# Existence and uniqueness of $t$-graphs of prescribed mean curvature in
Heisenberg groups

Julián Pozuelo  Dipartamento di Matematica “Tullio Levi-Civita”
Università di Padova
Via Trieste, 63, 35131, Padova, Italy<EMAIL_ADDRESS>and
Simone Verzellesi  Department of Mathematics, University of Trento, Via
Sommarive 14, 38123 Povo (Trento), Italy<EMAIL_ADDRESS>missing

###### Abstract.

We study the prescribed mean curvature equation for $t$-graphs in a Riemannian
Heisenberg group of arbitrary dimension. We characterize the existence of
classical solutions in a bounded domain without imposing Dirichlet boundary
data, and we provide conditions that guarantee uniqueness. Moreover, we extend
previous results to solve the Dirichlet problem when the mean curvature is
non-constant. Finally, by an approximation technique, we obtain solutions to
the sub-Riemannian prescribed mean curvature equation.

###### Key words and phrases:

Prescribed mean curvature, Heisenberg group, sub-Riemannian geometry

###### 1991 Mathematics Subject Classification:

53C17, 49Q10, 53A10

Acknowledgements. The authors would like to thank L. Capogna, G. Giovannardi,
A. Pinamonti, M. Ritoré, C. Rosales and G. Saracco for stimulating and
fruitful conversations about the contents of this paper. J. Pozuelo is
supported by the research grant PID2020-118180GB-I00 funded by
MCIN/AEI/10.13039/501100011033 and Fundación Ramón Areces grants _XXXV
convocatoria para la amplación de estudios en el extranjero en ciencias de la
vida y la materia_. S. Verzellesi is member of the Istituto Nazionale di Alta
Matematica (INdAM), Gruppo Nazionale per l’Analisi Matematica, la Probabilità
e le loro Applicazioni (GNAMPA), and is supported by INdAM–GNAMPA 2023 Project
_Equazioni differenziali alle derivate parziali di tipo misto o dipendenti da
campi di vettori_ and by MIUR-PRIN 2022 Project _Regularity problems in sub-
Riemannian structures_.

In Memory of Enrico Giusti

##  1\. Introduction

The Plateau problem has been a fundamental issue in geometry since the
pioneering works of Douglas (cf. [26]) and Radó (cf. [59]). The Euclidean
prescribed mean curvature equation of the graph of a function $u\in
C^{2}(\Omega)$ over a bounded domain $\Omega\subseteq{\mathbb{R}}^{n}$ with
$H\in C(\Omega)$ reads as

(PMC) $\operatorname{div}\left(\frac{Du}{\sqrt{1+|Du|^{2}}}\right)=H,$

where $D$ stands for the Euclidean gradient. Notice that we are taking the
mean curvature as the (not averaged) sum of the principal curvatures. When $H$
is constant and $\partial\Omega$ is of class $C^{2}$, Serrin (cf. [67])
characterized the existence of solutions to the Dirichlet problem for _any_
boundary datum $\varphi\in C^{2}(\partial\Omega)$ by the condition

(1.1) $|H|\leqslant H_{\partial\Omega}(z_{0})$

for any $z_{0}\in\partial\Omega$, where $H_{\partial\Omega}$ is the mean
curvature of the boundary of $\Omega$. In the proof, Serrin obtained Schauder
estimates for $C^{2}$ solutions first by providing height estimates for $|u|$,
and then, by means of a gradient maximum principle, showing that the maximum
of the gradient is attained at the boundary of $\Omega$. In the final step, he
estimated the gradient at the boundary exploiting the so-called _barriers_
(cf. [37]), whose construction relies on (1.1). When $H$ is not constant, an
approach based on the maximum principle typically fails. Therefore, in order
to deal both with non-constant sources and to allow merely continuous boundary
data, it is customary to rely on suitable interior and global gradient
estimates. Some references to these kind of estimates in the Euclidean space
are the works of Korevaar and Simon (cf. [46]) and Wang (cf. [71]). Beyond the
Euclidean framework, the Dirichlet problem for any sufficiently regular
boundary datum and constant source $H$ satisfying conditions analogous to
(1.1) has been studied in warped products with a particular lower bound on the
Ricci curvature (cf. [69]), in $3$-dimensional Heisenberg groups (cf. [3]) and
in higher dimensional Heisenberg groups (cf. [38]). When $H$ is not constant,
the previous results were later extended in Riemannian manifolds with a
Killing vector field and a lower bound on the Ricci curvature depending on
$\Omega$ (cf. [21, 20, 22]). When instead (1.1) fails, meaning that

(1.2) $|H|>H_{\partial\Omega}(z_{0})$

for some $z_{0}\in\partial\Omega$, we could lose control of the norm of the
gradient of a solution near the boundary, and hence of the existence of
solutions. More precisely, as shown in [37], when (1.2) holds there always
exists a boundary datum $\varphi$ for which the Dirichlet problem has no
solution. Nevertheless, the validity of (1.2) does not preclude _a priori_ the
existence of a _suitable_ boundary datum $\varphi$ for which the Dirichlet
problem is solvable. As an instance, taking as domain
$\Omega\subseteq{\mathbb{R}}^{n}$ the ball of radius $1$ centered at $0$, we
can write the half sphere in ${\mathbb{R}}^{n+1}$ centered at $0$ with radius
$1$ as a graph over $\Omega$. A simple computation reveals that it satisfies
(PMC) with $H=\frac{n}{n-1}H_{\partial\Omega}$ and with boundary datum
$\varphi\equiv 0$. In particular, (1.2) is verified for any
$z_{0}\in\partial\Omega$. In this regard, when $\Omega$ has Lipschitz
boundary, Giusti (cf. [40]) proved that the existence of solutions to (PMC)
with a suitable boundary condition, not imposed _a priori_ , is characterized
by

(*) $\left|\int_{\tilde{\Omega}}H(x)\,dx\right|<P(\tilde{\Omega})$

for any set $\tilde{\Omega}\subseteq\Omega$ such that
$\tilde{\Omega}\neq\emptyset$ and $\tilde{\Omega}\neq\Omega$, where
$P(\tilde{\Omega})$ is the perimeter of $\tilde{\Omega}$. Moreover, [40]
provides a characterization of those domains where (PMC) admits, up to
vertical translations, a unique solution. Precisely, the previous statement is
equivalent to each of the following conditions: there is no solution to (PMC)
in any domain $\Omega\subsetneq\hat{\Omega}$; there is a solution on $\Omega$
which is vertical at every point of $\partial\Omega$; (* ‣ 1) holds and

(**) $\left|\int_{\Omega}H(x)\,dx\right|=P(\Omega).$

In these cases, $\Omega$ is called an _extremal domain_. Otherwise, i.e. when
(* ‣ 1) also holds for $\tilde{\Omega}=\Omega$, then $\Omega$ is called a
_non-extremal domain_. The proof of the existence of solutions under condition
(* ‣ 1) relies on previous results by Giaquinta (cf. [35, 34]) and Miranda
(cf. [51]). In the non-extremal case the proof consists in showing the
existence of $BV$ minimizers of the penalized functional

(1.3)
$\mathcal{F}(u)=\int_{\Omega}\sqrt{1+|Du|^{2}}+\int_{\Omega}Hu\,dx+\int_{\partial\Omega}|u-\varphi|\,d\mathcal{H}^{n-1}$

for any $\varphi\in L^{1}(\partial\Omega)$, whose regularity is then gradually
improved in several steps. The more involved extremal case, i.e. when (** ‣ 1)
holds, follows by a compactness procedure. More precisely, in view of
condition (* ‣ 1), every domain $\tilde{\Omega}\subsetneq\Omega$ is itself a
non-extremal domain. Therefore, exploiting the existence result in the non-
extremal case, together with a compactness argument based on a notion of
_generalized solution_ first introduced by Miranda (cf. [52]), existence in
the extremal case follows. For similar results under weaker assumptions on the
boundary of $\Omega$ we refer the reader to [48].

Although the Dirichlet problem, as previously discussed, has been widely
studied beyond the Euclidean framework, as far as the authors are aware no
results in the spirit of [40] are available in the Riemannian setting. The aim
of this work, consequently, is to lay the groundwork for the study of
hypersurfaces with prescribed mean curvature outside the Euclidean setting and
overcoming conditions inspired by (1.1). As a relevant first instance, we
choose as ambient manifold the _$n$ -th dimensional Riemannian Heisenberg
group_. The Heisenberg group ${\mathbb{H}}^{n}$, for $n\geqslant 1$, is
${\mathbb{R}}^{2n+1}$ endowed with the non-Abelian group law

$(x,y,t)\ast(x,y^{\prime},t^{\prime})=\left(x+x^{\prime},y+y^{\prime},t+t^{\prime}\sum_{j=1}^{n}\left(x_{j}^{\prime}y_{j}-x_{j}y_{j}^{\prime}\right)\right),$

where $x=(x_{1},\ldots,x_{n})$,
$x^{\prime}=(x_{1}^{\prime},\ldots,x_{n}^{\prime})$, $y=(y_{1},\ldots,y_{n})$
and $y^{\prime}=(y_{1}^{\prime},\ldots,y_{n}^{\prime})$, which realizes it as
(the most relevant instance of) _Carnot group_ (cf. [9]). The Lie algebra of
${\mathbb{H}}^{n}$ is generated by the family of left-invariant vector fields

$X_{i}=\dfrac{\partial}{\partial x_{i}}+y_{i}\dfrac{\partial}{\partial
t},\qquad Y_{i}=\dfrac{\partial}{\partial
y_{i}}-x_{i}\dfrac{\partial}{\partial t},\qquad T=\dfrac{\partial}{\partial
t},$

for $i=1,\ldots,n$. A remarkable class of Riemannian structures can be defined
on ${\mathbb{H}}^{n}$ by choosing, for any $\varepsilon\neq 0$, the unique
Riemannian metric $g_{\varepsilon}$ which makes
$\\{X_{1},\ldots,X_{n},Y_{1},\ldots,Y_{n},\varepsilon T\\}$ an orthonormal
frame. The importance of $({\mathbb{H}}^{n},g_{\varepsilon})$ in the
Riemannian framework is supported by several reasons. For instance, it appears
in the classification of homogeneous $3$-spaces with isometry group of
dimension $4$, usually denoted by $Nil_{3}(-1/\tau)$ (cf. [1]). When
$\varepsilon$ goes to $0$, the space $(\mathbb{H}^{n},g_{\varepsilon})$
converges in Gromov-Hausdorff sense to the sub-Riemannian Heisenberg group
$({\mathbb{H}}^{n},\langle\cdot,\cdot\rangle)$, where
$\langle\cdot,\cdot\rangle$ is the restriction of any of the metric
$g_{\varepsilon}$ to the _horizontal distribution_
${\mathcal{H}}=\operatorname{span}\\{X_{1},\ldots,X_{n},Y_{1},\ldots,Y_{n}\\}$.
The sub-Riemannian Heisenberg group $\mathbb{H}^{n}$ is itself of fundamental
importance in various settings, since it constitutes the prototypical model in
the context of Carnot groups, sub-Riemannian manifolds (cf. [2]) and _CR
manifolds_ (cf. [14]). In the following, we deal with a relevant class of non-
parametric hypersurfaces, namely that of _vertical or ( $t$-) graphs_, i.e.
Euclidean graphs over the horizontal hyperplane
$\\{t=0\\}\equiv{\mathbb{R}}^{2n}$. The equation of prescribed mean curvature
of a $t$-graph over a bounded domain $\Omega\subseteq{\mathbb{R}}^{2n}$ for a
given source $H$ is formally given by

($\varepsilon$-PMC)
$\operatorname{div}\left(\frac{Du+X}{\sqrt{\varepsilon^{2}+|Du+X|^{2}}}\right)=H,$

where $X(x,y)=(-y,x)$ (cf. [16, 38]). Our main achievement, in the spirit of
[40], is an existence and regularity result for solutions to
($\varepsilon$-PMC), both in the non-extremal and in the extremal case.

######  Theorem 1.1.

Let $\Omega\subseteq{\mathbb{R}}^{2n}$ be a bounded domain with Lipschitz
boundary, and let $H\in\operatorname{Lip}(\Omega)\cap
C^{1,\gamma}_{loc}(\Omega)$ for some $\gamma\in(0,1)$. Then (* ‣ 1) holds if
and only if there exists $u\in C^{2}(\Omega)$ which is a classical solution to
($\varepsilon$-PMC) on $\Omega$. Moreover, if $H\in
C^{k,\gamma}_{loc}(\Omega)$ for some $k\in\mathbb{N}$, $k\geqslant 1$, then
$u\in C^{k+2,\gamma}_{loc}(\Omega)$. Finally, if $H\in C^{\infty}(\Omega)$,
then $u\in C^{\infty}(\Omega)$.

Our approach can be summarized in the following major points.

  * •

Following Giusti’s scheme, we will first prove existence of $BV$ minimizers of
a suitable penalized functional, analogous to (1.3), in the non-extremal case
(cf. Proposition 5.3). To improve the regularity of such minimizers, we first
rely on suitable variational properties of minimizers (cf. Theorem 5.5 and
Proposition 5.6) to infer that minimizers are locally bounded in $\Omega$ (cf.
Proposition 5.7).

  * •

Our second step relies on a generalization to the anisotropic setting (cf.
[68]) of some celebrated regularity results for almost-minimizers of the
perimeter (cf. e.g. [24, 60, 4, 64, 70, 8], and cf. [27, 49] and references
therein for recent developments). Exploiting some results from [68], we shall
see that the boundary $\partial E_{u}$ of the subgraph of a minimizer $u$ is
regular outside a singular portion with small Hausdorff dimension (cf.
Proposition 5.9). A crucial result then consists in translating these
regularity properties from $\partial E_{u}$ to $u$. More precisely, we show
that $u$ is regular outside a small set $\Omega_{u,0}\subseteq\Omega$ for
which $\mathcal{H}^{s}(\Omega_{u,0})=0$ for any $s>2n-7$ (cf Proposition
5.11). The proof of these results is based on a careful analysis of the
prescribed mean curvature equation for _intrinsic graphs_ in the sense of [6].

  * •

Next, owing to some structure properties of the Riemannian perimeter induced
by the metric $g_{\varepsilon}$ (cf. Lemma 2.4), we show that minimizers enjoy
Sobolev regularity (cf. Proposition 5.12).

  * •

In view of the previous steps, and exploiting a suitable existence result for
the Dirichlet problem in small balls (cf. Theorem 1.3), we provide local
Lipschitz regularity by means of an approximation procedure and a comparison
principle argument (cf. Proposition 5.13).

  * •

To pass from Lipschitz regularity to higher regularity, exploiting a well-
established approach, we write a linear uniformly elliptic equation for the
function

$u_{v}(z)=\frac{u(z+v)-u(z)}{|v|},$

so that both the De Giorgi-Nash-Moser theory for $C^{1,\alpha}$ regularity and
the classical Schauder theory for higher regularity apply (cf. Theorem 5.8).
This last step basically concludes the proof of Theorem 1.1 in the non-
extremal case.

  * •

Finally, the existence of classical solutions in the extremal case (cf.
Section 5.6) follows exploiting the aforementioned approximating procedure,
together with a suitable compactness argument and the extension to the
Riemannian setting of the Euclidean notion of generalized solution to (PMC).

As already mentioned, since our source $H$ may not be constant, a crucial step
in the proof of local Lipschitz regularity is the use of suitable interior
gradient estimates. More precisely, we extend the proof of Korevaar and Simon
(cf. [46]) to achieve the following result, which may be of independent
interest.

######  Theorem 1.2 (Interior gradient estimates).

Let $\Omega\subseteq{\mathbb{R}}^{2n}$ be a bounded domain. Let $H\in
C^{1}(\Omega)$. Let $u\in C^{3}(\Omega)$ be a solution to ($\varepsilon$-PMC)
and let $\tilde{\Omega}\Subset\Omega$ be a domain. For any domain
$\hat{\Omega}\Subset\tilde{\Omega}$ there exists a constant
$C=C\left(n,\varepsilon,d(\partial\hat{\Omega},\partial\tilde{\Omega}),\|u\|_{L^{\infty}(\tilde{\Omega})},\|H\|_{C^{1}(\tilde{\Omega})},\|X\|_{L^{\infty}(\tilde{\Omega})}\right)>0$
such that

(1.4) $\|Du\|_{L^{\infty}(\hat{\Omega})}\leqslant C,$

where $d(\partial\hat{\Omega},\partial\tilde{\Omega})$ the Euclidean distance
between $\partial\hat{\Omega}$ and $\partial\tilde{\Omega}$.

In the proof of Theorem 1.2, a crucial role is played by the identity

(1.5)
$\begin{split}\Delta_{S}(\nu^{\varepsilon}_{2n+1})&=g_{\varepsilon}(\nabla_{S}H,\varepsilon
T)-\nu^{\varepsilon}_{2n+1}\left(\operatorname{Ric}_{\varepsilon}(\nu^{\varepsilon},\nu^{\varepsilon})+|h^{\varepsilon}|^{2}\right),\end{split}$

which holds for an embedded orientable hypersurface of class $C^{3}$.
Exploiting (1.5), we can study the sign of the Laplace-Beltrami operator
$\Delta_{S}$ associated to $S=\text{graph}(u)$, applied to the vertical
coordinate of the Riemannian normal to the graph, $\nu^{\varepsilon}_{2n+1}$.
In (1.5), $\nabla_{S}$, $\operatorname{Ric}_{\varepsilon}$ and
$h^{\varepsilon}$ stand for the gradient in $S$, the Ricci curvature of
$(\mathbb{H}^{2n+1},g_{\varepsilon})$ and the second fundamental form of $S$
respectively. Remarkably, assuming the additional condition

(1.6) $|H|<H_{\partial\Omega}(z_{0})$

for any $z_{0}\in\partial\Omega$, i.e. the sub-optimal version of (1.1)
required in [3, 38], the proof of Theorem 1.2 can be adapted to provide global
gradient estimates (cf. Theorem 4.3). Consequently, we can improve the
existence results proved in [3, 38] for the Riemannian Dirichlet problem
associated to ($\varepsilon$-PMC). More precisely, we generalize [38, Theorem
6.1] to the case in which $H$ is not constant.

######  Theorem 1.3.

Let $\Omega\subseteq{\mathbb{R}}^{2n}$ be a bounded domain with $C^{2,\alpha}$
boundary, for some $\alpha\in(0,1)$. Let $\varphi\in
C^{2,\alpha}(\overline{\Omega})$ and let $H\in\operatorname{Lip}(\Omega)$.
Assume that (* ‣ 1) holds and that $\Omega$ is a non-extremal domain. Assume
in addition that (1.6) holds. Then, for any $\varepsilon\neq 0$ there exists
$u_{\varepsilon}\in\operatorname{Lip}(\Omega)\cap C^{2,\alpha}_{loc}(\Omega)$
which solves ($\varepsilon$-PMC) on $\Omega$ and such that
$u_{\varepsilon}=\varphi$ on $\partial\Omega$. If in addition $H\in
C^{1,\alpha}(\overline{\Omega})$, then $u_{\varepsilon}\in
C^{2,\alpha}(\overline{\Omega})$.

Regarding uniqueness, we shall prove that the extremal condition (** ‣ 1) is
equivalent to the maximality of the domain and the verticality of solutions to
($\varepsilon$-PMC), in a weak sense, at the boundary. Moreover, we show that
and any of these conditions implies uniqueness of solutions of
($\varepsilon$-PMC) up to vertical translations.

######  Theorem 1.4.

Let $\Omega\subseteq{\mathbb{R}}^{2n}$ be a bounded domain with Lipschitz
boundary. Let $H\in\operatorname{Lip}(\Omega)\cap C^{1,\gamma}_{loc}(\Omega)$
for some $\gamma\in(0,1)$ satisfy (* ‣ 1). The following statements are
equivalent.

  * (i)

If $\hat{\Omega}\subseteq{\mathbb{R}}^{2n}$ is any domain such that
$\Omega\subsetneq\hat{\Omega}$, then there is no solution $u\in
C^{2}(\hat{\Omega})$ to ($\varepsilon$-PMC) in $\hat{\Omega}$.

  * (ii)

(** ‣ 1) holds.

Moreover, if $\partial\Omega$ is of class $C^{2}$, then $(i)$ and $(ii)$ are
equivalent to the following condition.

  * (iii)

For any $u\in C^{2}(\Omega)$ which solves ($\varepsilon$-PMC) in $\Omega$, it
holds that

$\lim_{t\to
0^{+}}\left|\int_{\partial\Omega_{t}}\frac{\langle\nu_{t},Du+X\rangle}{\sqrt{\varepsilon^{2}+|Du+X|^{2}}}\,d\mathcal{H}^{2n-1}\right|=\mathcal{H}^{2n-1}(\partial\Omega),$

where $\Omega_{t}=\\{z\in\Omega\,:\,\min_{w\in\partial\Omega}|z-w|>t\\}$ is
defined for $t>0$ small enough and $\nu_{t}$ is its exterior unit normal.

Finally, if $\partial\Omega$ is of class $C^{2}$, each of the previous three
conditions implies that solutions to ($\varepsilon$-PMC) in $\Omega$ are
unique up to vertical translations.

With regard to the sub-Riemannian Heisenberg group ${\mathbb{H}}^{n}$, it is
well known (cf. [10, 16, 38]) that the equation of prescribed mean curvature
of a $t$-graph over $\Omega\subseteq{\mathbb{R}}^{2n}$ is formally given by

(${\mathcal{H}}$-PMC) $\operatorname{div}\left(\frac{Du+X}{|Du+X|}\right)=H.$

Due to the possible presence of _characteristic points_ , i.e. points in
$\Omega$ where $Du+X$ vanishes, (${\mathcal{H}}$-PMC) may be both degenerate
elliptic and singular (cf. [16, 38]). Therefore, this issue should be
understood in a weak sense. More precisely, the problem of finding $t$-graphs
with prescribed mean curvature can be formulated by looking for local
minimizers of the functional

(1.7) $\mathcal{I}(u)=\int_{\Omega}|Du+X|+\int_{\Omega}Hu\,dz.$

When $H=0$, the regularity of such minimizers has carried the attention of
many authors. In the papers of Ritoré (cf. [61]) and Serra Cassano and Vittone
(cf. [66]) there are examples in ${\mathbb{H}}^{1}$ both of entire area-
minimizing $t$-graphs with merely locally Lipschitz regularity and of even
discontinuous area-minimizing $t$-graph. Assuming that $H=0$ and that
minimizers have at least $C^{1}$ regularity, Cheng, Hwang and Yang (cf. [17])
proved the presence of a foliation by $C^{2}$ curves in the non-characteristic
set. Later on, Ritoré and Galli (cf. [31]) extended this result for a
continuous source $H$, and Ritoré and Giovannardi (cf. [39]) obtained the same
property under the weaker assumption of Lipschitz regularity and intrinsic
$C^{1}$ regularity in the sub-Finsler Heisenberg group. Regarding the
existence of such solutions, it is customary to rely on the aforementioned
Riemannian approximation (cf. [16, 55]). A similar approximation scheme was
considered in the sub-Riemannian setting in [12, 11] to study the Lipschitz
regularity for non-characteristic minimal surfaces. For a detailed analysis of
this approach, we refer to [10]. Accordingly, our existence result can be
applied to study minimization problems related to (1.7). More precisely, we
obtain solutions in $BV_{loc}(\Omega)\cap L^{\infty}_{loc}(\Omega)$ to the
sub-Riemannian prescribed mean curvature equation in the following sense.

######  Theorem 1.5.

Let $\Omega\subseteq{\mathbb{R}}^{2n}$ be an open and bounded set with
Lipschitz boundary. Let $H\in\operatorname{Lip}(\Omega).$ Assume that
$\eqref{Giusti}$ holds. Then, there exists $u\in BV_{loc}(\Omega)\cap
L^{\infty}_{loc}(\Omega)$ such that $u$ is an $H$-minimizer for
$P_{\mathbb{H}}$ on $\Omega\times{\mathbb{R}}$ in the sense of (7.1).
Moreover, there exist a sequence of open sets such that
$\Omega_{j}\Subset\Omega_{k}\Subset\Omega$ for any $j<k$ and
$\bigcup_{j=0}^{\infty}\Omega_{j}=\Omega$ and a sequence $(u_{j})_{j}\subseteq
C^{\infty}(\Omega_{j})$, such that each $u_{j}$ solves ($\varepsilon$-PMC) in
$\Omega_{j}$ and moreover

$u_{j}\to u\text{ almost everywhere in $\Omega$ }\qquad\text{and}\qquad
P_{\varepsilon_{j}}(E_{u_{j}},\cdot)\rightharpoonup^{*}P_{\mathbb{H}}(E_{u},\cdot)\text{
locally in }\Omega\times{\mathbb{R}},$

where $\rightharpoonup^{*}$ denotes the _weak- ∗ convergence_ of measures.

We point out that Theorem 1.5 generalizes the existence result proved in [66]
for minimal $t$-graphs allowing $H$ to be different from zero. Another
interesting difference with respect to [66] consists in the different approach
to the existence issue. Indeed, Serra Cassano and Vittone provided existence
essentially via direct methods. On the other hand, although we believe that
the same strategy could have worked as well in our framework, we preferred to
provide existence combining direct methods and approximation. Indeed, as
nicely explained in [12], sub-Riemannian minimizers arising as limit of
Riemannian minimizers are only a particular subfamily of all sub-Riemannian
minimizers, and they typically enjoy better regularity properties. Finally, as
a byproduct of the computations done to prove (1.5), we recover in Theorem 7.5
the interpretation of the second variation formula of a smooth non-
characteristic hypersurface in the sub-Riemannian Heisenberg group
${\mathbb{H}}^{n}$ as a Riemannian limit stated in [65, 33].

This paper is organized as follows. In Section 2, we fix the notation and we
provide some background, focusing mainly on some properties of the perimeter
of a subgraph. In Section 3 we provide some useful tools in order to deal with
embedded hypersurfaces in the Riemannian Heisenberg setting. Section 4 is
devoted to the proof of Theorem 1.2 and Theorem 4.3, i.e. interior and global
gradient estimates. Section 5 contains the existence and regularity results
for solutions to ($\varepsilon$-PMC), namely Theorem 1.1 and Theorem 1.3. In
Section 6 we prove the uniqueness result Theorem 1.4. In Section 7 we prove
Theorem 1.5 and Theorem 7.5. Finally, in Section 8 we prove (1.5), including
the proofs of some facts stated in Section 2 and Section 3.

##  2\. Preliminaries

###  2.1. Notation

Unless otherwise specified, we let $n\in\mathbb{N}$, $n\geqslant 1$ and we
denote by $\Omega$ a bounded domain in ${\mathbb{R}}^{2n}$ with Lipschitz
boundary. Given a measurable set $A\subseteq{\mathbb{R}}^{n}$, we denote by
$\overline{A}$ the closure of $A$ and by $\chi_{A}$ the characteristic
function of $A$. Given two open sets $A,B\subseteq{\mathbb{R}}^{n}$, we write
$A\Subset B$ whenever $\overline{A}\subseteq B$. The $n$-dimensional Lebesgue
measure in ${\mathbb{R}}^{n}$ is denoted by $|\cdot|$. The $s$-dimensional
Hausdorff measure with respect to the Euclidean distance is denoted by
$\mathcal{H}^{s}$. The Euclidean ball in ${\mathbb{R}}^{n}$ centered at
$p\in{\mathbb{R}}^{n}$ with radius $r>0$ is denoted by $B(p,r)$.

###  2.2. The Heisenberg group ${\mathbb{H}}^{n}$

We follow the notation and background given in [62]. We define a non-Abelian
group law $\cdot$ in ${\mathbb{R}}^{2n+1}$ by

$p\cdot
p^{\prime}=(x,y,t)\cdot(x^{\prime},y^{\prime},t^{\prime})=\left(x+x^{\prime},y+y^{\prime},t+t^{\prime}+\sum_{j=1}^{n}\left(x_{j}^{\prime}y_{j}-x_{j}y_{j}^{\prime}\right)\right),$

where we denote points $p\in\mathbb{R}^{2n+1}$ by
$p=(z,t)=(x,y,t)=(x_{1},\ldots,x_{n},y_{1},\ldots,y_{n},t)$. The Lie group
$({\mathbb{R}}^{2n+1},\cdot)$ is referred to as the _$n$ -th Heisenberg group_
and is denoted by ${\mathbb{H}}^{n}$. For any $p\in{\mathbb{H}}^{n}$, the
left-translation by $p$ is the diffeomorphism $\ell_{p}(q)=p\cdot q$. A basis
of left-invariant vector fields is given by

$X_{i}=\frac{\partial}{\partial x_{i}}+y_{i}\,\frac{\partial}{\partial
t},\qquad Y_{i}=\frac{\partial}{\partial
y_{i}}-x_{i}\,\frac{\partial}{\partial t},\qquad T=\frac{\partial}{\partial
t},$

where $i=1\ldots,n$. The horizontal distribution $\mathcal{H}$ in
${\mathbb{H}}^{n}$ is the smooth distribution generated by
$X_{1},Y_{1},\ldots,X_{n},Y_{n}$. A vector field $U$ is said _horizontal_ if
$U(p)\in\mathcal{H}_{p}$ for any $p\in{\mathbb{H}}^{n}$. Given a smooth
function $f$, we define the _horizontal gradient_ of $f$ by

$\nabla^{\mathbb{H}}f=\sum_{i=1}^{n}X_{i}(f)X_{i}+Y_{i}(f)Y_{i}.$

For any $\varepsilon\neq 0$, the matrix of change of basis from
$X_{1},\ldots,Y_{n},\varepsilon T$ to the canonical basis of
$T{\mathbb{R}}^{2n+1}$ at $p=(x_{1}\ldots x_{n},y_{1},\ldots,y_{n},t)$ is
given by

(2.1) $C_{\varepsilon}=\begin{pmatrix}1&\ldots&0&0&\ldots&0&0\\\
\vdots&\ddots&\vdots&\vdots&\ddots&\vdots&\vdots\\\ 0&\ldots&1&0&\ldots&0&0\\\
0&\ldots&0&1&\ldots&0&0\\\ \vdots&\ddots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
0&\ldots&0&0&\ldots&1&0\\\
y_{1}&\ldots&y_{n}&-x_{1}&\ldots&-x_{n}&\varepsilon\\\ \end{pmatrix}.$

In the following, we consider for any $\varepsilon\neq 0$ the left-invariant
Riemannian metric $g_{\varepsilon}=\langle\cdot\,,\cdot\rangle_{\varepsilon}$
on ${\mathbb{H}}^{n}$ such that $\\{X_{1},Y_{1},\ldots,X_{n},Y_{n},\varepsilon
T\\}$ is an orthonormal basis at every point. We shall drop the subindex in
the metric $\langle\cdot,\cdot\rangle$ when considering horizontal vectors. We
shall also use the compact notation $\\{Z_{1},\ldots Z_{2n+1}\\}$ to denote
the previous basis. Moreover, we denote by $\nabla^{\varepsilon}$ its
associated Levi-Civita connection. The norm of a vector field $U$ with respect
to $g_{\varepsilon}$ will be denoted by $|U|_{\varepsilon}$. By means of the
_Koszul formula_ (cf. [25]), the following relations hold.

$\displaystyle\nabla_{X_{i}}^{\varepsilon}X_{j}=0,\qquad\qquad\
\,\nabla^{\varepsilon}_{Y_{i}}Y_{j}=0,\qquad\ \,\nabla^{\varepsilon}_{T}T=0,$
(2.2)
$\displaystyle\nabla^{\varepsilon}_{X_{i}}Y_{j}=-\delta_{i,j}T,\qquad\,\nabla^{\varepsilon}_{X_{i}}\varepsilon
T=\frac{Y_{i}}{\varepsilon},\quad\ \ \nabla^{\varepsilon}_{Y_{i}}\varepsilon
T=-\frac{X_{i}}{\varepsilon},$
$\displaystyle\nabla^{\varepsilon}_{Y_{i}}X_{j}=\delta_{i,j}T,\qquad\ \
\,\,\nabla^{\varepsilon}_{\varepsilon T}X_{i}=\frac{Y_{i}}{\varepsilon},\ \
\quad\nabla^{\varepsilon}_{\varepsilon T}Y_{i}=-\frac{X_{i}}{\varepsilon}$

for any $1\leqslant i,j\leqslant n$, where $\delta_{i,j}$ is the Kronecker
delta. Setting $J(U)=\nabla^{1}_{U}T$ for any vector field $U$, we can easily
deduce that $J(X_{i})=Y_{i}$, $J(Y_{i})=-X_{i}$ and $J(T)=0$. The Riemannian
volume of a set $E$ for $g_{\varepsilon}$ is, up to a constant, the Lebesgue
measure in ${\mathbb{R}}^{2n+1}$. Given a norm $|\cdot|$ in
$T{\mathbb{H}}^{n}$, and $v\in T_{p}{\mathbb{H}}^{n}$, we will drop the
subindex $p$ and write $|v|$. We denote by $\operatorname{Ric}$ the quadratic
form associated with the Ricci tensor induced by $g_{\varepsilon}$, that is

$\begin{split}\operatorname{Ric}(U)&=\sum_{j=1}^{2n+1}g_{\varepsilon}\left(\nabla_{Z_{j}}\nabla_{U}U-\nabla_{U}\nabla_{Z_{j}}U+\nabla_{[U,Z_{j}]}U,Z_{j}\right)\end{split}$

for any vector field $U$ of class $C^{2}$. As we will show in Section 8,

(2.3)
$\operatorname{Ric}(U)=-\frac{2}{\varepsilon^{2}}g_{\varepsilon}\left(U,U\right)+(2n+2)\frac{u_{2n+1}^{2}}{\varepsilon^{2}}$

for any vector field $U=\sum_{j=1}^{2n+1}u_{j}Z_{j}\in
C^{2}(\mathbb{H}^{n};T\mathbb{H}^{n})$.

###  2.3. Carnot-Carathéodory structure on $\mathbb{H}^{n}$

If $\Gamma:[a,b]\longrightarrow\mathbb{H}^{n}$ is an absolutely continuous
curve, we say that it is _horizontal_ whenever
$\dot{\Gamma}(t)\in{\mathcal{H}}_{\Gamma(t)}$ for almost every $t\in[a,b]$,
and we say that it is _sub-unit_ whenever it is horizontal with
$|\dot{\Gamma}(t)|=1$ for almost every $t\in[a,b]$. Moreover, we define

(2.4)
$d^{\mathbb{H}}(p,p^{\prime}):=\inf\\{T\,:\,\Gamma:[0,T]\longrightarrow\mathbb{H}^{n}\text{
is sub-unit, $\Gamma(0)=p$ and $\Gamma(T)=p^{\prime}$}\\}$

which, by the Chow-Rashevskii theorem (cf. [19]), defines a distance on
$\mathbb{H}^{n}$, called _Carnot-Carathéodory distance_. The metric space
$(\mathbb{H}^{n},d^{\mathbb{H}})$ is then a prototype of _Carnot-Carathéodory
space_ (cf. [43]).

###  2.4. Embedded hypersurfaces in ${\mathbb{H}}^{n}$

We consider oriented hypersurfaces of class $C^{1}$ embedded in
${\mathbb{H}}^{n}$, and we choose a unit normal to $S$. In case $S$ is the
boundary of a domain in ${\mathbb{H}}^{n}$, we always choose the outer unit
normal $\nu^{\varepsilon}$. The _characteristic set_ of $S$, denoted by
$S_{0}$, is the set of points $p\in S$ where the tangent space $T_{p}S$
coincides with the horizontal distribution ${\mathcal{H}}_{p}$. The
_horizontal unit normal_ $\nu^{\mathbb{H}}$ is defined in $S\setminus S_{0}$
by

$\nu^{\mathbb{H}}=\frac{\nu^{\varepsilon}_{h}}{|\nu^{\varepsilon}_{h}|},$

where we denote by $U_{h}$ the orthogonal projection of a vector $U$ onto
$\mathcal{H}$.

###  2.5. Perimeters in ${\mathbb{H}}^{n}$

In this subsection, $\Omega\subseteq{\mathbb{R}}^{2n}$ is a bounded open set
and $\varepsilon\neq 0$. We denote the Euclidean norm, total variation,
perimeter and divergence in ${\mathbb{R}}^{2n+1}$ by $|\cdot|$, $Var$, $P$ and
$\operatorname{div}_{eu}$ respectively, and we denote the space of functions
of bounded variation by $BV_{eu}$. Given an open set
$A\subseteq{\mathbb{H}}^{n}$ and $f\in L^{1}(A)$, the _$\varepsilon$
-variation_ of $f$ in $A$ is denoted by

$Var_{\varepsilon}(f;A)=\sup\left\\{\int_{A}f\operatorname{div}_{\varepsilon}(U)\,dx:U\in
C_{c}^{1}(A;T{\mathbb{H}}^{n}),\ |U|_{\varepsilon,\infty}\leqslant
1\right\\},$

where $C_{c}^{1}(A;T{\mathbb{H}}^{n})$ is the space of $C^{1}$ compactly
supported vector fields in $A$, $|U|_{\varepsilon,\infty}=\sup_{p\in
A}|U(p)|_{\varepsilon}$ and $\operatorname{div}_{\varepsilon}$ is the
divergence associated to $g_{\varepsilon}$. The space of $L^{1}(A)$ functions
with bounded $\varepsilon$-variation is denoted by $BV_{\varepsilon}(A)$.
Given $E\subseteq{\mathbb{H}}^{n}$ a measurable set and
$A\subseteq{\mathbb{H}}^{n}$ an open set, the $\varepsilon$-perimeter of $E$
in $A$ is given by

$P_{\varepsilon}(E;A)=Var_{\varepsilon}(\chi_{E};A).$

In case $A={\mathbb{H}}^{n}$ we write
$P_{\varepsilon}(E;{\mathbb{H}}^{n})=P_{\varepsilon}(E)$. Given a vector field
$U\in C_{c}^{1}(A;T{\mathbb{H}}^{n})$, it follows from (2.1) that

(2.5) $\operatorname{div}_{eu}U=\operatorname{div}_{\varepsilon}U,$

so that from now on we shall simply write $\operatorname{div}$. There exist
constants $C(\Omega,\varepsilon)>0$ and $c=c(\Omega,\varepsilon)>0$ such that

(2.6) $C|v|\leqslant|v|_{\varepsilon}\leqslant c|v|$

for any $v\in T_{p}(\Omega\times{\mathbb{R}})$. Indeed, we assume for clarity
that $n=1$ and let $v=v_{1}\frac{\partial}{\partial
x}+v_{2}\frac{\partial}{\partial y}+v_{3}\frac{\partial}{\partial t}$. Using
(2.1) and recalling that $(a+b)^{2}\leqslant 2a^{2}+2b^{2}$ for any
$a,b\in{\mathbb{R}}$, we get

$\begin{split}|v|_{\varepsilon}^{2}\leqslant
v_{1}^{2}+v_{2}^{2}+\frac{2}{\varepsilon^{2}}v_{3}^{2}+\frac{4}{\varepsilon^{2}}y^{2}v_{1}^{2}+\frac{4}{\varepsilon^{2}}x^{2}v_{2}^{2}\leqslant
4\left(1+\frac{1+\max_{z\in\bar{\Omega}}|z|^{2}}{\varepsilon^{2}}\right)|v|^{2}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0},\end{split}$

On the other hand,

$\begin{split}|v|^{2}\leqslant
v_{1}^{2}+v_{2}^{2}+4x^{2}v_{2}^{2}+4y^{2}v_{1}^{2}+2(v_{3}-yv_{1}+xv_{2})^{2}\leqslant\max\left\\{1+4\max_{z\in\overline{\Omega}}|z|^{2},2\varepsilon^{2}\right\\}|v|_{\varepsilon}^{2}.\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\end{split}$

Notice that, in (2.6), $c$ can be chosen uniformly in $\varepsilon$ for
$|\varepsilon|$ big enough, while $C$ can be chosen uniformly in $\varepsilon$
for $|\varepsilon|$ small enough. Given an open set
$A\subseteq\Omega\times{\mathbb{R}}$ and a vector field $U\in
C_{c}^{1}(A;T{\mathbb{H}}^{n})$ with $|U|_{\varepsilon,\infty}\leqslant 1$, it
follows from (2.6) that $|CU|\leqslant|U|_{\varepsilon}\leqslant 1$ and

$\int_{A}f\operatorname{div}(U)dx=\frac{1}{C}\int_{A}f\operatorname{div}(CU)dx\leqslant\frac{1}{C}Var(f,A)$

for any $f\in L^{1}(\Omega\times{\mathbb{R}})$. Hence
$Var_{\varepsilon}(f,A)\leqslant\frac{1}{C}Var(f,A)$. Similarly,
$Var(f,A)\leqslant\frac{1}{c}Var_{\varepsilon}(f,A)$, so that

(2.7) $\frac{1}{c}Var(f,A)\leqslant
Var_{\varepsilon}(f,A)\leqslant\frac{1}{C}Var(f,A).$

Notice that (2.7) implies that $BV_{\varepsilon}(A)$ and $BV_{eu}(A)$ coincide
for $A\subseteq\Omega\times{\mathbb{R}}$, and shall be denoted by $BV(A)$.
Moreover, the perimeters $P$ and $P_{\varepsilon}$ are absolutely continuous
with respect to each other. Hence, the Euclidean reduced and essential
boundaries of a Caccioppoli set coincide with the ones induced by
$P_{\varepsilon}$. In the following, we will denote by $\partial^{*}E$ the
reduced boundary of $E$. Given $E\subseteq{\mathbb{H}}^{n}$ measurable and
$A\subseteq{\mathbb{H}}^{n}$ open, the _horizontal perimeter_ of $E$ in $A$ is
defined by

$P_{\mathbb{H}}(E,A)=\sup\left\\{\int_{E}\operatorname{div}U\,dx,\,U\in
C^{1}_{c}(A,\mathcal{H}),\,|U|_{1,\infty}\leqslant 1\right\\},$

where $C^{1}_{c}(A,\mathcal{H})$ is the space of $C^{1}$ compactly supported
horizontal vector fields in $A$. We refer to [30] for the main properties of
the horizontal perimeter. We point out that both $P_{\varepsilon}$ and
$P_{\mathbb{H}}$ behave like the Euclidean perimeter $P$ for vertical sets.
More precisely, given a Caccioppoli set $E\subseteq{\mathbb{R}}^{2n}$ and an
open set $A\subseteq\mathbb{H}^{n}$, arguing as in [13, (3.2)] and observing
that the last component of the measure theoretic Euclidean unit normal to
$E\times{\mathbb{R}}$ is zero, then

(2.8)
$P_{\varepsilon}(E\times{\mathbb{R}};A)=P_{\mathbb{H}}(E\times{\mathbb{R}};A)=P(E\times{\mathbb{R}},A).$

By the above definitions, the following relations between horizontal and
$\varepsilon$-perimeter hold.

######  Proposition 2.1.

Let $A\subseteq{\mathbb{H}}^{n}$ be an open bounded set and
$F\subseteq{\mathbb{H}}^{n}$ be a Caccioppoli set. Then

$P_{\mathbb{H}}(F,A)\leqslant
P_{\varepsilon}(F,A)\qquad\text{and}\qquad\lim_{\varepsilon\to
0}P_{\varepsilon}(F,A)=P_{\mathbb{H}}(F,A).$

Moreover, if $(\varepsilon_{j})_{j}\subseteq(0,1)$ satisfies
$\varepsilon_{j}\searrow 0$ as $j\to\infty$, and $E$ and $(E_{j})_{j}$ are
measurable sets such that $\chi_{E_{j}}\to\chi_{E}$ in $L^{1}_{loc}(A)$, then

(2.9)
$P_{\mathbb{H}}(E;A)\leqslant\liminf_{j\to\infty}P_{\varepsilon_{j}}(E_{j};A).$

Moreover, the following compactness result holds.

######  Proposition 2.2.

Let $A\subseteq\mathbb{H}^{n}$ be an open and $(E_{k})_{k}$ be a sequence of
finite $\mathbb{H}$-perimeter sets in $A$. Assume that there exists $M>0$ such
that $\sup_{k}P_{\mathbb{H}}(E_{k},A)<M$. Then there exists a finite
$\mathbb{H}$-perimeter set $E$ in $A$ such that $\chi_{E_{k}}\to\chi_{E}$ in
$L^{1}_{loc}(A)$.

###### Proof.

Let $A^{\prime}\Subset A$ open. Let $B_{1},\ldots,B_{k}$ be a covering of
$A^{\prime}$ of Carnot-Carathéodory balls, i.e. the metric balls with respect
to the distance introduced in (2.4), such that

$A^{\prime}\subseteq\bigcup_{j=1}^{s}B_{j}\Subset A.$

Notice that $\chi_{E_{k}}\in BV_{\mathbb{H}}(B_{j})$ for any $j=1,\ldots,k$
(cf. [29] for the definition of $BV_{\mathbb{H}}$). Let us consider first
$B_{1}$. In view of [29, Theorem 2.2.2], there exists $v_{k}\in
C^{\infty}(B_{1})\cap BV_{\mathbb{H}}(B_{1})$ such that

$\|v_{k}-\chi_{E_{k}}\|_{L^{1}(B_{1})}\leqslant\frac{1}{k}\qquad\text{and}\qquad\left|P_{\mathbb{H}}(E_{k},B_{1})-\int_{B_{1}}|\nabla^{\mathbb{H}}v_{k}|\,dx\right|\leqslant\frac{1}{k}$

Therefore

$\|v_{k}\|_{L^{1}(B_{1})}\leqslant|E_{k}\cap
B_{1}|+1\qquad\text{and}\qquad\int_{B_{1}}|\nabla^{\mathbb{H}}v_{k}|\,dx\leqslant\sup_{k}P_{\mathbb{H}}(E_{k},A)+1,$

so that $(v_{k})_{k}$ is bounded in $BV_{\mathbb{H}}(B_{1})$. Hence, [32]
implies that there exists $v\in BV_{\mathbb{H}}(B_{1})$ such that, up to a
subsequence, $v_{k}\to v$ in $L^{1}(B_{1})$. This fact trivially implies the
existence of a set $E^{1}\subseteq B_{1}$ such that, up to a subsequence,
$\chi_{E_{k}}\to\chi_{E^{1}}$ in $L^{1}(B_{1})$. The thesis then easily
follows by a diagonal process and the lower semicontinuity of the
$\mathbb{H}$-perimeter. ∎

###  2.6. $t$-graphs in ${\mathbb{H}}^{n}$

Given a bounded open set $\Omega\subseteq{\mathbb{R}}^{2n}$ and a measurable
function $u:\Omega\longrightarrow[-\infty,+\infty]$, we write the subgraph of
$u$ as

(2.10) $E_{u}=\\{(z,t)\in\Omega\times{\mathbb{R}}\,:\,t<u(z)\\}.$

A simple computation shows that, for $u\in W^{1,1}_{loc}(\Omega)$ and
$\tilde{\Omega}\subseteq\Omega$ open, the perimeter of $E_{u}$ in
$\tilde{\Omega}\times{\mathbb{R}}$ can be computed as

(2.11)
$\mathcal{A}_{\varepsilon}(u,\tilde{\Omega})=\int_{\tilde{\Omega}}\sqrt{\varepsilon^{2}+|Du+X|^{2}}\,dz,$

where $X:{\mathbb{R}}^{2n}\to{\mathbb{R}}^{2n}$ is defined by $X(x,y)=(-y,x)$
and $Du$ is the gradient of $u$. The $L^{1}$-relaxation of
$\mathcal{A}_{\varepsilon}$ for $u\in BV(\Omega)$ is

$\overline{\mathcal{A}}_{\varepsilon}(u,\tilde{\Omega})=\inf\left\\{\liminf_{k\to\infty}\mathcal{A}_{\varepsilon}(u_{k},\tilde{\Omega}):\,(u_{k})_{k}\subseteq
W^{1,1}(\Omega),\,u_{k}\to u\text{ in }L^{1}(\tilde{\Omega})\right\\}.$

We also define

$S_{\varepsilon}(u,\tilde{\Omega})=\sup\left\\{\int_{\tilde{\Omega}}(-u\operatorname{div}\tilde{g}+\langle
X,\tilde{g}\rangle+\varepsilon g_{2n+1})\,dz:\,g=(\tilde{g},g_{2n+1})\in
C^{1}_{c}(\tilde{\Omega},{\mathbb{R}}^{2n+1}),\,|g|\leqslant 1\right\\}.$

######  Lemma 2.3.

Given a bounded open set $\Omega\subseteq{\mathbb{R}}^{2n}$,
$\tilde{\Omega}\subseteq\Omega$ open and $u\in L^{1}(\Omega)$, then

$P_{\varepsilon}(E_{u},\tilde{\Omega}\times{\mathbb{R}})=\overline{\mathcal{A}}_{\varepsilon}(u,\tilde{\Omega})=S_{\varepsilon}(u,\tilde{\Omega}).$

###### Proof.

The proof follows exactly as the proof of [66, Theorem 3.2]. ∎

Recall that for any $u\in BV$, its distributional derivative $\tilde{D}u$ can
be decomposed as the sum of the two mutually singular measures
$Du\mathcal{L}^{2n}+(Du)_{s}$, where $Du\in L^{1}$ and $(Du)_{s}$ is singular
with respect to the Lebesgue measure $\mathcal{L}^{2n}$.

######  Lemma 2.4.

Given a bounded open set $\Omega\subseteq{\mathbb{R}}^{2n}$,
$\tilde{\Omega}\subseteq\Omega$ open and $u\in BV(\Omega)$, it holds that

$P_{\varepsilon}(E_{u},\tilde{\Omega}\times{\mathbb{R}})=(Du)_{s}(\tilde{\Omega})+\int_{\tilde{\Omega}}\sqrt{\varepsilon^{2}+|Du+X|^{2}}\,dz.$

###### Proof.

Let us define
$L:C^{1}_{c}(\Omega,{\mathbb{R}}^{2n+1})\longrightarrow{\mathbb{R}}$ by

$L(g)=\int_{A}(-u\operatorname{div}\bar{g}+\langle
X,\bar{g}\rangle+\varepsilon g_{2n+1})\,dz,$

where $g=(\bar{g},g_{2n+1})$. $L$ is clearly linear. Moreover, since by Lemma
2.3 $S_{\varepsilon}(u)<+\infty,$ then $L$ extends to a linear bounded
functional on $C^{0}_{c}(A,{\mathbb{R}}^{2n+1})$. Therefore, by Riesz Theorem
(cf. e.g. [5, Theorem 1.54]) there exists a unique $(2n+1)$-valued finite
Radon measure $\mu$ such that

$L(g)=\int_{A}g\cdot d\mu\qquad\text{and}\qquad S_{\varepsilon}(u,A)=|\mu|(A)$

for any $g\in C^{1}_{c}(A,{\mathbb{R}}^{2n+1})$. By the uniqueness of such a
measure it is easy to see that $\mu=(Du+Xd\mathcal{L}^{2n},\varepsilon
d\mathcal{L}^{2n})$, and so

$S_{\varepsilon}(u,A)=|(Du+Xd\mathcal{L}^{2n},\varepsilon
d\mathcal{L}^{2n})|(A).$

A trivial computation, together with Lemma 2.3, concludes the proof. ∎

##  3\. Geometry of hypersurfaces in $(\mathbb{H}^{n},g_{\varepsilon})$

Along this section, we fix $\varepsilon\neq 0$ and the metric
$g_{\varepsilon}$, and write as $\\{Z_{1},\ldots,Z_{2n},Z_{2n+1}\\}$ the frame
$\\{X_{1},\ldots,X_{n},Y_{1},\ldots,Y_{n},\varepsilon T\\}$ to keep a compact
notation. We let $S\subseteq{\mathbb{H}}^{n}$ be an embedded orientable
hypersurface of class $C^{3}$ with Riemannian unit normal $\nu^{\varepsilon}$.
We recall that

(3.1) $\sum_{j=1}^{2n+1}(Z_{i}\nu^{\varepsilon}_{j})\nu^{\varepsilon}_{j}=0$

for any $i=1,\ldots,2n+1$ for any unitary extension of $\nu^{\varepsilon}$,
being $(\nu^{\varepsilon}_{1},\ldots,\nu^{\varepsilon}_{2n+1})$ the
coordinates of $\nu^{\varepsilon}$ related to $\\{Z_{1},\ldots,Z_{2n+1}\\}$.
Moreover, if we denote by $d$ be the signed Riemannian distance from $S$, then
$d$ is of class $C^{3}$ near $S$, and satisfies the Eikonal equation $|\nabla
d|=1$ in a neighborhood of $S$, where we used the compact notation
$\nabla=\nabla^{\varepsilon}$. Therefore, $\nu^{\varepsilon}$ can be extended
to a suitable neighborhood of $S$ by letting $\nu^{\varepsilon}=\nabla d$.
With this extension, we have

(3.2) $Z_{i}(\nu^{\varepsilon}_{j})=Z_{j}(\nu^{\varepsilon}_{i})$

for any $i,j=1,\ldots,2n+1$ such that either $i=2n+1$, $j=2n+1$ or $|j-i|\neq
n$. Finally,

(3.3)
$X_{i}(\nu^{\varepsilon}_{n+i})=Y_{i}(\nu^{\varepsilon}_{i})-\frac{2\nu^{\varepsilon}_{2n+1}}{\varepsilon}\qquad\text{and}\qquad
Y_{i}(\nu^{\varepsilon}_{i})=X_{i}(\nu^{\varepsilon}_{n+i})+\frac{2\nu^{\varepsilon}_{2n+1}}{\varepsilon}$

for any $i=1,\ldots,n$. Hence

(3.4)
$\sum_{j=1}^{2n+1}(Z_{j}\nu^{\varepsilon}_{i})\nu^{\varepsilon}_{j}=-2\frac{\nu^{\varepsilon}_{2n+1}}{\varepsilon}J(\nu^{\varepsilon})_{i}$

for any $i=1,\ldots,2n+1$, where we recall that
$J(\nu^{\varepsilon})=(-\nu^{\varepsilon}_{n+1},\ldots,-\nu^{\varepsilon}_{2n},\nu^{\varepsilon}_{1},\ldots,\nu^{\varepsilon}_{n},0)$.
We denote by $h^{\varepsilon}$ and $H^{\varepsilon}$ the associated _second
fundamental form_ and _mean curvature_ of $S$ respectively, i.e., for a given
$p\in S$,
$h^{\varepsilon}_{p}(v,w)=g_{\varepsilon}(\nabla_{v}\nu^{\varepsilon},w)$ for
any $v,w\in T_{p}S$, and

(3.5)
$H^{\varepsilon}(p)=\sum_{i=1}^{2n}h^{\varepsilon}_{p}(e_{i},e_{i})=\sum_{i=1}^{2n}g_{\varepsilon}(\nabla_{e_{i}}\nu^{\varepsilon},e_{i})=\sum_{i=1}^{2n+1}Z_{i}\nu^{\varepsilon}_{i}(p)$

for any orthonormal basis $e_{1},\ldots,e_{2n}$ of $T_{p}S$. It is possible to
express the norm of $h^{\varepsilon}$ by

(3.6)
$|h^{\varepsilon}|^{2}=\sum_{l,s=1}^{2n+1}Z_{s}(\nu^{\varepsilon}_{l})Z_{l}(\nu^{\varepsilon}_{s})+4\left\langle
J(\nu^{\varepsilon}),\nabla\left(\frac{\nu^{\varepsilon}_{2n+1}}{\varepsilon}\right)\right\rangle+(2n-2)\frac{(\nu^{\varepsilon}_{2n+1})^{2}}{\varepsilon^{2}}+\frac{2}{\varepsilon^{2}},$

so that, combining (2.3) and (3.6), we infer that

(3.7)
$\operatorname{Ric}(\nu^{\varepsilon})+|h^{\varepsilon}|^{2}=\sum_{l,s=1}^{2n+1}Z_{s}(\nu^{\varepsilon}_{l})Z_{l}(\nu^{\varepsilon}_{s})+4\left\langle
J(\nu^{\varepsilon}),\nabla\left(\frac{\nu^{\varepsilon}_{2n+1}}{\varepsilon}\right)\right\rangle+4n\frac{(\nu^{\varepsilon}_{2n+1})^{2}}{\varepsilon^{2}}.$

Let us denote by $\nabla_{S}$ and $\Delta_{S}$ the gradient and the Laplace-
Beltrami operator in $(S,g_{\varepsilon}|_{S})$ respectively. A standard
computation shows that

(3.8) $\begin{split}&|\nabla_{S}f|^{2}=|\nabla f|^{2}-g_{\varepsilon}(\nabla
f,\nu^{\varepsilon})^{2}\\\
&\Delta_{S}f=\sum_{i,j=1}^{2n+1}g^{i,j}Z_{i}(Z_{j}f)-H^{\varepsilon}g_{\varepsilon}(\nabla
f,\nu^{\varepsilon})+\frac{2\nu^{\varepsilon}_{2n+1}}{\varepsilon}\langle\nabla
f,J(\nu^{\varepsilon})\rangle\end{split}$

for any $f\in C^{2}(S)$, where

(3.9) $g^{i,j}=\delta_{i,j}-\nu^{\varepsilon}_{i}\nu^{\varepsilon}_{j}.$

In particular, applying (3.8) to $\nu^{\varepsilon}_{2n+1}$ and exploiting
(3.7), we infer that

$\begin{split}\Delta_{S}(\nu^{\varepsilon}_{2n+1})&=g_{\varepsilon}\left(\nabla_{S}H^{\varepsilon},Z_{2n+1}\right)-\nu^{\varepsilon}_{2n+1}\left(\operatorname{Ric}(\nu^{\varepsilon},\nu^{\varepsilon})+|h^{\varepsilon}|^{2}\right).\end{split}$

##  4\. Interior and global gradient estimates

Throughout this section, we fix $\varepsilon\neq 0$ and the metric
$g_{\varepsilon}$, and we write as $\\{Z_{1},\ldots,Z_{2n},Z_{2n+1}\\}$ the
orthonormal frame $\\{X_{1},\ldots,X_{n},Y_{1},\ldots,Y_{n},\varepsilon T\\}$.
Moreover, we fix a bounded domain $\Omega\subseteq{\mathbb{R}}^{2n}$ and $H\in
C^{1}(\Omega)$. We shall provide interior and global gradient estimates for
$C^{3}$ solutions to ($\varepsilon$-PMC). Our approach follows the technique
developed in [45, 46]. Given $u\in C^{3}(\Omega)$, we denote by
$\nu^{\varepsilon}$ the Riemannian normal to the hypersurface
$S=\operatorname{graph}(u)$, which can be globally extended to
$\Omega\times{\mathbb{R}}$ through vertical translations by

(4.1)
$\nu^{\varepsilon}(z,t)=-\sum_{i=1}^{2n}\frac{D_{i}u(z)+X_{i}(z)}{\sqrt{\varepsilon^{2}+|Du(z)+X(z)|^{2}}}Z_{j}|_{(z,t)}+\frac{\varepsilon}{\sqrt{\varepsilon^{2}+|Du(z)+X(z)|^{2}}}Z_{2n+1}|_{(z,t)}$

for any $(z,t)\in\Omega\times{\mathbb{R}}$. Moreover, given $f\in
C^{3}(\Omega)$, we can consider $f$ as a $C^{3}$ function on $S$ or on
$\Omega\times{\mathbb{R}}$ by letting $f(z,u(z))=f(z)$ and $f(z,t)=f(z)$
respectively. In particular, it holds that $\nabla f=(Df,0)$. We begin with
the following preliminary result.

######  Proposition 4.1.

Let $u\in C^{3}(\Omega)$ be a classical solution to ($\varepsilon$-PMC) for
$H\in L^{\infty}(\Omega)\cap C^{1}(\Omega)$. Then

(4.2)
$\Delta_{S}u\geqslant-\|H\|_{\infty}(|\varepsilon|+\max_{\bar{\Omega}}|X|)-\frac{2}{|\varepsilon|}\max_{\overline{\Omega}}{|X|}.$

###### Proof.

Since $Z_{2n+1}u\equiv 0$, (4.1) implies that

$\begin{split}\operatorname{div}\left(\frac{Du+X}{\sqrt{\varepsilon^{2}+|Du+X|^{2}}}\right)&=\frac{\Delta
u}{\sqrt{\varepsilon^{2}+|Du+X|^{2}}}+\left<Du+X,D\left((\varepsilon^{2}+|Du+X|^{2})^{-\frac{1}{2}}\right)\right>\\\
&=\frac{\Delta
u}{\sqrt{\varepsilon^{2}+|Du+X|^{2}}}-\frac{\sum_{i,j=1}^{2n}D_{i}D_{j}u(Du+X)_{i}(Du+X)_{j}}{(\varepsilon^{2}+|Du+X|^{2})^{\frac{3}{2}}}\\\
&=\frac{1}{\sqrt{\varepsilon^{2}+|Du+X|^{2}}}\left(\sum_{i=1}^{2n+1}Z_{i}Z_{i}u-\sum_{i,j=1}^{2n+1}Z_{i}Z_{j}u\nu^{\varepsilon}_{i}\nu^{\varepsilon}_{j}\right)\\\
&=\frac{1}{\sqrt{\varepsilon^{2}+|Du+X|^{2}}}\left(\sum_{i,j=1}^{2n+1}g^{i,j}Z_{i}Z_{j}u\right),\end{split}$

and hence

$H\sqrt{\varepsilon^{2}+|Du+X|^{2}}=\sum_{i,j=1}^{2n+1}g^{i,j}Z_{i}Z_{j}u.$

Since $u$ solves ($\varepsilon$-PMC), then our choice of $\nu^{\varepsilon}$
in (4.1) implies that $H^{\varepsilon}=-H$. Hence, by (3.8),

$\begin{split}\Delta_{S}u&=H\sqrt{\varepsilon^{2}+|Du+X|^{2}}-\frac{H}{\sqrt{\varepsilon^{2}+|Du+X|^{2}}}\langle
Du,Du+X\rangle+\frac{2}{\sqrt{\varepsilon^{2}+|Du+X|^{2}}}\langle
Du,J(\nu^{\varepsilon})\rangle\\\
&=\frac{H}{\sqrt{\varepsilon^{2}+|Du+X|^{2}}}\left(\varepsilon^{2}+|Du+X|^{2}-\langle
Du,Du+X\rangle\right)+\frac{2}{\sqrt{\varepsilon^{2}+|Du+X|^{2}}}\langle
Du,J(\nu^{\varepsilon})\rangle\\\
&=\frac{H}{\sqrt{\varepsilon^{2}+|Du+X|^{2}}}\left(\varepsilon^{2}+\langle
X,Du+X\rangle\right)-\frac{2}{\sqrt{\varepsilon^{2}+|Du+X|^{2}}}\langle
X,J(\nu^{\varepsilon})\rangle\\\ \end{split}$

Finally, (4.2) follows at once from the previous computation. ∎

We are ready to prove Theorem 1.2.

###### Proof of Theorem 1.2.

Let $\tilde{\Omega}$ and $\hat{\Omega}$ be as in the statement, and let
$r\in(0,d(\partial\hat{\Omega},\partial\tilde{\Omega}))$ be fixed. In this
way, $B(z_{0},r)\Subset\tilde{\Omega}$ for any $z_{0}\in\hat{\Omega}$. Fix
then $z_{0}\in\hat{\Omega}$. We set
$\gamma_{1}=\|u\|_{L^{\infty}(\tilde{\Omega})}$ and
$\gamma_{2}=\|H\|_{C_{1}(\overline{\tilde{\Omega}})}$. Let $\varphi\in
C^{\infty}(\overline{B(z_{0},r)})$ be the paraboloid centered at $z_{0}$ such
that $\varphi(z_{0})=u(z_{0})-1$ and $\varphi(z)=\gamma_{1}$ for any
$z\in\partial B(z_{0},r)$, and let
$\gamma_{3}=\|\varphi\|_{C^{2}(\overline{B(z_{0},r)})}$. Notice that
$\gamma_{3}=\gamma_{3}(r,\gamma_{1})$. We define

$\eta(t)=\left(e^{Kt}-1\right)e^{-(\gamma_{1}+\gamma_{3})K}$

for any $t\in{\mathbb{R}}$, where $K>0$ is a constant to be chosen later.
Notice that $\eta\in C^{\infty}({\mathbb{R}})$ and
$0\leqslant\eta((u-\varphi)^{+}(z))\leqslant 1$ for any
$z\in\overline{B(z_{0},r)}$. Since the function
$\Phi:\overline{B(z_{0},r)}\longrightarrow{\mathbb{R}}$ defined by

$\Phi(z)=\frac{\varepsilon\cdot\eta((u-\varphi)^{+}(z))}{\nu^{\varepsilon}_{2n+1}(z)}$

is continuous, we can denote by $M$ its maximum over $\overline{B(z_{0},r)}$.
Moreover, by the choice of $\varphi$, it holds that $\Phi\equiv 0$ on
$\partial B(z_{0},r)$ and $\Phi(z_{0})>0$. Therefore the maximum $M$ is
achieved at some point $\tilde{z}\in B(z_{0},r)$. Since
$M\geqslant\Phi(z_{0})>0$, then $(u-\varphi)^{+}=u-\varphi$ locally near
$\tilde{z}$ and $\Phi$ is of class $C^{2}$ in a neighborhood of $\tilde{z}$.
In particular,

$\Psi(z):=\eta((u-\varphi)^{+}(z))-M\frac{\nu^{\varepsilon}_{2n+1}(z)}{\varepsilon}\leqslant
0$

for any $z\in\overline{B(z_{0},r)}$, and

(4.3)
$\Psi(\tilde{z})=0,\qquad\nabla_{S}\Psi(\tilde{z})=0\qquad\text{and}\qquad\Delta_{S}\Psi(\tilde{z})\leqslant
0.$

We claim that there exists
$M_{0}=M_{0}\left(\varepsilon,\|X\|_{L^{\infty}(\Omega)}\right)$ such that, if
$f\in C^{2}(\Omega)$ is any solution to ($\varepsilon$-PMC) satisfying

(4.4) $\|f\|_{L^{\infty}(B(z_{0},r))}\leqslant\gamma_{1},$

and $\varphi$, $M$ and $\tilde{z}$ are as above, then $M\geqslant M_{0}$
implies that

(4.5) $|\nabla_{S}(f-\varphi)|^{2}(\tilde{z})>\frac{\varepsilon^{2}}{2}.$

Indeed, let $(f_{k})_{k}$ be a sequence such that

(4.6) $M_{k}\geqslant
k\qquad\text{and}\qquad|\nabla_{S}(f_{k}-\varphi_{k})|^{2}(\tilde{z})\leqslant\frac{\varepsilon^{2}}{2},$

where $M_{k}$ and $\tilde{z}_{k}$ are defined as above. Notice that

$\sqrt{\varepsilon^{2}+|Df_{k}+X|^{2}(\tilde{z}_{k})}\geqslant\Phi(\tilde{z}_{k})\geqslant
k$

and, being $X$ bounded over $\overline{\Omega}$, $|Df_{k}(\tilde{z}_{k})|$
diverges to $+\infty$ as $k\to+\infty$. Moreover, by (3.8),

$\begin{split}|\nabla_{S}(f_{k}-\varphi_{k})|^{2}&=|\nabla(f_{k}-\varphi_{k})|^{2}-\langle\nabla(f_{k}-\varphi_{k}),\nu^{\varepsilon}\rangle^{2}\\\
&=|Df_{k}-D\varphi_{k}|^{2}-\frac{\langle
Df_{k}-D\varphi_{k},Df_{k}+X\rangle^{2}}{\varepsilon^{2}+|Df_{k}+X|^{2}}\\\
&\geqslant\frac{\varepsilon^{2}|Df_{k}-D\varphi_{k}|^{2}}{\varepsilon^{2}+|Df_{k}+X|^{2}}.\end{split}$

Hence, since $(D\varphi_{k}(\tilde{z}_{k}))_{k}$ bounded by (4.4), we get

$\liminf_{k\to+\infty}|\nabla_{S}(f_{k}-\varphi_{k})(\tilde{z}_{k})|^{2}\geqslant\lim_{k\to+\infty}\frac{\varepsilon^{2}|Df_{k}-D\varphi_{k}|^{2}(\tilde{z}_{k})}{\varepsilon^{2}+|Df_{k}+X|^{2}(\tilde{z}_{k})}=\varepsilon^{2},$

which contradicts (4.6). We claim that $M\leqslant M_{0}$ for suitable choices
of $K$. Indeed, for a fixed $K>0$, assume that $M>M_{0}$. Hence (4.5) holds.
Notice that $(u-\varphi)^{+}=u-\varphi$ locally around $\tilde{z}$. Notice
that $H^{\varepsilon}(z,t)=-H(z)$ for any $z\in\Omega$ since $u$ solves
($\varepsilon$-PMC), and $H$ is extended vertically on
$\Omega\times{\mathbb{R}}$. Exploiting (1.5), (2.3), (4.2), (4.3) and (4.5),
and recalling that $u$ solves ($\varepsilon$-PMC), we infer that, at
$\tilde{z}$,

(4.7)
$\begin{split}\Delta_{S}\Psi&=\Delta_{S}(\eta(u-\varphi))-\frac{M}{\varepsilon}\Delta_{S}\nu^{\varepsilon}_{2n+1}\\\
&=\eta^{\prime\prime}|\nabla_{S}(u-\varphi)|^{2}+\eta^{\prime}\left(\Delta_{S}u-\Delta_{S}\varphi\right)+\frac{M}{\varepsilon}\big{(}g_{\varepsilon}(\nabla_{S}H,\varepsilon
T)+\nu^{\varepsilon}_{2n+1}\left(\operatorname{Ric}_{\varepsilon}(\nu^{\varepsilon},\nu^{\varepsilon})+|h^{\varepsilon}|^{2}\right)\big{)}\\\
&=\eta^{\prime\prime}|\nabla_{S}(u-\varphi)|^{2}+\eta^{\prime}\left(\Delta_{S}u-\Delta_{S}\varphi\right)+\frac{M\nu^{\varepsilon}_{2n+1}}{\varepsilon}\big{(}-g_{\varepsilon}(\nabla
H,\nu^{\varepsilon})+\operatorname{Ric}_{\varepsilon}(\nu^{\varepsilon},\nu^{\varepsilon})+|h^{\varepsilon}|^{2}\big{)}\\\
&=\eta^{\prime\prime}|\nabla_{S}(u-\varphi)|^{2}+\eta^{\prime}\left(\Delta_{S}u-\Delta_{S}\varphi\right)+\eta\big{(}-g_{\varepsilon}(\nabla
H,\nu^{\varepsilon})+\operatorname{Ric}_{\varepsilon}(\nu^{\varepsilon},\nu^{\varepsilon})+|h^{\varepsilon}|^{2}\big{)}\\\
&\geqslant\frac{\varepsilon^{2}}{2}\eta^{\prime\prime}-C\eta^{\prime}-C\eta\end{split}$

for some $C=C(n,\varepsilon,r,\gamma_{2},\gamma_{3})>0$. Hence, since
$\Delta_{S}\Psi(\tilde{z})\leqslant 0$ and up to choosing a different constant
$C=C(n,\varepsilon,r,\gamma_{2},\gamma_{3})>0$, we conclude that

(4.8)
$\eta^{\prime\prime}((u-\varphi)(\tilde{z}))-C\eta^{\prime}((u-\varphi)(\tilde{z}))-C^{2}\eta((u-\varphi)(\tilde{z}))\leqslant
0.$

The choice $K=2C$ is in contradiction with (4.8), so that, for this choice of
$K$, we must have $M\leqslant M_{0}$. Hence

$\eta((u-\varphi)^{+}(z))-M_{0}\frac{\nu^{\varepsilon}_{2n+1}(z)}{\varepsilon}\leqslant
0.$

Since $(u-\varphi)^{+}(z_{0})=1$, we conclude that

$|Du|(z_{0})\leqslant\sqrt{\varepsilon^{2}+|Du+X|(z_{0})^{2}}+|X|(z_{0})\leqslant\frac{M_{0}}{\eta(1)}+|X|(z_{0}),$

whence the thesis follows. ∎

An approach as in Theorem 1.2 allows to reduce global gradient estimates for
solutions to ($\varepsilon$-PMC) to boundary gradient estimates. For future
convenience, we state the result for a slightly more general class of
equations.

######  Theorem 4.2 (From boundary to global gradient estimates).

Let $H\in C^{1}(\overline{\Omega})$ and $\sigma\in[0,1]$, and let $u\in
C^{3}(\Omega)\cap C^{1}(\overline{\Omega})$ be a solution to

(4.9) $\operatorname{div}\left(\frac{Du+\sigma
X}{\sqrt{\varepsilon^{2}+|Du+\sigma X|^{2}}}\right)=\sigma H(z)$

on $\Omega$. Then there exists
$C=C\left(n,\varepsilon,\|u\|_{L^{\infty}(\Omega)},\|H\|_{C^{1}(\overline{\Omega})},\|X\|_{L^{\infty}(\Omega)}\right)>0,$
thus independent of $\sigma\in[0,1]$, such that

$\|Du\|_{L^{\infty}(\Omega)}\leqslant
C{\Big{(}\|Du\|_{L^{\infty}(\partial\Omega)}+1\Big{)}}.$

###### Proof.

Let us set $\gamma_{1}=\|u\|_{L^{\infty}(\Omega)}$. Let
$\eta(t)=e^{K(t-\gamma_{1})}$, where $K>0$ is a fixed constant to be chosen
later. Notice that

(4.10) $e^{-2K\gamma_{1}}\leqslant\eta(u(z))\leqslant 1$

for any $z\in\overline{\Omega}$. Let $\Phi:\bar{\Omega}\to{\mathbb{R}}$ be
defined by

$\Phi(z)=\frac{\varepsilon(\eta(u(z))}{\nu^{\varepsilon}_{2n+1}(z)}.$

Since $u\in C^{1}(\overline{\Omega})$, then $\Phi\in C(\overline{\Omega})$, so
that $\Phi$ achieves its maximum $M$ at some point
$\tilde{z}\in\overline{\Omega}$. Suppose first $\tilde{z}\in\Omega$. Arguing
as in the proof of Theorem 1.2, and thanks to (4.10), there exists
$M_{0}=M_{0}\left(\varepsilon,\|X\|_{L^{\infty}(\Omega)}\right)$ such that, if
$f\in C^{2}(\Omega)$ is any solution to ($\varepsilon$-PMC) satisfying (4.4),
and $M$ and $\tilde{z}$ are as above, then $M\geqslant M_{0}$ implies that

(4.11) $|\nabla_{S}f|^{2}(\tilde{z})>\frac{\varepsilon^{2}}{2}.$

Repeating the computations of (4.7), and exploiting (4.11), we get that, for a
particular choice of
$K=K(n,\varepsilon,\|X\|_{L^{\infty}(\Omega)},\|H\|_{C^{1}(\bar{\Omega})})>0$,
it holds $M\leqslant M_{0}$. Hence, we get that

$\eta(u(z))\sqrt{\varepsilon^{2}+|Du+X|(z)^{2}}\leqslant\eta(u(\tilde{z}))\sqrt{\varepsilon^{2}+|Du+X|(\tilde{z})^{2}}\leqslant
M_{0}$

for any $z\in\overline{\Omega}$, so that, by (4.10),

$|Du|(z)\leqslant\sqrt{\varepsilon^{2}+|Du+X|(z)^{2}}+|X|(z)\leqslant
M_{0}e^{2K\gamma_{1}}+\|X\|_{L^{\infty}(\Omega)}$

for any $z\in\overline{\Omega}$. If instead $\tilde{z}\in\partial\Omega$, then

$\begin{split}|Du|(z)&\leqslant
e^{2K\gamma_{1}}\eta(u(z))\sqrt{\varepsilon^{2}+|Du+X|(z)^{2}}+|X|(z)\\\
&\leqslant
e^{2K\gamma_{1}}\eta(u(\tilde{z}))\sqrt{\varepsilon^{2}+|Du+X|(\tilde{z})^{2}}+\|X\|_{L^{\infty}(\Omega)}\\\
&\leqslant
e^{2K\gamma_{1}}\left(|\varepsilon|+\|Du\|_{L^{\infty}(\partial\Omega)}+\|X\|_{L^{\infty}(\Omega)}\right)+\|X\|_{L^{\infty}(\Omega)}\end{split}$

for any $z\in\overline{\Omega}$, whence the thesis follows. ∎

As a corollary of Theorem 4.2, we provide global gradient estimates for
solutions to the Dirichlet problem associated with ($\varepsilon$-PMC) when
(1.6) holds.

######  Theorem 4.3 (Global gradient estimates).

Let $\Omega$ be a bounded domain with boundary of class $C^{2}$. Let $H\in
C^{1}(\overline{\Omega})$ be such that (1.6) holds. Let $\varphi\in
C^{2}(\overline{\Omega})$, $\varepsilon\neq 0$ and $\sigma\in[0,1]$. Assume
that there exist a constant
$\gamma_{1}=\gamma_{1}(n,\varepsilon,\Omega,\varphi,X,H)>0$, independent of
$\sigma\in[0,1]$, such that any solution $u\in C^{2}(\overline{\Omega})$ to

(4.12) $\begin{cases}\operatorname{div}\left(\frac{Du+\sigma
X}{\sqrt{\varepsilon^{2}+|Du+\sigma X|^{2}}}\right)=\sigma H(z)&\text{ in
}\Omega\\\ u=\sigma\varphi&\text{ in }\partial\Omega\end{cases}$

satisfies $\|u\|_{L^{\infty}(\Omega)}\leqslant\gamma_{1}$. Then there exists
$C=C\left(n,\varepsilon,\|\varphi\|_{C^{2}(\overline{\Omega})},\gamma_{1},\|H\|_{C^{1}(\overline{\Omega})},\|X\|_{L^{\infty}(\Omega)}\right)>0,$
thus independent of $\sigma\in[0,1]$, such that any $u\in C^{3}(\Omega)\cap
C^{1}(\overline{\Omega})$ solution to (4.12) satisfies

$\|Du\|_{L^{\infty}(\Omega)}\leqslant C.$

###### Proof.

In view of Theorem 4.2 we are left to provide boundary gradient estimates.
Thanks to (1.6), and following [38, Section 6], the latter follow _verbatim_
as in [38, Proposition 4.8]. ∎

##  5\. Existence and regularity of $t$-graphs

###  5.1. Existence of minimizers: the non-extremal case

Throughout this subsection we fix a bounded domain
$\Omega\subseteq{\mathbb{R}}^{2n}$ with Lipschitz boundary, and we consider
$H\in L^{\infty}(\Omega)$ such that (* ‣ 1) holds and

(***) $\Big{|}\int_{\Omega}Hdz\Big{|}<P(\Omega).$

We stress that, even without imposing boundary conditions, (* ‣ 1) is a
necessary condition to the existence of a solution $u\in C^{2}(\Omega)$ to
($\varepsilon$-PMC) in $\Omega$. More precisely, arguing as in the Euclidean
setting, the following result holds.

######  Theorem 5.1.

Let $\Omega\subseteq{\mathbb{R}}^{2n}$ be a bounded domain with Lipschitz
boundary, let $H\in\operatorname{Lip}(\Omega)$ and assume that there exists
$u\in C^{2}(\Omega)$ which solves ($\varepsilon$-PMC) in $\Omega$. Then (* ‣
1) holds.

For any $\varphi\in L^{1}(\partial\Omega)$, we define the functional
$\mathcal{I}_{\varepsilon}:BV(\Omega)\to{\mathbb{R}}$ by

(5.1)
$\mathcal{I}_{\varepsilon}(v)=P_{\varepsilon}(E_{v},\Omega\times{\mathbb{R}})+\int_{\Omega}Hu\,dz+\int_{\partial\Omega}|v-\varphi|\,d\mathcal{H}^{2n-1},$

where $E_{v}$ is defined as in (2.10). The following result follows as [40,
Lemma 1.1] (cf. also [48, Lemma 3.2]).

######  Lemma 5.2.

Assume that (* ‣ 1) and (*** ‣ 5.1) hold. Then there exists $\delta>0$ such
that

(5.2)
$\Big{|}\int_{\tilde{\Omega}}H\,dz\Big{|}\leqslant(1-\delta)P(\tilde{\Omega})$

for every measurable set $\tilde{\Omega}\subseteq\Omega$.

The proof of the following proposition reproduces the argument of [35, Theorem
1.1].

######  Proposition 5.3.

Let $H\in L^{\infty}(\Omega)$ and assume that (* ‣ 1) and (*** ‣ 5.1) hold.
Then $\mathcal{I}_{\varepsilon}$ has a minimum in $BV(\Omega)$ for every
$\varphi\in L^{1}(\partial\Omega)$.

###### Proof.

Let $B\subseteq{\mathbb{R}}^{2n}$ be a ball containing $\Omega$ such that the
Euclidean distance between $\partial\Omega$ and $\partial B$ is positive, and
extend $H$ to $B$ by letting $H\equiv 0$ outside $\Omega$. Fix a function
$\phi\in W_{0}^{1,1}(B)$ with trace $\varphi$ on $\partial\Omega$. Then
minimizing $\mathcal{I}_{\varepsilon}$ is equivalent to minimize the
functional

(5.3)
$\mathcal{J}_{\varepsilon}(v)=P_{\varepsilon}(E_{v},\Omega\times{\mathbb{R}})+Var(v,B\setminus\Omega)+\int_{B}Hv\,dz$

in $K=\\{v\in BV(B):v=\phi\text{ in }B\setminus\Omega\\}$. Indeed, given
$v_{0}$ and $v$ in $K$ with $v_{0}$ minimum of $\mathcal{J}_{\varepsilon}$ and
writing $u_{0}=v_{0}|_{\Omega}$ and $u=v|_{\Omega}$, it follows from [41,
Remark 2.13] that

$\mathcal{I}_{\varepsilon}(u)-\mathcal{I}_{\varepsilon}(u_{0})=\mathcal{J}_{\varepsilon}(v)-\mathcal{J}_{\varepsilon}(v_{0})\geqslant
0.$

For a given $v\in K$, we define $v^{+},v^{-}\in BV(B)$ by

$v^{+}(z)=\displaystyle{\begin{cases}\max\\{0,v(z)\\}&\text{ if
}z\in\Omega,\\\ 0&\text{otherwise}\end{cases}}\qquad\text{and}\qquad
v^{-}(z)=\displaystyle{\begin{cases}\max\\{0,-v(z)\\}&\text{ if
}z\in\Omega,\\\ 0&\text{otherwise}\end{cases}}.$

Notice that

(5.4) $E_{t}:=\\{z\in
B\,:\,v^{+}(z)>t\\}=\\{z\in\Omega\,:\,v(z)>t\\}\subseteq\Omega$

and

(5.5) $F_{t}:=\\{z\in
B\,:\,v^{-}(z)>t\\}=\\{z\in\Omega\,:\,v(z)<-t\\}\subseteq\Omega$

for any $t>0$. Hence, thanks to Lemma 5.2, (5.4), (5.5), the _layer-cake
formula_ (cf. [50, Remark 13.6]) and the _Coarea formula_ (cf. [50, Theorem
13.1]), it follows that

$\begin{split}\int_{B}H(z)v(z)\,dz&=\int_{\Omega}H(z)v^{+}(z)\,dz-\int_{\Omega}H(z)v^{-}(z)\,dz\\\
&=\int_{0}^{+\infty}\,dt\int_{\Omega}\chi_{E_{t}}H(z)\,dz-\int_{0}^{+\infty}\,dt\int_{\Omega}\chi_{F_{t}}H(z)\,dz\\\
&\geqslant-(1-\delta)\int_{0}^{+\infty}P(E_{t})\,dt-(1-\delta)\int_{0}^{+\infty}P(F_{t})\,dt\\\
&=-(1-\delta)\int_{0}^{+\infty}P(E_{t},B)\,dt-(1-\delta)\int_{0}^{+\infty}P(F_{t},B)\,dt\\\
&\geqslant-(1-\delta)Var(v^{+},B)-(1-\delta)Var(v^{-},B).\end{split}$

Therefore, owing again to [41, Remark 2.13],

(5.6) $\begin{split}\int_{B}&H(z)v(z)dz\\\
&\geqslant-(1-\delta)(Var(v^{+},\Omega)+Var(v^{-},\Omega))-(1-\delta)\left(\int_{\partial\Omega}|v^{+}|\,d\mathcal{H}^{2n-1}+\int_{\partial\Omega}|v^{-}|\,d\mathcal{H}^{2n-1}\right)\\\
&=-(1-\delta)Var(v,\Omega)-(1-\delta)\int_{\partial\Omega}|v|\,d\mathcal{H}^{2n-1}\\\
&\geqslant-(1-\delta)Var(v,\Omega)-(1-\delta)\int_{\partial\Omega}|v-\varphi|\,d\mathcal{H}^{2n-1}-(1-\delta)\int_{\partial\Omega}|\varphi|\,d\mathcal{H}^{2n-1}\\\
&=-(1-\delta)Var(v,\overline{\Omega})-(1-\delta)\int_{\partial\Omega}|\varphi|\,d\mathcal{H}^{2n-1}\\\
&\geqslant-(1-\delta)Var(v,B)-\int_{\partial\Omega}|\varphi|\,d\mathcal{H}^{2n-1}\\\
\end{split}$

On the other hand, it holds that

(5.7) $P_{\varepsilon}(E_{v};\Omega\times{\mathbb{R}})\geqslant
Var(v,\Omega)-\int_{\Omega}|X|\,dz.$

Indeed, let $(v_{k})_{k}\subseteq W^{1,1}(\Omega)$ be such that $v_{k}\to v$
in $L^{1}(\Omega)$. Then, for any fixed ${k}\in\mathbb{N}$, (2.11) implies
that

$\begin{split}P_{\varepsilon}(E_{v_{k}};\Omega\times{\mathbb{R}})&=\int_{\Omega}\sqrt{\varepsilon^{2}+|Dv_{k}+X|^{2}}\,dz\geqslant\int_{\Omega}|Dv_{k}+X|\,dz\geqslant\int_{\Omega}|Dv_{k}|\,dz-\int_{\Omega}|X|\,dz\end{split}$

Hence, by the lower semicontinuity of $Var(\cdot,\Omega)$ with respect to the
$L^{1}$-convergence, we conclude that

$\int_{\Omega}|Dv|\,dz-\int_{\Omega}|X|\,dz\leqslant\liminf_{k\to+\infty}\left(\int_{\Omega}|Dv_{k}|\,dz-\int_{\Omega}|X|\,dz,\right)\leqslant\liminf_{k\to+\infty}P_{\varepsilon}(E_{v_{k}};\Omega\times{\mathbb{R}}),$

from which (5.7) follows by Lemma 2.3. Substituting (5.6) and (5.7) in (5.3),
we finally obtain

(5.8) $\mathcal{J}_{\varepsilon}(v)\geqslant\delta
Var(v,B)-\int_{\Omega}|X|\,dz-\int_{\partial\Omega}|\varphi|\,d\mathcal{H}^{2n-1}$

Let $\\{v_{k}\\}\subseteq K$ be a minimizing sequence for
$\mathcal{J}_{\varepsilon}$. By (5.8) and the Poincaré’s inequality,
$\\{v_{k}\\}$ is bounded in $BV(B)$, so that, up to a subsequence, there
exists $v_{0}\in BV(B)$ such that $v_{k}\to v_{0}$ in $L^{1}(B)$. Noticing
that $K$ is a closed with respect to the $L^{1}$ convergence on $B$, then
$v_{0}\in K$. By the lower semicontinuity of the perimeter with respect to
$L^{1}$ convergence, we get that $v_{0}$ is a minimizer of
$\mathcal{J}_{\varepsilon}$ and hence $\mathcal{I}_{\varepsilon}$ has a
minimizer in $BV(\Omega)$. ∎

###  5.2. Variational properties of minimizers

Throughout this subsection, we fix a bounded domain
$\Omega\subseteq{\mathbb{R}}^{2n}$ and $H\in L^{\infty}_{loc}(\Omega)$.

######  Definition 5.4.

We say that a Caccioppoli set $E\subseteq\Omega\times{\mathbb{R}}$ is a
(local) _$H$ -minimizer_ in $\Omega\times{\mathbb{R}}$ if

(5.9) $P_{\varepsilon}(E,A)+\int_{E\cap A}Hdx\leqslant
P_{\varepsilon}(F,A)+\int_{F\cap A}Hdx$

for any open set $A\Subset\Omega\times{\mathbb{R}}$ and any measurable set $F$
such that $E\Delta F\Subset A$, where $H(z,t)=H(z)$.

In the following theorem we prove that subgraphs of minimizers of
$\mathcal{I_{\varepsilon}}$ are $H$-minimizers (cf. [28] for an exhaustive
account of this and related results in the Euclidean setting). Notice that, if
$u\in BV(\Omega)$ is a minimizer for $\mathcal{I}_{\varepsilon}$, then

(5.10)
$\int_{\tilde{\Omega}}\sqrt{\varepsilon^{2}+|Du+X|^{2}}+\int_{\tilde{\Omega}}Hu\,dz\leqslant\int_{\tilde{\Omega}}\sqrt{\varepsilon^{2}+|Dv+X|^{2}}+\int_{\tilde{\Omega}}Hv\,dz$

for any open set $\tilde{\Omega}\Subset\Omega$ and any $v\in BV_{loc}(\Omega)$
such that $\\{u\neq v\\}\Subset\tilde{\Omega}$. When $u\in BV_{loc}(\Omega)$
satisfies (5.10), we refer to it as _(local) $H$-minimizer_. The ambiguity
with Definition 5.4 is motivated by the following result.

######  Theorem 5.5.

Let $u\in BV_{loc}(\Omega)$. Then $u$ satisfies (5.10) if, and only if,
$E_{u}$ is an $H$-minimizer in $\Omega\times{\mathbb{R}}$.

###### Proof.

Assume first that $u\in BV_{loc}(\Omega)$ satisfies (5.10). Let
$A\Subset\Omega\times{\mathbb{R}}$ be open, and let
$\tilde{\Omega}\Subset\Omega$ be an open set such that
$A\Subset\tilde{\Omega}\times{\mathbb{R}}$. let now $F$ be a measurable set
such that $F\Delta E_{u}\Subset A$. We can assume without loss of generality
that $F$ has finite perimeter in $A$. Since $E_{u}$ is a subgraph, we infer
that

$\lim_{t\to+\infty}\chi_{F}(z,t)=0\qquad\text{and}\qquad\lim_{t\to-\infty}\chi_{F}(z,t)=1$

for a.e. $z\in\Omega$. Inspired by [41, Lemma 14.7], we set $w\in
BV_{loc}(\Omega)$ the function

$w(z)=\lim_{k\to\infty}w_{k}(z)=\lim_{k\to\infty}\left(\int_{-k}^{k}\chi_{F}(z,t)\,dt-k\right).$

We claim that

(5.11) $P_{\varepsilon}(E_{w};\tilde{\Omega}\times{\mathbb{R}})\leqslant
P_{\varepsilon}(F,\tilde{\Omega}\times{\mathbb{R}})$

and

(5.12)
$\int_{\Omega^{\prime}\times{\mathbb{R}}}\left(\chi_{E_{w}}-\chi_{F}\right)\,dz\,dt=0$

for any open set $\Omega^{\prime}\Subset\tilde{\Omega}$. Indeed, assume first
that $\exists L>0$ such that

(5.13) $\Omega\times(-\infty,-L)\subseteq F\subseteq\Omega\times(-\infty,L).$

It is clear that $-L\leqslant w\leqslant L$ and
$w(z)=-L+\int_{-L}^{L}\chi_{F}(z,t)\,dt$. Let $\eta:{\mathbb{R}}\to[0,1]$ be a
smooth cut-off function between $[-L,L]$ and $[-L-1,L+1]$. From a direct
computation, we get that for a.e. $z\in\Omega$, it holds

(5.14)
$\int_{\mathbb{R}}\chi_{F}(z,t)\eta(t)\,dt=w(z)+\alpha\qquad\text{and}\qquad\int_{\mathbb{R}}\chi_{F}(z,t)\eta^{\prime}(t)\,dt=1,$

where $\alpha=L+\int_{-L-1}^{L}\eta(t)\,dt$. Let $g=(\tilde{g},g_{2n+1})\in
C_{c}^{1}(\Omega,{\mathbb{R}}^{2n+1})$ be such that $|g|\leqslant 1$. Let us
set

$W|_{(z,t)}=\sum_{j=1}^{2n}\eta(t)g_{j}(z)Z_{j}|_{(z,t)}.$

Then $W\in C_{c}^{1}(\Omega\times{\mathbb{R}},T{\mathbb{H}}^{n})$ and
$|W|_{\varepsilon}\leqslant 1$. Therefore (5.14) implies that

$\begin{split}P_{\varepsilon}(F,\Omega\times{\mathbb{R}})\geqslant&\int_{\Omega\times{\mathbb{R}}}\chi_{F}(z,t)\operatorname{div}W(z,t)\,dt\,dz\\\
=&\int_{\Omega}\int_{\mathbb{R}}\chi_{F}(z,t)\left(\eta(t)\operatorname{div}\tilde{g}(z)-\eta^{\prime}(t)\langle\tilde{g}(z),X(z)\rangle+\eta^{\prime}(t)\varepsilon
g_{2n+1}(z)\right)\,dt\,dz\\\
=&\int_{\Omega}\Big{(}\operatorname{div}\tilde{g}(z)\int_{\mathbb{R}}\chi_{F}(z,t)\eta(t)\,dt-\langle\tilde{g}(z),X(z)\rangle\int_{\mathbb{R}}\chi_{F}(z,t)\eta^{\prime}(t)\,dt\\\
&+\varepsilon
g_{2n+1}\int_{\mathbb{R}}\chi_{F}(z,t)\eta^{\prime}(t)\,dt\Big{)}\,dz\\\
=&\int_{\Omega}(w(z)+\alpha)\operatorname{div}\tilde{g}(z)-\langle\tilde{g}(z),X(z)\rangle+\varepsilon
g_{2n+1}(z)\,dz\\\
=&\int_{\Omega}w(z)\operatorname{div}\tilde{g}(z)-\langle\tilde{g}(z),X(z)\rangle+\varepsilon
g_{2n+1}(z)\,dz,\end{split}$

where we used the fact that $\text{supp}(\tilde{g})\subseteq\Omega$. Hence,
assuming (5.13), (5.11) holds by Lemma 2.3. Moreover,

(5.15)
$\begin{split}\int_{\tilde{\Omega}}H(u-w)dz=&\int_{\tilde{\Omega}\cap\\{u\geqslant
w\\}}H(u-w)dz-\int_{\tilde{\Omega}\cap\\{u<w\\}}H(w-u)dz\\\
=&\int_{\tilde{\Omega}\cap\\{u\geqslant w\\}}H(|E_{u}(z)\cap
A(z)|-|E_{w}(z)\cap A(z)|)dz\\\
&-\int_{\tilde{\Omega}\cap\\{u<w\\}}H(|E_{w}(z)\cap A(z)|-|E_{u}(z)\cap
A(z)|)dz\\\ =&\int_{(\tilde{\Omega}\cap\\{u\geqslant
w\\})\times{\mathbb{R}}}H(\chi_{E_{u}\cap A}-\chi_{E_{w}\cap
A})dx-\int_{\tilde{\Omega}\cap\\{u<w\\}}H(\chi_{E_{w}\cap A}-\chi_{E_{u}\cap
A})dx\\\ =&\int_{\tilde{\Omega}\times{\mathbb{R}}}H(\chi_{E_{u}\cap
A}-\chi_{E_{w}\cap A})dx,\end{split}$

where $E_{v}(z)=\\{t\in{\mathbb{R}}:v(z)>t\\}$ and
$A(z)=\\{t\in{\mathbb{R}}:(z,t)\in A\\}$. To drop (5.13), one can argue
exactly as in the proof of [41, Theorem 14.8]. Finally, (5.12) follows
_verbatim_ as in the proof of [51, Teorema 2.3]. In view of (5.11), (5.12) and
(5.15), arguing as in [41, Theorem 14.9] and [52] $E_{u}$ is an $H$-minimizer
in $\Omega\times{\mathbb{R}}$. Being the converse implication fairly
straightforward, the thesis follows. ∎

Clearly, since $H\in L^{\infty}_{loc}(\Omega)$, $H$-minimizer sets are _almost
minimizers_ in the sense of [4] for suitable _anisotropic energies_. More
precisely, if $E$ is an $H$-minimizer in $\Omega\times{\mathbb{R}}$,
$\tilde{\Omega}\Subset\Omega$ and $\|H\|_{L^{\infty}(\tilde{\Omega})}=H_{0}$,
then

(5.16) $P_{\varepsilon}(E,A)\leqslant P_{\varepsilon}(F,A)+H_{0}|E\Delta F|$

for any open set $A\Subset\tilde{\Omega}\times{\mathbb{R}}$ and any measurable
set $F\subseteq\Omega\times{\mathbb{R}}$ such that $E\Delta F\Subset A$. For
almost minimizers in the previous sense, arguing as in the Euclidean setting
(cf. [50]) it is possible to derive _uniform density estimates_ for both
volume and perimeter. In view of (2.7), the following well-known estimates
follow from for instance from [68, Proposition 4.5].

######  Proposition 5.6.

For any domain $\tilde{\Omega}\Subset\Omega$, there exist
$c_{0}=c_{0}(\varepsilon)>0$, $c_{1}=c_{1}(\varepsilon)>0$, $c_{2}>0$
independent of $\varepsilon\in(0,1]$ and $r_{0}>0$ such that for any
$H$-minimizer $E$, any
$p\in\overline{\partial^{*}E}\cap(\tilde{\Omega}\times{\mathbb{R}})$ and any
$r<r_{0}$, it holds

(5.17) $c_{0}r^{2n+1}\leqslant\min\\{|E\cap B(p,r)|,|E\setminus B(p,r)|\\}$

and

(5.18) $c_{1}r^{2n}\leqslant P_{\varepsilon}(E,B(p,r))\leqslant c_{2}r^{2n},$

where $B(p,r)$ is the Euclidean ball centered at $p$ of radius $r$.

The local boundedness of minimizers is a consequence of Theorem 5.5 and
Proposition 5.6, and it follows as its Euclidean counterpart (cf. [41, Theorem
14.10]).

######  Proposition 5.7.

Let $u\in BV(\Omega)$ be a minimizer of $\mathcal{I}_{\varepsilon}$. Then
$u\in L^{\infty}_{loc}(\Omega)$.

###  5.3. Higher regularity of Lipschitz continuous t-graphs

In the rest of this section, inspired by [41], we show classical regularity
for minimizers of $\mathcal{I_{\varepsilon}}$. The main difficulty consists in
obtaining Lipschitz regularity since, as soon as a minimizer is Lipschitz,
standard regularity results for uniformly elliptic equations apply. More
precisely, the following regularity property for Lipschitz weak solutions to
($\varepsilon$-PMC) holds.

######  Theorem 5.8.

Let $H\in\operatorname{Lip}_{loc}(\Omega)$ and
$u\in\operatorname{Lip}_{loc}(\Omega)$ be a weak solution on $\Omega$ to
($\varepsilon$-PMC). Then $u\in C^{2,\alpha}_{loc}(\Omega)$ for any
$\alpha\in(0,1)$ and is a classical solution to ($\varepsilon$-PMC). Moreover,
if $H\in C^{k,\gamma}_{loc}(\Omega)$ for some $k\geqslant 1$ and
$\gamma\in(0,1)$, then $u\in C^{k+2,\gamma}_{loc}(\Omega)$. Finally, if $H\in
C^{\infty}(\Omega)$, then $u\in C^{\infty}(\Omega)$.

###### Proof.

Let $\alpha\in(0,1)$. Let $\tilde{\Omega}\Subset\Omega$ be a bounded domain.
It suffices to show that $u\in C^{2,\alpha}(\tilde{\Omega})$. For $r>0,$ set

$\tilde{\Omega}_{r}=\left\\{z\in\tilde{\Omega}\,:\,\min_{w\in\partial\Omega}|z-w|>r\right\\}.$

Given $\psi\in C^{\infty}_{c}(\tilde{\Omega})$, we can assume that $\psi\in
C^{\infty}_{c}(\tilde{\Omega}_{r})$ for $r>0$ small enough. Let $v\in B(0,r)$.
Using $\psi$ and $\psi(\cdot-v)$ as test functions the weak formulation of
($\varepsilon$-PMC), we get

(5.19) $\int_{\tilde{\Omega}}\langle
A(z+v,Du(z+v))-A(z,Du(z)),D\psi(z)\rangle\,dz=\int_{\tilde{\Omega}}(H(z)-H(z+v))\psi(z)\,dz,$

where

$A(z,\xi)=\frac{\xi+X(z)}{\sqrt{\varepsilon^{2}+|\xi+X(z)|^{2}}}.$

Fixed $z_{0}\in\tilde{\Omega}_{r}$, the fundamental theorem of calculus
implies that

(5.20)
$A(z_{0}+v,Du(z_{0}+v))-A(z_{0},Du(z_{0}))=\int_{0}^{1}\frac{d(A(\alpha_{z_{0}}(s))}{ds}dt,$

where $\alpha_{z_{0}}:[0,1]\to{\mathbb{R}}^{2n}\times{\mathbb{R}}^{2n}$ is
given by

$\alpha_{z_{0}}(t)=(z_{0}+tv,Du(z_{0})+t(Du(z_{0}+v)-Du(z_{0}))).$

Writing $u_{v}(z)=\frac{u(z+v)-u(z)}{|v|}$, it follows from a direct
computation that

(5.21) $\begin{split}\frac{d(A(\alpha_{z}(t))}{dt}&=A_{z}(\alpha_{z}(t))\cdot
v+|v|A_{\xi}(\alpha_{z}(t))\cdot Du_{v}(z),\end{split}$

where $A_{\xi}$ is the matrix with entries $(A_{\xi})_{ij}=\frac{\partial
A_{j}}{\partial z_{i}}$ given by

(5.22)
$(A_{\xi})_{ij}(z,\xi)=\frac{\delta_{ij}(\varepsilon^{2}+|\xi+X(z)|^{2})-(\xi_{i}+X(z)_{i})(\xi_{j}+X(z)_{j})}{(\varepsilon^{2}+|\xi+X(z)|^{2})^{\frac{3}{2}}}.$

We set

$\begin{split}\tilde{A}(z)&=\int_{0}^{1}A_{\xi}(\alpha_{z}(t))\,dt,\\\
\tilde{B}_{v}(z)&=\left(\int_{0}^{1}A_{z}(\alpha_{z}(t))\,dt\right)\cdot\frac{v}{|v|}\\\
\tilde{H}_{v}(z)&=\frac{H(z)-H(z+v)}{|v|}.\end{split}$

Inserting (5.20) and (5.21) in (5.19) and dividing by $|v|$, we get

$\int_{\tilde{\Omega}}\langle\tilde{A}(z)\cdot
Du_{v}(z),D\psi(z)\rangle\,dz=\int_{\tilde{\Omega}}\tilde{H}_{v}(z)\psi(z)\,dz-\int_{\tilde{\Omega}}\left\langle\tilde{B}_{v}(z),D\psi(z)\right\rangle\,dz.$

In other words, $u_{v}$ is a weak solution on $\tilde{\Omega}$ to the linear
equation

$\operatorname{div}\left(\tilde{A}\cdot
Du_{v}\right)-\tilde{H}_{v}-\operatorname{div}\tilde{B}_{v}.$

Since $H\in\operatorname{Lip}(\tilde{\Omega})$, the coefficients of
$\tilde{A}$, $\tilde{B}_{v}$ and $\tilde{H}_{v}$ are uniformly bounded with
respect to $v$. Moreover, since $u\in\operatorname{Lip}(\tilde{\Omega})$,
(5.22) implies that $\tilde{A}$ is uniformly elliptic on $\tilde{\Omega}$.
Finally, since $u\in\operatorname{Lip}(\tilde{\Omega})$, then $u_{v}$ is
bounded in $L^{\infty}(\tilde{\Omega})$ uniformly for $v\in B_{r}$ with $r>0$
small enough. Therefore, using the celebrated De Giorgi-Nash-Moser method (cf.
[15, Chapter 4, Theorem 2.3] and cf. the proof of the Corollary right after
[15, Chapter 4, Theorem 2.2]), we conclude that, up to a smaller
$\tilde{\Omega}$, $u_{v}$ is bounded in $C^{0,\alpha}(\tilde{\Omega})$
uniformly for $v\in B_{r}$, so that $u\in C^{1,\alpha}(\tilde{\Omega})$ and
$u_{v}\in C^{1,\alpha}(\tilde{\Omega})$. In particular, $\tilde{A}$ and
$\tilde{B}_{v}$ are of class $C^{0,\alpha}$. Therefore, in view of [37,
Theorem 8,32] and up to a smaller $\tilde{\Omega}$, $u_{v}$ is bounded in
$C^{1,\alpha}(\tilde{\Omega})$ uniformly for $v\in B_{r}$, so that $u\in
C^{2,\alpha}(\tilde{\Omega})$. In particular, $u$ is a classical solution to
($\varepsilon$-PMC). Let $k\geqslant 1$ and assume $H\in
C^{k,\gamma}_{loc}(\Omega)$. Since $u$ is a weak solution to
($\varepsilon$-PMC), by means of [15, Chapter 12, Theorem 1.1] we infer that,
for any $j=1,\ldots,2n$, $g:=(Du)_{j}$ is a weak solution to the linear
equation

$\operatorname{div}\left(\frac{\partial A}{\partial\xi}(z,Du)\cdot
Dg\right)=D_{j}H-\operatorname{div}\left(\frac{\partial A}{\partial
z_{j}}(z,Du)\right).$

The thesis then follows exploiting the classical Schauder’s theory (cf. [36,
Theorem 5.20]). ∎

###  5.4. The Dirichlet problem for ($\varepsilon$-PMC)

As a corollary of the global gradient estimates, we extend the existence
result obtained in [38] for the sub-Finsler constant mean curvature equation
to the existence of solutions for the sub-Riemannian prescribed, but not
necessarily constant, mean curvature equation.

###### Proof of Theorem 1.3.

Assume first that $H\in C^{1,\alpha}(\overline{\Omega})$. Arguing exactly as
in [38], it suffices to provide _a priori_ estimates in
$C^{1}(\overline{\Omega})$ for solutions $u\in
C^{2,\alpha}(\overline{\Omega})$ to (4.9) with boundary datum $\varphi$ which
are independent of $u$ and $\sigma\in[0,1]$. First, by means of Lemma 5.2,
uniform estimates in $L^{\infty}(\Omega)$ follow as in [38, Proposition 4.4].
Moreover, arguing _verbatim_ as in Theorem 5.8, any solution $u\in
C^{2,\alpha}(\overline{\Omega})$ to (4.9) belongs to $C^{3}(\Omega)\cap
C^{1}(\overline{\Omega})$. Hence, Theorem 4.3 provides global gradient
estimates uniform with respect to $\sigma\in[0,1]$. Assume now that
$H\in\operatorname{Lip}(\Omega)$, and denote by $H_{0}$ its Lipschitz
constant. By McShane’s extension theorem, we can suppose that
$H\in\operatorname{Lip}({\mathbb{R}}^{2n})$ with the same Lipschitz constant
$H_{0}$. By a standard mollification argument, there exists a sequence
$(H_{j})_{j}\subseteq C^{\infty}(\overline{\Omega})$ such that

(5.23) $H_{j}\to H\text{ uniformly on
}\overline{\Omega}\qquad\text{and}\qquad\|H_{j}\|_{C^{1}(\overline{\Omega})}\leqslant
H_{0}+\|H\|_{L^{\infty}(\Omega)}+1$

for any $j\in\mathbb{N}$. Since $H$ and $\Omega$ satisfy (* ‣ 1) and (*** ‣
5.1), we let $\delta$ be as in the statement of Lemma 5.2. Let us denote by
$h(\Omega)$ the _Cheeger constant_ of $\Omega$ (cf. [47]), that is

$h(\Omega)=\inf\left\\{\frac{P(A)}{|A|}\,:\,A\subseteq\Omega,\,|A|>0\right\\}.$

Being $\Omega$ bounded and open, it is well known (cf. [47, Proposition 3.5])
that $h(\Omega)>0$. By (5.23), we can assume up to a subsequence that

$\|H-H_{j}\|_{L^{\infty}(\Omega)}\leqslant\frac{\delta h(\Omega)}{2},$

so that

(5.24)
$\Big{|}\int_{A}H_{j}dz\Big{|}\leqslant\Big{|}\int_{A}Hdz\Big{|}+|A|\|H-H_{j}\|_{L^{\infty}(\Omega)}\leqslant\left(1-\frac{\delta}{2}\right)P(A)$

for any $A\subseteq\Omega$ such that $|A|\neq 0$. On the other hand, by (1.6)
and (5.23), there exists $C=C(\Omega,\|H\|_{L^{\infty}(\Omega)})>0$ such that,
up to a subsequence,

(5.25) $|H_{j}(z_{0})|\leqslant H_{\partial\Omega}(z_{0})-C$

for any $z_{0}\in\partial\Omega$ and any $j\in\mathbb{N}$. Combining (5.24)
and (5.25), from the previous step we get a solution $u_{j}\in
C^{2,\alpha}(\overline{\Omega})$ to ($\varepsilon$-PMC) with boundary datum
$\varphi$ and source $H_{j}$. Again by (5.23), (5.24) and (5.25), and
following [38, Proposition 4.4], [38, Proposition 4.8] and Theorem 4.3,
$(u_{j})_{j}$ is uniformly bounded in $C^{1}(\overline{\Omega})$, so that, by
Ascoli-Arzelà Theorem, there exists $u\in\operatorname{Lip}(\Omega)$ such that
$u_{j}\to u$ uniformly on $\overline{\Omega}$. First, notice that $u=\varphi$
on $\partial\Omega$. Finally, a compactness argument as the forthcoming
Theorem 7.4, coupled with Theorem 5.5 and Theorem 5.8, implies that $u\in
C^{2,\alpha}_{loc}(\Omega)$ and that $u$ is a classical solution to
($\varepsilon$-PMC), whence the thesis follows. ∎

###  5.5. Lipschitz regularity of $t$-graphs

Throughout this subsection, we fix a bounded domain
$\Omega\subseteq{\mathbb{R}}^{2n}$ with Lipschitz boundary and $H\in
C^{1,\gamma}_{loc}(\Omega)$ for some $\gamma\in(0,1)$. We are left to show
that $BV$-minimizers of $\mathcal{I}_{\varepsilon}$ are locally Lipschitz
continuous.

######  Proposition 5.9.

Let $H\in C^{1,\gamma}_{loc}(\Omega)$ and let $u\in BV(\Omega)$ be such that
$E_{u}$ is an $H$-minimizer for $P_{\varepsilon}$ on
$\Omega\times{\mathbb{R}}$. Then $\partial^{*}E_{u}$ is a $C^{3,\gamma}$
manifold and $\mathcal{H}^{s}(\partial E\setminus\partial^{*}E)=0$ for any
$s>2n-7$.

###### Proof.

Let $u$ be as in the statement. Let $\tilde{\Omega}\Subset\Omega$ be an open
set. Fix $\alpha\in(0,1)$ and let $F\Subset\tilde{\Omega}\times{\mathbb{R}}$
be a finite perimeter set such that $E\Delta F\Subset
B(p,r)\Subset\tilde{\Omega}\times{\mathbb{R}}$ for some
$p\in\tilde{\Omega}\times{\mathbb{R}}$ and $r\in(0,1)$. Taking
$H_{0}=\|H\|_{L^{\infty}(\tilde{\Omega})}$, it follows that

$\begin{split}P_{\varepsilon}(E,B(p,r))&\leqslant
P_{\varepsilon}(F,B(p,r))+H_{0}|E\Delta F|\leqslant
P_{\varepsilon}(F,B(p,r))+H_{0}w_{n}r^{2n+\alpha}.\end{split}$

Moreover, for any Caccioppoli set $F$ in $\Omega\times{\mathbb{R}}$ and any
open set $A\subseteq\Omega\times{\mathbb{R}}$ it holds that

$P_{\varepsilon}(F,A)=\int_{\partial^{\star}F\cap
A}|C_{\varepsilon}(p)^{T}\nu(p)|\,d\mathcal{H}^{2n}(p)=\int_{\partial^{\star}F\cap
A}\sqrt{\langle
M_{\varepsilon}(p)\nu(p),\nu(p)\rangle}\,d\mathcal{H}^{2n}(p),$

where $C_{\varepsilon}$ is defined in (2.1),
$M_{\varepsilon}(p)=C_{\varepsilon}(p)^{T}\cdot C_{\varepsilon}(p)$ for any
$p\in\Omega\times{\mathbb{R}}$ and $\nu$ is the measure theoretic Euclidean
unit normal to $F$. It is easy to check that $M_{\varepsilon}$ is uniformly
positive definite and $\alpha$-Hölder continuous. Then [68, Theorem 1.1]
implies that $E_{u}$ is a manifold of class $C^{1,\frac{\alpha}{4}}$, and
moreover $\mathcal{H}^{s}(\partial E\setminus\partial^{*}E)=0$ for any
$s>2n-7$. Fix $0<\alpha<1$ such that $\frac{\alpha}{4}\leqslant\gamma$. Since
$S$ is $C^{1,\frac{\alpha}{4}}$, for any $p\in S$ we can consider an open
neighborhood $U$ of $p$ in the tangent hyperplane of $S$ at $p$ where $S$
coincides with $\Phi_{\tilde{u}}(U):=\\{q-\nu(p)\tilde{u}(q):q\in U\\}$ for a
suitable function $\tilde{u}$ of class $C^{1,\frac{\alpha}{4}}$. Given $v\in
W^{1,1}(U)$, we let $A(v)$ be the area of $\Phi_{v}(U)$ with respect to the
metric $g_{\varepsilon}$. By Theorem 5.5, $\tilde{u}$ is a critical point of
the functional

(5.26) $I(v)=A(v)+\int_{U}\tilde{H}v,$

for a suitable $\tilde{H}\in C_{loc}^{1,\frac{\alpha}{4}}(U)$. Writing $A$ in
local coordinates, and writing the Euler-Lagrange equation associated to
(5.26) for $\tilde{u}$, one can argue exactly as in the proof of Theorem 5.8
to conclude that $\tilde{u}\in C_{loc}^{2,\beta}(U)$ for any $\beta\in(0,1)$.
Since $\tilde{H}\in C_{loc}^{1,\gamma}(U)$, the thesis follows as in Theorem
5.8. ∎

Let $\pi:\mathbb{H}^{n}\to\Omega$ be the projection on the first $2n$
components and $u\in BV(\Omega)$. We denote

(5.27) $\Omega_{u,0}=\pi(\partial E_{u}\setminus\partial^{\star}E_{u}).$

We say that $S\subseteq{\mathbb{H}}^{n}$ is locally a _$Y_{1}$ -graph_ around
$p=(\bar{x}_{1},\ldots,\bar{x}_{n},\bar{y}_{1},\ldots,\bar{y}_{n},\bar{t})\in
S$ if there exists a neighborhood $U\subseteq{\mathbb{R}}^{2n}$ of
$\bar{w}=(\bar{x}_{1},\ldots,\bar{x}_{n},\bar{y}_{2},\ldots,\bar{y}_{n},\bar{t}+\bar{x}_{1}\bar{y}_{1})\in{\mathbb{R}}^{2n}$
and a function $\varphi\in C(U)$ such that

$S=\\{(x_{1},\ldots,x_{n},\varphi(w),y_{2},\ldots,y_{n},t-x_{1}\varphi(w))\,:\,w=(x_{1},\ldots,x_{n},y_{2},\ldots,y_{n},t)\in
U\\}$

(cf. [30]). Identifying ${\mathbb{R}}^{2n}$ with the hyperplane
$\\{y_{1}=0\\}\subseteq{\mathbb{H}}^{n}$, we follow [6, 7] and consider the
vector fields on $U$ given by

(5.28) $W^{\varphi,\varepsilon}_{1}=\frac{\partial}{\partial x_{1}}+2\varphi
T|_{U},\quad W^{\varphi,\varepsilon}_{j}=X_{j}|_{U},\quad
W^{\varphi,\varepsilon}_{n+j-1}=Y_{j}|_{U}\quad\text{and}\quad
W_{2n}^{\varphi,\varepsilon}=\varepsilon T|_{U},$

where $j=2,\ldots,n$. We denote by
$\left(W^{\varphi,\varepsilon}_{j}\right)^{\star}$ the adjoint operator of
$W^{\varphi,\varepsilon}_{j}$ with respect to $L^{2}({\mathbb{R}}^{2n})$, we
write
$W^{\varphi,\varepsilon}=(W^{\varphi,\varepsilon}_{1},\ldots,W^{\varphi,\varepsilon}_{2n})$
and we let $C^{\varphi}_{\varepsilon}$ be the coefficient matrix associated to
the family $W^{\varphi,\varepsilon}$. Notice that for any $j=2,\ldots,2n$, it
holds that

(5.29)
$\left(W^{\varphi,\varepsilon}_{1}\right)^{\star}\phi=-W^{\varphi,\varepsilon}_{1}\phi-2\phi
T|_{U}\varphi\qquad\text{and}\qquad\left(W^{\varphi,\varepsilon}_{j}\right)^{\star}\phi=-W^{\varphi,\varepsilon}_{j}\phi.$

and hence

$\left(W^{\varphi,\varepsilon}_{j}\right)^{\star}W^{\varphi,\varepsilon}_{2n}=-W^{\varphi,\varepsilon}_{2n}W^{\varphi,\varepsilon}_{j}.$

The following standard first variation formula follows as in [7, Section 3.1].

######  Lemma 5.10.

Let $u\in BV(\Omega)$ be such that $E_{u}$ is an $H$-minimizer for
$P_{\varepsilon}$ on $\Omega\times{\mathbb{R}}$, and set
$S=\operatorname{graph}(u)$. Assume that $S$ is the $Y_{1}$-graph of
$\varphi\in C^{2}(U)$, where $U\subseteq{\mathbb{R}}^{2n}$. Then

(5.30)
$-\sum_{j=1}^{2n}\int_{U}\frac{W^{\varphi,\varepsilon}_{j}\varphi\left(W^{\varphi,\varepsilon}_{j}\right)^{\star}\psi}{\sqrt{1+|W^{\varphi,\varepsilon}\varphi|^{2}}}\,dw=\int_{U}H(x_{1},,\ldots,x_{n},\varphi(w),y_{2},\ldots,y_{n})\psi\,dw$

for any $\psi\in C^{\infty}_{c}(U)$.

######  Proposition 5.11.

Let $H\in C^{1,\gamma}_{loc}(\Omega)$ and let $u\in BV(\Omega)$ be such that
$E_{u}$ is an $H$-minimizer for $P_{\varepsilon}$ on
$\Omega\times{\mathbb{R}}$. Then $u\in
C^{3,\gamma}_{loc}(\Omega\setminus\Omega_{u,0})$ and
$\mathcal{H}^{s}(\Omega_{u,0})=0$ for any $s>2n-7$.

###### Proof.

Let $z=(\bar{x},\bar{y})\in\Omega\setminus\Omega_{u,0}$, where
$\bar{x}=(\bar{x}_{1},\ldots,\bar{x}_{n})$ and
$\bar{y}=(\bar{y}_{1},\ldots,\bar{y}_{n})$. By Proposition 5.9,
$S=\operatorname{graph}(u)$ is a hypersurface of class $C^{3,\gamma}$ near
$p=(z,u(z))$, and $\mathcal{H}^{s}(\Omega_{u,0})=0$ for any $s>2n-7$. Hence it
suffices to show that $g_{\varepsilon}(T,\nu^{\varepsilon})\neq 0$ on
$\Omega\setminus\Omega_{u,0}$, where $\nu^{\varepsilon}$ is the outer unit
normal of $S$. Indeed, assume by contradiction that
$g_{\varepsilon}(T,\nu^{\varepsilon}(p))=0$. Then
$\nu^{\varepsilon}(p)\in{\mathcal{H}}_{p}$, and so $p$ is non-characteristic,
since otherwise $\nu^{\varepsilon}(p)\in{\mathcal{H}}_{p}=T_{p}S$. Therefore,
by the implicit function theorem for intrinsic graphs (cf. [30, Theorem 6.5])
$S$ is, locally near $p$, a $Y_{1}$-graph with respect to a continuous
function $\varphi$ defined on an open neighborhood
$U\subseteq{\mathbb{R}}^{2n}$ of
$\bar{w}=(\bar{x}_{1},\ldots,\bar{x}_{n},\bar{y}_{2},\ldots,\bar{y}_{n},\bar{t}+\bar{x}_{1}\bar{y}_{1})$.
Moreover, arguing as in [57, Proposition 6.1], $\varphi\in C^{3,\gamma}(U)$.
Since $S$ is a $t$-graph vertical at $p$, up to choosing a smaller $U$, we
infer that

(5.31) $T|_{U}\varphi(\bar{w})=0\qquad\text{and}\qquad
T|_{U}\varphi(w)\geqslant 0$

for any $w\in U$. If $W^{\varphi,\varepsilon}$ is defined as in (5.28), in the
following we drop the superscript for the sake of clarity. Taking
$\psi=W_{2n}\psi$ in (5.30) and using (5.29) and the definition of adjoint
operator, we infer that

$\begin{split}\int_{U}\frac{\partial H}{\partial
y_{1}}W_{2n}\varphi\psi\,dw&=-\int_{U}HW_{2n}\psi\,dw\\\
&=\sum_{j=1}^{2n}\int_{U}\frac{W_{j}\varphi
W_{j}^{\star}\big{(}W_{2n}\psi\big{)}}{\sqrt{1+|W\varphi|^{2}}}\,dw\\\
=&\sum_{j=1}^{2n}\int_{U}W_{2n}\left(\frac{W_{j}\varphi}{\sqrt{1+|W\varphi|^{2}}}\right)W_{j}\psi\,dw\\\
=&\sum_{j,k=1}^{2n}\int_{U}\left\\{\frac{W_{2n}\big{(}W_{j}\varphi\big{)}}{\sqrt{1+|W\varphi|^{2}}}-\frac{W_{2n}\big{(}W_{k}\varphi\big{)}W_{k}\varphi
W_{j}\varphi}{\left(1+|W\varphi|^{2}\right)^{\frac{3}{2}}}\right\\}W_{j}\psi\,dw.\end{split}$

Setting $g=W_{2n}\varphi$, we get

$0=\sum_{j,k=1}^{2n}\int_{U}\left\\{\left(\frac{W_{j}^{\star}g}{\sqrt{1+|W\varphi|^{2}}}-\frac{W_{k}^{\star}gW_{k}\varphi
W_{j}\varphi}{\left(1+|W\varphi|^{2}\right)^{\frac{3}{2}}}\right)W_{j}\psi+\frac{\partial
H}{\partial y_{1}}g\psi\right\\}\,dw,$

that can be rewritten as

$\int_{U}W\psi\cdot A\cdot\left(Wg\right)^{T}+\langle W\psi,B\rangle
g-\frac{\partial H}{\partial y_{1}}g\psi\,dw=0,$

where $A=\left(A_{jk}\right)_{jk}$ and $B=(B_{j})_{j}$ are defined by

$A_{jk}=\frac{\left(1+|W\varphi|^{2}\right)\delta_{jk}-W_{j}\varphi
W_{k}\varphi}{\left(1+|W\varphi|^{2}\right)^{\frac{3}{2}}}\qquad\text{and}\qquad
B_{j}=2T|_{U}\varphi A_{j1}$

for any $j,k=1,\ldots,2n$. Since $W\varphi$ is continuous, an easy computation
shows that, up to choosing a smaller $U$, there exists $\alpha>0$ such that
$\xi\cdot A(w)\cdot\xi^{T}\geqslant\alpha|\xi|^{2}$ for any $w\in U$ and any
$\xi\in{\mathbb{R}}^{2n}$. Moreover, as $Wf=Df\cdot C^{\varphi,\varepsilon}$
for any $f\in C^{1}(U)$, we infer that

$\int_{U}D\psi\cdot\tilde{A}\cdot(Dg)^{T}+\langle D\psi,\tilde{B}\rangle
g-\frac{\partial H}{\partial y_{1}}g\psi\,dw=0,$

where

$\tilde{A}=C^{\varphi,\varepsilon}\cdot
A\cdot\left(C^{\varphi,\varepsilon}\right)^{T}\qquad\text{and}\qquad\tilde{B}=B\cdot\left(C^{\varphi,\varepsilon}\right)^{T}.$

Being $C^{\varphi,\varepsilon}(w)$ invertible and continuous in $w$, up to
restricting $U$ there exists $\tilde{\alpha}>0$ such that

$\begin{split}\xi\cdot\tilde{A}\cdot\xi^{T}&=\xi\cdot
C^{\varphi,\varepsilon}\cdot
A\cdot\left(C^{\varphi,\varepsilon}\right)^{T}\cdot\xi^{T}=\xi\cdot
C^{\varphi,\varepsilon}\cdot A\cdot\left(\xi\cdot
C^{\varphi,\varepsilon}\right)^{T}\geqslant\alpha|\xi\cdot
C^{\varphi,\varepsilon}|^{2}\geqslant\tilde{\alpha}|\xi|^{2}\end{split}$

on $U$ for any $\xi\in{\mathbb{R}}^{2n}$. Hence $\tilde{A}$ is uniformly
elliptic on $U$. Therefore, recalling (5.31) and choosing a suitable smaller
neighborhood $U$, we can apply a strong maximum principle as in [58, page 73,
Theorem 10] to conclude that $T|_{U}\varphi\equiv 0$ on $U$. In particular,
$g_{\varepsilon}(T,\nu^{\varepsilon})=0$ in a neighborhood $O$ of $p$ and
$\mathcal{H}^{2n-1}(V)>0$, where $V=\pi(O)$. Arguing _verbatim_ as in [41,
page 169], it follows that $V\subseteq\Omega_{u,0}$, which is a contradiction
with $\mathcal{H}^{2n-1}(\Omega_{u,0})=0$. ∎

######  Proposition 5.12.

Let $H\in C^{1,\gamma}_{loc}(\Omega)$ and let $u\in BV(\Omega)$ be such that
$E_{u}$ is an $H$-minimizer for $P_{\varepsilon}$ on
$\Omega\times{\mathbb{R}}$. Then $u\in W^{1,1}(\Omega).$

###### Proof.

By Proposition 5.11, $u\in C^{3,\gamma}(\Omega\setminus\Omega_{u,0})$ and
$\mathcal{H}^{2n-1}(\Omega_{u,0})=0$, where $\Omega_{u,0}$ is defined in
(5.27). Let $\tilde{D}u$ be the distributional derivatives of $u$, and
consider the decomposition $\tilde{D}u=Du\mathcal{L}^{2n}+(Du)_{s}$ as in
Lemma 2.4. It is enough to show that $(Du)_{s}\equiv 0$. Since in particular
$u\in W^{1,1}_{loc}(\Omega\setminus\Omega_{u,0})$, (2.11) combined with Lemma
2.4 implies that

$\begin{split}P_{\varepsilon}(E_{u},\tilde{\Omega}\times{\mathbb{R}})&=\int_{\tilde{\Omega}}\sqrt{\varepsilon^{2}+|Du+X|^{2}}\,dz+(Du)_{s}(\tilde{\Omega})\\\
&=\int_{\tilde{\Omega}\setminus\Omega_{u,0}}\sqrt{\varepsilon^{2}+|Du+X|^{2}}\,dz+(Du)_{s}(\tilde{\Omega})\\\
&=P_{\varepsilon}(E_{u},(\tilde{\Omega}\setminus\Omega_{u,0})\times{\mathbb{R}})+(Du)_{s}(\tilde{\Omega}),\end{split}$

so that

$(Du)_{s}(\tilde{\Omega})=P_{\varepsilon}(E_{u},(\tilde{\Omega}\cap\Omega_{u,0})\times{\mathbb{R}}).$

Arguing as in Proposition 5.7, there exists $L>0$ such that
$\partial^{\star}E_{u}\cap(\tilde{\Omega}\times{\mathbb{R}})\subseteq\tilde{\Omega}\times[-L,L].$
Therefore, exploiting (2.7), there exists $\tilde{C}>0$ such that

$(Du)_{s}(\tilde{\Omega})\leqslant\tilde{C}P(E_{u},(\tilde{\Omega}\cap\Omega_{u,0})\times{\mathbb{R}})=\tilde{C}\mathcal{H}^{2n}(\partial^{*}E_{u}\cap(\tilde{\Omega}\cap\Omega_{u,0})\times{\mathbb{R}})\\\
\leqslant 2\tilde{C}L\mathcal{H}^{2n-1}(\Omega_{u,0})=0.\qed$

######  Proposition 5.13.

Let $\Omega\subseteq{\mathbb{R}}^{2n}$ be a bounded domain with Lipschitz
continuous boundary and let $H\in C^{1,\gamma}_{loc}(\Omega)$ for some
$\gamma\in(0,1)$. Let $u\in BV(\Omega)$ be such that $E_{u}$ is an
$H$-minimizer for $P_{\varepsilon}$ on $\Omega\times{\mathbb{R}}$. then
$u\in\operatorname{Lip}_{loc}(\Omega)$.

###### Proof.

Fix $z_{0}\in\Omega$. Let $r_{0}>0$ small enough to ensure that
$B_{r_{0}}=B(z_{0},r_{0})\Subset\Omega$. We claim that there exists
$r\in(r,r_{0})$ such that (* ‣ 1), (*** ‣ 5.1) and (1.6) hold for
$B_{r}=B(z_{0},r)$. Indeed, setting $H_{0}=\|H\|_{L^{\infty}(B_{r_{0}})}$,
then

$\left|\int_{A}H\,dz\right|\leqslant H_{0}|A|$

for any $r\in(0,r_{0})$ and any measurable set $A\subseteq B_{r}$. Therefore,
recalling that

$h(B_{r})=\frac{P(B_{r})}{|B_{r}|}=\frac{1}{r}\frac{r_{0}P(B_{r_{0}})}{|B_{r_{0}}|}$

(cf. [47]), the claim concerning (* ‣ 1) and (*** ‣ 5.1) follows by taking
$r<\frac{r_{0}P(B_{r_{0}})}{H_{0}|B_{r_{0}}|}$. Regarding (1.6), it suffices
to see that $H_{\partial B_{r}}=\frac{2n-1}{r}$, so that $H_{\partial B_{r}}$
can be made arbitrarily big as $r$ becomes small. Given $w\in W^{1,1}(B_{r})$,
we consider the extension of $w$ to a function $\tilde{w}\in BV(\Omega)$ by
letting $\tilde{w}\equiv u$ on $\Omega\setminus\overline{B_{r}}$. In
particular, $u$ and $\tilde{w}$ share the same trace on $\partial\Omega$.
Using that $u\in W^{1,1}(\Omega)$ is such that $E_{u}$ is an $H$-minimizer for
$P_{\varepsilon}$ on $\Omega\times{\mathbb{R}}$ and (2.8), we infer that
$u|_{B_{r}}$ minimizes the functional

(5.32)
$I(w)=\int_{B_{r}}\sqrt{\varepsilon^{2}+|Dw+X|^{2}}\,dz+\int_{B_{r}}Hwdz+\int_{\partial
B_{r}}|u-w|d\mathcal{H}^{2n-1}$

in $W^{1,1}(B_{r})$. Since $\mathcal{H}^{2n-1}(\Omega_{u,0})=0$ by Proposition
5.11, we can take a sequence of open sets $(\Omega_{k})_{k}$ such that
$\Omega_{u,0}\subseteq\Omega_{k+1}\subseteq\Omega_{k}\subseteq\Omega$ for any
$k\in\mathbb{N}$, $\Omega_{u,0}=\bigcap_{k\in\mathbb{N}}\Omega_{k}$ and
$\mathcal{H}^{2n-1}(\Omega_{k}\cap\partial B_{r})\to 0$ as $k\to\infty$. For
any $k\in\mathbb{N}$, let $\varphi_{k}\in C^{2,\gamma}(\overline{B_{r}})$ be
such that $\varphi_{k}\equiv u$ on $\partial B_{r}\setminus\Omega_{k}$ and

(5.33) $\sup_{\partial B_{r}}|\varphi_{k}|\leqslant 2\sup_{\partial
B_{r}}|u|.$

We apply Theorem 1.3 to get a classical solution $v_{k}\in
C^{2}(\overline{B_{r}})$ to ($\varepsilon$-PMC) such that
$v_{k}\equiv\varphi_{k}$ on $\partial B_{r}$ for any $k\in\mathbb{N}$. In
particular, being $\mathcal{I_{\varepsilon}}$ convex in $W^{1,1}(B_{r})$, we
infer that $v_{k}$ minimizes the functional

(5.34)
$I_{k}(w)=\int_{B_{r}}\sqrt{\varepsilon^{2}+|Dw+X|^{2}}\,dz+\int_{B_{r}}Hwdz+\int_{\partial
B_{r}}|\varphi_{k}-w|d\mathcal{H}^{2n-1}$

in $W^{1,1}(B_{r})$. Thanks to Lemma 5.2 we can apply [38, Proposition 4.4]
which, together with (5.33), implies that $(v_{k})_{k}$ is uniformly bounded
in $L^{\infty}(B_{r})$. By Theorem 5.8, $v_{k}\in C^{3,\gamma}_{loc}(B_{r})$,
and by Theorem 1.2 the sequence $(v_{k})_{k}$ is locally uniformly bounded in
$\operatorname{Lip}(B_{r})$. By Ascoli-Arzelà Theorem, there exists
$v\in\operatorname{Lip}_{loc}(B_{r})$ such that, up to a subsequence,
$v_{k}\to v$ locally uniformly on $B_{r}$. Since $(v_{k})_{k}$ is uniformly
bounded in $L^{\infty}(B_{r})$ and by the lower semicontinuity properties of
Lemma 2.3, we can pass to the limit in (5.34) with $w\equiv 0$ to infer that
$v\in W^{1,1}(B_{r})$. Let us check that $v$ is a minimum of $I$ in
$W^{1,1}(B_{r})$. Let $y\in\partial B_{r}$ be a regular point for $u$, $k$ big
enough so that $y\in\partial B\setminus\Omega_{k}$ and $V$ a neighborhood of
$y$ in $\partial B_{r}$. Let $\varphi^{\pm}\in C^{2,\gamma}(\overline{B_{r}})$
be such that

$\varphi^{\pm}=u\text{ in
}V\qquad\text{and}\qquad\varphi^{-}\leqslant\varphi_{k}\leqslant\varphi^{+}\text{
in }\partial B_{r},$

and let $v^{\pm}$ be the solutions to the Dirichlet problem in $B_{r}$ with
boundary datum $\varphi^{\pm}$. Then, using the maximum principle [37, Theorem
10.7], we get $v^{-}\leqslant v_{k}\leqslant v^{+}$, and $v=u$ at regular
points of $u$, which together with $\mathcal{H}^{2n-1}(\Omega_{u,0})$ implies
that $u$ and $v$ have the same trace on $\partial B_{r}$. Moreover, by the
lower semicontinuity, the local uniform convergence of $v_{h}\to v$ and that
$\varphi_{k}\to u$ in $L^{1}(\partial B_{r})$, we get

$I(v)\leqslant\liminf_{k\to+\infty}I_{k}(v_{k})\leqslant\liminf_{k\to+\infty}I_{k}(w)=I(w)$

for any $w\in W^{1,1}(B_{r})$. Therefore, $u$ and $v$ are minimizers in
$W^{1,1}(B_{r})$ of the functional $I$. Recalling that $u-v\in
W^{1,1}_{0}(B_{r})$, the conclusion easily follows by the strict convexity of
the functional. ∎

As a direct consequence of Proposition 5.3, Theorem 5.8 and Proposition 5.13,
we have the following result.

######  Theorem 5.14.

Let $\Omega\subseteq{\mathbb{R}}^{2n}$ be a bounded domain with Lipschitz
continuous boundary and let $H\in C^{1,\gamma}_{loc}(\Omega)\cap
L^{\infty}(\Omega)$ for some $\gamma\in(0,1)$. Assume that (* ‣ 1) and (*** ‣
5.1) holds. Then there exists $u\in C_{loc}^{3,\gamma}(\Omega)\cap
W^{1,1}(\Omega)$ which minimizes $\mathcal{I_{\varepsilon}}$ and solves
($\varepsilon$-PMC). Moreover, if $H\in C^{\infty}(\Omega)$ then $u\in
C^{\infty}(\Omega)$.

###  5.6. Existence of minimizers: the extremal case

Throughout this subsection we fix $H\in\operatorname{Lip}(\Omega)\cap
C^{1,\gamma}_{loc}(\Omega)$ for some $\gamma\in(0,1)$. To deal with solutions
to ($\varepsilon$-PMC) in the extremal case (** ‣ 1), we follow the approach
of [40]. To this aim, we generalize the notion of $H$-minimizer as in (5.10)
admitting merely measurable functions.

######  Definition 5.15.

A measurable function $u:\Omega\longrightarrow[-\infty,+\infty]$ is a
_generalized $H$-minimizer_ for $P_{\varepsilon}$ on
$\Omega\times{\mathbb{R}}$ if (5.9) holds.

According to the notation introduced in [41], if
$u:\Omega\longrightarrow[-\infty,+\infty]$ is a measurable function we set

(5.35) $N_{+}=\\{z\in\Omega\,:\,u(z)=+\infty\\}\qquad\text{and}\qquad
N_{-}=\\{z\in\Omega\,:\,u(z)=-\infty\\}.$

As in the Euclidean setting, we have the following minimization property.

######  Proposition 5.16.

Let $u$ be a generalized $H$-minimizer. Then

(5.36) $P(N_{\pm},A)\pm\int_{N_{\pm}\cap A}H\,dz\leqslant P(F,A)\pm\int_{F\cap
A}H\,dz$

for any open set $A\Subset\Omega$ and any measurable set $F\subseteq\Omega$
such that $N_{+}\Delta F\Subset A$.

###### Proof.

Let us check this property for $N_{+}$, being the case $N_{-}$ analogous. For
any $j\in\mathbb{N}$, set $u_{j}=u-j$. Then $u_{j}$ converges almost
everywhere to

$v(z)=\displaystyle{\begin{cases}+\infty&\text{ if }z\in N_{+}\\\
-\infty&\text{ if }z\notin N_{+}.\end{cases}}$

Arguing _verbatim_ as in [50, Section 21.5]. (cf. Section 7 for the proof of a
similar result), $E_{v}$ is an $H$-minimizer as in (5.9). Assume by
contradiction that there exists an open set $A\Subset\Omega$, a measurable set
$F$ such that $N_{+}\Delta F\Subset A$ and $\delta>0$ such that

$P(F;A)+\int_{F\cap A}H\,dz\leqslant P(N_{+};A)+\int_{N_{+}\cap
A}H\,dz-\delta.$

Given $L>0$, we set $A_{L}=A\times[-L,L]$ and $A_{2L}=A\times(-2L,2L)$, and

$F_{L}=\displaystyle{\begin{cases}F\times{\mathbb{R}}&\text{ in }A_{L}\\\
N_{+}\times{\mathbb{R}}&\text{ otherwise.}\end{cases}}$

Then, using (2.8), we have

$\begin{split}P_{\varepsilon}(F_{L};A_{2L})+\int_{F_{L}\cap
A_{2L}}H\,dx&\leqslant
4\int_{\Omega}\sqrt{\varepsilon^{2}+|X|^{2}}\,dz+2LP(N_{+},A)+2LP(F;A)\\\
&\quad+2L\int_{N_{+}\cap A}H\,dz+2L\int_{F\cap A}H\,dz\\\ &\leqslant
4\int_{\Omega}\sqrt{\varepsilon^{2}+|X|^{2}}\,dz+4LP(N_{+};A)+4L\int_{F_{+}\cap
A}H\,dz-2L\delta\\\ &=P_{\varepsilon}(E_{v};A_{2L})+\int_{E_{v}\cap
A_{2L}}H\,dx+4\int_{\Omega}\sqrt{\varepsilon^{2}+|X|^{2}}\,dz-2L\delta\\\
&<P_{\varepsilon}(E_{v};A_{2L})+\int_{E_{v}\cap A_{2L}}H\,dx,\end{split}$

where the last strict inequality follows provided that
$L>\frac{2}{\delta}\int_{\Omega}\sqrt{\varepsilon^{2}+|X|^{2}}\,dz$. Being
$E_{v}$ an $H$-minimizer, a contradiction follows. ∎

###### Proof of Theorem 1.1.

Let us assume (** ‣ 1), since otherwise the thesis follows by Theorem 5.14.
Let $(\varepsilon_{j})_{j}\subseteq(0,1)$ be such that
$\varepsilon_{j}\searrow 0$ as $j\to\infty$. Let $(\Omega_{j})_{j}$ be a
sequence of open domains with Lipschitz boundary such that
$\Omega_{j}\Subset\Omega_{k}\Subset\Omega$ for any $j<k$m
$\bigcup_{j=0}^{\infty}\Omega_{j}=\Omega$ and $P(\Omega_{j})\to P(\Omega)$ as
$j\to\infty$ (cf. [63]). By hypothesis

$\Big{|}\int_{\Omega_{j}}H\Big{|}<P(\Omega_{j})$

for any $j\in\mathbb{N}$. Hence, for any fixed boundary datum $\varphi_{j}\in
L^{1}(\partial\Omega_{j})$, Proposition 5.3 implies the existence of $u_{j}\in
BV(\Omega_{j})$ which minimizes $\mathcal{I}_{\varepsilon}$ on $\Omega_{j}$.
In view of Theorem 5.5, $E_{u_{j}}$ is an $H_{j}$-minimizer for
$P_{\varepsilon_{j}}$ on $\Omega_{j}\times{\mathbb{R}}$. Arguing as in [40,
48], up to vertical translations we assume that

(5.37) $\min\\{|\\{z\in\Omega_{j}\,:\,u_{j}(z)\leqslant
0\\}|,\\{z\in\Omega_{j}\,:\,u_{j}(z)\geqslant
0\\}|\\}\geqslant\frac{|\Omega|}{4}$

for any $j\in\mathbb{N}$. Applying again a compactness argument as in [50,
Section 21.5] and Section 7, there exists a generalized $H$-minimizer $u$ for
$P_{\varepsilon}$ on $\Omega\times{\mathbb{R}}$ as in Definition 5.15 such
that $u_{j}\to u$ almost everywhere on $\Omega$. We are left to show that
$u\in L^{\infty}_{loc}(\Omega)$. Indeed, in this case, since $E_{u}$ is a
Caccioppoli set, Lemma 2.3 would imply that $u\in BV_{loc}(\Omega)$, whence
Proposition 5.13 and Theorem 5.8 guarantee the requested regularity. To show
that $u\in L^{\infty}_{loc}(\Omega)$, we let $N_{+}$ and $N_{-}$ be as in
(5.35). We claim that $|N_{+}|=|N_{-}|=0$. In this case, arguing _verbatim_ as
in [42, Proposition 16.7], volume density estimates as in Proposition 5.6
allow to conclude that $u\in L^{\infty}_{loc}(\Omega)$. We prove that
$|N_{+}|=0$, being the other case analogous. To this aim, we apply Proposition
5.16 to infer that $N_{+}$ is a minimizer as in (5.36). But then, (** ‣ 1)
allows to apply [40, Lemma 1.2], whence either $|N_{+}|=0$ or
$|N_{-}|=|\Omega|$. Being the latter possibility in contradiction with (5.37),
we conclude that $|N_{+}|=0$, from which the thesis follows. ∎

##  6\. Essential uniqueness of solutions

Similarly to what happens in the Euclidean setting (cf. [40]), the extremal
case (** ‣ 1) describes those maximal configurations $\Omega$ for which
($\varepsilon$-PMC) admits a classical solution. Moreover, in these cases,
solutions are unique up to vertical translations.

###### Proof of Theorem 1.4.

The equivalence between (i) and (ii), thanks to Theorem 1.1, follows word-by-
word the proof of [40, Proposition 2.2]. To prove that (i) is equivalent to
(iii) it suffices to notice that, if $u\in C^{2}(\Omega)$ solves
($\varepsilon$-PMC) in $\Omega$, then

$\int_{\Omega_{t}}H\,dz=\int_{\Omega_{t}}\operatorname{div}\left(\frac{Du+X}{\sqrt{\varepsilon^{2}+|Du+X|^{2}}}\right)\,dz=\int_{\partial\Omega_{t}}\frac{\langle\nu_{t},Du+X\rangle}{\sqrt{\varepsilon^{2}+|Du+X|^{2}}}\,d\mathcal{H}^{2n-1},$

so that

$\int_{\Omega}H\,dz=\lim_{t\to
0^{+}}\int_{\partial\Omega_{t}}\frac{\langle\nu_{t},Du+X\rangle}{\sqrt{\varepsilon^{2}+|Du+X|^{2}}}\,d\mathcal{H}^{2n-1}.$

To show uniqueness, assume that $u,v\in C^{2}(\Omega)$ satisfy
($\varepsilon$-PMC) in $\Omega$. Up to changing the sign of $u$ and $v$, we
can assume that $\int_{\Omega}H\,dz\geqslant 0$. Being ($\varepsilon$-PMC)
invariant under vertical translations, we fix $z_{0}\in\Omega$ and we assume
that $v(z_{0})=u(z_{0})$. In order to simplify the notation, we let

$W_{\varepsilon}u=\frac{Du+X}{\sqrt{\varepsilon^{2}+|Du+X|^{2}}},$

and $W_{\varepsilon}v$ accordingly. Notice in particular that
$|W_{\varepsilon}u|,|W_{\varepsilon}v|\leqslant 1$. Notice that, for any
$\varphi\in\operatorname{Lip}(\Omega)$ such that $\varphi\geqslant 0$ on
$\Omega$, and for any sufficiently small $t\in(0,1)$, it holds that

$\begin{split}\int_{\Omega_{t}}\langle
W_{\varepsilon}u-W_{\varepsilon}v,D\varphi\rangle\,dz=\int_{\partial\Omega_{t}}\varphi\left(\langle\nu_{t},W_{\varepsilon}u\rangle-\langle\nu_{t},W_{\varepsilon}v\rangle\right)\,dz\geqslant\int_{\partial\Omega_{t}}\varphi\left(\langle\nu_{t},W_{\varepsilon}u\rangle-1\right)\,dz.\end{split}$

For any $k>0$, let $\varphi_{k}=\max\\{0,\min\\{v-u,k\\}\\}$. Then
$\varphi_{k}\in\operatorname{Lip}(\Omega)$, $\varphi_{k}\geqslant 0$ on
$\Omega$ and $\langle
W_{\varepsilon}u-W_{\varepsilon}v,D\varphi_{k}\rangle\leqslant 0$ on $\Omega$,
so that

$0\geqslant\int_{\Omega_{t}}\langle
W_{\varepsilon}u-W_{\varepsilon}v,D\varphi_{k}\rangle\,dz\geqslant-k\left(P(\Omega_{t})-\int_{\partial\Omega_{t}}\langle\nu_{t},W_{\varepsilon}u\rangle\,dz\right).$

Therefore, in view of (iii), we let first $t\to 0^{+}$ and then $k\to\infty$
to obtain

(6.1) $\int_{\Omega}\langle
W_{\varepsilon}u-W_{\varepsilon}v,D\varphi\rangle\,dz=0,$

where $\varphi:=\max\\{0,v-u\\}$ verifies again $\langle
W_{\varepsilon}u-W_{\varepsilon}v,D\varphi\rangle\leqslant 0$ on $\Omega$.
Hence (6.1) implies that $\langle
W_{\varepsilon}u-W_{\varepsilon}v,D\varphi\rangle=0$ on $\Omega$. In view of
the definition of $\varphi$, a simple computation shows that $D\varphi=0$ on
$\Omega$, so that $\varphi(z)=\varphi(z_{0})=0$ for any $z\in\Omega$. Hence we
conclude that $u\equiv v$ on $\Omega$. ∎

##  7\. Some applications to the sub-Riemannian Heisenberg group

###  7.1. Existence of local minimizers via Riemannian approximation

Throughout this subsection, we fix a bounded domain
$\Omega\subseteq{\mathbb{R}}^{2n}$ with Lipschitz boundary and $H\in
L^{\infty}_{loc}(\Omega)$. It is well-known (cf. e.g. [16, 38]) that
($\varepsilon$-PMC) arises naturally as an elliptic approximation of the _sub-
Riemannian prescribed mean curvature equation_ (${\mathcal{H}}$-PMC). In this
section we provide existence of solutions to (${\mathcal{H}}$-PMC) in a broad
sense by means of our previous results coupled with the aforementioned
Riemannian approximation scheme (cf. e.g. [44, 55, 16, 14] for further
insights). To describe our approach, assume that $u\in W^{1,1}(\Omega)$ is a
weak solution to (${\mathcal{H}}$-PMC). A standard variational argument,
together with [66, Theorem 3.2] shows that

(7.1)
$\int_{\tilde{\Omega}}|Du+X|+\int_{\tilde{\Omega}}Hu\,dz\leqslant\int_{\tilde{\Omega}}|Dv+X|+\int_{\tilde{\Omega}}Hv\,dz$

for any open set $\tilde{\Omega}\Subset\Omega$ and any $v\in BV_{loc}(\Omega)$
such that $\\{u\neq v\\}\Subset\tilde{\Omega}$, where, following [66], we have
set

(7.2)
$\int_{\tilde{\Omega}}|Dv+X|:=P_{\mathbb{H}}(E_{v},\tilde{\Omega}\times{\mathbb{R}})$

for any open set $\tilde{\Omega}\Subset\Omega$ and any $v\in
BV_{loc}(\Omega)$. A function $u\in BV_{loc}(\Omega)$ satisfying (7.1) is
called a _$H$ -minimizer for $P_{\mathbb{H}}$_. This definition is motivated
by the fact that, arguing as in the proof of Theorem 5.5 (cf. also [66,
Theorem 3.15] and [66, Corollary 3.16]), the subgraph $E_{u}$ of an
$H$-minimizer satisfies

(7.3) $P_{\mathbb{H}}(E_{u},A)+\int_{E_{u}\cap A}H\,dz\leqslant
P_{\mathbb{H}}(F,A)+\int_{F\cap A}H\,dz$

for any open set $A\Subset\Omega\times{\mathbb{R}}$ and any measurable set $F$
such that $E\Delta F\Subset A$. On the other hand, a truncation argument as in
[41, Theorem 14.8] implies that if $E_{u}$ satisfies (7.3), then $u$ is an
$H$-minimizer for $P_{\mathbb{H}}$. We stress that, in light of [66, Theorem
1.2], the sub-Riemannian area functional in (7.2) is finite. Therefore, for
any $\varphi\in L^{1}(\partial\Omega)$, we define the functional
$\mathcal{I}_{\mathbb{H}}:BV(\Omega)\to{\mathbb{R}}$ by

$\mathcal{I}_{\mathbb{H}}(v)=P_{\mathbb{H}}(E_{v},\Omega\times{\mathbb{R}})+\int_{\Omega}Hudz+\int_{\partial\Omega}|v-\varphi|d\mathcal{H}^{2n-1}.$

######  Proposition 7.1.

Let $H\in L^{\infty}(\Omega)$ and assume that (* ‣ 1) and (*** ‣ 5.1) hold.
Then $\mathcal{I}_{\mathbb{H}}$ has a minimum in $BV(\Omega)\cap
L^{\infty}_{loc}(\Omega)$ for every $\varphi\in L^{1}(\partial\Omega)$.

###### Proof.

Let $B\subseteq{\mathbb{R}}^{2n}$ be a ball containing $\Omega$ such that the
Euclidean distance between $\partial\Omega$ and $\partial B$ is positive, and
extend $H$ to $B$ by letting $H\equiv 0$ outside $\Omega$. As done in the
proof of Proposition 5.3, minimizing $\mathcal{I}_{\mathbb{H}}$ is equivalent
to minimize

$\mathcal{J}_{\mathbb{H}}(v)=P_{\mathbb{H}}(E_{v},\Omega\times{\mathbb{R}})+Var(v,B\setminus\Omega)+\int_{B}Hvdz$

in $K=\\{v\in BV(B):v=\phi\text{ in }B\setminus\Omega\\}$, where $\phi$ is a
fixed function in $W_{0}^{1,1}(B)$ with trace $\varphi$ on $\partial\Omega$.
Let $(\varepsilon_{j})_{j}\subseteq(0,1)$ be such that
$\varepsilon_{j}\searrow 0$ as $j\to\infty$. Let $v_{j}$ be a minimizer of the
functional $\mathcal{J}_{\varepsilon_{j}}$ defined in (5.3). By (5.8) and
(2.7), we have

$\delta
Var(v_{j},B)\leqslant\mathcal{J}_{\varepsilon}(v_{j})+\tilde{C}\leqslant\mathcal{J}_{\varepsilon_{j}}(\phi)+\tilde{C}\leqslant
C^{\prime}P(E_{\phi},\Omega\times{\mathbb{R}})+Var(\phi,B\setminus\Omega)+\int_{B}H\phi
dz+\tilde{C},$

where
$\tilde{C}=\int_{\Omega}|X|dz+\int_{\partial\Omega}|\varphi|d\mathcal{H}^{2n-1}$
and $C^{\prime}=C^{\prime}(\Omega)$. Hence, arguing as in Proposition 5.3
$(v_{j})_{j}$ is bounded in $BV(B)$ and it converges in $L^{1}(B)$ to
$v_{0}\in K$. By (2.9), $v_{0}$ is a minimizer of $\mathcal{J}_{\mathbb{H}}$,
whence $\mathcal{I}_{\mathbb{H}}$ has a minimizer in $BV(\Omega)$. Finally,
the same arguments of Section 5.2 can be carried out thanks to the sub-
Riemannian density estimates (c.f. [56]) to prove that $u\in
L^{\infty}_{loc}(\Omega)$. ∎

As in the Riemannian setting, in the extremal case (** ‣ 1) we rely again on
the notion of generalized solution.

######  Definition 7.2.

A measurable function $u:\Omega\longrightarrow[-\infty,+\infty]$ is a
_generalized $H$-minimizer_ for $P_{\mathbb{H}}$ on $\Omega\times{\mathbb{R}}$
if (7.3) holds.

If $N_{+}$ and $N_{-}$ are defined as in (5.35), in view of (2.8) the
following analogous to (5.16) holds.

######  Proposition 7.3.

Let $u$ be a generalized $H$-minimizer. Then

(7.4) $P(N_{\pm},A)\pm\int_{N_{\pm}\cap A}H\,dz\leqslant P(F,A)\pm\int_{F\cap
A}H\,dz$

for any open set $A\Subset\Omega$ and any measurable set $F\subseteq\Omega$
such that $N_{+}\Delta F\Subset A$.

Before proving Theorem 1.5, we need the following compactness argument, whose
proof follows the approach of [50].

######  Theorem 7.4.

Let $(\varepsilon_{j})_{j}\subseteq(0,1)$ be such that
$\varepsilon_{j}\searrow 0$ as $j\to\infty$. Let $(\Omega_{j})_{j}$ be a
sequence of open sets such that $\Omega_{j}\Subset\Omega_{k}\Subset\Omega$ for
any $j<k$ and $\bigcup_{j=0}^{\infty}\Omega_{j}=\Omega$. Let
$(H_{j})_{j}\subseteq L^{\infty}(\Omega)$ be such that $H_{j}\to H$ uniformly
on $\Omega$. For any $j\in\mathbb{N}$, assume that $u_{j}\in
BV_{loc}(\Omega_{j})$ is such that $E_{u_{j}}$ is an $H_{j}$-minimizer for
$P_{\varepsilon_{j}}$ on $\Omega_{j}\times{\mathbb{R}}$ (cf. Definition 5.4).
Then, up to a subsequence, $(u_{j})_{j}$ converges almost everywhere to a
generalized $H$-minimizer for $P_{\mathbb{H}}$ on $\Omega\times{\mathbb{R}}$.
If in addition $E_{u}$ is a Caccioppoli set, then

(7.5)
$P_{\varepsilon_{j}}(E_{u_{j}},\cdot)\rightharpoonup^{*}P_{\mathbb{H}}(E_{u},\cdot)$

locally on $\Omega\times{\mathbb{R}}$.

###### Proof.

We set $E_{j}=E_{u_{j}}$ and $P_{\varepsilon_{j}}=P_{j}$ for any
$j\in\mathbb{N}$. First we show that there exists an $\mathbb{H}$-Caccioppoli
set $E$ on $\Omega\times{\mathbb{R}}$ such that, up to a subsequence,

(7.6) $\chi_{E_{j}}\to\chi_{E}$

in $L^{1}(A^{\prime})$ for any open set
$A^{\prime}\Subset\Omega\times{\mathbb{R}}$. Since $\bar{A}^{\prime}$ is
compact, there exists $k\in\mathbb{N}$, $p_{1},\ldots,p_{k}\in A^{\prime}$,
$r_{1},\ldots,r_{k}>0$ and $\tilde{\Omega}\Subset\Omega$ such that

$\bar{A}^{\prime}\Subset\bigcup_{i=1}^{k}B(p_{i},r_{i})\Subset\tilde{\Omega}\times{\mathbb{R}}.$

Therefore, in view of Proposition 2.1 and (5.18), we have

$\begin{split}P_{\mathbb{H}}\left(E_{j},\bigcup_{i=1}^{k}B(p_{i},r_{i})\right)\leqslant
P_{\varepsilon_{j}}\left(E_{j},\bigcup_{i=1}^{k}B(p_{i},r_{i})\right)\leqslant\sum_{i=1}^{k}P_{\varepsilon_{j}}(E_{j},B(p_{i},r_{i}))\leqslant
c_{2}\sum_{i=1}^{k}r_{i}^{2n}.\end{split}$

Hence we can apply Proposition 2.2 to infer the existence of a finite
$\mathbb{H}$-perimeter set $F$ in $\bar{A}^{\prime}$ such that, up to a
subsequence, $\chi_{E_{j}}\to\chi_{F}$ in $L^{1}(A^{\prime})$. Taking a
sequence of relative compact sets that covers $\Omega\times{\mathbb{R}}$ and
using a standard diagonal argument, (7.6) follows. Thanks to [41, Lemma 16.3],
$E=E_{u}$ for a measurable function
$u:\Omega\longrightarrow[-\infty,+\infty]$, and moreover, up to a subsequence,
$u_{j}\to u$ almost everywhere on $\Omega$. Arguing as above,

$\sup_{j\in\mathbb{N}}P_{j}(E_{j},K)<\infty$

for any $K\Subset\Omega\times{\mathbb{R}}$. Therefore, [5, Theorem 1.59]
implies the existence of a $(2n+1)$-valued Radon measure $\mu$ on
$\Omega\times{\mathbb{R}}$ and a scalar Radon measure $\lambda$ on
$\Omega\times{\mathbb{R}}$ such that, up to a subsequence,

(7.7) $D_{j}\chi_{E_{j}}\rightharpoonup^{*}\mu\qquad\text{and}\qquad
P_{j}(E_{j},\cdot)\rightharpoonup^{*}\lambda$

locally on $\Omega\times{\mathbb{R}}$ as $j\to\infty$. We claim that

(7.8) $\mu=(D_{\mathbb{H}}\chi_{E},0).$

Indeed, fix $K\Subset\Omega\times{\mathbb{R}}$ and $g\in
C^{1}_{c}(K,{\mathbb{R}}^{2n+1})$, and set

$V_{j}=\sum_{i=1}^{n}(g_{i}X_{i}+g_{n+i}Y_{i})+g_{2n+1}\varepsilon_{j}T=:V+g_{2n+1}\varepsilon_{j}T.$

Then, since $\chi_{E_{j}}\to\chi_{E}$ in
$L^{1}_{loc}(\Omega\times{\mathbb{R}})$, we infer that

$\begin{split}\left|\int_{K}g\cdot dD_{j}\chi_{E_{j}}-\int_{K}g\cdot
d(D_{\mathbb{H}}\chi_{E},0)\right|&=\left|\int_{K\cap
E_{j}}\operatorname{div}V_{j}\,dx-\int_{K\cap
E}\operatorname{div}V\,dx\right|\\\ &\leqslant\int_{K\cap(E\Delta
E_{j})}|\operatorname{div}V|\,dx+\varepsilon_{j}\int_{K}\left|\frac{\partial
g_{2n+1}}{\partial t}\right|\,dx\\\ &\leqslant|K\cap(E\Delta
E_{j})|\|\operatorname{div}V\|_{L^{\infty}(K)}+\varepsilon_{j}|K|\left\|\frac{\partial
g_{2n+1}}{\partial t}\right\|_{L^{\infty}(K)}\\\ &\to 0\end{split}$

as $j\to\infty$. An easy approximation argument allows to extend the previous
convergence to any $g\in C^{0}_{c}(K,{\mathbb{R}}^{2n+1})$, thus proving
(7.8). Therefore, combining (7.7), (7.8) and [5, Proposition 1.62], we
conclude that

(7.9) $P_{\mathbb{H}}(E,\cdot)\leqslant\lambda(\cdot).$

Let $A$ be an open set such that $A\Subset\Omega\times{\mathbb{R}}$, and let
$F$ be a Caccioppoli set on $\Omega\times{\mathbb{R}}$ such that $F\Delta
E\Subset A$. We claim that

(7.10) $P_{\mathbb{H}}(E,A)+\int_{E\cap A}H\,dx\leqslant
P_{\mathbb{H}}(F,A)+\int_{F\cap A}H\,dx.$

For any $j\in\mathbb{N}$, Let $A_{j}$ be an open set with Lipschitz boundary
such that $E\Delta F\Subset A_{0}\Subset A_{j}\Subset A_{j+1}\Subset A$ and
$\bigcup_{j\in\mathbb{N}}A_{j}=A$. Up to a further subsequence, we assume that
$\chi_{E_{j}}\to\chi_{E}$ almost everywhere on $\Omega\times{\mathbb{R}}$.
This fact allows to choose $(A_{j})_{j}$ in such a way that

(7.11) $\chi_{E_{j}}\to\chi_{E}$

$\mathcal{H}^{2n}$-almost everywhere on $\partial A_{j}$. Moreover, being
$E_{j}$ and $F$ are Caccioppoli sets,

(7.12) $\mathcal{H}^{2n}(\partial^{*}E_{j}\cap\partial
A_{j})=0\qquad\text{and}\qquad\mathcal{H}^{2n}(\partial^{*}F\cap\partial
A_{j})=0$

for any $j\in\mathbb{N}$. For any $j\in\mathbb{N}$, we define $F_{j}:=(F\cap
A_{j})\cup(E_{j}\setminus A_{j}).$ It is clear that $F_{j}$ is a Caccioppoli
set such that $F_{j}\Delta E_{j}\Subset A_{j}\Subset A$. Therefore in
particular

(7.13) $F_{j}\cap(A_{j}\setminus A_{0})=E_{j}\cap(A_{j}\setminus A_{0}).$

Notice that, thanks to (2.7),(7.11) and (7.12) and arguing as in [50, Theorem
21.14],

(7.14) $\lim_{j\to\infty}P_{j}(F_{j},\partial A_{j})=0.$

We are able to prove (7.10). Indeed, exploiting the $H$-minimality of $E_{j}$
for $P_{j}$, together with (7.13), we see that

$\begin{split}P_{j}(E_{j},A)+\int_{E_{j}\cap A}H_{j}\,dx&\leqslant
P_{j}(F_{j},A)+\int_{F_{j}\cap A}H_{j}\,dx\\\
&=P_{j}(F_{j},A_{j})+P_{j}(F_{j},\partial
A_{j})+P_{j}(F_{j},A\setminus\overline{A_{j}})+\int_{F_{j}\cap A}H_{j}\,dx\\\
&=P_{j}(F,A_{j})+P_{j}(F_{j},\partial
A_{j})+P_{j}(E_{j},A\setminus\overline{A_{j}})+\int_{F_{j}\cap A}H_{j}\,dx\\\
&\leqslant P_{j}(F,A)+P_{j}(F_{j},\partial
A_{j})+P_{j}(E_{j},A\setminus\overline{A_{j}})+\int_{F_{j}\cap
A}H_{j}\,dx,\end{split}$

which implies that

$P_{j}(E_{j},A_{j})+\int_{E_{j}\cap A}H_{j}\,dx\leqslant
P_{j}(F,A)+P_{j}(F_{j},\partial A_{j})+\int_{F_{j}\cap A}H_{j}\,dx.$

Therefore, exploiting Proposition 2.1 together with (7.6) and (7.13), we can
pass to the limit and obtain

$\lambda(A)+\int_{E\cap A}H\,dx\leqslant P_{\mathbb{H}}(F,A)+\int_{F\cap
A}H\,dx.$

Notice that (7.9) allows to achieve (7.10). Finally, if $E_{u}$ is a
Caccioppoli set, then we can choose $F=E$ in the previous inequality, so that,
recalling (7.9), (7.5) follows. ∎

###### Proof of Theorem 1.5.

Being the non-extremal case already covered by (7.1), we can assume (** ‣ 1).
Let $(\varepsilon_{j})_{j}$ and $(\Omega_{j})_{j}$ be as in the statement of
Theorem 7.4. Assume in addition that $P(\Omega_{j})\to P(\Omega)$ as
$j\to\infty$. Assume that, for any $j\in\mathbb{N}$, $\Omega_{j}$ has
Lipschitz boundary. By hypothesis

$\Big{|}\int_{\tilde{\Omega}_{j}}H\Big{|}<P(\Omega_{j})$

for any $j\in\mathbb{N}$. Arguing as in the proof of Theorem 1.3, there exists
a sequence $(H_{j})_{j}\subseteq C^{\infty}(\overline{\Omega})$ such that

$H_{j}\to H\text{ uniformly on
}\Omega\qquad\text{and}\qquad\|H_{j}\|_{C^{1}(\Omega)}\leqslant
H_{0}+\|H\|_{L^{\infty}(\Omega)}+1$

for any $j\in\mathbb{N}$, being $H_{0}$ the Lipschitz constant of $H$ on
$\Omega$. Arguing again as in the proof of Theorem 1.3, up to a subsequence
each $H_{j}$ satisfies the hypotheses of Proposition 5.3 on $\Omega_{j}$.
Hence, for any $j\in\mathbb{N}$, there exists $u_{j}\in BV(\Omega_{j})$ which
minimizes $\mathcal{I}_{\varepsilon_{j}}$, with source $H_{j}$, for any
$j\in\mathbb{N}$. In view of Theorem 5.5, $E_{u_{j}}$ is an $H_{j}$-minimizer
for $P_{\varepsilon_{j}}$ on $\Omega_{j}\times{\mathbb{R}}$. Arguing as in the
proof of Theorem 1.1, Theorem 7.4 implies the existence of a generalized
$H$-minimizer for $P_{\mathbb{H}}$ on $\Omega$ as in Definition 7.2 such that,
up to vertical translations, $u_{j}\to u$ almost everywhere on $\Omega$.
Moreover, arguing again as in Theorem 1.1, in view of Proposition 7.3 and
exploiting sub-Riemannian volume density estimates as in [56], we conclude
that $u\in L^{\infty}_{loc}(\Omega)$. Hence, being $E_{u}$ an $H$-Caccioppoli
set, then [66, Theorem 1.2] implies that $u\in BV_{loc}(\Omega)$. In this
case, $E_{u}$ is a Caccioppoli set, so that (7.5) follows. ∎

###  7.2. An application to the sub-Riemannian second variation formula

Let $S=\partial E$ be a smooth non-characteristic hypersurface, let
$A\subseteq{\mathbb{H}}^{n}$ be an open set such that $A\cap S\neq\emptyset$,
and let $\xi\in C^{\infty}_{c}(A)$. Then it is well-known (cf. e.g. [53, 54])
that

$\frac{d^{2}}{dt^{2}}|\partial
E_{t}|_{\mathbb{H}}(A)\Big{|}_{t=0}=\int_{S}\left(|\nabla^{\mathbb{H},S}\xi|^{2}-\xi^{2}\left(q-(H^{\mathbb{H}})^{2}\right)\right)\,d|\partial
E|_{\mathbb{H}},$

where

(7.15)
$q=\sum_{h,k=1}^{2n}Z_{h}(\nu^{\mathbb{H}}_{k})Z_{k}(\nu^{\mathbb{H}}_{h})+4\langle
J(\nu^{\mathbb{H}}),\nabla^{\mathbb{H}}(Td^{\mathbb{H}})\rangle+4n(Td^{\mathbb{H}})^{2}$

and where by $E_{t}$ we mean a smooth variation along the vector field
$\xi\nu^{\mathbb{H}}$. Observe that $q$ does not depend on the chosen unitary
extension of $\nu^{\mathbb{H}}|_{S}$. Here $d|\partial E|_{\mathbb{H}}$ is the
horizontal perimeter measure, $d^{\mathbb{H}}$ is the horizontal Carnot-
Carathéodory distance defined in (2.4), $\nabla^{\mathbb{H}}$ is the
_horizontal gradient_ , $\nabla^{\mathbb{H},S}$ is the _horizontal tangential
gradient_ of $S$ and $H^{\mathbb{H}}$ is the _horizontal mean curvature_ of
$S$ (cf. [23]). In the following theorem, we recover the interpretation of $q$
given in [65, 33] as the limit of
$|h^{\varepsilon}|^{2}+\operatorname{Ric}_{\varepsilon}(\nu^{\varepsilon})$ as
$\varepsilon\to 0$, using the fact that the family of Riemannian structures
$(\mathbb{H}^{n},g_{\varepsilon})$ approximate in a suitable sense the sub-
Riemannian manifold $(\mathbb{H}^{n},\langle\cdot,\cdot\rangle$) (cf. [14]).
Notice that both terms $|h^{\varepsilon}|^{2}$ and
$\operatorname{Ric}_{\varepsilon}(\nu^{\varepsilon})$ diverge, while the sum
is controlled thanks to (3.7).

######  Theorem 7.5.

Let $S=\partial E$ be a hypersurface of class $C^{3}$. Then

$\lim_{\varepsilon\to
0}\left(|h^{\varepsilon}|^{2}+\operatorname{Ric}_{\varepsilon}(\nu^{\varepsilon})\right)=q$

locally uniformly in the non-characteristic part of $S$, where $q$ is as in
(7.15).

###### Proof.

It suffices to observe that

$\nu^{\varepsilon}=\frac{\nu^{\mathbb{H}}}{\sqrt{1+\varepsilon^{2}(Td^{\mathbb{H}})^{2}}}+\frac{\varepsilon
Td^{\mathbb{H}}}{\sqrt{1+\varepsilon^{2}(Td^{\mathbb{H}})^{2}}}\varepsilon T.$

Therefore, noticing that

$\lim_{\varepsilon\to
0}\nu^{\varepsilon}_{j}=\nu^{\mathbb{H}}_{j}\qquad\text{and}\qquad\lim_{\varepsilon\to
0}\frac{\nu^{\varepsilon}_{2n+1}}{\varepsilon}=Td^{\mathbb{H}}$

locally uniformly for any $j=1,\ldots,2n$, and moreover

$\lim_{\varepsilon\to
0}Z_{i}\nu^{\varepsilon}_{j}=Z_{i}\nu^{\mathbb{H}}_{j}\qquad\text{and}\qquad\lim_{\varepsilon\to
0}Z_{2n+1}\nu^{\varepsilon}_{k}=\lim_{\varepsilon\to
0}Z_{k}\nu^{\varepsilon}_{2n+1}=0$

locally uniformly for any $i,j=1,\ldots,2n$ and $k=1,\ldots,2n+1$, the thesis
follows from (3.7). ∎

##  8\. Proofs of Section 2 and Section 3

###### Proof of (2.3).

Fix $j=1,\ldots,n$ and $i=1,\ldots,2n+1$. From (2.2), it easily follows that

$\begin{split}&g_{\varepsilon}\big{(}\nabla_{Z_{i}}U,X_{j}\big{)}=Z_{i}u_{j}-\frac{1}{\varepsilon}(\delta_{i,j+n}u_{2n+1}+\delta_{i,2n+1}u_{j+n})\\\
&g_{\varepsilon}\big{(}\nabla_{U}U,X_{j}\big{)}=Uu_{j}-\frac{2}{\varepsilon}u_{n+j}u_{2n+1}\\\
&g_{\varepsilon}\big{(}\nabla_{X_{j}}U,\nabla_{U}X_{j}\big{)}=\frac{1}{\varepsilon}X_{j}\big{(}u_{n+j}u_{2n+1}\big{)}+\frac{1}{\varepsilon^{2}}\big{(}u^{2}_{2n+1}-u_{n+j}^{2}\big{)}.\end{split}$

Hence

$\begin{split}g_{\varepsilon}\left(\nabla_{X_{j}}\nabla_{U}U,X_{j}\right)&=X_{j}g_{\varepsilon}\big{(}\nabla_{U}U,X_{j}\big{)}=X_{j}(Uu_{j})-\frac{2}{\varepsilon}X_{j}\big{(}u_{n+j}u_{2n+1}\big{)}\\\
-g_{\varepsilon}\left(\nabla_{U}\nabla_{X_{j}}U,X_{j}\right)&=-Ug_{\varepsilon}\left(\nabla_{X_{j}}U,X_{j}\right)+g_{\varepsilon}\left(\nabla_{X_{j}}U,\nabla_{U}X_{j}\right)\\\
&=-U(X_{j}u_{j})+\frac{1}{\varepsilon}X_{j}\big{(}u_{n+j}u_{2n+1}\big{)}+\frac{1}{\varepsilon^{2}}\big{(}u^{2}_{2n+1}-u_{n+j}^{2}\big{)}.\end{split}$

Moreover, from

$[U,X_{j}]=\sum_{k=1}^{2n+1}u_{k}[Z_{k},X_{j}]-X_{j}(u_{k})Z_{k}=2u_{n+j}T-\sum_{k=1}^{2n+1}X_{j}(u_{k})Z_{k},$

we get

$\begin{split}g_{\varepsilon}\left(\nabla_{[U,X_{j}]}U,X_{j}\right)&=\frac{2}{\varepsilon}u_{n+j}g_{\varepsilon}\left(\nabla_{\varepsilon
T}U,X_{j}\right)-\sum_{k=1}^{2n+1}X_{j}(u_{k})g_{\varepsilon}\left(\nabla_{Z_{k}}U,X_{j}\right)\\\
&=2u_{n+j}T(u_{j})-\frac{2}{\varepsilon^{2}}u_{n+j}^{2}-\sum_{k=1}^{2n+1}X_{j}(u_{k})Z_{k}(u_{j})+\frac{1}{\varepsilon}X_{j}\big{(}u_{n+j}u_{2n+1}\big{)}\\\
&=[U,X_{j}](u_{j})-\frac{2}{\varepsilon^{2}}u_{n+j}^{2}+\frac{1}{\varepsilon}X_{j}\big{(}u_{n+j}u_{2n+1}\big{)},\end{split}$

so that

$g_{\varepsilon}\left(\nabla_{X_{j}}\nabla_{U}U-\nabla_{U}\nabla_{X_{j}}U+\nabla_{[U,X_{j}]}U,X_{j}\right)=\frac{1}{\varepsilon^{2}}(u_{2n+1}^{2}-3u_{n+j}^{2}).$

A similar computation shows that

$\begin{split}&g_{\varepsilon}\left(\nabla_{Y_{j}}\nabla_{U}U-\nabla_{U}\nabla_{Y_{j}}U+\nabla_{[U,Y_{j}]}U,Y_{j}\right)=\frac{1}{\varepsilon^{2}}(u_{2n+1}^{2}-3u_{j}^{2})\\\
&g_{\varepsilon}\left(\nabla_{\varepsilon
T}\nabla_{U}U-\nabla_{U}\nabla_{\varepsilon T}U+\nabla_{[U,\varepsilon
T]}U,\varepsilon
T\right)=\frac{1}{\varepsilon^{2}}\sum_{j=1}^{n}(u_{j}^{2}+u_{n+j}^{2}).\end{split}$

In view of the previous computations, (2.3) follows. ∎

###### Proof of (3.6).

Consider the extension of $\nu^{\varepsilon}$ to a neighborhood of $S$ as
$\nu^{\varepsilon}=\nabla d$. By a continuity argument and since $S\setminus
S_{0}$ is dense in $S$, it suffices to show (3.6) in the non-characteristic
part of $S$. Let us fix $p\in S\setminus S_{0}$. It is well known (cf. e.g.
[18]) that there exist a local orthonormal frame
$e_{1},\ldots,e_{n-1},e_{n+1},\ldots,e_{2n-1}$ of $\mathcal{H}\cap T_{p}S$
such that $e_{n+j}=J(e_{j})$ at $p$ for any $j=1,\ldots,n-1$. We consider the
extension to a local orthonormal frame of $T_{p}S$ by letting

$e_{n}=\frac{1}{\sqrt{1-(\nu^{\varepsilon}_{2n+1})^{2}}}J(\nu^{\varepsilon})\qquad\text{and}\qquad
e_{2n}=-\frac{\nu^{\varepsilon}_{2n+1}}{\sqrt{1-(\nu^{\varepsilon}_{2n+1})^{2}}}(\nu^{\varepsilon})_{h}+\sqrt{1-(\nu^{\varepsilon}_{2n+1})^{2}}\varepsilon
T,$

where
$(\nu^{\varepsilon})_{h}=\nu^{\varepsilon}-\nu^{\varepsilon}_{2n+1}\varepsilon
T$ is the horizontal projection. Moreover, for any $i,j=1,\ldots,2n$ we set
$e_{i}=\sum_{k=1}^{2n+1}\alpha^{i}_{k}Z_{k}$. From our choice it follows that

(8.1) $1=-\langle
e_{i},J(e_{i+n})\rangle=\sum_{k=1}^{n}(\alpha^{n+i}_{n+k}\alpha^{i}_{k}-\alpha^{n+i}_{k}\alpha^{i}_{n+k})$

for any $i=1,\ldots,n-1$. We set

$a_{i,j}=-\sum_{l,s=1}^{2n+1}\alpha^{i}_{s}\alpha^{j}_{l}Z_{s}(\nu^{\varepsilon}_{l}).$

Using (2.2), it follows that

$\begin{split}&g_{\varepsilon}\left(\nabla_{e_{i}}e_{n+i},\nu^{\varepsilon}\right)=a_{i,n+i}-\frac{\nu^{\varepsilon}_{2n+1}}{\varepsilon}\qquad\qquad
g_{\varepsilon}\left(\nabla_{e_{n}}e_{2n},\nu^{\varepsilon}\right)=a_{n,2n}-\frac{1}{\varepsilon}\\\
&g_{\varepsilon}\left(\nabla_{e_{n+i}}e_{i},\nu^{\varepsilon}\right)=a_{n+i,i}+\frac{\nu^{\varepsilon}_{2n+1}}{\varepsilon}\qquad\qquad
g_{\varepsilon}\left(\nabla_{e_{2n}}e_{n},\nu^{\varepsilon}\right)=a_{2n,n}-\frac{1}{\varepsilon}+\frac{2(\nu^{\varepsilon}_{2n+1})^{2}}{\varepsilon}\end{split}$

for any $i=1,\ldots,2n-1$, and

$g_{\varepsilon}\left(\nabla_{e_{i}}e_{j},\nu^{\varepsilon}\right)=a_{i,j}$

for any $i,j=1,\ldots,2n,\ |i-j|\neq n$. Writing

$\begin{split}b_{i,n+i}&=-\frac{\nu^{\varepsilon}_{2n+1}}{\varepsilon}\qquad\qquad\qquad
b_{n,2n}=-\frac{1}{\varepsilon},\qquad\\\
b_{n+i,i}&=\frac{\nu^{\varepsilon}_{2n+1}}{\varepsilon}\qquad\qquad\qquad
b_{2n,n}=-\frac{1}{\varepsilon}+\frac{2(\nu^{\varepsilon}_{2n+1})^{2}}{\varepsilon}\end{split}$

for any $i=1,\ldots,n-1$, and $b_{i,j}=0$ for any $i,j=1,\ldots,2n$ with
$|i-j|\neq n$, it follows that

(8.2)
$|h_{p}|^{2}=\sum_{i,j=1}^{2n}h_{i,j}h_{j,i}=\sum_{i,j=1}^{2n}a_{i,j}a_{j,i}+2\sum_{i=1}^{n}(a_{i,n+i}b_{n+i,i}+a_{n+i,i}b_{i,n+i})+2\sum_{i=1}^{n}b_{i,n+i}b_{n+i,i},$

where $h_{i,j}$ are the entries of $h$ with respect to $e_{1},\ldots,e_{2n}$.
First, by (3.1) and arguing as in [57],

(8.3)
$\begin{split}\sum_{i,j=1}^{2n}a_{i,j}a_{j,i}=\sum_{l,s=1}^{2n+1}Z_{s}(\nu^{\varepsilon}_{l})Z_{l}(\nu^{\varepsilon}_{s}).\end{split}$

Moreover,

(8.4)
$\sum_{i=1}^{n}b_{i,n+i}b_{n+i,i}=-(n-1)\frac{(\nu^{\varepsilon}_{2n+1})^{2}}{\varepsilon^{2}}+\frac{1}{\varepsilon}\left(\frac{1}{\varepsilon}-\frac{2(\nu^{\varepsilon}_{2n+1})^{2}}{\varepsilon}\right)=-(n+1)\frac{(\nu^{\varepsilon}_{2n+1})^{2}}{\varepsilon^{2}}+\frac{1}{\varepsilon^{2}}.$

Finally,

(8.5)
$\sum_{i=1}^{n}(a_{i,n+i}b_{n+i,i}+a_{n+i,i}b_{i,n+i})=\frac{\nu^{\varepsilon}_{2n+1}}{\varepsilon}\sum_{i=1}^{n-1}\left(a_{i,n+i}-a_{n+i,i}\right)-\frac{1}{\varepsilon}\big{(}a_{2n,n}+(1-2(\nu^{\varepsilon}_{2n+1})^{2})a_{n,2n}\big{)}.$

From one hand, in view of (3.1), (3.2), (3.3) and (8.1),

(8.6)
$\begin{split}\frac{\nu^{\varepsilon}_{2n+1}}{\varepsilon}\sum_{i=1}^{n-1}\left(a_{i,n+i}-a_{n+i,i}\right)&=\frac{\nu^{\varepsilon}_{2n+1}}{\varepsilon}\sum_{i=1}^{n-1}\left(\sum_{h,k=1}^{2n}\alpha^{n+i}_{k}\alpha^{i}_{h}Z_{k}(\nu^{\varepsilon}_{h})-\sum_{h,k=1}^{2n}\alpha^{n+i}_{k}\alpha^{i}_{h}Z_{h}(\nu^{\varepsilon}_{k})\right)\\\