beta_{p,q}$ is a simple curve joining $p$ and $q$.
Let $B_{1}\subset\mathbb{R}^{2}$ be the open unit disc centred at the origin
and let $\Phi=(\Phi_{1},\Phi_{2},\Phi_{3})\colon\overline{B}_{1}\to
S\subset\mathbb{R}^{3}$ be a parametrization of $S$ with $\Phi(\partial
B_{1})=\Gamma$, that is harmonic, conformal, and therefore analytic in
$B_{1}$, continuous up to $\partial B_{1}$. Further by $(1)$ it follows that
$\Phi$ is an embedding and hence injective (see [18] and also [10, page 343]).
By assumption (5.2) we have $\\{w\in\partial
B_{1}\colon\Phi_{3}(w)=0\\}=\\{\Phi^{-1}(p,0),\Phi^{-1}(q,0)\\}$, so that
$\Phi_{3}$ changes sign only twice on $\partial B_{1}$. By applying Rado’s
lemma (see e.g. [10, Lemma 2, page 295]) to the harmonic function $\Phi_{3}$
we deduce that $\nabla\Phi_{3}\neq 0$ in $B_{1}$ and in particular $\\{w\in
B_{1}\colon\Phi_{3}(w)>0\\}$ and $\\{w\in B_{1}\colon\Phi_{3}(w)<0\\}$ are
connected, and $\\{w\in B_{1}\colon\Phi_{3}(w)=0\\}$ is a simple smooth curve
in $B_{1}$ joining $\Phi^{-1}(p,0)$ and $\Phi^{-1}(q,0)$. By the injectivity
of $\Phi$ we have that $S\cap(\mathbb{R}^{2}\times\\{0\\})=\Phi(\\{w\in
B_{1}\colon\Phi_{3}(w)=0\\})$ is a simple smooth curve joining $p$ and $q$.

Step 2: $S$ is symmetric with respect to the horizontal plane
$\mathbb{R}^{2}\times\\{0\\}$.
By step 1 the sets $\\{w\in\overline{B}_{1}\colon\Phi_{3}(w)\geq 0\\}$ and
$\\{w\in\overline{B}_{1}\colon\Phi_{3}(w)\leq 0\\}$ are simply connected and
the two surfaces

$S^{+}:=\Phi(\\{w\in\overline{B}_{1}\colon\Phi_{3}(w)\geq 0\\})\,,\quad
S^{-}:=\Phi(\\{w\in\overline{B}_{1}\colon\Phi_{3}(w)\leq 0\\})$

have the topology of the disc. We assume without loss of generality that
$\mathcal{H}^{2}(S^{+})\leq\mathcal{H}^{2}(S^{-})$. Let

${\rm Sym}(S^{+}):=\\{(x^{\prime},x_{3})\colon(x^{\prime},-x_{3})\in
S^{+}\\}\,,\quad\widetilde{S}:=S^{+}\cup{\rm Sym}(S^{+})\,.$

Then $\widetilde{S}$ is symmetric surface of disc-type with
$\partial\widetilde{S}=\Gamma$ and

$\mathcal{H}^{2}(\widetilde{S})=2\mathcal{H}^{2}(S^{+})\leq\mathcal{H}^{2}(S^{+})+\mathcal{H}^{2}(S^{-})=\mathcal{H}^{2}(S)\,.$

In particular $\widetilde{S}$ is a symmetric solution to the Plateau problem
for $\Gamma$. Further $S=\widetilde{S}$ on a relatively open subset of $S$;
hence, since they are real analytic surfaces, they must coincide,
$S=\widetilde{S}$.

Step 3: $S^{+}$ is the graph of a function $\widetilde{\psi}\in
W^{1,1}(U_{p,q})\cap C^{0}(\overline{U}_{p,q})$.
To show this it is enough to check the validity of the following

Claim: Every vertical plane $\Pi$ is tangent to ${\rm int}(S)$ at most at one
point.

In fact by step 2 this readily implies that ${\rm int}(S^{+})$ has no points
with vertical tangent plane and hence we can conclude. We prove the claim
arguing by contradiction as in [6, page 97], that is we assume there is a
vertical plane $\Pi$ tangent to ${\rm int}(S)$ at $x^{\prime}$ and
$x^{\prime\prime}$ with $x^{\prime}\neq x^{\prime\prime}$. We define the
linear map $d_{\nu}(x):=(x-x^{\prime})\cdot\nu$ with $\nu$ a unit normal to
$\Pi$, so that clearly $\Pi=\\{x\in\mathbb{R}^{3}\colon d_{\nu}(x)=0\\}$.
Since $F$ is convex $\Pi\cap(\partial F\times\\{0\\})$ contains at most two
points. By properties $(1)$-$(3)$ each of these points is either the
projection on the horizontal plane of one or two points of $\Pi\cap\Gamma$, or
the projection on the horizontal plane of one of the vertical segments
$\\{p\\}\times[0,f(p)]$ and $\\{q\\}\times[0,f(q)]$. Hence $\Pi\cap\Gamma$
contains either:

  * •

at most two points and a segment;

  * •

two segments;

  * •

four points.

Without loss of generality we restrict our analysis to the last case (the
others are simpler to treat), namely we assume that there are four (clockwise
ordered) points $w_{1},\dots,w_{4}\in\partial B_{1}$ such that
$\Pi\cap\Gamma=\\{\Phi(w_{1}),\dots,\Phi(w_{4})\\}$, that is
$d_{\nu}\circ\Phi(w_{i})=0$ for $i=1,\dots,4$. We may also assume
$d_{\nu}\circ\Phi>0$ on $\wideparen{w_{1}w_{2}}\cup\wideparen{w_{3}w_{4}}$ and
$d_{\nu}\circ\Phi<0$ on $\wideparen{w_{2}w_{3}}\cup\wideparen{w_{4}w_{1}}$.
Here $\wideparen{w_{i}w_{j}}$ denotes the relatively open arc in $\partial
B_{1}$ joining $w_{i}$ and $w_{j}$ for $i,j\in\\{1,\dots,4\\}$.
Notice that the function $d_{\nu}\circ\Phi\colon\overline{B}_{1}\to\mathbb{R}$
is harmonic in $B_{1}$, continuous up to $\partial B_{1}$ and vanishes at
$w_{1},\dots,w_{4}$; hence, by classical arguments [19, Section 437] we see
that the set $\\{w\in B_{1}\colon d_{\nu}\circ\Phi=0\\}$, in a neighbourhood
of $w^{\prime}:=\Phi^{-1}(x^{\prime})$ (respectively
$w^{\prime\prime}:=\Phi^{-1}(x^{\prime\prime})$), is the union of a number
$m\geq 2$ of analytic curves crossing at $w^{\prime}$ (respectively
$w^{\prime\prime}$). Thus near $w^{\prime}$ and $w^{\prime\prime}$ the set
$\\{w\in B_{1}\colon d_{\nu}\circ\Phi(w)>0\\}$ is the union of at least two
disjoint open regions $A_{1,1}$, $A_{1,2}$ and $A_{2,1}$, $A_{2,2}$
respectively such that
$\overline{A}_{1,1}\cap\overline{A}_{1,2}=\\{w^{\prime}\\}$,
$\overline{A}_{2,1}\cap\overline{A}_{2,2}=\\{w^{\prime\prime}\\}$. Moreover
each $A_{i,j}$ belongs either to the connected component of $\\{w\in
B_{1}\colon d_{\nu}\circ\Phi(w)>0\\}$ containing $\wideparen{w_{1}w_{2}}$ or
to the one containing $\wideparen{w_{3}w_{4}}$. Up to relabelling the indices
we have two possibilities.

  1. Case 1:

$A_{1,1}$ and $A_{1,2}$ belong to the same connected component containing
$\wideparen{w_{1}w_{2}}$. Then we can find two simple curves $\alpha_{1}$,
$\alpha_{2}$ contained in $A_{1,1}$ and $A_{1,2}$ respectively, that connect
$w^{\prime}$ to a point in $\wideparen{w_{1}w_{2}}$ and such that the region
enclosed by the curve $\alpha_{1}\cup\alpha_{2}$ intersects $\\{w\in
B_{1}\colon d_{\nu}\circ\Phi(w)<0\\}$. Since $d_{\nu}\circ\Phi>0$ on
$\alpha_{1}\cup\alpha_{2}$ by the maximum principle we have a contradiction.

  2. Case 2:

$A_{1,1}$ and $A_{2,1}$ belong to the connected component containing
$\wideparen{w_{1}w_{2}}$ while $A_{1,2}$ and $A_{2,2}$ belong to the connected
component containing $\wideparen{w_{3}w_{4}}$. Then we can find four simple
curves $\alpha_{i,j}$ (with $i,j=1,2$) contained respectively in $A_{i,j}$,
such that $\alpha_{1,1}$ (respectively $\alpha_{2,1}$) connects $w^{\prime}$
(respectively $w^{\prime\prime}$) to a point in $\wideparen{w_{1}w_{2}}$ and
$\alpha_{1,2}$ (respectively $\alpha_{2,2}$) connects $w^{\prime}$
(respectively $w^{\prime\prime}$) to $\wideparen{w_{3}w_{4}}$. Then the region
enclosed by the curve $\cup_{i,j}\alpha_{i,j}$ intersects $\\{w\in B_{1}\colon
d_{\nu}\circ\Phi(w)<0\\}$, while $d_{\nu}\circ\Phi>0$ on
$\cup_{i,j}\alpha_{i,j}$, which again by the maximum principle gives a
contradiction.

Thus the claim follows.

Step 4: The curve $\beta_{p,q}$ is contained in the closed convex hull of
$\Gamma$, and the set $F\setminus U_{p,q}$ is convex.
Let $\pi(\Gamma)\subset\partial F$ be the projection of $\Gamma$ onto the
plane $\mathbb{R}^{2}\times\\{0\\}$. By [10, Theorem 3, pag. 343] the relative
interior of $S$ is strictly contained in the convex hull of $\Gamma$, thus in
particular the curve $\beta_{p,q}$ (respectively
$\beta_{p,q}\setminus\\{p,q\\}$) is contained (respectively strictly
contained) in the same half-plane (with respect to the line $\overline{pq}$)
that contains $\pi(\Gamma)$.

Now, assume by contradiction that $F\setminus U_{p,q}$ is not convex. Then
there are $p^{\prime},q^{\prime}\in\beta_{p,q}$ with the following properties:

  * •

The open region $U^{\prime}$ enclosed by $\beta_{p,q}$ and the segment
$\overline{p^{\prime}q^{\prime}}$ is non-empty and contained in $U_{p,q}$;

  * •

the points $p$ and $q$ and the set $U^{\prime}$ lie on the same side with
respect to the line containing $\overline{p^{\prime}q^{\prime}}$.

Let then $d_{W}\colon\mathbb{R}^{3}\to\mathbb{R}$ be an affine function that
vanishes on the vertical plane containing $\overline{p^{\prime}q^{\prime}}$
and is positive on the half-space $W^{+}$ containing $p,q$ and $U^{\prime}$.
We now observe that $\Gamma\cap W^{+}$ is the union of two connected subcurves
$\Gamma_{1}$ and $\Gamma_{2}$, containing $p$ and $q$ respectively. As a
consequence $\Phi^{-1}(\Gamma_{1})=\wideparen{w_{1}w_{2}}$ and
$\Phi^{-1}(\Gamma_{2})=\wideparen{w_{3}w_{4}}$ for some
$w_{1},w_{2},w_{3},w_{4}\in\partial B_{1}$ (clockwise oriented).
On the other hand since $d_{W}>0$ on $U^{\prime}$ we can find
$t^{\prime}\in\partial U^{\prime}\setminus\overline{p^{\prime}q^{\prime}}$
such that $d_{W}\circ\Phi(\Phi^{-1}(t^{\prime}))=d_{W}(t^{\prime})>0$ with
$\Phi^{-1}(t^{\prime})\in B_{1}$.
Once again by the harmonicity of
$d_{W}\circ\Phi\colon\overline{B}_{1}\to\mathbb{R}$ we deduce the existence of
a curve $\alpha\subset\\{w\in B_{1}\colon d_{W}\circ\Phi(w)>0\\}$ joining
$\Phi^{-1}(t^{\prime})$ to one of $\wideparen{w_{1}w_{2}}$ and
$\wideparen{w_{3}w_{4}}$. Hence
$\Phi(\alpha)\subset\Phi(B_{1})=\widetilde{\psi}(U_{p,q})$ is a curve joining
$t^{\prime}$ to one of $\Gamma_{1}$ and $\Gamma_{2}$, say $\Gamma_{1}$. This
implies that the projection $\pi(\Phi(\alpha))$ of $\Phi(\alpha)$ onto the
horizontal plane $\mathbb{R}^{2}\times\\{0\\}$ is a curve contained in
$U_{p,q}$ that connects $t^{\prime}$ to $\pi(\Gamma_{1})$. So in particular,
the curve $\pi(\Phi(\alpha))$ cannot be included in the half-plane $W^{+}$.
But this contradicts the fact that $\alpha\subset\\{w\in B_{1}\colon
d_{W}\circ\Phi(w)>0\\}$ (this is because the values of $d_{W}$ at a point $x$
and $\pi(x)$ are the same). ∎

We need also the following technical results on the distance function $\,{\rm
d}_{F}$ from a convex set $F$.

######  Lemma 5.6.

Let $F\subset\mathbb{R}^{2}$ be bounded, closed and convex. Then $\Delta\,{\rm
d}_{F}\in L^{\infty}_{\rm loc}(\mathbb{R}^{2}\setminus F)\cap L^{1}(B\setminus
F)$ for every ball $B$ with $F\subset\subset B$.

###### Proof.

By [8, Theorem 3.6.7 pag. 75] it follows that $\,{\rm d}_{F}\in C^{1,1}_{\rm
loc}(\mathbb{R}^{2}\setminus F)$, hence $\nabla^{2}\,{\rm d}_{F}\in
L^{\infty}_{\rm loc}(\mathbb{R}^{2}\setminus F;\mathbb{R}^{2\times 2})$.
Therefore we only have to check that $\Delta\,{\rm d}_{F}\in L^{1}(B\setminus
F)$.
Let $\eta>0$ be fixed sufficiently small. Select
$(f_{k})_{k\in\mathbb{N}}\subset C^{1}_{c}(\mathbb{R}^{2};\mathbb{R}^{2})$
such that $f_{k}\to\nabla\,{\rm d}_{F}$ in $W^{1,1}(B\setminus
F_{\eta/2}^{+})$ as $k\to+\infty$. By the divergence theorem we have

$\int_{B\setminus F^{+}_{\eta}}\textrm{div}f_{k}\;dx=\int_{\partial
B\cup\partial(F^{+}_{\eta})}f_{k}\cdot\nu_{\eta}\;d\mathcal{H}^{1},$ (5.3)

with $\nu_{\eta}$ the outer unit normal to $\partial
B\cup\partial(F^{+}_{\eta})$. By taking the limit as $k\rightarrow\infty$ we
get

$\lim_{k\to+\infty}\int_{B\setminus
F^{+}_{\eta}}\textrm{div}f_{k}\;dx=\int_{B\setminus F^{+}_{\eta}}\Delta\,{\rm
d}_{F}\;dx\,,$ (5.4)

and

$\lim_{k\to+\infty}\int_{\partial
B\cup\partial(F^{+}_{\eta})}f_{k}\cdot\nu_{\eta}\;d\mathcal{H}^{1}=\int_{\partial
B\cup\partial(F^{+}_{\eta})}\nabla\,{\rm
d}_{F}\cdot\nu_{\eta}\;d\mathcal{H}^{1}\,,$ (5.5)

where (5.5) follows by using that $\partial(F^{+}_{\eta})$ is of class
$C^{1,1}$ and hence $f_{k}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits(\partial
B\cup\partial(F^{+}_{\eta}))\to\nabla\,{\rm d}_{F}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits(\partial
B\cup\partial(F^{+}_{\eta}))$ in $L^{1}(\partial
B\cup\partial(F^{+}_{\eta}))$. Since $\,{\rm d}_{F}$ is convex we have
$\Delta\,{\rm d}_{F}\geq 0$ a.e. in $\mathbb{R}^{2}\setminus F$, moreover
$|\nabla\,{\rm d}_{F}|=1$ in $\mathbb{R}^{2}\setminus F$; then gathering
together (5.3), (5.4), (5.5) we have

$\int_{B\setminus F^{+}_{\eta}}|\Delta\,{\rm d}_{F}|\;dx=\int_{B\setminus
F^{+}_{\eta}}\Delta\,{\rm d}_{F}\;dx=\int_{\partial
B\cup\partial(F^{+}_{\eta})}\nabla\,{\rm
d}_{F}\cdot\nu_{\eta}\;d\mathcal{H}^{1}\leq\mathcal{H}^{1}(\partial
B\cup\partial(F^{+}_{\eta}))\leq C,$

with $C>0$ independent of $\eta$. By the arbitrariness of $\eta>0$, the thesis
follows. ∎

######  Corollary 5.7.

Let $U\subset\mathbb{R}^{2}$ be a bounded open set with Lipschitz boundary.
Let $F\subset\mathbb{R}^{2}$ be closed and convex such that $U\cap F=\mathchar
31\relax$ and let $\psi\in W^{1,1}(U)\cap L^{\infty}(U)\cap C^{0}(U)$. Then
the following formula holds:

$\begin{split}-\int_{U}\psi\Delta\,{\rm
d}_{F}\,dx=\int_{U}\nabla\psi\cdot\nabla\,{\rm d}_{F}\,dx-\int_{\partial
U}\psi\,\gamma\,d\mathcal{H}^{1},\end{split}$

where $\nu$ is the outer normal to $\partial U$ and $\gamma$ denotes the
normal trace of $\nabla\,{\rm d}_{F}$ on $\partial U$.

###### Proof.

We have $|\nabla\,{\rm d}_{F}|=1$ in $\mathbb{R}^{2}\setminus F$, moreover
since $U\cap F=\mathchar 31\relax$, by Lemma 5.6 we deduce also $\Delta\,{\rm
d}_{F}\in L^{1}(U)$. Therefore the thesis readily follows by applying [3,
Theorem 1.9]. ∎

######  Remark 5.8.

The normal trace $\gamma$ of $\nabla\,{\rm d}_{F}$ on $\partial F$ equals $1$
$\mathcal{H}^{1}$-a.e. on $\partial F$. Indeed, from Corollary 5.7 we have
that for all $\varphi\in C^{1}_{c}(\mathbb{R}^{2};\mathbb{R}^{2})$ it holds

$\displaystyle-\int_{\mathbb{R}^{2}\setminus F^{+}_{\eta}}\varphi\Delta\,{\rm
d}_{F}\,dx$ $\displaystyle=\int_{\mathbb{R}^{2}\setminus
F^{+}_{\eta}}\nabla\varphi\cdot\nabla\,{\rm
d}_{F}\,dx-\int_{\partial(F_{\eta}^{+})}\varphi\,\gamma\,d\mathcal{H}^{1}$
$\displaystyle=\int_{\mathbb{R}^{2}\setminus
F^{+}_{\eta}}\nabla\varphi\cdot\nabla\,{\rm
d}_{F}\,dx-\int_{\partial(F_{\eta}^{+})}\varphi\,d\mathcal{H}^{1},$

where we have used that $\partial(F_{\eta}^{+})$ being a level set of $\,{\rm
d}_{F}$, it results $\nabla\,{\rm d}_{F}=\nu_{\eta}$ on it. Letting
$\eta\rightarrow 0$ and using that $\Delta\,{\rm d}_{F}\in L^{1}(B\setminus
F)$ for all balls $B$, we infer

$\displaystyle-\int_{\mathbb{R}^{2}\setminus F}\varphi\Delta\,{\rm d}_{F}\,dx$
$\displaystyle=\int_{\mathbb{R}^{2}\setminus F}\nabla\varphi\cdot\nabla\,{\rm
d}_{F}\,dx-\int_{\partial F}\varphi\,d\mathcal{H}^{1}.$

By the arbitrariness of $\varphi$ and again by Corollary 5.7, the claim
follows.

######  Lemma 5.9.

Let $F\subset\overline{\Omega}$ be closed and convex with non-empty interior,
and let $\delta>0$. Let $\psi\in W^{1,1}((F^{+}_{\delta}\setminus
F)\cap\Omega)\cap L^{\infty}((F^{+}_{\delta}\setminus F)\cap\Omega)\cap
C^{0}((F^{+}_{\delta}\setminus F)\cap\Omega)$. Then

$\lim_{\varepsilon\to
0^{+},\;\varepsilon<\delta}\int_{\Omega\cap\partial(F^{+}_{\varepsilon})}\psi\,d\mathcal{H}^{1}=\int_{\Omega\cap\partial
F}\psi\,d\mathcal{H}^{1}\,.$ (5.6)

###### Proof.

Let $\varepsilon\in(0,\delta)$ and
$T_{\varepsilon}:=(F^{+}_{\varepsilon}\setminus F)\cap\Omega$. Since
$T_{\varepsilon}\cap F=\mathchar 31\relax$, by Corollary 5.7 we get

$\begin{split}-\int_{T_{\varepsilon}}\psi\Delta\,{\rm
d}_{F}\,dx=\int_{T_{\varepsilon}}\nabla\psi\cdot\nabla\,{\rm
d}_{F}\,dx-\int_{\partial
T_{\varepsilon}}\psi\,\gamma\,d\mathcal{H}^{1}\,.\end{split}$ (5.7)

By Remark 5.8 we have

$\begin{split}-\int_{T_{\varepsilon}}\psi\Delta\,{\rm
d}_{F}\,dx=&\int_{T_{\varepsilon}}\nabla\psi\cdot\nabla\,{\rm d}_{F}\,dx\\\
&+\int_{\Omega\cap\partial
F}\psi\,d\mathcal{H}^{1}-\int_{\Omega\cap\partial(F_{\varepsilon}^{+})}\psi\,d\mathcal{H}^{1}-\int_{((F^{+}_{\varepsilon})\setminus
F)\cap\partial\Omega}\psi\,\gamma\,d\mathcal{H}^{1}\,.\end{split}$ (5.8)

Now

$\lim_{\varepsilon\to
0^{+}}\Big{|}\int_{T_{\varepsilon}}\nabla\psi\cdot\nabla\,{\rm
d}_{F}\,dx\Big{|}\leq\lim_{\varepsilon\to
0^{+}}\int_{T_{\varepsilon}}|\nabla\psi|\,dx=0\,,$ (5.9)

and

$\lim_{\varepsilon\to 0^{+}}\Big{|}\int_{(F^{+}_{\varepsilon}\setminus
F)\cap\partial\Omega}\psi\,\gamma\,d\mathcal{H}^{1}\Big{|}\leq\lim_{\varepsilon\to
0^{+}}\int_{(F^{+}_{\varepsilon}\setminus
F)\cap\partial\Omega}\psi\,d\mathcal{H}^{1}=0\,.$ (5.10)

Moreover, since $\Delta\,{\rm d}_{F}\in L^{1}(T_{\varepsilon})$ by Lemma 5.6,
we deduce also

$\lim_{\varepsilon\to 0^{+}}\Big{|}\int_{T_{\varepsilon}}-\psi\,\Delta\,{\rm
d}_{F}\,dx\Big{|}\leq\|\psi\|_{L^{\infty}}\lim_{\varepsilon\to
0^{+}}\int_{T_{\varepsilon}}|\Delta\,{\rm d}_{F}|\,dx=0\,.$ (5.11)

Finally gathering together (5.8)-(5.11) we infer (5.6). ∎

######  Remark 5.10.

Let $F$, $\delta$ and $\psi$ be as in Lemma 5.9. Let $\alpha$ be any connected
component of $\Omega\cap\partial F$, and for every $0<\varepsilon<\delta$ let
$\alpha_{\varepsilon}$ be the corresponding component of
$\Omega\cap\partial(F^{+}_{\varepsilon})$; namely, if $\pi_{F}$ is the
orthogonal projection onto the convex closed set $F$, setting

$\widehat{\alpha}_{\varepsilon}:=\\{x\in\partial(F^{+}_{\varepsilon}):\pi_{F}(x)\in\alpha\\},$

then one has $\alpha_{\varepsilon}:=\widehat{\alpha}_{\varepsilon}\cap\Omega$.
Arguing as in Lemma 5.9, we can show that

$\lim_{\varepsilon\to
0^{+}}\int_{\alpha_{\varepsilon}}\psi\,d\mathcal{H}^{1}=\int_{\alpha}\psi\,d\mathcal{H}^{1}\,.$

######  Lemma 5.11.

Let $(\sigma,\psi)\in{\mathcal{W}_{\rm conv}}$ be a minimizer as in Theorem
4.1. Then there is a minimizer
$(\widehat{\sigma},\widehat{\psi})\in{\mathcal{W}_{\rm conv}}$ with the
following properties:

  1. 1.

$\partial E(\widehat{\sigma})\cap\partial\Omega=\partial
E(\sigma)\cap\partial\Omega$;

  2. 2.

$\widehat{\psi}$ is continuous and null on $\Omega\cap\partial
E(\widehat{\sigma})$.

The second condition means essentially that $\widehat{\psi}$ vanishes on
$\Omega\cap\partial E(\widehat{\sigma})$ when considering its trace from the
side of $\Omega\setminus E(\widehat{\sigma})$.

###### Proof.

We know by Lemma 5.3 that $(\sigma,\psi)$ satisfies the following properties:

  * •

Each connected component of $E(\sigma)$ is convex;

  * •

$\psi$ is positive and real analytic in $\Omega\setminus E(\sigma)$;

  * •

$\psi=\varphi$ on $\partial^{D}\Omega\setminus\partial E(\sigma)$.

In what follows we are going to modify $(\sigma,\psi)$ near each arc of
$\partial E(\sigma)$ using an iterative argument in order to get a new
minimizer $(\widehat{\sigma},\widehat{\psi})\in{\mathcal{W}_{\rm conv}}$ that
satisfies 1-2. To this aim we denote by $F_{1},\dots,F_{k}$ with $1\leq k\leq
n$ the closed connected components of $E(\sigma)$; we also set
$\delta_{0}:=\min_{i\neq j}\textrm{dist}(F_{i},F_{j})>0$. Moreover by the
first property we deduce that $\Omega\cap\partial E(\sigma)$ is the union of
an at most countable family of pairwise disjoint arcs with endpoints in
$\partial\Omega$, i.e.,

$\Omega\cap\partial
E(\sigma)=\bigcup_{i=1}^{k}\bigcup_{j=1}^{\infty}\alpha_{i,j}\,,$

where $\alpha_{i,j}$ is a connected component of $\Omega\cap\partial F_{i}$
for $i\in\\{1,\dots,k\\}$, $j\geq 1$999Notice that at this stage we do not
have any information about the geometry of the set $\partial\Omega\cap\partial
E(\sigma)$, and $\Omega\cap\partial F_{i}$ could a priori be the union of
infinitely many connected components..

Step 1: Base case. Let $\alpha$ be one of the connected components of
$\Omega\cap\partial F$, with $F:=F_{i}$ for some $i\in\\{1,\dots,k\\}$. In
this step we construct a new minimizer
$(\sigma^{\alpha},\psi^{\alpha})\in{\mathcal{W}_{\rm conv}}$ such that
$\partial E(\sigma^{\alpha})\cap\partial\Omega=\partial
E(\sigma)\cap\partial\Omega$ and $\psi^{\alpha}$ is continuous and null on
$\alpha^{\prime}$, where $\alpha^{\prime}\subset\Omega\cap\partial
E(\sigma^{\alpha})$ is a suitable curve that replaces $\alpha$ and has the
same endpoints as $\alpha$.

For $\varepsilon\in(0,\delta_{0}/2)$ we define the stripe

$\widehat{T}_{\varepsilon}(\alpha):=\\{x\in\Omega\setminus
F\colon\textrm{dist}(x,\alpha)<\varepsilon\\}\subset
F^{+}_{\varepsilon}\setminus F\,,$

and consider the planar curve $\alpha_{\varepsilon}$ in $\overline{\Omega}$
defined as in Remark 5.10. Let $T_{\varepsilon}(\alpha)$ be the connected
component of $\widehat{T}_{\varepsilon}(\alpha)$ whose boundary contains
$\alpha_{\varepsilon}$. Let $L_{\varepsilon}$ be defined as

$L_{\varepsilon}:=\partial T_{\varepsilon}(\alpha)\cap\partial\Omega,$

so that in particular $\partial
T_{\varepsilon}(\alpha)=\alpha\cup\alpha_{\varepsilon}\cup L_{\varepsilon}\,.$
Let $p,q\in\partial\Omega$ be the endpoints of $\alpha$ (and then also the
endpoints of $\alpha_{\varepsilon}\cup L_{\varepsilon}$, which are independent
of $\varepsilon$). We define the curves

$\Gamma_{\varepsilon}:=\Gamma_{\varepsilon}^{+}\cup\Gamma_{\varepsilon}^{-}\,,\quad\Gamma_{\varepsilon}^{+}:=\mathcal{G}_{\psi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\alpha_{\varepsilon}}\cup\mathcal{G}_{\varphi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits L_{\varepsilon}}\cup
l^{+}\,,\quad\Gamma_{\varepsilon}^{-}:=\mathcal{G}_{-\psi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\alpha_{\varepsilon}}\cup\mathcal{G}_{-\varphi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits L_{\varepsilon}}\cup l^{-}\,,$

where

$l^{+}:=(\\{p\\}\times[0,\varphi(p)])\cup(\\{q\\}\times[0,\varphi(q)])\,,\quad
l^{-}:=(\\{p\\}\times[-\varphi(p),0])\cup(\\{q\\}\times[-\varphi(q),0])\,.$

By observing that $L_{\varepsilon}\subset\partial^{D}\Omega\setminus\partial
E(\sigma)$ and recalling that $\psi=\varphi$ on
$\partial^{D}\Omega\setminus\partial E(\sigma)$ we deduce that
$\Gamma_{\varepsilon}$ is a closed non-planar curve in $\mathbb{R}^{3}$ that
satisfies assumptions $(1)$-$(3)$ of Lemma 5.4. In particular a solution
$S_{\varepsilon}$ to the classical Plateau problem corresponding to
$\Gamma_{\varepsilon}$ is a disc-type surface such that:

  1. 1.

$\beta_{p,q}^{\varepsilon}:=S_{\varepsilon}\cap(\mathbb{R}^{2}\times\\{0\\})$
is a simple curve of class $C^{\infty}$ joining $p$ and $q$;

  2. 2.

$S_{\varepsilon}$ is symmetric with respect to the horizontal plane;

  3. 3.

the surface $S^{+}_{\varepsilon}:=S_{\varepsilon}\cap\\{x_{3}\geq 0\\}$ is the
graph of a function $\psi_{p,q}^{\varepsilon}\in
W^{1,1}(U_{p,q}^{\varepsilon})\cap C^{0}(\overline{U}_{p,q}^{\varepsilon})$,
where $U_{p,q}^{\varepsilon}\subset F\cup T_{\varepsilon}(\alpha)$ is the open
region enclosed between $\alpha_{\varepsilon}\cup L_{\varepsilon}$ and
$\beta_{p,q}^{\varepsilon}$;

  4. 4.

the curve $\beta_{p,q}^{\varepsilon}$ is contained in the closed convex hull
of $\Gamma_{\varepsilon}$ and $(F\cup T_{\varepsilon}(\alpha))\setminus
U_{p,q}^{\varepsilon}$ is convex.

We would like to compare the area of $S_{\varepsilon}^{+}$ with the area of
the generalized graph of $\psi$ on $\overline{T_{\varepsilon}(\alpha)}$. This
is not immediate since, due to the fact that $\psi$ is just $BV$, we cannot, a
priori, conclude that this generalized graph is of disc-type101010This is due
to the jump of $\psi$ on $\partial F$ which is, in general, not regular
enough.. Hence we proceed as follows. We fix
$\bar{\varepsilon}\in(0,\delta_{0}/2)$; we claim that

$\mathcal{A}(\psi_{p,q}^{\bar{\varepsilon}};U_{p,q}^{\bar{\varepsilon}})\leq\mathcal{A}(\psi;T_{\bar{\varepsilon}}(\alpha))+\int_{\alpha}\psi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits{T_{\bar{\varepsilon}}(\alpha)}\,d\mathcal{H}^{1}\,.$
(5.12)

Since $\psi$ is analytic in
$T_{\bar{\varepsilon}}(\alpha)\subset\Omega\setminus E(\sigma)$, by Lemma 5.9
and Remark 5.10 it follows that

$\lim_{\varepsilon\to
0^{+},\;\varepsilon<\bar{\varepsilon}}\int_{\alpha_{\varepsilon}}\psi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits{T_{\bar{\varepsilon}}(\alpha)}\,d\mathcal{H}^{1}=\int_{\alpha}\psi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits{T_{\bar{\varepsilon}}(\alpha)}\,d\mathcal{H}^{1}\,.$
(5.13)

We take

$T_{\varepsilon}^{\bar{\varepsilon}}(\alpha):={T_{\bar{\varepsilon}}(\alpha)\setminus\overline{T_{\varepsilon}(\alpha)}}\quad\text{and}\quad
Y_{\bar{\varepsilon}}:=S_{\varepsilon}\cup\mathcal{G}_{\psi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits
T_{\varepsilon}^{\bar{\varepsilon}}(\alpha)}\cup\mathcal{G}_{-\psi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits
T_{\varepsilon}^{\bar{\varepsilon}}(\alpha)}\,.$

Since $S_{\varepsilon}$ is a disc-type surface and $\psi$ is analytic in
$T_{\varepsilon}^{\bar{\varepsilon}}(\alpha)$ it turns out that
$Y_{\bar{\varepsilon}}$ is also a disc-type surface satisfying $\partial
Y_{\bar{\varepsilon}}=\Gamma_{\bar{\varepsilon}}$. Therefore using that
$S_{\bar{\varepsilon}}$ and $S_{\varepsilon}$ are solutions to the Plateau
problems corresponding to $\Gamma_{\bar{\varepsilon}}$ and
$\Gamma_{\varepsilon}$ respectively, we have

$\begin{split}\mathcal{H}^{2}(S_{\bar{\varepsilon}})\leq\mathcal{H}^{2}(Y_{\bar{\varepsilon}})&=2\mathcal{H}^{2}(\mathcal{G}_{\psi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits
T_{\varepsilon}^{\bar{\varepsilon}}(\alpha)})+\mathcal{H}^{2}(S_{\varepsilon})\\\
&\leq 2\mathcal{H}^{2}(\mathcal{G}_{\psi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits
T_{\bar{\varepsilon}}(\alpha)})+2\int_{\alpha_{\varepsilon}\cup
L_{\varepsilon}}\psi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits{T_{\bar{\varepsilon}}(\alpha)}\,d\mathcal{H}^{1}\\\
&=2\mathcal{H}^{2}(\mathcal{G}_{\psi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits
T_{\bar{\varepsilon}}(\alpha)})+2\int_{\alpha_{\varepsilon}}\psi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits{T_{\bar{\varepsilon}}(\alpha)}\,d\mathcal{H}^{1}+2\int_{L_{\varepsilon}}\psi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits{T_{\bar{\varepsilon}}(\alpha)}\,d\mathcal{H}^{1}\,.\end{split}$

Passing to the limit as $\varepsilon\to 0^{+}$, by (5.13) and the fact that
$\mathcal{H}^{1}(L_{\varepsilon})\to 0$, we obtain

$\mathcal{H}^{2}(S_{\bar{\varepsilon}})\leq
2\mathcal{H}^{2}(\mathcal{G}_{\psi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits
T_{\bar{\varepsilon}}(\alpha)})+2\int_{\alpha}\psi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits{T_{\bar{\varepsilon}}(\alpha)}\,d\mathcal{H}^{1},$

which yields

$\mathcal{A}(\psi_{p,q}^{\bar{\varepsilon}};U_{p,q}^{\bar{\varepsilon}})=\mathcal{H}^{2}(S_{\bar{\varepsilon}}^{+})\leq\mathcal{H}^{2}(\mathcal{G}_{\psi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits
T_{\bar{\varepsilon}}(\alpha)})+\int_{\alpha}\psi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits{T_{\bar{\varepsilon}}(\alpha)}\,d\mathcal{H}^{1}=\mathcal{A}(\psi;T_{\bar{\varepsilon}}(\alpha))+\int_{\alpha}\psi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits{T_{\bar{\varepsilon}}(\alpha)}\,d\mathcal{H}^{1},$

and (5.12) is proved.

We now define $E^{\alpha}:=(E(\sigma)\cup
T_{\bar{\varepsilon}}(\alpha))\setminus U^{\bar{\varepsilon}}_{p,q}$ and

$\psi^{\alpha}:=\begin{cases}0&\text{in}\ E^{\alpha}\\\
\psi^{\bar{\varepsilon}}_{p,q}&\text{in}\ U^{\bar{\varepsilon}}_{p,q}\\\
\psi&\text{otherwise}\,.\end{cases}$

By (5.12) and using that $U^{\bar{\varepsilon}}_{p,q}\cup
E^{\alpha}=E(\sigma)\cup T_{\bar{\varepsilon}}(\alpha)$ we derive

$\begin{split}\mathcal{A}(\psi^{\alpha};\Omega)-|E^{\alpha}|&=\mathcal{A}(\psi^{\bar{\varepsilon}}_{p,q};U^{\bar{\varepsilon}}_{p,q})+\mathcal{A}(\psi;\Omega\setminus(U^{\bar{\varepsilon}}_{p,q}\cup
E^{\alpha}))\\\
&=\mathcal{A}(\psi^{\bar{\varepsilon}}_{p,q};U^{\bar{\varepsilon}}_{p,q})+\mathcal{A}(\psi;\Omega\setminus(T_{\bar{\varepsilon}}(\alpha)\cup
E(\sigma)))\\\
&\leq\mathcal{A}(\psi;T_{\bar{\varepsilon}}(\alpha))+\int_{\alpha}\psi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits{T_{\bar{\varepsilon}}(\alpha)}\,d\mathcal{H}^{1}+\mathcal{A}(\psi;\Omega\setminus
T_{\bar{\varepsilon}}(\alpha))-|E(\sigma)|\\\
&=\mathcal{A}(\psi;\Omega)-|E(\sigma)|\,.\end{split}$ (5.14)

It remains to construct $\sigma^{\alpha}\in\Sigma_{\rm conv}$. Without loss of
generality we may assume

$\sigma_{1}([0,1]),\dots,\sigma_{h}([0,1])\subset
F\quad\text{and}\quad\sigma_{h+1}([0,1]),\dots,\sigma_{n}([0,1])\not\subset F$

for some $h\leq n$, notice that if $h=n$ the second family of curves is empty.
Then we define
$\sigma^{\alpha}:=(\sigma^{\alpha}_{1},\dots,\sigma_{h}^{\alpha},\sigma_{h+1},\dots,\sigma_{n})\in{\rm
Lip}([0,1];\overline{\Omega})^{n}$ where if $h>1$

$\sigma^{\alpha}_{i}([0,1])=\begin{cases}\overline{q_{i}p_{i+1}}&\text{for}\quad
i=1,\dots,h-1\\\ \partial(F\cup T_{\bar{\varepsilon}}(\alpha)\setminus
U^{\bar{\varepsilon}}_{p,q})\setminus\Bigl{(}(\cup_{i=1}^{h}\partial^{0}_{i}\Omega)\cup(\cup_{i=1}^{h-1}\overline{q_{i}p_{i+1}})\Bigr{)}&\text{for}\quad
i=h,\end{cases}$

where $\overline{q_{i}p_{i+1}}$ is the segment joining $q_{i}$ to $p_{i+1}$;
if instead $h=1$ we simply set

$\sigma^{\alpha}_{1}([0,1])=\partial(F\cup
T_{\bar{\varepsilon}}(\alpha)\setminus
U^{\bar{\varepsilon}}_{p,q})\setminus\partial_{1}^{0}\Omega.$

Clearly the pair $(\sigma^{\alpha},\psi^{\alpha})$ belongs to
${\mathcal{W}_{\rm conv}}$, and by (5.14) it satisfies

$\begin{split}{\mathcal{F}}(\sigma^{\alpha},\psi^{\alpha})={\mathcal{F}}(\sigma,\psi)\,.\end{split}$

Moreover $\partial E(\sigma^{\alpha})\cap\partial\Omega=\partial
E(\sigma)\cap\partial\Omega$ and $\psi^{\alpha}$ is continuous and null on
$\alpha^{\prime}$, where

$\alpha^{\prime}:=\beta_{p,q}^{\bar{\varepsilon}}\subset\Omega\cap\partial
E(\sigma^{\alpha})\,.$ (5.15)

Summarizing, we have replaced the curve $\alpha$ with $\alpha^{\prime}$,
ensuring that the new function $\psi^{\alpha}$ is now continuous and null on
$\alpha^{\prime}$.

Step 2: Iterative case. In this step we construct a minimizer
$(\widehat{\sigma},\widehat{\psi})\in{\mathcal{W}_{\rm conv}}$ that satisfies
the thesis by iterating step one at most a countable number of times.
We first consider $F=F_{1}$ and apply step 1 for each $\alpha_{1,j}$ with
$j\geq 1$. More precisely we define the pair
$(\sigma_{1,j},\psi_{1,j})\in{\mathcal{W}_{\rm conv}}$ as follows:

  * •

if $j=1$ we set

$(\sigma_{1,1},\psi_{1,1}):=(\sigma^{\alpha_{1,1}},\psi^{\alpha_{1,1}})\,,$

where $(\sigma^{\alpha_{1,1}},\psi^{\alpha_{1,1}})\in{\mathcal{W}_{\rm conv}}$
is a minimizer constructed as in step 1 with $\alpha=\alpha_{1,1}$;

  * •

if $j>1$ we set

$(\sigma_{1,j},\psi_{1,j}):=(\sigma_{1,j-1}^{\alpha_{1,j}},\psi_{1,j-1}^{{\alpha}_{1,j}})\,,$

where
$(\sigma_{1,j-1}^{\alpha_{1,j}},\psi_{1,j-1}^{{\alpha}_{1,j}})\in{\mathcal{W}_{\rm
conv}}$ is a minimizer constructed as in step 1 with
$(\sigma,\psi)=(\sigma_{1,j-1},\psi_{1,j-1})$ and $\alpha=\alpha_{1,j}$.

Since $\mathcal{F}(\sigma_{1,j},\psi_{1,j})=\mathcal{F}(\sigma,\psi)$ for all
$j\geq 1$, by Lemma 3.4 it follows that $(\sigma_{1,j},\psi_{1,j})$ converges
to $(\sigma_{1},\psi_{1})\in{\mathcal{W}_{\rm conv}}$ in the sense of
Definition 3.2. Moreover by construction we have that for every $j\geq 1$ the
pair $(\sigma_{1,j},\psi_{1,j})$ satisfies

$\partial E(\sigma_{1,j})\cap\partial\Omega=\partial
E(\sigma)\cap\partial\Omega\,,$

and $\psi_{1,j}$ is continuous and null on
$\cup_{h=1}^{j}\alpha^{\prime}_{1,h}\subset\Omega\cap\partial
E(\sigma_{1,j})\cap\partial F_{1}$, where $\alpha^{\prime}_{1,h}$ are defined
as in (5.15). As a consequence $(\sigma_{1},\psi_{1})$ satisfies

$\partial E(\sigma_{1})\cap\partial\Omega=\partial
E(\sigma)\cap\partial\Omega\,,$

and $\psi_{1}$ is continuous and null on
$\cup_{j=1}^{\infty}\alpha^{\prime}_{1,j}\subset\Omega\cap\partial
E(\sigma_{1})\cap\partial F_{1}$. Moreover

$\Omega\cap\partial
E(\sigma_{1})=(\cup_{j=1}^{\infty}\alpha^{\prime}_{1,j})\cup(\cup_{i=2}^{k}\cup_{j=1}^{\infty}\alpha_{i,j})\,,$

Now repeating the argument above for the pair $(\sigma_{1},\psi_{1})$ and
$i=2$ we obtain a new minimizer $(\sigma_{2},\psi_{2})\in{\mathcal{W}_{\rm
conv}}$ satisfying

$\partial E(\sigma_{2})\cap\partial\Omega=\partial
E(\sigma)\cap\partial\Omega\,,$

$\psi_{2}$ is continuous and null on
$\cup_{j=1}^{\infty}(\alpha^{\prime}_{1,j}\cup\alpha^{\prime}_{2,j})\subset\Omega\cap\partial
E(\sigma_{1})\cap(\partial F_{1}\cup\partial F_{2})$ and

$\Omega\cap\partial
E(\sigma_{2})=(\cup_{i=1}^{2}\cup_{j=1}^{\infty}\alpha^{\prime}_{i,j})\cup(\cup_{i=3}^{k}\cup_{j=1}^{\infty}\alpha_{i,j})\,.$

Iterating this process a finite number of times we finally get a minimizer
$(\widehat{\sigma},\widehat{\psi})\in{\mathcal{W}_{\rm conv}}$ with the
required properties. ∎

We are finally in the position to conclude the proof of Theorem 5.1.

###### Proof of Theorem 5.1.

Let $(\sigma,\psi)\in{\mathcal{W}_{\rm conv}}$ be any minimizer as in Theorem
4.1. By Lemma 5.3 we know that $(\sigma,\psi)$ satisfies properties 1, 2 and

$\psi=\varphi\quad\text{on}\quad\partial^{D}\Omega\setminus\partial
E(\sigma)\,.$

Moreover by Lemma 5.11 there is a minimizer
$(\widehat{\sigma},\widehat{\psi})\in{\mathcal{W}_{\rm conv}}$ such that

$\partial E(\widehat{\sigma})\cap\partial\Omega=\partial
E(\sigma)\cap\partial\Omega\,,$ (5.16)

and $\widehat{\psi}$ is continuous and null on $\Omega\cap\partial
E(\widehat{\sigma})$.

It remains to show that if $\partial^{D}_{i}\Omega$ is not straight for some
$i=1,\dots,n$, then

$\partial E(\sigma)\cap\partial^{D}_{i}\Omega=\partial
E(\widehat{\sigma})\cap\partial^{D}_{i}\Omega=\mathchar 31\relax\,.$

If instead $\partial^{D}_{i}\Omega$ is straight for some $i=1,\dots,n$ we
prove that property 4 holds. Eventually we show that there is a minimizer that
satisfies property 5. This will be achieved in a number of steps.

Step 1: Assuming that there is $i\in\\{1,\dots,n\\}$ such that
$\partial_{i}^{D}\Omega$ is not straight, we show that
$\partial_{i}^{D}\Omega\cap E(\widehat{\sigma})=\mathchar 31\relax$. To prove
this we proceed by analysing three different cases.

Case A: Suppose, to the contrary, that there is a non-straight111111Namely,
$\wideparen{ab}$ is not contained in a line. arc $\wideparen{ab}$ (with
endpoints $a\neq b$) in $\partial_{i}^{D}\Omega\cap\partial
E(\widehat{\sigma})$. Thus in particular
$\wideparen{ab}\subset\cup_{j=1}^{n}\widehat{\sigma}_{j}([0,1])$. We may
assume without loss of generality that
$\wideparen{ab}\subset\widehat{\sigma}_{1}([0,1])$.
Then we consider the curves

$\Gamma:=\Gamma^{+}\cup\Gamma^{-}\,,\quad\Gamma^{+}:=\mathcal{G}_{\varphi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\wideparen{ab}}\cup
l^{+}\,,\quad\Gamma^{-}:=\mathcal{G}_{-\varphi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\wideparen{ab}}\cup l^{-}\,,$
(5.17)

where

$l^{+}:=(\\{a\\}\times[0,\varphi(a)])\cup(\\{b\\}\times[0,\varphi(b)])\,,\quad
l^{-}:=(\\{a\\}\times[-\varphi(a),0])\cup(\\{b\\}\times[-\varphi(b),0])\,.$

In this way $\Gamma$ satisfies the assumptions of Lemma 5.4 and hence a
solution $S$ to the Plateau problem spanning $\Gamma$ is a disc-type surface
such that:

  1. i.

$\beta_{a,b}:=S\cap(\mathbb{R}^{2}\times\\{0\\})$ is a simple curve of class
$C^{\infty}$ joining $a$ and $b$;

  2. ii.

$S$ is symmetric with respect to $\mathbb{R}^{2}\times\\{0\\}$;

  3. iii.

the surface $S^{+}:=S\cap\\{x_{3}\geq 0\\}$ is the graph of a function
$\psi_{a,b}\in W^{1,1}(U_{a,b})\cap C^{0}(\overline{U}_{a,b})$, where
$U_{a,b}\subset E(\widehat{\sigma}_{1})$ is the open region enclosed between
$\wideparen{ab}$ and $\beta_{a,b}$;

  4. iv.

the curve $\beta_{a,b}$ is contained in the closed convex hull of $\Gamma$ and
$E(\widehat{\sigma}_{1})\setminus U_{a,b}$ is convex.

The inclusion $U_{a,b}\subset E(\widehat{\sigma}_{1})$ follows since
$\wideparen{ab}\subset\widehat{\sigma}_{1}([0,1])$, $E(\widehat{\sigma}_{1})$
is convex, and $S$ is contained in the convex envelope of $\Gamma$.
Furthermore by the minimality of $S$ one has

$\mathcal{A}(\psi_{a,b};U_{a,b})=\mathcal{H}^{2}(S^{+})<\int_{\wideparen{ab}}\varphi\,d\mathcal{H}^{1}=\int_{\wideparen{ab}}|\widehat{\psi}-\varphi|\,d\mathcal{H}^{1}\,.$
(5.18)

Here the strict inequality follows since the vertical wall spanning $\Gamma$
given by $\\{(x^{\prime},x_{3})\colon x^{\prime}\in\wideparen{ab},\
x_{3}\in[-\varphi(x^{\prime}),\varphi(x^{\prime})]\\}$ is a disc-type surface
but, since $\wideparen{ab}$ is not a segment, cannot be a solution to the
Plateau problem. We now consider the pair
$(\widetilde{\sigma},\widetilde{\psi})\in{\mathcal{W}_{\rm conv}}$ given by

$\widetilde{\sigma}:=(\widetilde{\sigma}_{1},\widehat{\sigma}_{2},\dots,\widehat{\sigma}_{n})\,,\qquad\widetilde{\psi}:=\begin{cases}0&\text{in
}\widetilde{E}\,,\\\ \psi_{a,b}&\text{in }U_{a,b}\,,\\\
\widehat{\psi}&\text{otherwise}\,,\end{cases}$ (5.19)

where $\widetilde{\sigma}_{1}$ is such that
$\widetilde{\sigma}_{1}([0,1])=(\widehat{\sigma}_{1}([0,1])\setminus\wideparen{ab})\cup\beta_{a,b}$
and $\widetilde{E}:=E(\widehat{\sigma})\setminus
U_{a,b}=E(\widetilde{\sigma})$. Then noticing that $\widehat{\psi}=0$ in
$U_{a,b}$, $E(\widehat{\sigma})=E(\widetilde{\sigma})\cup U_{a,b}$, and
recalling (5.18), we get

$\begin{split}{\mathcal{F}}(\widetilde{\sigma},\widetilde{\psi})&=\mathcal{A}(\widetilde{\psi};\Omega)-|E(\widetilde{\sigma})|+\int_{\partial\Omega}|\widetilde{\psi}-\varphi|\,d\mathcal{H}^{1}\\\
&=\mathcal{A}(\widehat{\psi};\Omega\setminus
U_{a,b})+\mathcal{A}(\psi_{a,b};U_{a,b})-|E(\widetilde{\sigma})|+\int_{\partial\Omega}|\widetilde{\psi}-\varphi|\,d\mathcal{H}^{1}\\\
&=\mathcal{A}(\widehat{\psi};\Omega)+\mathcal{A}(\psi_{a,b};U_{a,b})-|E(\widehat{\sigma})|+\int_{\partial\Omega}|\widehat{\psi}-\varphi|\,d\mathcal{H}^{1}\\\
&<\mathcal{A}(\widehat{\psi};\Omega)-|E(\widehat{\sigma})|+\int_{\partial\Omega}|\widetilde{\psi}-\varphi|\,d\mathcal{H}^{1}+\int_{\wideparen{ab}}|\widehat{\psi}-\varphi|\,d\mathcal{H}^{1}\\\
&=\mathcal{A}(\widehat{\psi};\Omega)-|E(\widehat{\sigma})|+\int_{\partial\Omega}|\widehat{\psi}-\varphi|\,d\mathcal{H}^{1}={\mathcal{F}}(\widehat{\sigma},\widehat{\psi})\,,\end{split}$

where the penultimate equality follows from the fact that $\widetilde{\psi}$
is continuous and equal to $\varphi$ on $\wideparen{ab}$ while the traces of
$\widetilde{\psi}$ and $\widehat{\psi}$ coincide on
$\partial\Omega\setminus\wideparen{ab}$. This contradicts the minimality of
$(\widehat{\sigma},\widehat{\psi})$.

Case B: Suppose by contradiction that the set
$\partial_{i}^{D}\Omega\cap\partial E(\widehat{\sigma})$ contains an isolated
point $c$ or has a straight segment $\overline{cc^{\prime}}$ as isolated
connected component. Then there are two arcs
$\wideparen{ab}\subset\partial_{i}^{D}\Omega$ and
$\wideparen{a^{\prime}b^{\prime}}\subset\partial E(\widehat{\sigma})$ with
either $a\neq a^{\prime}$ or $b\neq b^{\prime}$ (and with endpoints $a\neq b$
and $a^{\prime}\neq b^{\prime}$) such that
$\overline{aa^{\prime}}\cap\overline{bb^{\prime}}=\mathchar 31\relax$ and
$\wideparen{ab}\cap\wideparen{a^{\prime}b^{\prime}}=\\{c\\}$ (respectively
$\wideparen{ab}\cap\wideparen{a^{\prime}b^{\prime}}=\overline{cc^{\prime}}$).
Notice also that, since $\partial_{i}^{D}\Omega$ is not straight, the segment
$\overline{cc^{\prime}}$ does not coincide with $\partial_{i}^{D}\Omega$ and
hence the arc $\wideparen{ab}$ can be chosen so that it properly contains the
segment $\overline{cc^{\prime}}$. We consider the curves

$\Gamma:=\Gamma^{+}\cup\Gamma^{-}\,,\quad\Gamma^{+}:=\mathcal{G}_{\varphi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\wideparen{ab}}\cup\mathcal{G}_{\widehat{\psi}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\overline{aa^{\prime}}}\cup\mathcal{G}_{\widehat{\psi}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\overline{bb^{\prime}}}\,,\quad\Gamma^{-}:=\mathcal{G}_{-\varphi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\wideparen{ab}}\cup\mathcal{G}_{-\widehat{\psi}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\overline{aa^{\prime}}}\cup\mathcal{G}_{-\widehat{\psi}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\overline{bb^{\prime}}}\,.$
(5.20)

Notice that $\Gamma^{\pm}$ connect $a^{\prime}$ to $b^{\prime}$. By applying
again Lemma 5.4 to the nonplanar curve $\Gamma$ and arguing as in case A we
obtain the contradiction also in this case.

Case C: More generally, assume by contradiction that both the sets
$\partial_{i}^{D}\Omega\cap\partial E(\widehat{\sigma})$ and
$\partial_{i}^{D}\Omega\setminus\partial E(\widehat{\sigma})$ are nonempty.
Then we can find a not flat arc $\wideparen{ab}\subset\partial_{i}^{D}\Omega$
such that the following holds121212This is a consequence of the fact that
$\wideparen{ab}\setminus\partial E(\widehat{\sigma})$ is relatively open in
$\wideparen{ab}$, so it is an at most countable union of disjoint relatively
open arcs.: there are pairs of points
$\\{c_{j},d_{j}\\}_{j\in\mathbb{N}}\subset\partial_{i}^{D}\Omega\cap\partial
E(\widehat{\sigma})$ such that the arcs $\wideparen{ad_{0}}$,
$\wideparen{c_{0}b}$, and $\\{\wideparen{c_{j}d_{j}}\\}_{j=1}^{\infty}$ are
mutually disjoint and

$\wideparen{ab}\setminus\partial
E(\widehat{\sigma})=\wideparen{ad_{0}}\cup(\cup_{j=1}^{\infty}\wideparen{c_{j}d_{j}})\cup\wideparen{c_{0}b}\,.$

Without loss of generality, we might assume that all the points
$c_{j},d_{j}\in\widehat{\sigma}_{1}([0,1])$. For all $j\geq 1$ we denote by
$V_{j}$ the region enclosed by $\wideparen{c_{j}d_{j}}$ and $\partial
E(\widehat{\sigma})$131313These regions are simply connected since
$c_{j},d_{j}\in\widehat{\sigma}_{1}([0,1])$.. We now argue as in case B and
choose $a^{\prime},b^{\prime}\in\widehat{\sigma}_{1}([0,1])$. Additionally,
let $V_{0}=V_{0}^{a}\cup V_{0}^{b}$, with $V_{0}^{a}$ (respectively
$V_{0}^{b}$) be the region enclosed between $\partial E(\widehat{\sigma})$ and
$\overline{aa^{\prime}}\cup\wideparen{ad_{0}}$ ($\partial E(\widehat{\sigma})$
and $\overline{bb^{\prime}}\cup\wideparen{c_{0}b}$, respectively). We finally
define $\Gamma$ correspondingly, as in (5.20). Again by Lemma 5.4 the solution
$S$ to the Plateau problem corresponding to $\Gamma$ satisfies properties
i.-iv. with $a^{\prime}$ and $b^{\prime}$ in place of $a$ and $b$
respectively. Moreover by the minimality of $S$ for every $N\geq 1$ there
holds141414The right-hand side is the area of the surface given by the
(positive) subgraph of $\varphi$ on
$\wideparen{ab}\setminus\cup_{j=1}^{N}\wideparen{c_{j}d_{j}}$ and the graph of
$\widehat{\psi}$ on the region $\cup_{j=0}^{N}V_{j}$, which is of disc-type.
To see this we use that the trace of $\widehat{\psi}$ on the subarcs of
$\partial E(\widehat{\sigma})$ between the points $c_{j}$ and $d_{j}$ is zero
(and between $a^{\prime}$ and $d_{0}$, and $d_{0}$ and $b^{\prime}$).

$\mathcal{A}(\psi_{a^{\prime},b^{\prime}};U_{a^{\prime},b^{\prime}})=\mathcal{H}^{2}(S^{+})\leq\int_{\wideparen{ab}}\varphi\,d\mathcal{H}^{1}-\int_{\wideparen{ad_{0}}\cup\wideparen{c_{0}b}}\varphi\,d\mathcal{H}^{1}-\sum_{j=1}^{N}\int_{\wideparen{c_{j}d_{j}}}\varphi\,d\mathcal{H}^{1}+\sum_{j=0}^{N}\mathcal{A}(\psi,V_{j})\,.$
(5.21)

In particular by taking the limit as $N\to\infty$ in (5.21) we get

$\mathcal{A}(\psi_{a^{\prime},b^{\prime}};U_{a^{\prime},b^{\prime}})=\mathcal{H}^{2}(S^{+})\leq\int_{\wideparen{ab}\setminus\partial
E(\widehat{\sigma})}\varphi\,d\mathcal{H}^{1}+\mathcal{A}(\widehat{\psi},\cup_{j=0}^{\infty}V_{j})\,.$
(5.22)

Let $(\widetilde{\sigma},\widetilde{\psi})\in{\mathcal{W}_{\rm conv}}$ be
defined as in (5.19), then observing that $\widehat{\psi}=0$ in
$U_{a^{\prime},b^{\prime}}\setminus(\cup_{j=0}^{\infty}V_{j})$,
$E(\widehat{\sigma})=E(\widetilde{\sigma})\cup(U_{a^{\prime},b^{\prime}}\setminus\cup_{j=0}^{\infty}V_{j})$
and using (5.22) we deduce

$\begin{split}{\mathcal{F}}(\widetilde{\sigma},\widetilde{\psi})&=\mathcal{A}(\widehat{\psi};\Omega\setminus
U_{a^{\prime},b^{\prime}})+\mathcal{A}(\psi_{a^{\prime},b^{\prime}};U_{a^{\prime},b^{\prime}})-|E(\widetilde{\sigma})|+\int_{\partial\Omega}|\widetilde{\psi}-\varphi|\,d\mathcal{H}^{1}\\\
&=\mathcal{A}(\widehat{\psi};\Omega\setminus(\cup_{j=0}^{\infty}V_{j}))+\mathcal{A}(\psi_{a^{\prime},b^{\prime}};U_{a^{\prime},b^{\prime}})-|E(\widehat{\sigma})|+\int_{\partial\Omega}|\widetilde{\psi}-\varphi|\,d\mathcal{H}^{1}\\\
&\leq\mathcal{A}(\widehat{\psi};\Omega\setminus(\cup_{j=0}^{\infty}V_{j}))-|E(\widehat{\sigma})|+\int_{\partial\Omega}|\widetilde{\psi}-\varphi|\,d\mathcal{H}^{1}+\int_{\wideparen{ab}\cap\partial
E(\widehat{\sigma})}\varphi\,d\mathcal{H}^{1}+\mathcal{A}(\widehat{\psi};\cup_{j=0}^{\infty}V_{j})\\\
&=\mathcal{A}(\widehat{\psi};\Omega)-|E(\widehat{\sigma})|+\int_{\partial\Omega}|\widehat{\psi}-\varphi|\,d\mathcal{H}^{1}={\mathcal{F}}(\widehat{\sigma},\widehat{\psi})\,,\end{split}$

which in turn implies

${\mathcal{F}}(\widetilde{\sigma},\widetilde{\psi})\leq{\mathcal{F}}(\widehat{\sigma},\widehat{\psi})\,.$
(5.23)

To conclude we need to show that the inequality in (5.23) is strict. To this
aim we choose $c\in\\{c_{j}\\}_{j=1}^{\infty}$. Consider the curves
$\Gamma_{1}$ and $\Gamma_{2}$ defined as follows

$\Gamma_{1}:=\Gamma_{1}^{+}\cup\Gamma_{1}^{-}\,,\quad\Gamma_{1}^{+}:=\mathcal{G}_{\varphi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\wideparen{ac}}\cup\mathcal{G}_{\widehat{\psi}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\overline{aa^{\prime}}}\cup
l^{+}\,,\quad\Gamma_{1}^{-}:=\mathcal{G}_{-\varphi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\wideparen{ac}}\cup\mathcal{G}_{-\widehat{\psi}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\overline{aa^{\prime}}}\cup
l^{-}\,,$
$\Gamma_{2}:=\Gamma_{2}^{+}\cup\Gamma_{2}^{-}\,,\quad\Gamma_{2}^{+}:=\mathcal{G}_{\varphi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\wideparen{cb}}\cup\mathcal{G}_{\widehat{\psi}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\overline{bb^{\prime}}}\cup
l^{+}\,,\quad\Gamma_{2}^{-}:=\mathcal{G}_{-\varphi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\wideparen{cb}}\cup\mathcal{G}_{-\widehat{\psi}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\overline{bb^{\prime}}}\cup
l^{-}\,,$

where

$l^{+}:=(\\{c\\}\times[0,\varphi(c)])\,,\quad
l^{-}:=(\\{c\\}\times[-\varphi(c),0])\,.$

Let $S_{1}$ and $S_{2}$ be the solutions to the Plateau problem corresponding
to $\Gamma_{1}$ and $\Gamma_{2}$ respectively, so that properties i.-iv. are
satisfied with $c$ in place of $b^{\prime}$ and $a^{\prime}$ respectively. By
the minimality of $S$ we have

$\mathcal{A}(\psi_{a^{\prime},b^{\prime}},U_{a^{\prime},b^{\prime}})<\mathcal{A}(\psi_{a^{\prime},c},U_{a^{\prime},c})+\mathcal{A}(\psi_{c,b^{\prime}},U_{c,b^{\prime}})\,.$
(5.24)

On the other hand by arguing as above151515With the arc $\wideparen{ac}$
($\wideparen{cb}$, respectively) in place of $\wideparen{ab}$. we conclude

$\mathcal{A}(\psi_{a^{\prime},c},U_{a^{\prime},c})\leq\int_{\wideparen{ac}\cup\partial
E(\widehat{\sigma})}\varphi\,d\mathcal{H}^{1}+\mathcal{A}(\widehat{\psi},\cup_{j\in
I_{1}}V_{j}\cup V^{a}_{0})\,,$ (5.25)

and

$\mathcal{A}(\psi_{c,b^{\prime}},U_{c,b^{\prime}})\leq\int_{\wideparen{cb}\cup\partial
E(\widehat{\sigma})}\varphi\,d\mathcal{H}^{1}+\mathcal{A}(\widehat{\psi},\cup_{j\in
I_{2}}V_{i}\cup V^{b}_{0})\,,$ (5.26)

where $I_{1}:=\\{j\colon\wideparen{c_{j}d_{j}}\subset\wideparen{ac}\\}$ and
$I_{2}:=\\{j\colon\wideparen{c_{j}d_{j}}\subset\wideparen{cb}\\}$. Gathering
together (5.24)-(5.26) we derive

$\mathcal{A}(\psi_{a^{\prime},b^{\prime}},U_{a^{\prime},b^{\prime}})<\int_{\wideparen{ab}\cup\partial
E(\widehat{\sigma})}\varphi\,d\mathcal{H}^{1}+\mathcal{A}(\widehat{\psi},\cup_{j=0}^{\infty}V_{j})\,,$

which in turn implies

${\mathcal{F}}(\widetilde{\sigma},\widetilde{\psi})<{\mathcal{F}}(\widehat{\sigma},\widehat{\psi})\,,$

and thus the contradiction.

Step 2: Assuming there is $i\in\\{1,\dots,n\\}$ such that
$\partial_{i}^{D}\Omega$ is a straight segment, and we show that either
$\partial E(\widehat{\sigma})\cap\partial_{i}^{D}\Omega=\mathchar 31\relax$ or
$\partial
E(\widehat{\sigma})\cap\partial_{i}^{D}\Omega=\partial_{i}^{D}\Omega$.
Suppose by contradiction that $\partial
E(\widehat{\sigma})\cap\partial_{i}^{D}\Omega\neq\mathchar 31\relax$ and also
$\partial_{i}^{D}\Omega\setminus\partial E(\widehat{\sigma})\neq\mathchar
31\relax$. Without loss of generality we can restrict to the case $\partial
E(\widehat{\sigma})\cap\partial_{i}^{D}\Omega=\partial
F\cap\partial_{i}^{D}\Omega$ with $F$ any connected component of
$E(\widehat{\sigma})$. Since $F$ is convex and $\partial_{i}^{D}\Omega$ is a
segment $\partial F\cap\partial_{i}^{D}\Omega$ has to be connected, i.e., it
is either a single point $a$ or a segment
$\overline{aa^{\prime}}\neq\partial_{i}^{D}\Omega$.
In both cases we then consider a (small enough) ball $B$ centred at $a$ such
that $B\cap E(\widehat{\sigma})=B\cap F$ (in the second case we also require
that the radius of $B$ is smaller than $\overline{aa^{\prime}}$).

If $\partial F\cap\partial_{i}^{D}\Omega=\\{a\\}$ we let $\\{p,q\\}:=\partial
B\cap\partial F$ and $\\{b,c\\}:=\partial B\cap\partial_{i}^{D}\Omega$ (with
$b,p$ and $c,q$ lying on the same side with respect to $a$). Then we define
the curves

$\Gamma:=\Gamma^{+}\cup\Gamma^{-}\,,\quad\Gamma^{+}:=\mathcal{G}_{\varphi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\overline{bc}}\cup\mathcal{G}_{\psi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\wideparen{bp}}\cup\mathcal{G}_{\psi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\wideparen{cq}}\,,\quad\Gamma^{-}:=\mathcal{G}_{-\varphi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\overline{bc}}\cup\mathcal{G}_{-\psi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\wideparen{bp}}\cup\mathcal{G}_{-\psi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\wideparen{cq}}\,,$

where $\wideparen{bp}$, $\wideparen{cq}$ denote the arcs in $\partial B$
joining $b$ to $p$ and $c$ to $q$ respectively.

If $\partial F\cap\partial_{i}^{D}\Omega=\overline{aa^{\prime}}$ we let
$\\{p,q\\}:=\partial B\cap\partial F$ and $\\{b,c\\}:=\partial
B\cap\partial_{i}^{D}\Omega$ where we identify $q$ and $c$. Then we consider
the curves

$\Gamma:=\Gamma^{+}\cup\Gamma^{-}\,,\quad\Gamma^{+}:=\mathcal{G}_{\varphi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\overline{bc}}\cup\mathcal{G}_{\psi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\wideparen{bp}}\cup(\\{c\\}\times[0,\varphi(c)])\,,\quad\Gamma^{-}:=\mathcal{G}_{-\varphi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\overline{bc}}\cup\mathcal{G}_{-\psi\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\wideparen{bp}}\cup(\\{c\\}\times[-\varphi(c),0])\,.$

By applying again Lemma 5.4 to $\Gamma$ and arguing as above we get the
contradiction.

Step 3: We show that there is a minimizer
$(\widetilde{\sigma},\widetilde{\psi})$ that satisfies property 5.
We first notice that $\widehat{\psi}$ is continuous and null on $\partial
E(\widehat{\sigma})\setminus\partial^{D}\Omega$. Moreover by steps 1 and 2 it
follows that $\partial E(\widehat{\sigma})\cap\Omega$ is the union of a finite
number of pairwise disjoint Lipschitz curves each of them joining each $p_{i}$
for $i=1,\dots,n$ to each of the $q_{j}$ for some $j=1,\dots,n$. To conclude
it is enough to replace each curve, without increasing the energy, with a
smooth one having the same endpoints. More precisely, let $\gamma$ be any of
such curves. Reasoning as in the proof of Lemma 5.11 step 1, we can replace
$(\widehat{\sigma},\widehat{\psi})$ with a new minimizer
$(\sigma^{\gamma},\psi^{\gamma})\in{\mathcal{W}_{\rm conv}}$ such that
$\partial E(\sigma^{\gamma})\cap\partial\Omega=\partial
E(\sigma)\cap\partial\Omega$ and $\psi^{\gamma}=0$ on $\gamma^{\prime}$, where
$\gamma^{\prime}\subset\partial E(\sigma^{\gamma})\cap\Omega$ is a suitable
smooth curve that replaces $\gamma$ and has the same endpoints of $\gamma$. In
particular $\psi^{\gamma}$ is continuous and null on $\partial
E(\sigma^{\gamma})\setminus\partial^{D}\Omega$. Eventually iterating this
procedure for each curve in $\partial
E(\widehat{\sigma})\setminus\partial\Omega$ we can construct a new minimizer
$(\widetilde{\sigma},\widetilde{\psi})$ with the required properties. ∎

###  5.1 The example of the catenoid containing a segment

Consider the setting depicted in Figure 4. Here
$\Omega=R_{2l}=(0,2l)\times(-1,1)$, $n=1$,
$\partial^{D}\Omega=(\\{0,2l\\}\times(-1,1))\cup((0,2l)\times\\{-1\\})$ and
$\partial^{0}\Omega=(0,2l)\times\\{1\\}$, $p=(0,1)$, $q=(2l,1)$. The map
$\varphi$ is $\varphi(w_{1},w_{2})=\sqrt{1-w_{2}^{2}}$, and thus vanishes on
$[0,2l]\times\\{-1\\}$; for this reason this case is not covered by our
analysis. However we can find a solution as in Theorem 1.1 also in this case,
by an approximation procedure.

Precisely, for $\varepsilon>0$ and consider an approximating sequence
$(\varphi)_{\varepsilon}$ of continuous Dirichlet data, with
$\mathcal{G}_{\varphi_{\varepsilon}}$ Lipschitz, which tends to $\varphi$
uniformly and satisfy $\varphi_{\varepsilon}=0$ on $\partial^{0}\Omega$,
$\varphi_{\varepsilon}>0$ on $\partial^{D}\Omega$. Let
$(\sigma_{\varepsilon},\psi_{\varepsilon})$ be a solution as in Theorem 1.1
corresponding to the boundary datum $\varphi_{\varepsilon}$; as
$\mathcal{F}(\sigma_{\varepsilon},\psi_{\varepsilon})$ is equibounded161616We
can indeed always bound it from above by
$|\Omega|+\int_{\partial_{D}\Omega}|\varphi_{\varepsilon}|d\mathcal{H}^{1}$.,
arguing as in the proof of Lemma 3.4, we can see that, up to a subsequence,
$((\sigma_{\varepsilon},\psi_{\varepsilon}))$ tends to some
$(\sigma,\psi)\in\mathcal{W}_{\textrm{conv}}$, which minimizes the functional
$\mathcal{F}$ with Dirichlet condition $\varphi$. In this case however we
cannot guarantee that $\sigma$ does not touch $\partial^{D}\Omega$, even if
this is not a straight segment. This is essentially due to the presence of the
portion $[0,2l]\times\\{-1\\}$ of $\partial\Omega$ where $\varphi$ is zero,
which does not allow to apply the arguments used in the proof of Theorem 5.1.

In particular, it can be seen that if $l$ is large enough, the solution
$(\sigma,\psi)$ splits and becomes degenerate, being $\psi\equiv 0$ and the
functional $\mathcal{F}$ pays only the area of two vertical half discs of
radius $1$. Under a certain threshold instead the solution satisfies the
regularity properties stated in Theorem 5.1, and in particular $\psi=\varphi$
on $\partial^{D}\Omega$, and $\sigma$ is the graph of a smooth convex function
passing through $p$ and $q$. We refer to [6] for details and comprehensive
proofs of these facts.

##  6 Comparison with the parametric Plateau problem: The case $n=1,2$

In this section we compare the solutions provided by Theorems 3.1 and 5.1 with
the solutions to the classical Plateau problem in parametric form.
Specifically, motivated by the example of the catenoid, we will restrict our
analysis to the classical disc-type and annulus-type Plateau problem. These
configurations correspond to the cases $n=1$ and $n=2$ respectively, i.e., the
Dirichlet boundary $\partial^{D}\Omega$ is either an open arc or the union of
two open arcs of $\partial\Omega$ with disjoint closure. Due to the highly
involved geometric arguments, we do not discuss the case $n>2$, which requires
further investigation.