# Multiplicity results for mass constrained Allen-Cahn equations on Riemannian
manifolds with boundary

D. Corona, S. Nardulli, R. Oliver-Bonafoux, G. Orlandi and P. Piccione

###### Abstract

We present multiplicity results for mass constrained Allen-Cahn equations on a
Riemannian manifold with boundary, considering both Neumann and Dirichlet
conditions. These results hold under the assumptions of small mass constraint
and small diffusion parameter. We obtain lower bounds on the number of
solutions according to the Lusternik–Schnirelmann category of the manifold in
case of Dirichlet boundary conditions and of its boundary in the case of
Neumann boundary conditions. Under generic non-degeneracy assumptions on the
solutions, we obtain stronger results based on Morse inequalities. Our
approach combines topological and variational methods with tools from
Geometric Measure Theory.

_2020 Mathematics Subject Classification:_ 35A15, 49Q20, 58J32.

_Key words and phrases:_ Variational methods, $\Gamma$-convergence,
isoperimetric-type problems, Allen-Cahn equations, manifolds with boundary.

##  1 Introduction

Let $(M,g)$ be a smooth, compact $n$–dimensional Riemannian manifold with
boundary, for $n\geq 2$. Let
$W\in\mathcal{C}^{3}_{\mathrm{loc}}({\mathbb{R}},[0,+\infty))$ a double-well
potential such that $W^{-1}(0)=\\{0,1\\}$. For fixed $\varepsilon,m>0$,
consider the following Allen-Cahn type equations with a mass constraint:

$\begin{dcases}-\varepsilon\Delta
u_{\varepsilon,m}+\frac{1}{\varepsilon}W^{\prime}(u_{\varepsilon,m})=\lambda_{\varepsilon,m},&\mbox{
on }M,\\\ \int_{M}u_{\varepsilon,m}\mathrm{d}v_{g}=m,&\end{dcases}$ (1.1)

where the unknown parameter $\lambda_{\varepsilon,m}\in{\mathbb{R}}$ is a
Lagrange multiplier related to the mass constraint
$\int_{M}u_{\varepsilon,m}\mathrm{d}v_{g}=m$. We study (1.1) under both
Neumann and Dirichlet boundary conditions, which leads respectively to the two
following problems:

$\begin{dcases}-\varepsilon\Delta
u_{\varepsilon,m}+\frac{1}{\varepsilon}W^{\prime}(u_{\varepsilon,m})=\lambda_{\varepsilon,m},&\mbox{
on }M,\\\ \int_{M}u_{\varepsilon,m}=m,&\\\ \frac{\partial
u_{\varepsilon,m}}{\partial\nu}=0,&\mbox{ on }\partial M,\end{dcases}$ (1.2)

where $\nu\in TM\big{|}_{\partial M}$ is the unit inner normal vector to
$\partial M$ in $M$, and

$\begin{dcases}-\varepsilon\Delta
u_{\varepsilon,m}+\frac{1}{\varepsilon}W^{\prime}(u_{\varepsilon,m})=\lambda_{\varepsilon,m},&\mbox{
on }M,\\\ \int_{M}u_{\varepsilon,m}=m,&\\\ u_{\varepsilon,m}=0,&\mbox{ on
}\partial M.\end{dcases}$ (1.3)

A straightforward computation shows that solutions of the Neumann problem
(1.2) can be found among the critical points of the functional

$\mathcal{E}_{\varepsilon}(u)\coloneqq\int_{M}\left(\varepsilon\,\frac{\lvert\nabla
u\rvert^{2}}{2}+\frac{1}{\varepsilon}W(u)\right)\mathrm{d}v_{g},$ (1.4)

in the function space

$\mathcal{H}_{m}\coloneqq\left\\{u\in
H^{1}(M,{\mathbb{R}}):\int_{M}u\,\mathrm{d}v_{g}=m\right\\},$

while critical points of the restriction of $\mathcal{E}_{\varepsilon}$ to the
subspace

$\mathcal{H}_{m,0}\coloneqq\left\\{u\in
H^{1}_{0}(M,{\mathbb{R}}):\int_{M}u\,\mathrm{d}v_{g}=m\right\\},$

are solutions of the Dirichlet problem (1.3).

The functional $\mathcal{E}_{\varepsilon}$ is widely used in the modelling of
phase transition phenomena. For small $\varepsilon>0$, critical points of
$\mathcal{E}_{\varepsilon}$ in $\mathcal{H}_{m}$ and $\mathcal{H}_{m,0}$
develop transition layers whose profiles are close to solutions of
isoperimetric-type problems. This link was first noticed (in the minimizing
case, using the ideas of $\Gamma$-convergence) in the seminal work of Modica
[40] in the case of Neumann boundary conditions and by Owen, Rubinstein and
Sternberg [44] in the case of Dirichlet boundary conditions. Extensions to
arbitrary critical points have been given by Hutchinson and Tonegawa [33].
Analogously, in the absence of a mass constraint critical points of Allen-Cahn
functionals are linked to the min-max theory of minimal surfaces (and, more
generally, submanifolds) which was initiated by Almgren [3, 4] and improved
later by Pitts [50]. This theory has been object of an extensive study in the
past years after the works of Marques and Neves (e.g., [38, 37]). In
particular, a min-max theory for isoperimetric-type surfaces was developed by
Zhou and Zhu [58]. As a consequence, there has been an increased and renewed
interest in the study of existence and multiplicity of critical points for
functionals of Allen-Cahn type and their asymptotic behavior as
$\varepsilon\to 0$, departing from results of Gaspar and Guaraco [29, 31]. In
the presence of a mass constraint, first studies in this direction have been
carried out in [14] in the case of manifolds without boundary, while related
vectorial problems have been recently studied in [5]. In a related direction,
Bellettini and Wickramaseckera [9] studied the existence of solutions for a
more general equation of Allen-Cahn-type associated to surfaces of prescribed
mean curvature.

The purpose of this paper is to extend the main results of [14] to the case of
manifolds with boundary. More precisely, we establish lower bounds on the
number of solutions for (1.2) and (1.3) in terms of certain topological
invariants of the underlying manifold. These lower bounds hold true for values
of the parameters $\varepsilon$ and $m$ which are relatively small. We believe
that these results on manifolds with boundary are of interest because they
include the case of Euclidean domains in ${\mathbb{R}}^{n}$ (the one
originally considered in classical works such as [40, 44]), which is left out
in the setting of manifolds without boundary. Moreover, they require a fine
understanding of relative isoperimetric-type problems for small volumes on
Riemannian manifolds with boundary, which actually depend heavily on the
boundary conditions (Dirichlet or Neumann) that are imposed. Overall,
significant differences appear when one moves from the setting without
boundary to the setting with boundary.

In order to state and prove our main results, we consider the following
typical assumptions on the potential $W$:

######  (H1).

The wells are non-degenerate global minimizers, i.e.
$W^{\prime\prime}(0),W^{\prime\prime}(1)>0$.

######  (H2).

The potential $W$ is coercive, i.e. there exist $R>0$ and $\alpha>0$ such that

$W^{\prime}(u)u\geq\alpha\lvert u\rvert^{2},\quad\text{ if }\lvert u\rvert\geq
R.$ (1.5)

######  (H3).

The potential $W$ has subcritical growth at infinity, i.e. there exist
$p\in[2,2^{*})$, $\alpha^{\prime}>0$ and $R^{\prime}>0$ such that

$\lvert W^{\prime\prime}(u)\rvert\leq\alpha^{\prime}\lvert
u\rvert^{p-2},\quad\text{ if }\lvert u\rvert\geq R^{\prime}.$

Here $2^{*}$ denotes the critical Sobolev exponent, i.e. $2^{*}=2n/(n-2)$ if
$n\geq 3$ and $2^{*}=+\infty$ if $n=2$.

For $n=2,3$, a standard example of function $W:{\mathbb{R}}\to[0,+\infty)$
satisfying (H1), (H2) and (H3) is the quartic potential:

$W_{q}(u)=u^{2}(u-1)^{2}.$

We will denote by $\mathcal{E}_{\varepsilon,m}$ the restriction of
$\mathcal{E}_{\varepsilon}$ to $\mathcal{H}_{m}$, and by
$\overline{\mathcal{E}}_{\varepsilon,m}$ the restriction to
$\mathcal{H}_{m,0}$, that is

$\mathcal{E}_{\varepsilon,m}\coloneqq\mathcal{E}_{\varepsilon}\big{|}_{\mathcal{H}_{m}}\quad\text{and}\quad\overline{\mathcal{E}}_{\varepsilon,m}\coloneqq\mathcal{E}_{\varepsilon}\big{|}_{\mathcal{H}_{m,0}}.$

Our multiplicity results for critical points of $\mathcal{E}_{\varepsilon,m}$
and $\overline{\mathcal{E}}_{\varepsilon,m}$ are established in term of
certain topological invariants of $M$ and $\partial M$ that we quickly recall
here, referring for instance to Benci [10] for more details on definitions.
Given a topological space $X$, denote by $\mathrm{cat}(X)$ its _Lusternik-
Schnirelmann category_ , i.e. the smallest number of sets, open and
contractible in $X$, needed to cover $X$ ($\mathrm{cat}(X)=+\infty$ in case
such number does not exist). Denote moreover by $\mathscr{P}_{1}(X)$ the sum
of the _Betti numbers_ of $X$ or, equivalently, the value at $1$ of the
_Poincaré polynomial_ of $X$.

The main results of the paper are the following ones.

######  Theorem A.

Assume that (H1), (H2) and (H3) hold. Then, there exists $m^{*}>0$ such that
for all $m\in(0,m^{*})$ there exists $\varepsilon_{m},c_{m}>0$ such that for
any $\varepsilon\in(0,\varepsilon_{m})$ the Neumann problem (1.2) has at least
$\mathrm{cat}(\partial M)$ solutions $u_{\varepsilon,m}$ with
$\mathcal{E}_{\varepsilon}(u_{\varepsilon,m})\leq c_{m}$ and at least one
solution $v_{\varepsilon,m}$ with
$\mathcal{E}_{\varepsilon}(v_{\varepsilon,m})>c_{m}$. Moreover, if $m$ and
$\varepsilon$ as above are such that all critical points of
$\mathcal{E}_{\varepsilon,m}$ are non-degenerate, then (1.2) has at least
$\mathscr{P}_{1}(\partial M)$ solutions $u_{\varepsilon,m}$ with
$\mathcal{E}_{\varepsilon}(u_{\varepsilon,m})\leq c_{m}$ and at least
$\mathscr{P}_{1}(\partial M)-1$ solutions $v_{\varepsilon,m}$ with
$\mathcal{E}_{\varepsilon}(v_{\varepsilon,m})>c_{m}$.

######  Theorem B.

Assume that (H1), (H2) and (H3) hold. Then, there exists $m^{*}>0$ such that
for all $m\in(0,m^{*})$ there exist $\varepsilon_{m},c_{m}>0$ such that for
any $\varepsilon\in(0,\varepsilon_{m})$ the Dirichlet problem (1.3) has at
least $\mathrm{cat}(M)$ solutions $u_{\varepsilon,m}$ with
$\mathcal{E}_{\varepsilon}(u_{\varepsilon,m})\leq c_{m}$ and, if
$\mathrm{cat}(M)>1$, there exists at least one solution $v_{\varepsilon,m}$
with $\mathcal{E}_{\varepsilon}(v_{\varepsilon,m})>c_{m}$. Moreover, if $m$
and $\varepsilon$ as above are such that all critical points of
$\overline{\mathcal{E}}_{\varepsilon,\varepsilon}$ are non-degenerate, then
(1.3) has at least $\mathscr{P}_{1}(M)$ solutions $u_{\varepsilon,m}$ with
$\mathcal{E}_{\varepsilon}(u_{\varepsilon,m})\leq c_{m}$ and at least
$\mathscr{P}_{1}(M)-1$ solutions $v_{\varepsilon,m}$ with
$\mathcal{E}_{\varepsilon}(v_{\varepsilon,m})>c_{m}$.

In section 5 we discuss some (essentially known) results which state that
solutions of (1.2) are non-degenerate in suitable generic situations. Roughly
speaking, given the equation (1.2), one can perform an arbitrary small
perturbation of a key parameter (more precisely, either the Riemannian metric
or the boundary of the domain) so that the resulting problem is such that all
solutions are non-degenerate for a dense set of $\varepsilon$ and $m$. That
is, the second parts of Theorems A and B, which are stronger than the first
ones, apply in almost all cases.

######  Remark 1.1.

Notice that while the lower bound given in Theorem B for the Dirichlet problem
(1.3) is formulated in term of the topology of the whole of $M$ (as in the
case without boundary), the lower bound on the number of solutions given by
Theorem A for the Neumann problem (1.2) depends solely on the topology of
$\partial M$. This fact is related to the different nature of the limiting
isoperimetric-type problem linked to each case in the $m,\varepsilon\to 0$
asymptotics.

######  Remark 1.2.

It is easy to check that one always has $\mathrm{cat}(\partial M)>1$, as
$\partial M$ is a closed and compact manifold. However, this is not always
true for $M$ (consider, for instance, a closed ball in ${\mathbb{R}}^{n}$).

######  Example 1.3.

Let $M$ be a closed ball in $\mathbb{R}^{n}$. Then we have
$\mathrm{cat}(\partial M)=\mathscr{P}_{1}(\partial M)=2$, so that by Theorem A
we have at least 3 solutions for the Neumann problem (1.2) (in the appropriate
parameter regime). In particular, we do not obtain a better result in the non-
degenerate case when $M$ is a ball. As $\mathrm{cat}(M)=1$ and
$\mathscr{P}_{1}(M)=1$, Theorem B only guarantees the existence of one
solution for the Dirichlet problem (1.3).

######  Example 1.4.

The situation is different if we take $M$ to be equal to an annulus in
${\mathbb{R}}^{2}$. In this case, $\partial M$ is the union of two disjoint
circles and this implies that $\mathrm{cat}(\partial
M)=\mathscr{P}_{1}(\partial M)=4$, so Theorem A gives in general at least 5
solutions for the Neumann problem (1.2), and at least 7 solutions in the non-
degenerate case. Moreover, as $\mathrm{cat}(M)=2$, Theorem B ensures the
existence of at least three solutions for (1.3).

######  Remark 1.5.

The non-degeneracy assumption (H1) for the potential $W$ is standard and,
moreover, generic. The coercivity assumption (H2) is also standard and
guarantees that the $\Gamma$-convergence results for the energies
$(\mathcal{E}_{\varepsilon})_{\varepsilon}$ holds, see, e.g., Fonseca and
Tartar [28], Owen, Rubinstein and Sternberg [44] (in the Euclidean case).
Regarding the assumption (H3), it is of more technical nature but also
standard in variational problems concerning nonlinear elliptic PDEs or
differential geometry. It ensures that these problems are compact or, more
precisely, that both $\mathcal{E}_{\varepsilon,m}$ and
$\overline{\mathcal{E}}_{\varepsilon,m}$ satisfy the Palais-Smale condition.
As we prove in Section 5, it is possible to drop (H3) and obtain some partial
results. However, it is not clear to us whether Theorems A and B still hold
true if (H3) is dropped. Indeed, the coercivity assumption (H2) could be used
to construct suitable energy decreasing projections or truncation-type
operators in order to restore strong compactness of sublevel sets of
$\mathcal{E}_{\varepsilon}$, but such projections or operators would not
respect in general the mass constraint. One could also wonder about the
validity of our result without the compactness assumption on the manifold $M$.
In any case, these questions, although interesting, go beyond the scope of
this paper.

######  Remark 1.6.

Theorem A and Theorem B extend to vectorial double-well potentials
$W\colon{\mathbb{R}}^{k}\to{\mathbb{R}}$ without substantial modifications on
the proofs. For simplicity, we have chosen to present the results for scalar
potentials. Nevertheless, as noticed in [5] and also as discussed below, the
extension to vectorial potentials with three or more phases is not
straightforward.

Our approach to prove Theorem A and Theorem B combines some topological and
variational methods with $\Gamma$-convergence properties of
$\mathcal{E}_{\varepsilon}$ and tools from Geometric Measure Theory.
Variational methods relating the multiplicity of solutions of an elliptic
problem with the topology of the underlying domain were already used in
earlier works by Benci, Cerami and Passaseo [11, 12, 13], see also Benci [10]
and references therein. The results contained in this paper are mainly
inspired by the recent contribution [14], where the manifold $M$ is supposed
to be _closed_ and compact (cf. also the earlier result [15]).

Roughly speaking, the idea of the proofs both here and in [14] consists on
using the $\Gamma$-convergence properties of the functionals
$\mathcal{E}_{\varepsilon,m}$ and $\overline{\mathcal{E}}_{\varepsilon,m}$ in
order to show that under small mass constraint and for $\varepsilon>0$ small
enough the sublevel sets

$\mathcal{E}^{c}_{\varepsilon,m}\coloneqq\left\\{u\in\mathcal{H}_{m}:\mathcal{E}_{\varepsilon}(u)\leq
c\right\\}\quad\text{and}\quad\overline{\mathcal{E}}^{c}_{\varepsilon,m}\coloneqq\left\\{u\in\mathcal{H}_{m,0}:\mathcal{E}_{\varepsilon}(u)\leq
c\right\\}$

are, for suitable $c>0$, homotopically equivalent to $\partial M$ and $M$,
respectively. The topological complexity of the above sublevel sets is hence
comparable to that of $\partial M$ and $M$, and we may apply the infinite-
dimensional versions of Lusternik-Schnirelman and Morse theories (see Palais
[46, 47]) to establish the stated lower bounds on the number of critical
points.

There are other results in related settings which are also obtained by similar
strategies. For instance, Jerrard and Sternberg [35] provided an abstract
framework which allows to use “critical points” (in a non-smooth sense) of a
$\Gamma$-limit functional in order to prove existence of critical points for
the $\Gamma$-converging functionals when they are “close” enough to the
$\Gamma$-limit. They applied this result to the 2D Allen-Cahn and 3D Ginzburg-
Landau (with and without magnetic field) functionals. Improvements of this
abstract approach in the concrete setting of Ginzburg-Landau functionals on
manifolds without boundary have been made recently by Colinet, Jerrard and
Sternberg [24] in 3D and by De Philippis and Pigati [25] in arbitrary
dimension. Similar ideas, but also using different methods, had been applied
before by Pacard and Ritoré [45] and Kowalczyk [36] in the Allen-Cahn setting.
In the same spirit, Gaspar and Guaraco [29, 31] proved that the number of
solutions of the Allen-Cahn equation (without volume constraint) in a closed
manifold goes to infinity as $\varepsilon\to 0$. Their proof is restricted to
even potentials, see also Passaseo [48]. By similar methods, Bellettini and
Wickramaseckera [9] proved existence of solutions for a more general equation
of Allen-Cahn type. Related results for the Ginzburg-Landau equations on
manifolds were obtained by Stern [54], and by Pigati and Stern [49] in the
case with magnetic field, see also [20, 21]. Moreover, earlier work of this
type exists also in the context of the 2D Ginzburg-Landau functional
considered by Bethuel, Brezis and Helein [17]. Almeida and Bethuel [1, 2]
proved that there exists a lower bound on the number of critical points
according to the topological degree of the boundary data as long as the
perturbation parameter $\varepsilon>0$ is small enough (see also Zhou and Zhou
[57] for improvements).

In our case, the $\Gamma$-limit functionals related to
$\mathcal{E}_{\varepsilon,m}$ and $\overline{\mathcal{E}}_{\varepsilon,m}$
correspond to different isoperimetric-type problems on $M$ in the small volume
regime, for which there are plenty of available results which provide a quite
complete picture. In [16], Bérard and Meyer showed that the isoperimetric
profile of a $n$-dimensional compact and smooth Riemannian manifold becomes
comparable to the isoperimetric profile of ${\mathbb{R}}^{n}$ as the volume
tends to zero. Morgan and Johnson [42] took one step further by proving that
solutions to the isoperimetric problem with small volume are close to small
spheres, so that their diameter tends to zero for vanishing volumes. More
precisely, a result in [43] implies that these spheres are exactly those
centered at a point of maximal scalar curvature (of the manifold), see also
Druet [26]. Analogous results are available in the setting of compact
Riemannian manifolds with boundary (without taking the boundary into account):
Bayle and Rosales [8] proved the asymptotic equivalence with the relative
isoperimetric profile of the _half_ -plane and Fall [27] showed that relative
isoperimetric regions of small volume are close to half-spheres centered at
the boundary points of maximal mean curvature. Moreover, one might also
consider variants of the isoperimetric problem in which the boundary part of
the manifold is also taken into account, still obtaining an asymptotic
equivalence with the isoperimetric problem in ${\mathbb{R}}^{n}$.

One natural question concerns the extension of the results obtained in Theorem
A and Theorem B to systems of Allen-Cahn-type. It turns out that the main
obstacle for such extensions lies in the fine understanding of the resulting
$\Gamma$-limit problem. It is known since the work of Baldo [7] that the
$\Gamma$-limit functional for vectorial multi-well potentials corresponds to a
minimization problem for the perimeter of clusters (with cardinality
corresponding to the number of wells of the potential) under volume
constraints. These problems can be seen as vectorial versions of the
isoperimetric problem, and they are still poorly understood in the small
volume regime. In particular, as pointed out in [5], in the vectorial case it
is not known in general whether the diameters of the different cluster
components of minimizers are uniformly controlled by their respective volume,
which is a crucial ingredient in our approach. However, properties of that
kind are available for clusters of three sets on the plane, allowing to extend
the results of [14] to the case of triple-well potentials on 2D closed
manifolds (see [5]).

In a different direction, most of the recent results quoted above concerning
the existence of critical points of Allen-Cahn and Ginzburg-Landau functionals
(e.g., [5, 9, 24, 25, 29, 31, 49, 54]) are proven in the setting of manifolds
without boundary. It is natural to wonder about their validity when one
considers an underlying manifold which has a non-empty boundary as we do in
this paper. Moreover, not much is known concerning Ginzburg-Landau functionals
with a flux constraint on the vorticity, which are naturally associated to a
limiting isoperimetric-type problem in codimension 2. To our knowledge, the
only results in this direction were proven in [18] and by Chiron [23], where
they were applied to problems of existence of traveling waves for the Gross-
Pitaevskii equation. It is possible that these results could be extended to
some degree and then used to obtain multiplicity results in the spirit of the
present paper.

###### Contents

  1. 1 Introduction
  2. 2 Sketch of the proofs
  3. 3 Proof of Theorem A
    1. 3.1 The relative isoperimetric problem
    2. 3.2 $\Gamma$-convergence
    3. 3.3 Photography map
    4. 3.4 Barycenter map
    5. 3.5 Conclusion of the proof
  4. 4 Proof of Theorem B
    1. 4.1 The isoperimetric problem
    2. 4.2 Gamma convergence
    3. 4.3 Photography map
    4. 4.4 Barycenter map
    5. 4.5 Conclusion of the proof
  5. 5 Generic non-degeneracy
    1. 5.1 Abstract transversality result
    2. 5.2 Genericity with respect to the Riemannian metric
    3. 5.3 Genericity with respect to the domain
  6. 6 A partial result without the subcritical growth assumption

##  2 Sketch of the proofs

In this section we outline the main ingredients used in the proofs of Theorems
A and B. We recall that a $C^{1}$ functional $f$ on a Banach manifold $X$ is
said to satisfy the Palais-Smale condition if every Palais-Smale sequence for
$f$ admits a converging subsequence, where $(x_{n})_{n\in{\mathbb{N}}}$ is a
Palais-Smale sequence for $f$ if $(f(x_{n}))_{n\in{\mathbb{N}}}$ is bounded
and $(\lVert\mathrm{d}f(x_{n})\rVert_{X^{*}})_{n\in{\mathbb{N}}}$ tends to $0$
as $n\to\infty$. The Palais-Smale condition is a classical and useful
sufficient condition in critical point theory. When it holds one only needs to
detect a change of topology in the level sets of the functional in order to
prove the existence of critical points. The interested reader can find more
details in [47]. In our case, the Palais-Smale condition is fulfilled:

######  Lemma 2.1.

Assume that (H3) holds. Then, for all $\varepsilon>0$ and $m\in{\mathbb{R}}$
the functional $\mathcal{E}_{\varepsilon}$ is well defined as a $C^{2}$
functional on $H^{1}(M,\mathbb{R})$ and it verifies the Palais-Smale
condition. In particular, $\mathcal{E}_{\varepsilon}$ is a $C^{2}$ functional
satisfying the Palais-Smale condition on both $\mathcal{H}_{m}$ and
$\mathcal{H}_{m,0}$. Moreover, if $u_{\varepsilon,m}\in\mathcal{H}_{m}$ is a
critical point of $\mathcal{E}_{\varepsilon,m}$, then there exists
$\lambda_{\varepsilon,m}$ such that
$(u_{\varepsilon,m},\lambda_{\varepsilon,m})$ solves (1.2) and if
$u_{\varepsilon,m}\in\mathcal{H}_{m,0}$ is a critical point of
$\overline{\mathcal{E}}_{\varepsilon,m}$, then there exists
$\lambda_{\varepsilon,m}$ such that
$(u_{\varepsilon,m},\lambda_{\varepsilon,m})$ solves (1.3).

The proof of Lemma 2.1 is standard. In particular, compactness of the Palais-
Smale sequences for $\mathcal{E}_{\varepsilon}$ follows from the compactness
of $M$ along with the subcritical behavior of the potential provided by
assumption (H3) (for more details see, for instance, [14, Proposition 4.12]).

According to the previous discussion, the proofs of Theorems A and B boil down
to determining changes of topology in the sublevel sets of the functionals
$\mathcal{E}_{\varepsilon,m}$ and $\overline{\mathcal{E}}_{\varepsilon,m}$. To
do this, we employ the so-called _photography method_ , which allows to
understand some topological invariants of a sublevel of a functional through a
homotopy equivalence with another topological space for which these invariants
are more easily computable. More precisely, we will use the following result
(see, e.g., [10, 11]).

######  Theorem C.

Let $X$ be a topological space, $\mathfrak{M}$ be a $C^{2}$-Hilbert manifold,
$f\colon\mathfrak{M}\to\mathbb{R}$ be a $C^{1}$-functional. For
$c\in\mathbb{R}$, let $f^{c}\coloneqq\\{u\in\mathfrak{M}:f(u)\leq c\\}$ be the
$c$-sublevel set of $f$. Assume that

  1. $($$\mathrm{E1}$$)$

$f$ is bounded below;

  2. $($$\mathrm{E2}$$)$

$f$ satisfies the Palais–Smale condition;

  3. $($$\mathrm{E3}$$)$

There exists $c\in{\mathbb{R}}$ and two continuous maps $\Psi_{R}\colon
X\rightarrow f^{c}$ and $\Psi_{L}\colon f^{c}\rightarrow X$ such that
$\Psi_{L}\circ\Psi_{R}$ is homotopic to the identity map of $X$.

Then, the number of critical points in $f^{c}$ is greater than
$\mathrm{cat}(X)$. If $\mathfrak{M}$ is contractible and $\mathrm{cat}(X)>1$,
there is at least another critical point of $f$ outside $f^{c}$. Moreover,
there exists $c_{0}\in(c,\infty)$ such that one of the two following
conditions holds:

  * (i)

$f^{c_{0}}$ contains infinitely many critical points;

  * (ii)

$f^{c}$ contains $\mathscr{P}_{1}(X)$ critical points and $f^{c_{0}}\setminus
f^{c}$ contains $\mathscr{P}_{1}(X)-1$ critical points if counted with their
multiplicity. More exactly, we have the following relation:

$\sum_{u\in{\rm
Crit}(f)}t^{\mu(u)}=t\mathscr{P}_{t}(X)+t^{2}[\mathscr{P}_{t}(X)-1]+t(1+t)\mathcal{Q}(t),$
(2.1)

where ${\rm Crit}(f)$ is the set of critical points of $f$, $\mu(u)$ is the
Morse index of the solution $u$ and $\mathcal{Q}(t)$ is a polynomial with
nonnegative integer coefficients. In particular, if all the critical points
are non-degenerate, there are at least $\mathscr{P}_{1}(X)$ critical points in
$f^{c_{0}}$ and $\mathscr{P}_{1}(X)-1$ critical points in $f^{c_{0}}\setminus
f^{c}$.

Since $\mathcal{E}_{\varepsilon}$ is bounded below (as it is nonnegative) and
satisfies the Palais-Smale condition by Lemma 2.1, Theorem C implies Theorem A
if we are able to show that for all $m$ and $\varepsilon$ sufficiently small
there exist $c>0$ and two continuous maps
$\mathcal{P}_{\varepsilon,m}\colon\partial
M\to\mathcal{E}^{c}_{\varepsilon,m}$ (the _photography_ map) and
$\mathcal{B}\colon\mathcal{E}^{c}_{\varepsilon,m}\to\partial M$ (the
_barycenter_ map) such that
$\mathcal{B}\circ\mathcal{P}_{\varepsilon,m}\colon\partial M\to\partial M$ is
homotopic to the identity map. Similarly, Theorem B will follow by showing the
existence of two continuous maps $\mathcal{P}_{\varepsilon,m}\colon
M\to\overline{\mathcal{E}}^{c}_{\varepsilon,m}$ and
$\mathcal{B}\colon\overline{\mathcal{E}}^{c}_{\varepsilon,m}\to M$ such that
$\mathcal{B}\circ\mathcal{P}_{\varepsilon,m}\colon M\to M$ is homotopic to the
identity map of $M$. Analogously to [14], the two main ingredients that allow
to construct the maps $\mathcal{P}_{\epsilon,m}$ and $\mathcal{B}$ are the
following:

  1. 1.

The $\Gamma$-convergence, as $\varepsilon\to 0^{+}$, of the energies
$\mathcal{E}_{\varepsilon}$ towards some form of isoperimetric problem, the
latter depending one the chosen boundary condition. In the case of Neumann
boundary conditions, the $\Gamma$-limit is the classical relative
isoperimetric problem, i.e., the boundary of $M$ is not taken into account on
the computation of the perimeter. If one considers homogeneous Dirichlet
boundary conditions, then the $\Gamma$-limit takes into account the perimeter
intersecting the boundary $\partial M$.

  2. 2.

The fact that the isoperimetric profile and the relative isoperimetric profile
on compact manifolds with boundary are asymptotic as $m\to 0^{+}$ to the
isoperimetric profiles on the Euclidean space and half-space, respectively. As
a consequence, for sufficiently small volumes any “almost minimizer” (a
concept that will be formally defined later) of the relative isoperimetric
problem concentrates almost its entire volume around a point on the boundary
of $M$. For the isoperimetric problem, the same concentration phenomena occurs
but for points on $M$. See Figure 1 for an explanatory picture.

Combining this two ingredients, we find that in the Neumann case any function
that belongs to the smallest sublevel $\mathcal{E}_{\varepsilon,m}^{c}$ that
contains the image of the photography map has almost all its mass in a
geodesic ball centered in a point of $\partial M$. By composing with the
nearest point projection on $\partial M$, it follows that the barycenter map
$\mathcal{B}\colon\mathcal{E}_{\varepsilon,m}^{c}\to\partial M$ is well
defined and continuous. In the Dirichlet case, we have that an almost
minimizer of $\overline{\mathcal{E}}_{\epsilon,m}$ has almost all its mass in
a geodesic ball centered in a point of $M$, so that the barycenter map
$\mathcal{B}\colon\mathcal{E}_{\varepsilon}^{c}\to M$ is well defined and
continuous. Conversely, one finds the photography maps
$\mathcal{P}_{\varepsilon,m}$ by constructing almost minimizers of
$\mathcal{E}_{\varepsilon}$ supported on geodesic balls centered at arbitrary
points of $\partial M$ in the Neumann case and of $M$ in the Dirichlet case.

$\partial M$$M$$M$ Figure 1: The filled regions represent an arbitrary almost
minimizer of the relative isoperimetric problem (left) and the isoperimetric
problem (right). In the former case, most of the volume is concentrated inside
a ball (dotted line) a half-ball centered at the boundary. In the latter case,
concentration occurs around some point of $M$. However, some of the volume
might be away from these balls, but it is always a small portion of the volume
which can be controlled. In particular, half-balls and balls are almost
minimizers of the relative isoperimetric and isoperimetric problems,
repectively.

######  Remark 2.2.

To apply Theorem C in the proof of Theorem A, one needs that
$\mathrm{cat}(\partial M)>1$ and that $\mathcal{H}_{m}$ is contractible. Since
these two conditions are always satisfied (in fact, $\mathcal{H}_{m}$ is
convex), they are omitted in the statement of Theorem A. This is also the
reason why the condition $\mathrm{cat}(M)>1$, not necessarily fulfilled a
priori, appears in the statement of Theorem B.

##  3 Proof of Theorem A

###  3.1 The relative isoperimetric problem

We recall some standard notations: let $BV(M,{\mathbb{R}})$ denotes the set of
functions of bounded variation defined on $M$; for any $N\subset M$, we denote
by $\mathbf{1}_{N}$ its characteristic function; $\mathcal{H}^{n}$ the
$n$–dimensional Hausdorff measure on ${\mathbb{R}}^{\tilde{n}}$; for any
measurable set $A\subset M$, we denote by $\partial^{*}A$ the reduced boundary
relative to $M$, which is the subset of $\partial A$ whose notion of measure-
theoretic normal vector exists and has length equal to one. We denote by
$\mathcal{C}_{g}(M)$ the set of Caccioppoli subsets of $M$, hence

$\mathcal{C}_{g}(M)\coloneqq\Big{\\{}\Omega\subset M:\,\Omega\text{ is
measurable and }\mathcal{H}^{n-1}(\partial^{*}\Omega)<+\infty\Big{\\}}.$

Moreover, let $\sigma>0$ be the following constant, which depends only on $W$:

$\sigma\coloneqq
2\,\inf\left\\{\int_{0}^{1}\sqrt{W(\gamma(s))}|\gamma^{\prime}(s)|\mathrm{d}s:\gamma\in
C^{1}([0,1],\mathbb{R}),\gamma(0)=0,\gamma(1)=1\right\\}.$ (3.1)

For $u\in L^{1}(M,{\mathbb{R}})$ we set

$\mathcal{E}_{0}(u)\coloneqq\begin{cases}\sigma\mathcal{H}^{n-1}(\partial^{*}\Omega\cap\mathrm{int}(M)),&\mbox{
if }u=\mathbf{1}_{\Omega},\\\ +\infty,&\mbox{ otherwise}.\end{cases}$

Alternatively, one can define $\mathcal{E}_{0}$ as the relative perimeter of
$\Omega$, which is defined as the total measure of
$\lvert\nabla\mathbf{1}_{\Omega}\rvert$. The fact $\mathbf{1}_{\Omega}\in
BV(M,{\mathbb{R}})$ implies that both definitions are equivalent, see Giusti
[30] (see also Sternberg [55] for more details).

For fixed $m\in[0,\mathrm{vol}(M)]$, we define

$I_{M}(m)\coloneqq\inf\left\\{\mathcal{H}^{n-1}(\partial^{*}\Omega\cap\mathrm{int}(M)):\Omega\in\mathcal{C}_{g}(M)\text{
and }\int_{M}\mathbf{1}_{\Omega}\mathrm{d}v_{g}=m\right\\}.$ (3.2)

Notice that $I_{M}\colon[0,\mathrm{vol}(M)]\to\mathbb{R}$ is the
_isoperimetric profile function_ of $M$ and that

$I_{M}(m)=\frac{1}{\sigma}\inf\left\\{\mathcal{E}_{0}(u):u\in
L^{1}(M,\mathbb{R}),\,\int_{M}u\,\mathrm{d}v_{g}=m\right\\}.$ (3.3)

The same problem can be stated in the euclidean semi-space

$\mathbb{R}^{n}_{+}=\left\\{z\in\mathbb{R}^{n}:z^{n}\geq
0\right\\}=\mathbb{R}^{n-1}\times[0,+\infty[,$

so that the function $I_{\mathbb{R}^{n}_{+}}\colon(0,+\infty)\to\mathbb{R}$ is
defined analogously and there exists a dimensional constant $c_{n}^{+}>0$ such
that

$I_{R^{n}_{+}}(m)=c_{n}^{+}m^{\frac{n-1}{n}}.$ (3.4)

When $m$ is small, there exists a link between the isoperimetric function
$I_{M}$ and its euclidean counterpart. More formally, Bayle and Rosales [8]
(see also Fall [27]) have shown that

$I_{M}(m)=\big{(}1+\mathcal{O}(m^{1/n})\big{)}I_{\mathbb{R}^{n}_{+}}(m).$
(3.5)

###  3.2 $\Gamma$-convergence

We now state the two main $\Gamma$-convergence results we need, starting with
the the most celebrated one, first proven by Modica and Mortola [40, 41], and
later extended to the vector-valued setting by several authors: Baldo [7],
Fonseca and Tartar [28] and Sternberg [55]. In [28], the double-well case was
considered without the subcritical growth condition (H3), meaning that their
result is the most well adapted to our situation. We also point out that in
the above mentioned papers the results are proven in the Euclidean case, but
it was later shown (see Benci et al. [14] and the references therein) that
they can be extended to the Riemannian setting.

######  Theorem 3.1 (cf. [28], Theorem 3.4 and [14], Proposition 3.3).

Assume that (H1) and (H2) hold. Then, the following statements hold:

  1. i)

($\Gamma$-$\liminf$ inequality): If
$(\varepsilon_{k})_{k\in{\mathbb{N}}}\subset]0,+\infty[$ is such that
$\varepsilon_{k}\to 0^{+}$ and
$(u_{\varepsilon_{k}})_{k\in{\mathbb{N}}}\subset H^{1}(M,{\mathbb{R}})$ is
such that $u_{\varepsilon_{k}}\to u_{0}$ in $L^{1}(M,{\mathbb{R}})$, then
$\liminf_{k\to\infty}\mathcal{E}_{\varepsilon_{k}}(u_{\varepsilon_{k}})\geq\mathcal{E}_{0}(u_{0})$;

  2. ii)

($\Gamma$-$\limsup$ (in)equality): For any $u_{0}\in L^{1}(M,{\mathbb{R}})$
such that $u_{0}=\mathbf{1}_{\Omega}$ for some $\Omega\subset M$ and for every
sequence $(\varepsilon_{k})_{k\in{\mathbb{N}}}\subset]0,+\infty[$ such that
$\varepsilon_{k}\to 0^{+}$, there exists
$(u_{\varepsilon_{k}})_{k\in{\mathbb{N}}}\subset H^{1}(M,{\mathbb{R}})$ such
that $u_{\varepsilon_{k}}\to u_{0}$ in $L^{1}(M,{\mathbb{R}})$,

$\int_{M}u_{\varepsilon_{k}}\mathrm{d}v_{g}=\int_{M}u_{0}\mathrm{d}v_{g}\quad\text{and}\quad\limsup_{k\to\infty}\mathcal{E}_{\varepsilon_{k}}(u_{\varepsilon_{k}})\leq\mathcal{E}_{0}(u_{0}).$

######  Remark 3.2.

The interested reader can find in [14, Proposition 3.3] a detailed and
explicit construction of the functions $u_{\varepsilon}$ that approximate
$u_{0}$ as required by the $\Gamma$-$\limsup$ statement of Theorem 3.1, which
is the Riemannian version of the construction given by Modica in [40] in the
Euclidean setting. We give here just the idea of this construction, following
[40]. For every $A\subset M$ we denote by $d_{A}\colon M\to\mathbb{R}$ the
following function:

$d_{A}(x)\coloneqq\begin{dcases}-\mathrm{dist}_{g}(x,\partial A),&\mbox{if
$x\in A$},\\\ \phantom{-}\mathrm{dist}_{g}(x,\partial A),&\mbox{if $x\notin
A$}.\end{dcases}$ (3.6)

For every $\varepsilon>0$, let us consider an increasing and Lipschitz
continuous function $\widetilde{q}_{\varepsilon}\colon\mathbb{R}\to[0,1]$ such
that $\widetilde{q}_{\varepsilon}(t)=0$ if $t<0$, and
$\widetilde{q}_{\varepsilon}(t)=1$ if $t\geq\eta_{\varepsilon}>0$, where
$\eta_{\varepsilon}\to 0$ as $\varepsilon\to 0$. We remind that such a
function is linked with the energy functional $\mathcal{E}_{\varepsilon}$
since it is a solution of the following differential equation:

$\frac{\mathrm{d}}{\mathrm{d}t}\widetilde{q}_{\varepsilon}=\frac{1}{\varepsilon}\sqrt{\varepsilon^{3/2}+2W(\widetilde{q}_{\varepsilon})}.$
(3.7)

Finally, for every $u_{0}=\mathbf{1}_{\Omega}$, $u_{\varepsilon}$ is obtained
as

$u_{\varepsilon}(x)=\widetilde{q}_{\varepsilon}(d_{M\setminus\Omega}(x)+\delta_{\varepsilon,\Omega}),$
(3.8)

where $\delta_{\varepsilon,\Omega}\in[0,\eta_{\varepsilon}]$ is a correction
term such that

$\int_{M}u_{\varepsilon}\,\mathrm{d}v_{g}=\int_{M}\widetilde{q}_{\varepsilon}(d_{M\setminus\Omega}(x)+\delta_{\varepsilon,\Omega})\mathrm{d}v_{g}=\int_{M}u_{0}\,\mathrm{d}v_{g}.$
(3.9)

######  Theorem 3.3 (cf. Theorem 4.1 of [28]).

Assume that (H1) and (H2) hold. If $(\varepsilon_{k})_{k}\subset\mathbb{R}$ is
a sequence of positive numbers such that $\varepsilon_{k}\to 0^{+}$ and
$(u_{\varepsilon_{k}})_{k}\subset H^{1}(M,\mathbb{R})$ is a sequence of
functions such that

$\mathcal{E}_{\varepsilon_{k}}(u_{\varepsilon_{k}})\leq c,\qquad\forall
k\in\mathbb{N},$ (3.10)

for some constant $c>0$, then there exists a subsequence, still denoted by
$(\varepsilon_{k})_{k}$, such that $(u_{\varepsilon_{k}})_{k}$ converges to a
function $u_{0}\in L^{1}(M,{\mathbb{R}})$ with respect to the $L^{1}$–norm.

######  Remark 3.4.

As a direct consequence of Theorem 3.3 and the lim-inf property of Theorem
3.1, if $\mathcal{E}_{\epsilon_{k}}(u_{\varepsilon_{k}})\leq c$ for every
$k\in\mathbb{N}$, then the function $u_{0}\in L^{1}(M,{\mathbb{R}})$ such that
$u_{\varepsilon_{k}}\to u_{0}$ up to subsequences is such that
$\mathcal{E}_{0}(u_{0})\leq E^{*}$, there exists a measurable set
$\Omega\subset M$ such that $u_{0}=\mathbf{1}_{\Omega}$ and

$\lim_{k\to\infty}\int_{M}u_{k}\,\mathrm{d}v_{g}=\int_{M}u_{0}\,\mathrm{d}v_{g}.$

###  3.3 Photography map

In order to obtain the map $\mathcal{P}_{\varepsilon,m}$, we combine the
$\Gamma$-convergence result given by Theorem 3.1 with the characterization of
the isoperimetric regions in a manifold with boundary given by Fall [27].
Roughly, as long as $m$ and $\varepsilon$ are sufficiently small, for every
point $p\in\partial M$ we define $\mathcal{P}_{\varepsilon,m}(p)$ as the
$\varepsilon$–approximation ensured by the lim-sup part of Theorem 3.1 of the
characteristic function of a small perturbation of a half-ball centered at $p$
and with volume $m$, denoted by $E_{p,m}\subset M$. Such a set is defined by
employing the normal coordinates at the point $p$, and by using the implicit
function theorem to construct the perturbation that minimizes the perimeter.
The details of the construction of $E_{p,m}$ can be explored in [27]. However,
for the purposes of this article, we will provide a concise overview of the
essential details.

Let $N_{\partial M}$ the unit interior normal vector field along $\partial M$
and, for any $p\in\partial M$, let $\mathrm{exp}^{\partial M}_{p}$ be the
exponential map of $\partial M$ at $p$. Let us consider an orthonormal frame
field $(e_{1},\dots,e_{n-1},N_{\partial M})$ of $M$ along $\partial M$. We
define the map $f^{p}\colon\mathbb{R}^{n-1}\to\partial M$ as follows:

$f^{p}(z)\coloneqq\mathrm{exp}^{\partial M}_{p}(z^{i}e_{i}),$ (3.11)

where we used the Einstein summation convention. For every
$z\in\mathbb{R}^{n-1}$ and $t\in\mathbb{R}$, with $t\geq 0$ and sufficiently
small, let us define also

$F^{p}(z,t)=\exp_{f^{p}(z)}(tN_{\partial M})\in M,$ (3.12)

so that $F^{p}$ provides a local parametrization of a neighborhood of $p\in
M$. Our aim is to define a “semi-sphere” on $M$ with center in $p$. To this
aim, let us denote by $B^{n-1}$ the unit ball in $\mathbb{R}^{n-1}$ and by
$S^{n-1}_{+}\in\mathbb{R}^{n}$ the upper hemisphere of $S^{n-1}$, hence

$S^{n-1}_{+}\coloneqq\big{\\{}z\in\mathbb{R}^{n}:\lVert z\rVert=1,z^{n}\geq
0\big{\\}}.$

Let us denote by $\Theta\colon B^{n-1}\to S^{n-1}_{+}$ the inverse of the
stereographic projection from the south pole
$(0,\dots,0,-1)\in\mathbb{R}^{n}$, formally seeing $B^{n-1}$ as
$\\{z\in\mathbb{R}^{n}:\lVert z\rVert\leq 1,z^{n}=0\\}$. Let
$\widetilde{\Theta}\colon B^{n-1}\to B^{n-1}$ and $t\colon B^{n-1}\to[0,1]$ be
such that

$\Theta(z)=\big{(}\widetilde{\Theta}(z),t(z)\big{)}.$

For every $r$ sufficiently small, the hypersurface defined by

$\left\\{F^{p}\big{(}r\widetilde{\Theta}(z),rt(z)\big{)}:z\in
B^{n-1}\right\\}\subset M$

can be seen as an semi-sphere centered at $p$. All the hypersurfaces nearby
this semi-sphere centered at $p\in\partial M$ can be obtained by applying a
small perturbation $\omega\colon S^{n-1}_{+}\to\mathbb{R}$, and so we define
the hypersurface $\Sigma_{p,r,\omega}\subset M$ as follows:

$\Sigma_{p,r,\omega}\coloneqq\left\\{F^{p}\Big{(}\big{(}r+\omega(\Theta(z))\big{)}\,\widetilde{\Theta}(z),\big{(}r+\omega(\Theta(z))\big{)}t(z)\Big{)}:z\in
B^{n-1}\right\\}.$ (3.13)

Since $t(z)=0$ for every $z\in\partial B^{n-1}$,
$\partial\Sigma_{p,r,\omega}\subset\partial M$, so that $\Sigma_{p,r,\omega}$
and $\partial M$ enclose a set that we denote by $E_{p,r,\omega}$.

By [27, Lemma 4.6], there exists $r_{0}>0$ such that for any $p\in\partial M$
and $r\in(0,r_{0})$ there exists a unique smooth $\omega^{p,r}\in
C^{2,\alpha}(S^{n-1}_{+})$ such that $\Sigma_{p,r,\omega^{p,r}}$ has a mean
curvature function that is, modulo an additive constant, an eigenfunction of
the first strictly positive eigenvalue of the Laplacian on the hemisphere. In
this way, $\Sigma_{p,r,\omega^{p,r}}$ represents an almost CMC (constant mean
curvature) half-sphere centered at $p$, thereby implying that it is the unique
competitor among all hypersurfaces $\Sigma_{p,r,\omega}$ to serve as a
solution to the isoperimetric problem. Moreover,
$\lVert\omega^{p,r}\rVert_{{}_{C}^{1,\alpha}(S^{n-1}_{+})}\to 0$ as $r\to 0$
and, since it is defined by using the Implicit Function Theorem, the map
$(p,r)\mapsto\omega^{p,r}$ is continuous. We denote as _pseudo half-bubbles_
the hypersurfaces $\Sigma_{p,r,\omega^{p,r}}$, in analogy with the _pseudo-
bubbles_ introduced in [43].

Following the notation in [27], we will use the simpler notation $E_{p,r}$ to
refer to $E_{p,r,\omega^{p,r}}$. Furthermore, for any $p\in\partial M$ and
sufficiently small $m$, we denote by $r_{p,m}>0$ the unique radius such that
$E_{p,r_{p,m}}$ has volume $m$, i.e.,
$\int_{M}\mathbf{1}_{E_{p,r_{p,m}}}\mathrm{d}v_{g}=m$. For more detailed
information on the existence of such $r_{p,m}$, please refer to [27, Lemma
4.7].

######  Remark 3.5.

By the compactness assumption of $M$ and $\partial M$, there exists $m_{0}>0$
such that for every $p\in\partial M$ and $m\in(0,m_{0})$ we have that
$r_{p,m}$ is well defined and less than $r_{0}$. As a consequence,
$\omega^{p,r_{p,m}}$ is always well defined by [27, Lemma 4.6] for every
$p\in\partial M$ and $m\in(0,m_{0})$.

For the sake of notation, for every $p\in\partial M$ and $m\in(0,m_{0})$ we
set

$E_{p,m}\coloneqq
E_{p,r_{p,m}}\quad\text{and}\quad\Sigma_{p,m}\coloneqq\Sigma_{p,r_{p,m},\omega^{p,r_{p,m}}}=\partial
E_{p,m}\cap M.$

Let us notice that, since $m<m_{0}$, the hypersurface $\Sigma_{p,m}$ is
smooth. Finally, we will denote by $u^{m}_{0,p}\in L^{1}(M,\mathbb{R})$ the
characteristic function of $E_{p,m}$, hence

$u^{m}_{0,p}(x)\coloneqq\begin{dcases}1,&\mbox{if }x\in E_{p,m},\\\
0,&\mbox{otherwise}.\end{dcases}$

Before defining the photography map, let us state [27, Lemma 4.8] in
accordance with our notation.

######  Lemma 3.6.

For every $p\in\partial M$, let us denote by $H_{\partial M}(p)$ the mean
curvature of $\partial M$ at $p$. Then, there exists a dimensional constant
$\gamma_{n}>0$ such that

$\mathcal{E}_{0}(u^{m}_{0,p})=\sigma\left(I_{\mathbb{R}^{n}_{+}}(m)-\gamma_{n}H_{\partial
M}(p)m+\mathcal{O}(m^{\frac{n+1}{n}})\right),\qquad\forall m\in(0,m_{0}),$
(3.14)

and, moreover, the following holds:

$I_{M}(m)=I_{\mathbb{R}^{n}_{+}}(m)-\gamma_{n}\max_{p\in\partial
M}\\{H_{\partial M}(p)\\}m+\mathcal{O}(m^{\frac{n+1}{n}}).$ (3.15)

######  Remark 3.7.

For our specific purposes, it is possible to avoid the use of the perturbation
$\omega^{p,r}$ by setting $\omega\equiv 0$ in the definitions of
$\Sigma_{p,r,\omega}$, $E_{p,r}$, and $u^{m}_{0,p}$. This choice is viable
because the primary tool we will employ in the subsequent construction is the
estimate (3.14), which remains valid even when $\omega\equiv 0$. However, in
order to avoid a detailed proof of this last assertion, as it falls outside
the scope of our work, we rely on the estimate provided by [27].

######  Definition 3.8.

For every $m\in(0,m_{0})$ and $\varepsilon>0$,
$\mathcal{P}_{\varepsilon,m}\colon\partial M\to\mathcal{H}_{m}$ is the map
that links $p\in\partial M$ to $u^{m}_{\varepsilon,p}$, that is the
$\varepsilon$–approximation of $u^{m}_{0,p}\in L^{1}(M,\mathbb{R})$ given by
the _lim-sup_ property of Theorem 3.1 (see Remark 3.2).

$\partial M$$\partial
M$$M$$M$$p$$p$$0$$0$$1$$1$$u_{0,p}^{m}$$u_{\varepsilon,p}^{m}$ Figure 2:
Depiction of the map $\mathcal{P}_{\varepsilon,m}$ introduced in Definition
3.8. On the left, we see $u_{0,p}^{m}$, the indicator function of the set
$E_{p,m}$, which is the set enclosed by the semi-sphere $\Sigma_{p,m}$ and
$\partial M$. The map $\mathcal{P}_{\varepsilon,m}$ returns
$u^{m}_{\varepsilon,m}$, depicted on the right, which is the smooth
approximation of $u_{0,p}^{m}$ given by the $\Gamma$-convergence result
(Theorem 3.1).

See Figure 2 for a picture aimed at clarifying Definition 3.8. The main result
of this section is the following, which provides an estimate on the smallest
sublevel of $\mathcal{E}_{\varepsilon,m}$ that contains the image of the
photography map.

######  Proposition 3.9.

Assume that (H1) and (H2) hold. There exists a constant
$\theta=\theta(M,g,W)>0$ such that there exists
$m_{1}=m_{1}(M,g,W,\theta)\in]0,m_{0}[$ such that for every $m\in]0,m_{1}[$
there exists $\varepsilon_{1}=\varepsilon_{1}(M,g,W,\theta,m)>0$ such that for
every $\varepsilon\in]0,\varepsilon_{1}[$ we have

$\mathcal{E}_{\varepsilon,m}(\mathcal{P}_{\varepsilon,m}(p))\leq\sigma
I_{M}(m)+\theta m,\qquad\forall p\in\partial M.$ (3.16)

In other words, the sublevel $\mathcal{E}_{\varepsilon,m}^{\sigma
I_{M}(m)+\theta m}$ contains the whole image of the photography map
$\mathcal{P}_{\epsilon,m}$.

###### Proof.

Combining (3.14) and (3.15), we get

$\mathcal{E}_{0}(u^{m}_{0,p})=\sigma\left(I_{M}(m)+\gamma_{n}\big{(}\max_{p\in\partial
M}\\{H_{\partial M}(p)\\}-H_{\partial
M}(p)\big{)}m\right)+\mathcal{O}(m^{\frac{n+1}{n}}).$ (3.17)

Hence, setting $\theta=\theta(M,g)>0$ as

$\theta=\sigma\gamma_{n}\left(\max_{p\in\partial M}\\{H_{\partial
M}(p)\\}-\min_{p\in\partial M}\\{H_{\partial M}(p)\\}+1\right),$ (3.18)

we have

$\mathcal{E}_{0}(u^{m}_{0,p})<\sigma I_{M}(m)+\theta
m+\mathcal{O}(m^{\frac{n+1}{n}}),\qquad\forall p\in\partial M.$ (3.19)

As a consequence, there exists $m_{1}=m_{1}(M,g,\theta)\in(0,m_{0})$ such that
for every $m\in(0,m_{1})$ we have

$\mathcal{E}_{0}(u^{m}_{0,p})<\sigma I_{M}(m)+\theta m,\qquad\forall
p\in\partial M.$ (3.20)

By Theorem 3.1 and Remark 3.2, we also know that

$\limsup_{\varepsilon\to
0^{+}}\mathcal{E}_{\varepsilon}(u^{m}_{\varepsilon,p})=\limsup_{\varepsilon\to
0^{+}}\mathcal{E}_{\varepsilon}(\mathcal{P}_{\varepsilon,m}(p))\leq\mathcal{E}_{0}(u^{m}_{0,p}),\qquad\forall
p\in\partial M.$ (3.21)

By the strict inequality in (3.20) and since $M$ is a compact manifold, by
(3.21) we infer the existence of
$\varepsilon_{1}=\varepsilon_{1}(M,g,W,\theta,m)>0$ such that (3.16) holds for
every $\varepsilon\in(0,\varepsilon_{1})$. ∎

######  Remark 3.10.

By (3.4) and (3.5), namely that $I_{M}(m)\approx m^{\frac{n-1}{n}}$ as $m$
goes to zero, we have

$\lim_{m\to 0^{+}}\frac{m}{I_{M}(m)}=0,$ (3.22)

hence the “width” of the sublevel that contains the photography goes rapidly
to $0$ as $m\to 0^{+}$.

In order to use $\mathcal{P}_{\varepsilon,m}$ as a photography map in the
sense of Theorem C, it remains to prove its continuity.

######  Proposition 3.11.

Assume that (H1) and (H2) hold. Let $m_{1}>0$ and $\varepsilon_{1}>0$ be
defined as in Proposition 3.9. For every $m\in(0,m_{1})$ and
$\varepsilon\in(0,\varepsilon_{1})$, the map
$\mathcal{P}_{\varepsilon,m}\colon\partial M\to\mathcal{H}_{m}$ is continuous.

###### Proof.

The proof follows the same line of the one of [14, Proposition 4.14], and we
report here the main steps.

We recall that for every $p\in\partial M$ and $\varepsilon>0$ the function
$\mathcal{P}_{\varepsilon,m}(p)$ is defined by using (3.8), hence we have

$\big{(}\mathcal{P}_{\varepsilon,m}(p)\big{)}(x)=\widetilde{q}_{\varepsilon}\big{(}d_{M\setminus
E_{p,m}}(x)+\delta_{\varepsilon,E_{p,m}}\big{)},$

where $d_{M\setminus E_{p,m}}$ is defined by using (3.6) and
$\widetilde{q}_{\varepsilon}$ is the Lipschitz continuous function that solves
(3.7). As a consequence, there exists a constant $C>0$, which depends on
$M,g,\varepsilon$ and $W$, such that the following inequality holds:

$\left\lVert\mathcal{P}_{\varepsilon,m}(p_{1})-\mathcal{P}_{\varepsilon,m}(p_{2})\right\rVert_{H^{1}(M,\mathbb{R})}\\\
\leq C\Big{[}\lVert d_{M\setminus E_{p_{1},m}}-d_{M\setminus
E_{p_{2},m}}\rVert_{H^{1}(M,\mathbb{R})}+|\delta_{\varepsilon,E_{p_{1},m}}-\delta_{\varepsilon,E_{p_{2},m}}|\Big{]}.$

The interested reader can find more details on this last inequality in [14,
Proposition 4.14]. Let us observe that, as long as $m<m_{0}$, the hypersurface
$\Sigma_{p,m}=\partial E_{p,m}\cap M$ is smooth and diffeomorphic to a
hemisphere. The proof ends by observing that, as $p_{1}\to p_{2}$, we have

$\lVert d_{M\setminus E_{p_{1},m}}-d_{M\setminus
E_{p_{2},m}}\rVert_{H^{1}(M,\mathbb{R})}\to
0\quad\text{and}\quad|\delta_{\varepsilon,E_{p_{1},m}}-\delta_{\varepsilon,E_{p_{2},m}}|\to
0,$

the last convergence obtained as a consequence of the Implicit Function
Theorem applied to (3.9), where $E_{p,m}$ substitutes $\Omega$, to define the
$C^{1}$ map $p\mapsto\delta_{\varepsilon,E_{p,m}}$. ∎

###  3.4 Barycenter map

The map $\mathcal{B}\colon\mathcal{E}_{\varepsilon,m}^{c}\to\partial M$ is
obtained by combining the $\Gamma$-convergence result with the information of
the isoperimetric problem given by Proposition 3.15, which ensures that the
“almost” minimizers of $\mathcal{E}_{0}$ have almost all their mass inside a
small ball centered on a point of the boundary $\partial M$. While the map
$\mathcal{P}_{\varepsilon,m}$ has been constructed by using the lim-sup part
of the $\Gamma$-convergence result, namely Theorem 3.1, to define
$\mathcal{B}$ we rely on the lim-inf statement and on the compactness result
ensured by Theorem 3.3.

Let us begin by introducing some new notation. Let $\mathrm{dist}_{g}\colon
M\times M\to\mathbb{R}$ be the distance function induced by $g$, hence

$\mathrm{dist}_{g}(x,y)\coloneqq\inf\left\\{\left(\int_{0}^{1}g(\dot{\gamma},\dot{\gamma})\mathrm{d}s\right)^{1/2}:\gamma\in
H^{1}([0,1],M),\gamma(0)=x,\gamma(1)=y\right\\}.$ (3.23)

With this notation, we set
$\mathrm{diam}_{g}(M)=\max\\{\mathrm{dist}_{g}(x,y):x,y\in M\\}$, and for any
$p\in M$ and $r\in(0,\mathrm{diam}_{g}(M))$, let

$B(p,r)\coloneqq\left\\{x\in M:\mathrm{dist}_{g}(p,x)\leq r\right\\}.$ (3.24)

Without loss of generality, from now on we assume that $M$ is isometrically
embedded in ${\mathbb{R}}^{\tilde{n}}$, for some $\tilde{n}\geq n$. This will
allow us to construct the map $\mathcal{B}$ as the composition of the
“extrinsic” barycenter map and the nearest point projection on $\partial M$.
More formally, let $(\partial M)_{r}\subset\mathbb{R}^{\tilde{n}}$ be the
following set

$(\partial
M)_{r}\coloneqq\left\\{z\in\mathbb{R}^{\tilde{n}}:\mathrm{dist}_{\mathbb{R}^{\tilde{n}}}(z,\partial
M)\leq r\right\\},$ (3.25)

and let us denote by $\pi_{\partial M}\colon(\partial M)_{r}\to\partial M$ the
nearest point projection onto $\partial M$. Notice that $\pi_{\partial M}$ is
well defined when $r$ is sufficiently small, thanks to the compactness
assumption on $\partial M$. We will combine $\pi_{\partial M}$ with the
following map

$\beta^{*}\colon u\in L^{1}(M,{\mathbb{R}})\to\frac{\int_{M}x\lvert
u(x)\rvert\mathrm{d}v_{g}}{\int_{M}\lvert
u(x)\rvert\mathrm{d}v_{g}}\in{\mathbb{R}}^{\tilde{n}},$ (3.26)

which gives the center of mass of a function in $\mathbb{R}^{\tilde{n}}$.

######  Lemma 3.12.

The map $\beta^{*}\colon L^{1}(M,\mathbb{R})\to\mathbb{R}^{\tilde{n}}$ is
continuous.

###### Proof.

For every $u_{1},u_{2}\in L^{1}(M,R)$, setting
$v_{1}=\int_{M}|u_{1}|\mathrm{d}v_{g}$ and
$v_{2}=\int_{M}|u_{2}|\mathrm{d}v_{g}$, we have

$\left\lVert\beta^{*}(u_{1})-\beta^{*}(u_{2})\right\rVert_{\mathbb{R}^{\tilde{n}}}=\left\lVert\frac{\int_{M}x|u_{1}(x)|\mathrm{d}v_{g}}{v_{1}}-\frac{\int_{M}x|u_{2}(x)|\mathrm{d}v_{g}}{v_{2}}\right\rVert_{\mathbb{R}^{\tilde{n}}}\\\
\leq\frac{2\lVert
x\rVert_{\infty}}{v_{1}}\int_{M}\left||u_{1}|-\frac{v_{1}}{v_{2}}|u_{2}|\right|\mathrm{d}v_{g},$

where $\lVert x\rVert_{\infty}=\max\\{\lVert
x\rVert_{\mathbb{R}^{\tilde{n}}}:x\in M\\}$. By using Lebesgue’s dominated
convergence and Hölder’s inequality, we get

$\frac{2\lVert
x\rVert_{\infty}}{v_{1}}\int_{M}\left||u_{1}|-\frac{v_{1}}{v_{2}}|u_{2}|\right|\mathrm{d}v_{g}\to
0\quad\text{as}\quad\lVert u_{1}-u_{2}\rVert_{L^{1}(M,\mathbb{R})}\to 0,$

so we obtain that

$\left\lVert\beta^{*}(u_{1})-\beta^{*}(u_{2})\right\rVert_{\mathbb{R}^{\tilde{n}}}\to
0,\quad\text{as}\quad\lVert u_{1}-u_{2}\rVert_{L^{1}(M,\mathbb{R})}\to 0,$

which concludes the proof. ∎

In order to define $\mathcal{B}\colon\mathcal{E}_{\varepsilon,m}^{\sigma
I_{M}(m)+\theta m}\to\partial M$, one needs to show that for every $r>0$ there
exist $m$ and $\varepsilon$ sufficiently small such that

$\beta^{*}(\mathcal{E}_{\varepsilon,m}^{\sigma I_{M}(m)+\theta
m})\subset(\partial M)_{r}.$ (3.27)

In other words, we need to prove that if $m$ and $\varepsilon$ are
sufficiently small, then the barycenter of every function in the sublevel
$\mathcal{E}_{\varepsilon,m}^{\sigma I_{M}(m)+\theta m}$ lies in a small
tubular neighbourhood of $\partial M$. Then, choosing $r$ sufficiently small
as well, one can define $\mathcal{B}\colon\mathcal{E}^{\sigma I_{M}(m)+\theta
m}_{\varepsilon,m}\to\partial M$ as follows:

$\mathcal{B}=\pi_{\partial M}\circ\beta^{*}.$ (3.28)

To this end, we need to ensure that if the volume $m$ is sufficiently small
then any function in the sublevel $\mathcal{E}_{\varepsilon,m}^{\sigma
I_{M}(m)+\theta m}$ has almost all its mass in a small ball centered on a
point of the boundary $\partial M$ (see Proposition 3.15). The proof of this
result is based on [6, Theorem 1.2], which gives a compactness result for
sequences of sets with uniformly bounded volume and perimeter. For the sake of
presentation, we restate here that theorem. If
$(X,\mathrm{d},\mathcal{H}^{n})$ is a metric measure space (where
$\mathcal{H}^{n}$ denots the $n$–dimensional Hausdorff measure) and
$\Omega\subset X$ is a Borel set, we remind that the definition of perimeter
of $\Omega$ used in the following theorem is

$P(\Omega,X)\coloneqq\inf\left\\{\liminf_{i}\int_{X}\mathrm{lip}f_{i}\mathrm{d}\mathcal{H}^{n}:f_{i}\in\mathrm{Lip}_{\mathrm{loc}}(X),f_{i}\to\mathbf{1}_{\Omega}\text{
in }L^{1}_{\text{loc}}(X,\mathcal{H}^{n})\right\\},$

where $\mathrm{lip}f(x)\coloneqq\limsup_{y\to
x}\frac{|f(y)-f(x)|}{\mathrm{d}(x,y)}$. Moreover, we recall that a metric
space $(X,\mathrm{d},\mathcal{H}^{n})$ is an $\mathrm{RCD}(\kappa,n)$ space
if, loosely speaking, it has synthetic notions of Ricci curvature bounded
below by $\kappa\in\mathbb{R}$ and dimension bounded above by
$n\in(0,+\infty]$ (see [6] for more details).

######  Theorem 3.13 (Theorem 1.2 of [6]).

Let $\kappa\in\mathbb{R}$ and $n\geq 2$. Let
$(X_{k},\mathrm{d}_{k},\mathcal{H}^{n}_{k})$ be a sequence of
$\mathrm{RCD}(\kappa,n)$ spaces and let $\Omega_{k}\subset X_{k}$ be bounded
sets of finite perimeter such that
$\sup_{k}\big{(}P(\Omega_{k},X_{k})+\mathcal{H}^{n}_{k}(\Omega_{k})\big{)}<+\infty$.
Assume there exists $v_{0}>0$ such that $\mathcal{H}^{n}_{k}(B(x,1))\geq
v_{0}$ for every $x\in X_{k}$ and for every $k\in\mathbb{N}$. Then, up to
subsequences, there exists a nondecreasing sequence
$(N_{k})_{k\in\mathbb{N}}\subset\mathbb{N}$, $N_{k}\geq 1$, points $p_{k,i}\in
X_{k}$, with $1\leq i\leq N_{k}$ and pairwise disjoint subsets
$\Omega_{k,i}\subset\Omega_{k}$ such that:

  1. i)

$\lim_{k\to\infty}\mathrm{d}_{k}(p_{k,i},p_{k,j})=+\infty$ for any $i\neq
j\leq\bar{N}$, where
$\overline{N}\coloneqq\lim_{k\to\infty}N_{k}\in\mathbb{N}\cup\\{+\infty\\}$;

  2. ii)

For every $1\leq i<\overline{N}+1$, the sequence
$(X_{k},\mathrm{d}_{k},\mathcal{H}^{n},p_{k,i})$ converges in the pointed
measured Gromov-Hausdorff topology to a pointed $\mathrm{RCD}(\kappa,n)$ space
$(Y_{i},\mathrm{dist}_{Y_{i}},\mathcal{H}^{n},p_{i})$ as $k\to\infty$;

  3. iii)

There exists $F_{i}\subset Y_{i}$ such that $\Omega_{k,i}\to F_{i}$ as
$k\to\infty$ in $L^{1}$–strong topology and the following relations hold:

$\lim_{k\to\infty}\mathcal{H}^{n}_{k}(\Omega_{k})=\sum_{i=1}^{\overline{N}}\mathcal{H}^{n}(F_{i}),$

and

$\sum_{i=1}^{\overline{N}}P(F_{i},Y_{i})\leq\liminf_{k\to\infty}P(\Omega_{k},X_{k}).$

Moreover, if $\Omega_{k}$ is an isoperimetric set in $X_{k}$ for any $k$, then
$F_{i}$ is an isoperimetric set in $Y_{i}$ for any $i<\overline{N}+1$ and

$P(F_{i},Y_{i})=\lim_{k\to\infty}P(\Omega_{k,i},X_{k})$ (3.29)

for any $i<\overline{N}+1$.

######  Remark 3.14.

In the following, we use the previous theorem on the more simple case of
Riemannian manifolds. We want to highlight that when the $\mathrm{RCD}$ space
is $(\mathrm{int}(M),g,\mathcal{H}^{n})$, the perimeter of a measurable set
$\Omega\subset\mathrm{int}(M)$ coincides with the $(n-1)$–dimensional
Hausdorff measure of its relative boundary, hence

$P(\Omega,\mathrm{int}(M))=\mathcal{H}^{n-1}(\partial^{*}\Omega\cap\mathrm{int}(M)).$

When we consider the space $(M,g,\mathcal{H}^{n})$, the whole perimeter of the
set $\Omega\subset M$ is considered, namely

$P(\Omega,M)=\mathcal{H}^{n-1}(\partial^{*}\Omega).$

######  Proposition 3.15.

There exists $\mu=\mu(M,g)>0$ such that the following property holds. For
every almost isoperimetric sequence $(u_{k})_{k\in\mathbb{N}}\subset
L^{1}(M,\mathbb{R})$ with volumes $m_{k}=\int_{M}u_{k}\mathrm{d}v_{g}\to 0$,
i.e.,

$\lim_{k\to\infty}\frac{\mathcal{E}_{0}(u_{k})}{\sigma I_{M}(m_{k})}=1,$
(3.30)

there exists a sequence $(p_{k})_{k\in\mathbb{N}}\subset\partial M$ such that

$\lim_{k\to+\infty}\frac{1}{m_{k}}\left(\int_{M\setminus B(p_{k},\mu
m_{k}^{1/n})}u_{k}\mathrm{d}v_{g}\right)=0.$ (3.31)

###### Proof of Proposition 3.15.

For every fixed $k\in\mathbb{N}$, let $\Omega_{k}\in\mathcal{C}_{g}(M)$ be
such that $u_{k}=\textbf{1}_{\Omega_{k}}$. Note that, given that $u_{k}\in
L^{1}(M,\mathbb{R})$, we can select any $\Omega_{k}$ in such a way that
$\Omega_{k}$ is a subset of the interior of $M$. Let us define the following
sequence of manifolds of bounded geometry:

$(X_{k},g_{k})\coloneqq(\mathrm{int}(M),m_{k}^{-1/n}g).$

In other words, we rescale the metric on $M$ by a factor that depends on
$m_{k}$ in such a way that

$\mathcal{H}_{k}^{n}(\Omega_{k})=\int_{X_{k}}u_{k}\mathrm{d}v_{g_{k}}=1,\qquad\forall
k\in\mathbb{N},$

where $\mathcal{H}_{k}^{n}$ denotes the $n$–dimensional Hausdorff measure on
$(X_{k},g_{k})$. Moreover, by (3.5) and (3.30) and recalling that
$\mathcal{E}_{0}(u_{k})=\sigma\mathcal{H}^{n-1}(\partial^{*}\Omega_{k}\cap\mathrm{int}(M))$,
we obtain the existence of a constant $C$ such that

$P(\Omega_{k},X_{k})=\mathcal{H}_{k}^{n-1}(\partial^{*}\Omega_{k}\cap\mathrm{int}(M))<I_{\mathbb{R}^{n}_{+}}(1)+C,\qquad\forall
k\in\mathbb{N}.$

Moreover, we have also the following equality:

$\lim_{k\to\infty}\mathcal{H}_{k}^{n-1}(\partial^{*}\Omega_{k}\cap\mathrm{int}(M))=I_{\mathbb{R}^{n}_{+}}(1).$
(3.32)

This means that the sequence
$\\{(X_{k},g_{k},\mathcal{H}^{n}_{k})\\}_{k\in\mathbb{N}}$ and
$\Omega_{k}\subset M_{k}$ satisfy the hypotheses of Theorem 3.13 and so, up to
subsequences, for every $k$ there exist $N_{k}\in\mathbb{N}$, with $N_{k}\geq
1$, and points $p_{k,i}\in M_{k}$ such that for every $i\in 1,\dots,N_{k}$ the
space $(X_{k},\mathrm{dist}_{g_{k}},\mathcal{H}^{n}_{k},p_{k,i})$ converges in
the Gromov-Hausdorff topology to a pointed $\mathrm{RCD}(\kappa,n)$ as
$k\to\infty$, that we denote by
$(M^{i}_{\infty},\mathrm{d}^{i}_{\infty},\mathcal{H}^{n,i}_{\infty},p^{i}_{\infty})$.
This space could be either the Euclidean semi-space
$(\mathbb{R}^{n}_{+},\mathrm{d},\mathcal{H}^{n},0)$ or the Euclidean space
$(\mathbb{R}^{n},\mathrm{d},\mathcal{H}^{n},0)$, depending on if
$(p_{k,i})_{k}$ converges to a point in $\partial M$ or not, respectively.
Moreover, setting $\overline{N}=\lim_{k\to\infty}N_{k}$, for every
$i=1,\dots,\overline{N}$ there exists $\Omega_{\infty}^{i}\subset
M^{i}_{\infty}$ such that

$\sum_{i=1}^{\overline{N}}\mathcal{H}^{n,i}_{\infty}(\Omega_{\infty}^{i})=\lim_{k\to\infty}\mathcal{H}^{n}_{k}(\Omega_{k})=1$
(3.33)

and, using also (3.32), we have

$\sum_{i=1}^{\overline{N}}\mathcal{H}^{n-1,i}_{\infty}(\partial^{*}\Omega_{\infty}^{i}\cap\mathrm{int}(M^{i}_{\infty}))\leq\liminf_{k\to\infty}\mathcal{H}_{k}^{n-1}(\partial^{*}\Omega_{k}\cap\mathrm{int}(M))=I_{\mathbb{R}^{n}_{+}}(1).$
(3.34)

By using contradiction arguments, this last inequality implies both that
$\overline{N}=1$ and that
$(M^{1}_{\infty},\mathrm{d}^{1}_{\infty},\mathcal{H}^{n}_{\infty},p^{1}_{\infty})$
is actually the Euclidean semispace
$(\mathbb{R}^{n}_{+},\mathrm{d},\mathcal{H}^{n},0)$, since
$I_{\mathbb{R}^{n}_{+}}(1)<I_{\mathbb{R}^{n}}(1)$. As a consequence, by using
(3.33), we get the existence of $\mu>0$ and a sequence $p_{k}\in\partial M$
such that (3.31) holds. ∎

Using the same constants $\theta$ and $\mu$ given by Proposition 3.9 and
Proposition 3.15, respectively, we obtain the following result, which ensures
that all the functions in the sublevel that contains the image of the
photography map have their mass concentrated near the boundary.

######  Lemma 3.16.

Assume that (H1) and (H2) hold. For every $\alpha\in(0,1)$, there exists
$m_{\alpha}=m_{\alpha}(M,g,W,\theta,\alpha)>0$ such that for every
$m\in(0,m_{\alpha})$ there exists
$\varepsilon_{\alpha}=\varepsilon_{\alpha}(M,g,W,\theta,\alpha,m)>0$ such that
for every $\varepsilon\in(0,\varepsilon_{\alpha})$ and for any
$u\in\mathcal{E}_{\varepsilon,m}^{\sigma I_{M}(m)+\theta m}$ there exists a
point $p_{u}\in\partial M$ such that

$\int_{M\setminus B(p_{u},\mu m^{1/n})}|u|\mathrm{d}v_{g}\leq\alpha m.$ (3.35)

###### Proof.

Arguing by contradiction, there exists a sequence $(m_{i})_{i\in\mathbb{N}}$
such that $m_{i}\to 0^{+}$ and for every $i\in\mathbb{N}$ there exist two
sequences $(\varepsilon_{i,j})_{j\in\mathbb{N}}\in\mathbb{R}$ and
$(u_{i,j})_{j\in\mathbb{N}}\in\mathcal{E}_{\varepsilon_{i,j},m_{i}}^{c_{i}}$,
with $c_{i}=I_{M}(m_{i})+\theta m_{i}$, such that for every $i\in\mathbb{N}$
we have $\varepsilon_{i,j}\to 0^{+}$ as $j\to\infty$ and

$\int_{M\setminus B_{g}(p,\mu m_{i}^{1/n})}|u_{i,j}|\mathrm{d}v_{g}>\alpha
m_{i},\qquad\forall p\in\partial M,\,\forall j\in\mathbb{N}.$ (3.36)

Since for any fixed $i\in\mathbb{N}$ we have
$\mathcal{E}_{\varepsilon_{i,j},m_{i}}(u_{i,j})\leq c_{i}$ for every
$j\in\mathbb{N}$, we can apply Theorem 3.3 to obtain a sequence of
characteristic functions $(u_{0,i})_{i}\subset L^{1}(M,\mathbb{R})$ such that,
for every $i$, we have $\lim_{j\to\infty}\lVert
u_{i,j}-u_{0,i}\rVert_{L^{1}(M,\mathbb{R})}=0$, up to subsequences (see also
Remark 3.4). Hence, for every $i$ there exists $j_{i}$ such that

$\int_{M}\big{(}|u_{i,j_{i}}|-u_{0,i}\big{)}\mathrm{d}v_{g}\leq\frac{\alpha}{4}m_{i},$
(3.37)

and we have also $\int_{M}u_{0,i}\mathrm{d}v_{g}=m_{i}$ and
$\mathcal{E}_{0}(u_{0,i})\leq c_{i}$, namely

$I_{M}(m_{i})\leq\frac{\mathcal{E}_{0}(u_{0,i})}{\sigma}\leq
I_{M}(m_{i})+\theta m_{i}.$

Using also (3.22), we obtain

$\lim_{i\to\infty}\frac{\mathcal{E}_{0}(u_{0,i})}{\sigma I_{M}(m_{i})}=1,$

so we can apply Proposition 3.15 to obtain the existence of a sequence
$(p_{i})_{i\in\mathbb{N}}\subset\partial M$ such that

$\lim_{i\to\infty}\frac{1}{m_{i}}\left(\int_{M\setminus B(p_{i},\mu
m_{i}^{1/n})}u_{0,i}\,\mathrm{d}v_{g}\right)=0,$

so there exists $i_{0}$ such that

$\int_{M\setminus B(p_{i},\mu
m_{i}^{1/n})}u_{0,i}\,\mathrm{d}v_{g}\leq\frac{\alpha}{4}m_{i},\qquad\forall
i\geq i_{0}.$ (3.38)

As a consequence, combining (3.37) and (3.38), for every $i>i_{0}$ we obtain

$\int_{M\setminus B(p_{i},\mu
m_{i}^{1/n})}|u_{i,j_{i}}|\mathrm{d}v_{g}=\int_{M\setminus B(p_{i},\mu
m_{i}^{1/n})}\Big{(}|u_{i,j_{i}}|-u_{0,i}\Big{)}\mathrm{d}v_{g}\\\
+\int_{M\setminus B(p_{i},\mu m_{i}^{1/n})}u_{0,i}\,\mathrm{d}v_{g}\\\
\leq\int_{M}\big{(}|u_{i,j_{i}}|-u_{0,i}\big{)}\mathrm{d}v_{g}+\int_{M\setminus
B(p_{i},\mu m_{i}^{1/n})}u_{0,i}\,\mathrm{d}v_{g}\leq\frac{\alpha}{2}m_{i},$

which contradicts (3.36). ∎

Using the previous result, we can ensure that if $m$ and $\varepsilon$ are
sufficiently small, the extrinsic barycenter of any function in the sublevel
of $\mathcal{E}_{\varepsilon,m}$ that contains the image of the photograpy map
is near the boundary of the manifold. More formally, setting

$\mathrm{diam}_{\mathbb{R}^{\tilde{n}}}(M)\coloneqq\max\big{\\{}\lVert
x-y\rVert_{\mathbb{R}^{\tilde{n}}}:x,y\in M\big{\\}},$

we have the following result.

######  Lemma 3.17.

Assume that (H1) and (H2) hold. For every $r>0$, there exists
$m_{2}=m_{2}(M,g,r,\mathrm{diam}_{\mathbb{R}^{\tilde{n}}}(M))>0$ such that for
every $m\in(0,m_{2})$ there exists
$\varepsilon_{2}=\varepsilon_{2}(M,g,r,m)>0$ such that for every
$\varepsilon\in(0,\varepsilon_{2})$ and any
$u\in\mathcal{E}_{\varepsilon,m}^{\sigma I_{M}(m)+\theta m}$ we have
$\beta^{*}(u)\in(\partial M)_{r}$.

###### Proof.

Recalling the definition of $\beta^{*}$ given by (3.26), for every $u\in
H^{1}(M,\mathbb{R})$ and $y\in M$, let us define

$\rho(u,y)=\frac{|u(y)|}{\int_{M}|u(x)|\mathrm{d}v_{g}},$

so that

$\beta^{*}(u)=\int_{M}x\rho(u,x)\mathrm{d}v_{g}\quad\text{and}\quad\int_{M}\rho(u,x)\mathrm{d}v_{g}=1,\,\forall
u\in H^{1}(M,\mathbb{R}).$

Let us choose $\alpha>0$ such that

$\alpha\leq\frac{r}{2\mathrm{diam}_{\mathbb{R}^{\tilde{n}}}(M)},$

and let $m_{\alpha}$ be given by Lemma 3.16. Let us choose
$m\in(0,m_{\alpha})$ and let $\varepsilon\in(0,\varepsilon_{\alpha})$. By
Lemma 3.16, for every $u\in\mathcal{E}_{\varepsilon,m}^{\sigma I_{M}(m)+\theta
m}$ there exists $p_{u}\in\partial M$ such that (3.35) holds. As a
consequence, for every $u\in\mathcal{E}_{\varepsilon,m}^{\sigma
I_{M}(m)+\theta m}$ we obtain

$\lVert\beta^{*}(u)-p_{u}\rVert_{\mathbb{R}^{\tilde{n}}}=\left\lVert\int_{M}\big{(}x-p_{u}\big{)}\rho(u,x)\mathrm{d}v_{g}\right\rVert_{\mathbb{R}^{\tilde{n}}}\\\
\leq\left\lVert\int_{B(p_{u},\mu
m^{1/n})}\big{(}x-p_{u}\big{)}\rho(u,x)\mathrm{d}v_{g}\right\rVert_{\mathbb{R}^{\tilde{n}}}+\left\lVert\int_{M\setminus
B(p_{u},\mu
m^{1/n})}\big{(}x-p_{u}\big{)}\rho(u,x)\mathrm{d}v_{g}\right\rVert_{\mathbb{R}^{\tilde{n}}}\\\
\leq\mu m^{1/n}+\alpha\mathrm{diam}_{\mathbb{R}^{\tilde{n}}}(M)\leq\mu
m^{1/n}+\frac{r}{2}.$

As a consequence, choosing $m_{2}\in(0,m_{\alpha})$ such that

$\mu m_{2}^{1/n}\leq\frac{r}{2},$

for all $m\in(0,m_{2})$ and for all $\varepsilon\in(0,\varepsilon_{2})$, with
$\varepsilon_{2}=\varepsilon_{\alpha}$, we obtain

$\lVert\beta^{*}(u)-p_{u}\rVert_{\mathbb{R}^{\tilde{n}}}\leq r,\qquad\forall
u\in\mathcal{E}_{\varepsilon,m}^{I_{M}(m)+\theta m},$

and we are done. ∎

######  Remark 3.18.

By Lemma 3.17 and the compactness of $\partial M$, there exists $r_{1}>0$ such
that, choosing $m_{2}(M,g,r_{1},\mathrm{diam}_{\mathbb{R}^{\tilde{n}}})>0$ and
$\varepsilon_{2}(M,g,r_{1},m)>0$ as in the lemma, for every $m\in(0,m_{2})$
and $\varepsilon\in(0,\varepsilon_{2})$, the map $\mathcal{B}=\pi_{\partial
M}\circ\beta^{*}\colon\mathcal{E}_{\varepsilon,m}^{\sigma I_{M}(m)+\theta
m}\to\partial M$ is well defined and continuous.

###  3.5 Conclusion of the proof

The following two results, namely, Lemma 3.19 and Lemma 3.20, demonstrate that
the barycenter of a function obtained by applying the photography map to a
point $p\in\partial M$ is in close proximity to $p$ itself when $m$ and
$\varepsilon$ are sufficiently small. As a result, taking into account the
definition of $\mathcal{B}$ provided in (3.28), the composition
$\mathcal{B}\circ\mathcal{P}_{\varepsilon,m}\colon\partial M\to\partial M$
closely approximates the identity map.

######  Lemma 3.19.

Assume that (H1) and (H2) hold. For every $r\in]0,r_{1}[$ there exists
$m_{3}=m_{3}(M,g,r)>0$ such that for every $m\in(0,m_{3})$ there exists
$\varepsilon_{3}=\varepsilon_{3}(M,g,r,m)>0$ such that

$\lVert\beta^{*}(\mathcal{P}_{\varepsilon,m}(p))-p\rVert_{\mathbb{R}^{\tilde{n}}}\leq
r,\qquad\forall p\in\partial M.$ (3.39)

###### Proof.

We recall that for every $p\in\partial M$ and every $m\in(0,\mathrm{vol}(M))$,
the radius $r_{p,m}$ is the one such that

$\int_{E_{p,r_{p,m}}}1\mathrm{d}v_{g}=m.$

By the compactness of $\partial M$, there exists $m_{3}=m_{3}(M,g,r)>0$ such
that for every $m\in(0,m_{3})$ and for every $p\in\partial M$ we have
$r_{p,m}<r/2$. As a consequence, for every $x\in E_{p,r_{p,m}}\subset M$ we
have

$\lVert x-p\rVert_{\mathbb{R}^{\tilde{n}}}<\frac{r}{2}.$ (3.40)

Moreover, we recall that for every $m,\varepsilon>0$ the function
$\mathcal{P}_{\varepsilon,m}(p)$ is defined as the $\varepsilon$–approximation
of $u_{0,x}^{m}=\mathbf{1}_{E_{p,m}}$ and that $u_{\varepsilon,p}^{m}$
converges to $u_{0,x}^{m}$ in the $L^{1}(M,\mathbb{R})$ norm as
$\varepsilon\to 0$. Hence, using also the continuity of $\beta^{*}$ with
respect to the $L^{1}$–norm ensured by Lemma 3.12 and again the compactness of
$\partial M$, we obtain the existence of a constant
$\varepsilon_{3}=\varepsilon_{3}(M,g,r,m)>0$ such that for every
$\varepsilon\in(0,\varepsilon_{3})$ and for every $p\in\partial M$ the
following holds:

$\lVert\beta^{*}(\mathcal{P}_{\varepsilon,m}(p))-\beta^{*}(u_{0,p}^{m})\rVert_{\mathbb{R}^{\tilde{n}}}=\lVert\beta^{*}(u_{\varepsilon,p}^{m})-\beta^{*}(u_{0,p}^{m})\rVert_{\mathbb{R}^{\tilde{n}}}\leq\frac{r}{2}.$
(3.41)

As a consequence, using both (3.40) and(3.41), for every $m\in(0,m_{3})$ and
for every $\varepsilon\in(0,\varepsilon_{3})$ we obtain

$\lVert\beta^{*}(\mathcal{P}_{\varepsilon,m}(p))-p\rVert_{\mathbb{R}^{\tilde{n}}}\leq\lVert\beta^{*}(\mathcal{P}_{\varepsilon,m}(p))-\beta^{*}(u_{0,p}^{m})\rVert_{\mathbb{R}^{\tilde{n}}}+\lVert\beta^{*}(u_{0,p}^{m})-p\rVert_{\mathbb{R}^{\tilde{n}}}\\\
\leq\frac{r}{2}+\frac{1}{\int_{M}|u_{0,p}^{m}|\mathrm{d}v_{g}}\int_{M}\Big{(}\left\lVert
x-p\right\rVert_{\mathbb{R}^{\tilde{n}}}|u_{0,p}^{m}|\Big{)}\mathrm{d}v_{g}\leq
r,\qquad\forall p\in\partial M,$

which concludes the proof. ∎

######  Lemma 3.20.

Assume that (H1) and (H2) hold. There exists $m^{*}=m^{*}(M,g,W)>0$ such that
for any $m\in(0,m^{*})$ there exists
$\varepsilon^{*}=\varepsilon^{*}(M,g,W,m)>0$ such that for any
$\varepsilon\in(0,\varepsilon^{*})$ the composition map

$\mathcal{B}\circ\mathcal{P}_{\varepsilon,m}\colon\partial M\to\partial M$

is well defined and homotopic to the identity map.

###### Proof.

For $p\in\partial M$, let $\mathrm{exp}^{\partial M}_{p}$ be the exponential
map of $\partial M$ at $p$, and let $B^{\partial M}_{g}(p,{R})$ be the
geodesic ball of center $p$ and radius $R$ on $\partial M$. We denote by
$\mathrm{inj}(\partial M)$ the _injectivity radius_ of $\partial M$, that is

$\mathrm{inj}(\partial M)\coloneqq\inf_{p\in\partial M}\sup\big{\\{}R>0\text{
s.t. }\mathrm{exp}^{\partial M}_{p}\colon B^{\partial M}_{g}(p,{R})\to\partial
M\mbox{ is a diffeomorphism}\big{\\}}$

which is positive since $\partial M$ is a compact manifold. Moreover, the
compactness of $\partial M$ implies that there exists a constant $C_{\partial
M}>0$ such that

$\mathrm{dist}_{g}(p,q)\leq C_{\partial M}\lVert
p-q\rVert_{\mathbb{R}^{\tilde{n}}}\qquad\forall p,q\in\partial M.$

Recalling the definition of $r_{1}>0$ given in Remark 3.18, let

$r^{*}=\min\big{\\{}r_{1},\mathrm{inj}(\partial M)/(4C_{\partial
M})\big{\\}},$

let $m_{2}=m_{2}(M,g,r^{*},\mathrm{diam}_{\mathbb{R}^{\tilde{n}}}(M))>0$ be
defined by Lemma 3.17, and for every $m\in(0,m_{2})$, set
$\varepsilon_{2}=\varepsilon_{2}(M,g,r^{*},m)>0$ in the same way. Moreover,
let $m_{3}=m_{3}(M,g,r^{*})>0$ and
$\varepsilon_{3}=\varepsilon_{3}(M,g,r^{*},m)$ be similarly defined by Lemma
3.19. Recalling Proposition 3.9, let $m^{*}=\min\\{m_{1},m_{2},m_{3}\\}$ and
for every $m\in(0,m^{*})$ set
$\varepsilon^{*}=\min\\{\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}\\}$.
By Proposition 3.9, for any $p\in\partial M$ we have
$\mathcal{P}_{\varepsilon,m}(p)\in\mathcal{E}_{\varepsilon,m}^{\sigma
I_{M}(m)+\theta m}$, so we obtain

$\mathrm{dist}_{g}\big{(}\mathcal{B}(\mathcal{P}_{\varepsilon,m}(p)),p\big{)}\leq
C_{\partial M}\lVert\pi_{\partial
M}(\beta^{*}(\mathcal{P}_{\varepsilon,m}(p)))-p\rVert_{\mathbb{R}^{\tilde{n}}}\\\
\leq C_{\partial M}\big{(}\lVert\pi_{\partial
M}(\beta^{*}(\mathcal{P}_{\varepsilon,m}(p)))-\beta^{*}(\mathcal{P}_{\varepsilon,m}(p))\rVert_{\mathbb{R}^{\tilde{n}}}+\lVert\beta^{*}(\mathcal{P}_{\varepsilon,m}(p))-p\rVert_{\mathbb{R}^{\tilde{n}}}\big{)}.$
(3.42)

Since $m^{*}$ and $\varepsilon^{*}$ are, respectively, less than $m_{2}$ and
$\varepsilon_{2}$ defined as in Lemma 3.17, we have

$\lVert\pi_{\partial
M}(\beta^{*}(\mathcal{P}_{\varepsilon,m}(p)))-\beta^{*}(\mathcal{P}_{\varepsilon,m}(p))\rVert_{\mathbb{R}^{\tilde{n}}}\leq
r^{*},$

and, by applying in an analogous manner Lemma 3.19, we obtain also

$\lVert\beta^{*}(\mathcal{P}_{\varepsilon,m}(p))-p\rVert_{\mathbb{R}^{\tilde{n}}}\leq
r^{*}.$

Hence, from (3.42) we infer

$\mathrm{dist}_{g}\big{(}\mathcal{B}(\mathcal{P}_{\varepsilon,m}(p)),p\big{)}\leq
2C_{\partial M}r^{*}\leq\frac{1}{2}\mathrm{inj}(\partial M).$

As a consequence, the map $F\colon[0,1]\times\partial M\to\partial M$, given
by

$F(t,p)\coloneqq\mathrm{exp}^{\partial
M}_{p}\Big{(}t\big{(}\mathrm{exp}^{\partial
M}_{p}\big{)}^{-1}\big{(}\mathcal{B}(\mathcal{P}_{\varepsilon,m}(p))\big{)}\Big{)},$

is well-defined, continuous and gives a homotopy equivalence between
$\mathcal{B}\circ\mathcal{P}_{\varepsilon,m}$ and the identity map in
$\partial M$. ∎

Finally, we are ready to prove Theorem A.

###### Proof of Theorem A.

For every $m,\varepsilon>0$, the functional $\mathcal{E}_{\varepsilon,m}$ is
clearly bounded below and, by Lemma 2.1, it is of class $C^{1}$ and satisfies
the Palais-Smale condition.

Let us choose $m^{*}$ as in Lemma 3.20 and, for every $m\in(0,m^{*})$, let
$\varepsilon_{m}$ be equal to $\varepsilon^{*}$ of the same lemma and set
$c_{m}=\sigma I_{M}(m)+\theta m$. By Proposition 3.11 and Remark 3.18, for
every $\varepsilon\in(0,\varepsilon_{m})$ both
$\mathcal{P}_{\varepsilon,m}\colon\partial
M\to\mathcal{E}_{\varepsilon,m}^{c_{m}}$ and
$\mathcal{B}\colon\mathcal{E}_{\varepsilon,m}^{c_{m}}\to\partial M$ are
continuous and, by Lemma 3.20, their composition is homotopic to the identity.
Then, Theorem A directly follows from Theorem C. In particular, there exist at
least $\mathrm{cat}(\partial M)$ critical points of
$\mathcal{E}_{\varepsilon,m}$ in $\mathcal{E}_{\varepsilon,m}^{c_{m}}$ and,
recalling that $\mathrm{cat}(\partial M)>1$ and $\mathcal{H}_{m}$ is a
contractible set (see Remark 2.2), there exist at least one critical point
with energy larger than $c_{m}$. By Lemma 2.1, these critical points are
solutions of (1.2). Moreover, if all the critical points of
$\mathcal{E}_{\varepsilon,m}$ are non-degenerate, then Theorem C and Lemma 2.1
ensure that (1.2) has at least $\mathscr{P}_{1}(\partial M)$ solutions with
energy less than $c_{m}$ and $\mathscr{P}_{1}(\partial M)-1$ solutions with
energy larger than $c_{m}$, and we are done. ∎

##  4 Proof of Theorem B

The proof of Theorem B follows essentially the same structure of the proof of
Theorem A, by applying again Theorem C. Relying on some $\Gamma$–convergence
results for $\overline{\mathcal{E}}_{\varepsilon,m}$ ([44, 34]), we can
construct the photography map $\mathcal{P}_{\varepsilon,m}\colon
M\to\overline{\mathcal{E}}_{\varepsilon,m}^{c}$, where the value of $c$ can be
estimated exploiting the convergence of the isoperimetric problem on $M$ that
takes into account also the boundary $\partial M$ to the standard Euclidean
case as $m$ goes to $0$.

###  4.1 The isoperimetric problem

We define the functional $\overline{\mathcal{E}}_{0}\colon
L^{1}(M,\mathbb{R})\to\mathbb{R}$ as

$\overline{\mathcal{E}}_{0}(u)\coloneqq\begin{cases}\sigma\mathcal{H}^{n-1}(\partial^{*}\Omega),&\mbox{
if }u=\mathbf{1}_{\Omega},\\\ +\infty,&\mbox{ otherwise},\end{cases}$

where $\sigma\in\mathbb{R}$ is given by (3.1). Let
$\overline{I}\colon]0,\mathrm{vol}(M)]\to\mathbb{R}$ be the following
function:

$\overline{I}_{M}(m)\coloneqq\inf\left\\{\mathcal{H}^{n-1}(\partial^{*}\Omega):\Omega\in\mathcal{C}_{g}(M)\text{
and }\int_{M}\mathbf{1}_{\Omega}\mathrm{d}v_{g}=m\right\\}.$ (4.1)

The quantity $\overline{I}_{M}$ is the isoperimetric profile of $M$ that takes
also into account the boundary $\partial M$. For small volumes, the function
$\overline{I}_{M}$ converges to its Euclidean counterpart, which is
$I_{\mathbb{R}^{n}}(m)=c_{n}m^{\frac{n-1}{n}}$, where $c_{n}$ is the Euclidean
isoperimetric constant. More formally, we have the following result.

######  Lemma 4.1.

The following equality holds:

$\lim_{m\to 0^{+}}\frac{\overline{I}_{M}(m)}{I_{\mathbb{R}^{n}}(m)}=\lim_{m\to
0^{+}}\frac{\overline{I}_{M}(m)}{c_{n}m^{\frac{n-1}{n}}}=1.$ (4.2)

###### Proof.

For every $p\in\mathrm{int}(M)$, if $m$ is sufficiently small we have that the
geodesic ball centered at $p$ and with volume $m$ does not intersect the
boundary. Setting $u_{0,p}^{m}\in L^{1}(M,\mathbb{R})$ the characteristic
function of this ball, by [43, Lemma 3.10] we have

$\overline{\mathcal{E}}_{0}(u_{0,p}^{m})=\sigma
c_{n}m^{\frac{n-1}{n}}+o(m^{\frac{n-1}{n}}).$

As a consequence, if $m$ is sufficiently small we have
$\overline{I}_{M}(m)\leq c_{n}m^{\frac{n-1}{n}}+o(m^{\frac{n-1}{n}})$, so we
obtain

$\limsup_{m\to 0}\frac{\overline{I}_{M}(m)}{I_{\mathbb{R}^{n}}(m)}\leq 1.$

So it remains to prove that

$\liminf_{m\to 0}\frac{\overline{I}_{M}(m)}{I_{\mathbb{R}^{n}}(m)}\geq 1.$

Arguing by contradiction, there exists a sequence $(m_{k})_{k\in\mathbb{N}}$
which goes to $0$, a sequence of isoperimetric sets
$(\Omega_{k})_{k\in\mathbb{N}}\subset\mathcal{C}_{g}(M)$ such that
$\mathrm{vol}(\Omega_{k})=m_{k}$ for every $k\in\mathbb{N}$ and
$\alpha\in(0,1)$ such that

$\overline{I}_{M}(m_{k})=\mathcal{H}^{n-1}(\partial^{*}\Omega)\leq\alpha
c_{n}m_{k}^{\frac{n-1}{n}},\qquad\forall k\in\mathbb{N}.$

Now, we rely on Theorem 3.13 to obtain the desired contradiction. We consider
the sequence of manifolds of bounded geometry given by
$(M_{k},g_{k},\mathcal{H}_{k}^{n})=(M,m_{k}^{-1/n}g,\mathcal{H}_{k}^{n})$. We
remind that this sequence is such that $\mathcal{H}_{k}^{n}(\Omega_{k})=1$ for
every $k\in\mathbb{N}$, where $\mathcal{H}_{k}^{n}$ denotes the
$n$–dimensional Hausdorff measure on $(M_{k},g_{k})$. Moreover, the rescaling
factor is such that

$\mathcal{H}^{n-1}_{k}(\partial^{*}\Omega_{k})\leq\alpha c_{n},\qquad\forall
k\in\mathbb{N}.$

This means that $(\Omega_{k})_{k\in\mathbb{N}}$ is a sequence of sets with
uniformly bounded volumes and perimeters and we can apply Theorem 3.13. Hence,
there exists a non-decreasing sequence
$(N_{k})_{k\in\mathbb{N}}\subset\mathbb{N}$ and for every $k$ we have a set of
points $\\{p_{k,i}\in M_{k},i=1,\dots,N_{k}\\}$ such that
$(M_{k},\mathrm{dist}_{g_{k}},\mathcal{H}^{n}_{k},p_{k,i})$ converges in the
Gromov-Hausdorff topology as $k\to\infty$ to either the Euclidean semi-space
$(\mathbb{R}^{n}_{+},\mathrm{d},\mathcal{H}^{n},0)$ or the Euclidean space
$(\mathbb{R}^{n},\mathrm{d},\mathcal{H}^{n},0)$. For every $k\in\mathbb{N}$ we
have a set of pairwise disjoint sets $\Omega_{k}^{i}\subset\Omega_{k}\subset
M$, for $i=1,\dots,N_{k}$, such that, setting
$\bar{N}=\lim_{k\to\infty}N_{k}$, we have
$\lim_{k\to\infty}\Omega_{k}^{i}=\Omega_{\infty}^{i}\subset\mathbb{R}^{n}$ for
every $i=1,\dots,\bar{N}$ and

$\sum_{i=1}^{\bar{N}}\mathcal{H}^{n}(\Omega_{\infty}^{i})=\lim_{k\to\infty}\mathcal{H}^{n}_{k}(\Omega_{k})=1.$

Moreover, we obtain the following inequality

$\sum_{i=1}^{\bar{N}}\mathcal{H}^{n-1}(\partial^{*}\Omega_{\infty}^{i})\leq\liminf_{k\to\infty}\mathcal{H}_{k}^{n-1}(\partial^{*}\Omega_{k})<c_{n},$
(4.3)

and every $\Omega^{i}_{\infty}$ is an isoperimetric set in
$\mathbb{R}^{n}_{+}$ or in $\mathbb{R}^{n}$. As a consequence, for every $i$
we know that $\Omega^{i}_{\infty}$ is a bounded set, so we can see them as a
family of pairwise disjoint sets in $\mathbb{R}^{n}$. This means that the set
$\Omega_{\infty}=\cup_{i=1}^{\bar{N}}\Omega_{\infty}^{i}$ is a subset of
$\mathbb{R}^{n}$ with volume $1$ and, by (4.3), its perimeter is strictly less
than $c_{n}$, which is the perimeter of the ball with the same volume,
obtaining the desired contradiction. ∎

###  4.2 Gamma convergence

######  Proposition 4.2 (cf. Proposition A and Proposition B of [34]).

Assume that (H1) and (H2) hold. Then, the following statements hold:

  1. i)

Lim-inf: If $(\varepsilon_{k})_{k\in{\mathbb{N}}}\subset]0,+\infty[$ is such
that $\varepsilon_{k}\to 0^{+}$ and
$(u_{\varepsilon_{k}})_{k\in{\mathbb{N}}}\subset H^{1}(M,{\mathbb{R}})$ is
such that $u_{\varepsilon_{k}}\to u_{0}$ in $L^{1}(M,{\mathbb{R}})$, then
$\liminf_{k\to\infty}\overline{\mathcal{E}}_{\varepsilon_{k}}(u_{\varepsilon_{k}})\geq\overline{\mathcal{E}}_{0}(u_{0})$;

  2. ii)

Lim-sup: For any $u_{0}\in L^{1}(M,{\mathbb{R}})$ such that
$u_{0}=\mathbf{1}_{\Omega}$ for some finite perimeter measurable set
$\Omega\subset M$ and for every sequence
$(\varepsilon_{k})_{k\in{\mathbb{N}}}\subset]0,+\infty[$ such that
$\varepsilon_{k}\to 0^{+}$, there exists
$(u_{\varepsilon_{k}})_{k\in{\mathbb{N}}}\subset H^{1}(M,{\mathbb{R}})$ such
that $u_{\varepsilon_{k}}\to u_{0}$ in $L^{1}(M,{\mathbb{R}})$,

$\int_{M}u_{\varepsilon_{k}}\mathrm{d}v_{g}=\int_{M}u_{0}\mathrm{d}v_{g}\quad\text{and}\quad\limsup_{k\to\infty}\overline{\mathcal{E}}_{\varepsilon_{k}}(u_{\varepsilon_{k}})\leq\overline{\mathcal{E}}_{0}(u_{0}).$

######  Remark 4.3.

Noticing that the compactness result ensured by Theorem 3.3 refers to the
functional $\mathcal{E}_{\varepsilon}\colon H^{1}(M,\mathbb{R})\to\mathbb{R}$,
as for the case of Neumann boundary condition we have that if
$(\varepsilon_{k})_{k\in\mathbb{N}}\subset]0,+\infty[$ is a sequence such that
$\varepsilon_{k}\to 0^{+}$ and if a sequence
$(u_{\varepsilon_{k}})_{k\in\mathbb{N}}\subset\mathcal{H}_{m,0}$ satisfies
$\overline{\mathcal{E}}_{\varepsilon_{k}}(u_{\varepsilon_{k}})\leq E^{*}$ for
some constant $E^{*}$, then, up to subsequences,
$(u_{\varepsilon_{k}})_{k\in\mathbb{N}}$ converges to a function $u_{0}\in
L^{1}(M,\mathbb{R})$. By applying the lim-inf property of Proposition 4.2, we
have that $\overline{\mathcal{E}}_{0}(u_{0})\leq E^{*}$, hence there exists a
measurable and finite perimeter set $\Omega\subset M$ such that
$u_{0}=\mathbf{1}_{\Omega}$ and
$\mathrm{vol}(\Omega)=\int_{M}u_{0}\mathrm{d}v_{g}=m$.

###  4.3 Photography map

Since now we are dealing with the Dirichlet boundary condition, namely
$u\equiv 0$ on $\partial M$, the photography map is defined by utilizing a
_boundary layer_. The main idea is to associate to each point of the manifold
the $\varepsilon$–approximation of the characteristic function of the geodesic
ball with prescribed volume centered in the point. However, when the point is
near the boundary, such a ball may intersect with the boundary and the
$\varepsilon$–approximation does not satisfy the boundary condition. To avoid
this problem, we construct a boundary layer map $\mathcal{L}\colon M\to M$
that slightly moves inside the points that lie on tubular neighbourhood of
$\partial M$. In this way, the geodesic ball of prescribed volume $m$ centered
at $\mathcal{L}(p)$ is in the interior of $M$ for every $p\in M$, provided $m$
sufficiently small.

Let us proceed with a formal construction. Since the boundary of $M$ is smooth
and compact, there exists $\delta_{M}>0$ such that the map

$\partial M\times[0,\delta_{M}]\ni(Q,t)\mapsto\exp_{Q}(tN_{\partial M})$

provides a coordinate system in a neighbourhood of $\partial M$, where we
recall that $N_{\partial M}$ stands for the unit interior normal vector field
along $\partial M$. Now, for any $p\in M$ such that $\mathrm{dist}(p,\partial
M)\leq\delta_{M}$, we denote by $(Q_{p},t_{p})$ the unique element in
$\partial M\times[0,\delta_{M}]$ such that

$p=\exp_{Q_{p}}(t_{p}N_{\partial M}).$

In other words, $t_{p}$ is the distance of $p$ to the boundary, while $Q_{p}$
is its projection on it. Now, let us choose a $C^{\infty}$ function
$h\colon[0,\delta_{M}]\to[0,\delta_{M}]$ that is strictly increasing (hence
invertible), $h(0)=\delta_{M}/2$, $h(\delta_{M})=\delta_{M}$ and
$h^{\prime}(\delta_{M})=1$. Our boundary-layer function $\mathcal{L}\colon
M\to M$ is defined as follows:

$\mathcal{L}(p)\coloneqq\begin{dcases}\exp_{Q_{p}}\big{(}h(t_{p})N_{\partial
M}\big{)},&\mbox{if }\mathrm{dist}(p,\partial M)\in[0,\delta_{M}],\\\
p,&\mbox{if }\mathrm{dist}(p,\partial M)\geq\delta_{M}.\end{dcases}$ (4.4)

Notice that $\mathcal{L}$ is a $C^{1}$–map homotopic to the identity.

Since $\mathrm{dist}(\mathcal{L}(p),\partial(M))\geq\delta_{M}/2$ for every
$p\in M$ and $M$ is compact, there exists a sufficiently small volume, say
$m_{0}>0$, such that for any $p\in M$ and $m\in(0,m_{0})$ the geodesic ball
centered at $\mathcal{L}(p)$ and with volume $m$ doesn’t intersect the
boundary of the manifold. More formally, denoting by $r_{q,m}>0$ the radius of
the geodesic ball centered at $q\in M$ with volume $m$, hence
$\int_{B(q,r_{q,m})}1\mathrm{d}v_{g}=m$, we have

$B(\mathcal{L}(p),r_{\mathcal{L}(p),m})\cap\partial M=\emptyset,\qquad\forall
p\in M,m\in(0,m_{0}).$ (4.5)

For any $p\in M$ and $m\in(0,m_{0})$ let us denote by
$u_{0,\mathcal{L}(p)}^{m}$ the characteristic function of
$B(\mathcal{L}(p),r_{\mathcal{L}(p),m})$. Moreover, for every $\varepsilon>0$
let $u_{\varepsilon,\mathcal{L}(p)}^{m}\in\mathcal{H}_{m}$ be
$\varepsilon$–approximation of $u_{0,\mathcal{L}(p)}^{m}$ given by the lim-sup
property of Proposition 4.2. Since $M$ is compact and
$\mathrm{supp}\,u_{0,\mathcal{L}(p)}^{m}\subset\joinrel\subset\mathrm{int}(M)$
for every $p\in M$ and $m\in(0,m_{0})$, there exists $\varepsilon_{0}>0$ such
that for every $\varepsilon\in]0,\varepsilon_{0}[$ we have
$u_{\varepsilon,\mathcal{L}(p)}^{m}\in\mathcal{H}_{m,0}$, hence
$u_{\varepsilon,\mathcal{L}(p)}\equiv 0$ on $\partial M$ (see Remark 3.2 for
the details about the construction of the $\varepsilon$–approximation).

######  Definition 4.4.

For every $m\in(0,m_{0})$ and $\varepsilon\in(0,\varepsilon_{0})$, we define
the photography map $\mathcal{P}_{\varepsilon,m}\colon M\to\mathcal{H}_{m,0}$
as follows:

$\mathcal{P}_{\varepsilon,m}(p)=u_{\varepsilon,\mathcal{L}(p)}^{m}.$

See Figure 3 for a representation of Definition 4.4.

$p$$\mathcal{L}(p)$$\partial
M$$M$$B(\mathcal{L}(p),r_{\mathcal{L}(p),m})$$0$$1$$u^{m}_{0,\mathcal{L}(p)}$$p$$\mathcal{L}(p)$$\partial
M$$M$$B(\mathcal{L}(p),r_{\mathcal{L}(p),m})$$0$$1$$u^{m}_{\varepsilon,\mathcal{L}(p)}$••••
Figure 3:  The map introduced in Definition 4.4 turns the indicator function
$u_{0,\mathcal{L}(p)}^{m}$ into the smooth approximation
$u_{\varepsilon,\mathcal{L}(p)}^{m}$ given by $\Gamma$-convergence
(Proposition 4.2). The functions are supported on the balls
$B(\mathcal{L}(p),r_{\mathcal{L}(p),m})$, whose center $\mathcal{L}(p)$ is
obtained by sending a point $p$ away from the boundary through the boundary
layer map $\mathcal{L}$.

######  Proposition 4.5.

Assume that (H1) and (H2) hold. There exists a function
$\tau\colon\mathbb{R}^{+}\to\mathbb{R}$ such that

$\lim_{m\to 0^{+}}\frac{\tau(m)}{m^{\frac{n-1}{n}}}=0$

and there exists $m_{1}=m_{1}(M,g,W,\tau)\in(0,m_{0})$ such that for every
$m\in(0,m_{1})$ there exists
$\varepsilon_{1}=\varepsilon_{1}(M,g,W,m)\in(0,\varepsilon_{0})$ such that for
every $\varepsilon\in(0,\varepsilon_{1})$ we have

$\overline{\mathcal{E}}_{\varepsilon,m}(\mathcal{P}_{\varepsilon,m}(p))\leq\sigma\overline{I}_{M}(m)+\tau(m),\qquad\forall
p\in M.$ (4.6)

###### Proof.

By applying some standard results (see, e.g., [43, Lemma 3.10]), since
$u_{0,\mathcal{L}(p)}^{m}$ has compact support in the interior of $M$ for
every $m\in(0,m_{0})$ we have

$\overline{\mathcal{E}}_{0}(u_{0,\mathcal{L}(p)}^{m})=\sigma\left(c_{n}m^{\frac{n-1}{n}}-\gamma_{n}\mathrm{Sc}_{g}(\mathcal{L}(p))m^{\frac{n+1}{n}}\right)+\mathcal{O}(m^{\frac{n+3}{n}}),\qquad\text{as
}m\to 0^{+},$ (4.7)

where $c_{n}$ is the Euclidean isoperimetric constant, $\gamma_{n}$ is a
constant which depends only on the dimension of the manifold and
$\mathrm{Sc}_{g}(\mathcal{L}(p))$ denotes the scalar curvature of the metric
tensor $g$ at the point $\mathcal{L}(p)$. Since $M$ is compact, there exists a
constant $\omega>0$ and $m_{1}\in(0,m_{0})$ such that for every
$m\in(0,m_{1})$ we have

$\overline{\mathcal{E}}_{0}(u_{0,\mathcal{L}(p)}^{m})<\sigma
c_{n}m^{\frac{n-1}{n}}+\omega m^{\frac{n+1}{n}},\qquad\forall p\in M.$

Since the last inequality is strict and using again the compactness of $M$, by
the lim-sup property of Proposition 4.2 there exists
$\varepsilon_{1}=\varepsilon_{1}(M,g,m)\in(0,\varepsilon_{0})$ such that for
every $\varepsilon\in(0,\varepsilon_{1})$ we have

$\overline{\mathcal{E}}_{\varepsilon,m}(u_{\varepsilon,\mathcal{L}(p)}^{m})\leq\sigma
c_{n}m^{\frac{n-1}{n}}+\omega m^{\frac{n+1}{n}},\qquad\forall p\in M.$ (4.8)

By (4.2), the function $\tau_{0}\colon\mathbb{R}^{+}\to\mathbb{R}$ given by
$\tau_{0}(m)=c_{n}m^{\frac{n-1}{n}}-\overline{I}_{M}(m)$ is a
$o(m^{\frac{n-1}{n}})$ as $m$ goes to $0$. As a consequence, setting
$\tau(m)=\sigma\tau_{0}(m)+\omega m^{\frac{n+1}{n}}$, (4.8) is equivalent to
(4.6). ∎

Since $\mathcal{L}\colon M\to M$ is a continuous function, one can employ the
same construction of the proof of Proposition 3.11 to obtain the continuity of
the photography map, which is stated by the next result.

######  Proposition 4.6.

Assume that (H1) and (H2) hold. Let $m_{1}>0$ and $\varepsilon_{1}>0$ be
defined as in Proposition 4.5. For every $m\in(0,m_{1})$ and
$\varepsilon\in(0,\varepsilon_{1})$ the map $\mathcal{P}_{\varepsilon,m}\colon
M\to\mathcal{H}_{m,0}$ is continuous.

###### Proof.

Cf. Proposition 3.11. ∎

###  4.4 Barycenter map

The construction of the barycenter map is analogous to the case of Neumann
boundary condition. In particular, we will use again the function
$\beta^{*}\colon L^{1}(M,\mathbb{R})\to\mathbb{R}^{\tilde{n}}$ defined in
(3.26), which gives the “extrinsic” center of mass of a function in the
Euclidean space $\mathbb{R}^{\tilde{n}}$ where the manifold $M$ is
isometrically embedded. In this case, we will compose this map with the
nearest point projection map $\pi_{M}\colon\mathbb{R}^{\tilde{n}}\to M$. We
need to prove that, when $m$ and $\varepsilon$ are sufficiently small,
$\beta^{*}(u)$ is near the manifold for any function $u$ that belongs to the
sublevel of $\overline{\mathcal{E}}_{\varepsilon,m}$ that contains the image
of the photography map, namely for every
$u\in\overline{\mathcal{E}}_{\varepsilon,m}^{c}$ with
$c=\sigma\overline{I}_{M}(m)+\tau(m)$ (see Proposition 4.5). To obtain the
last result, we rely on the “concentration property”, namely that as $m$ goes
to $0$ the support of every function in
$\overline{\mathcal{E}}_{\varepsilon,m}^{c}$ is inside a small ball, up to a
negligible part.

######  Proposition 4.7.

There exists $\mu=\mu(M,g)>0$ such that the following property holds. For
every almost isoperimetric sequence $(u_{k})_{k\in\mathbb{N}}\subset
L^{1}(M,\mathbb{R})$ with volumes $m_{k}=\int_{M}u_{k}\mathrm{d}v_{g}\to 0$,
i.e.,

$\lim_{k\to\infty}\frac{\overline{\mathcal{E}}_{0}(u_{k})}{\sigma\overline{I}_{M}(m_{k})}=1,$
(4.9)

there exists a sequence $(p_{k})_{k\in\mathbb{N}}\subset M$ such that

$\lim_{k\to+\infty}\frac{1}{m_{k}}\left(\int_{M\setminus B(p_{k},\mu
m_{k}^{1/n})}u_{k}\mathrm{d}v_{g}\right)=0.$ (4.10)

###### Proof.

The proof is analogous to the one of Proposition 3.15, and it is based on the
compactness result ensured by Theorem 3.13.

Let $(\Omega_{k})_{k\in\mathbb{N}}\subset\mathcal{C}_{g}$ such that
$u_{k}=\mathbf{1}_{\Omega_{k}}$ for every $k$. Let
$(X_{k},g_{k})=(M,m^{-1/k}g)$, so that $\mathcal{H}^{n}_{k}(\Omega_{k})=1$ for
every $k$, where $\mathcal{H}^{n}_{k}$ denotes the $n$–dimensional Hausdorff
measure on $(M_{k},g_{k})$. By (4.2) and (4.9), we have also that

$\lim_{k\to\infty}P(\Omega_{k},X_{k})=\lim_{k\to\infty}\mathcal{H}_{k}^{n-1}(\partial^{*}\Omega_{k})=I_{\mathbb{R}^{n}}(1).$

This means that, seen as subsets of $(M_{k},g_{k})$, the sequence
$(\Omega_{k})_{k\in\mathbb{N}}$ has uniformly bounded volumes and perimeters
and we can apply Theorem 3.13 to obtain a number
$\overline{N}\in\mathbb{N}\cup\\{+\infty\\}$, a family of pointed
$\mathrm{RCD}(\kappa,n)$ spaces
$(M^{i}_{\infty},\mathrm{d}^{i}_{\infty},\mathcal{H}^{n}_{\infty},p^{i})$, for
$i=1,\dots,\overline{N}$, and $\Omega^{i}_{\infty}\subset M^{i}_{\infty}$ such
that

$\sum_{i=1}^{\overline{N}}\mathcal{H}^{n}_{\infty}(\Omega^{i}_{\infty})=\lim_{k\to\infty}\mathcal{H}^{n}_{k}(\Omega_{k})=1,$
(4.11)

and

$\sum_{i=1}^{\overline{N}}\mathcal{H}^{n-1}_{\infty}(\partial^{*}\Omega^{i}_{\infty})\leq\liminf_{k\to\infty}\mathcal{H}^{n-1}_{k}(\partial^{*}\Omega_{k})=I_{\mathbb{R}^{n}}(1).$
(4.12)

By construction, each
$(M^{i}_{\infty},\mathrm{d}^{i}_{\infty},\mathcal{H}^{n}_{\infty},p^{i})$ is
either the Euclidean semi-space
$(\mathbb{R}^{n}_{+},\mathrm{d},\mathcal{H}^{n},0)$ or the Euclidean space
$(\mathbb{R}^{n},\mathrm{d},\mathcal{H}^{n},0)$. Since
$I_{\mathbb{R}^{n}}(m)=c_{n}m^{\frac{n-1}{n}}$ is a strictly concave function
and $I_{\mathbb{R}^{n}}(0)=0$, it is strictly subadditive and by using a
contradiction argument (4.11) and (4.12) show that $\overline{N}=1$. Using
again (4.11), this implies the existence of $\mu>0$ and of a sequence of
points $(p_{k})_{k\in\mathbb{N}}\subset M$ such that (4.10) holds. ∎

######  Lemma 4.8.

Assume that (H1) and (H2) hold and let $\tau\colon\mathbb{R}^{+}\to\mathbb{R}$
be given by Proposition 4.5. For every $\alpha\in(0,1)$, there exists
$m_{\alpha}=m_{\alpha}(M,g,W,\theta,\alpha)>0$ such that for every
$m\in(0,m_{\alpha})$ there exists
$\varepsilon_{\alpha}=\varepsilon_{\alpha}(M,g,W,\theta,\alpha,m)>0$ such that
for every $\varepsilon\in(0,\varepsilon_{\alpha})$ and for any
$u\in\mathcal{E}_{\varepsilon,m}^{\sigma\overline{I}_{M}(m)+\tau(m)}$ there
exists a point $p_{u}\in M$ such that

$\int_{M\setminus B(p_{u},\mu m^{1/n})}|u|\mathrm{d}v_{g}\leq\alpha m.$ (4.13)

###### Proof.

Relying on the concentration result ensured by Proposition 4.7, the proof is
analogous to the one of Lemma 3.16. For the sake of clarity, we report here
the main steps.

Arguing by contradiction, there exists a sequence $(m_{i})_{i\in\mathbb{N}}$
that goes to $0$ and for every $i\in\mathbb{N}$ there exists two sequences
$(\varepsilon_{i,j})_{j\in\mathbb{N}}$ and
$(u_{i,j})_{j\in\mathbb{N}}\subset\overline{\mathcal{E}}_{\varepsilon_{i,j},m_{i}}^{c_{i}}$,
with $c_{i}=\sigma\overline{I}_{M}(m_{i})+\tau(m_{i})$ such that
$\varepsilon_{i,j}\to 0^{+}$ as $j\to\infty$ and

$\int_{M\setminus B_{g}(p,\mu m_{i}^{1/n})}|u_{i,j}|\mathrm{d}v_{g}>\alpha
m_{i},\qquad\forall p\in M,\,\forall j\in\mathbb{N}.$ (4.14)

Since $\overline{\mathcal{E}}_{\varepsilon_{i,j},m_{i}}(u_{i,j})\leq c_{i}$
for every $j$, by Theorem 3.3 we have that for every fixed $i\in\mathbb{N}$
there exists a characteristic function $u_{0,i}\in L^{1}(M,\mathbb{R})$ such
that $u_{i,j}$ converges to it, up to subsequences. Therefore, for every
$i\in\mathbb{N}$ there exists $j_{i}$ sufficiently large such that

$\int_{M}\left(|u_{i,j_{i}}-u_{0,i}|\right)\mathrm{d}v_{g}\leq\frac{\alpha}{4}m_{i}.$
(4.15)

Moreover, by Remark 4.3 we have

$\overline{I}_{M}(m_{i})\leq\frac{\overline{\mathcal{E}}_{0}(u_{0,i})}{\sigma}\leq\overline{I}_{M}(m_{i})+\tau(m_{i}).$

Since $\tau(m)=o(m^{\frac{n-1}{n}})$ and (4.2) holds, the last chain of
inequality implies that

$\lim_{i\to\infty}\frac{\overline{\mathcal{E}}_{0}(u_{0,i})}{\sigma\overline{I}_{M}(m_{i})}=1,$

so we can apply Proposition 4.7 and obtain a sequence of points
$(p_{i})_{i\in\mathbb{N}}\subset M$ such that

$\lim_{i\to+\infty}\frac{1}{m_{i}}\left(\int_{M\setminus B(p_{i},\mu
m_{i}^{1/n})}u_{0,i}\mathrm{d}v_{g}\right)=0.$ (4.16)

Utilizing the same reasoning employed in the concluding section of the proof
for Lemma 3.16, (4.15) and (4.16) lead us to the intended contradiction. ∎

As we have done for the case of Neumann conditions, now that we have the
“concentration” result ensured by Lemma 4.8, it is possible to prove that if
$m$ and $\varepsilon$ are sufficiently small for any
$u\in\overline{\mathcal{E}}_{\varepsilon,m}^{\sigma\overline{I}_{M}(m)+\tau(m)}$
we have that $\beta^{*}(u)$ is in a neighbourhood of
$M\subset\mathbb{R}^{\tilde{n}}$. More formally, setting

$M_{r}=\left\\{x\in\mathbb{R}^{\tilde{n}}:\mathrm{dist}_{\mathbb{R}^{\tilde{n}}}(x,M)<r\right\\},\qquad\forall
r>0,$

we have the following result.

######  Lemma 4.9.

Assume that (H1) and (H2) hold. For any $r>0$, there exists
$m_{2}=m_{2}(M,g,r,\mathrm{diam}_{\mathbb{R}^{\tilde{n}}}(M))>0$ such that for
every $m\in(0,m_{2})$ there exists another positive constant
$\varepsilon_{2}=\varepsilon_{2}(M,g,r,m)$ such that for every
$\varepsilon\in(0,\varepsilon_{2})$ and any
$u\in\overline{\mathcal{E}}_{\varepsilon,m}^{\sigma\overline{I}_{M}(m)+\tau(m)}$
we have $\beta^{*}(u)\in M_{r}$.

###### Proof.

Cf. Lemma 3.17. ∎

######  Remark 4.10.

By the previous lemma and the compactness of $M$, there exists $r_{1}>0$ such
that for every $r\in]0,r_{1}[$, setting
$m_{2}=m_{2}(M,g,r,\mathrm{diam}_{\mathbb{R}^{\tilde{n}}}(M))$ and
$\varepsilon_{2}=\varepsilon_{2}(M,g,r,m)$ as in the lemma, the map
$\mathcal{B}=\pi_{M}\circ\beta^{*}\colon\overline{\mathcal{E}}_{\varepsilon,m}^{c}\to
M$ is well defined and continuous, with $c=\sigma\overline{I}_{M}(m)+\tau(m)$.

###  4.5 Conclusion of the proof

In this section we show that if $m$ and $\varepsilon$ are sufficiently small
then $\mathcal{B}\circ\mathcal{P}_{\varepsilon,m}\colon M\to M$ is homotopic
to the identity map. Then, Theorem B follows as an application of Theorem C.

Recalling the definition of $\delta_{M}>0$ given in Section 4.3 for the
purpose of defining the photography map $\mathcal{P}_{\varepsilon,m}\colon
M\to\mathcal{H}_{m,0}$, let us also define the map
$\widetilde{\mathcal{P}}_{\varepsilon,m}\colon
M^{\delta_{M}/2}\to\mathcal{H}_{m,0}$ as follows:

$\widetilde{\mathcal{P}}_{\varepsilon,m}(p)=u_{\varepsilon,p}^{m},$ (4.17)

where $M^{\delta_{M}/2}=\\{p\in M:\mathrm{dist}_{g}(p,\partial
M)\geq\delta_{M}/2\\}$. With this notation, $\mathcal{P}_{\varepsilon,m}$ is
given by $\widetilde{\mathcal{P}}_{\varepsilon,m}\circ\mathcal{L}$.

######  Lemma 4.11.

Assume that (H1) and (H2) hold. For every $r\in]0,\delta_{M}/2[$ there exists
$m_{3}=m_{3}(M,g,r)>0$ such that for every $m\in(0,m_{3})$ there exists
$\varepsilon_{3}=\varepsilon_{3}(M,g,r,m)>0$ such that

$\lVert\beta^{*}(\widetilde{\mathcal{P}}_{\varepsilon,m}(p))-p\rVert_{\mathbb{R}^{\tilde{n}}}\leq
r,\qquad\forall p\in M.$ (4.18)

###### Proof.

Cf. Lemma 3.19. ∎

######  Lemma 4.12.

Assume that (H1) and (H2) hold, let $\mathcal{P}_{\varepsilon,m}\colon
M\to\mathcal{H}_{m,0}$ be given by Definition 4.4 and let $\mathcal{B}\colon
H^{1}(M,\mathbb{R})\to M$ be defined as $\pi_{M}\circ\beta^{*}$. There exists
$m^{*}=m^{*}(M,g,W)>0$ such that for any $m\in(0,m^{*})$ there exists
$\varepsilon^{*}=\varepsilon^{*}(M,g,W,m)>0$ such that for any
$\varepsilon\in(0,\varepsilon^{*})$ the composition map

$\mathcal{B}\circ\mathcal{P}_{\varepsilon,m}\colon M\to M$

is well defined and homotopic to the identity map.

###### Proof.

The proof is very similar to the one of Lemma 3.20, but it has to take into
account the action of the boundary layer map $\mathcal{L}$. However, since
$\mathcal{L}$ is homotopic to the identity map, it suffices to show that
$\mathcal{B}\circ\widetilde{\mathcal{P}}_{\varepsilon,m}\colon
M^{\delta_{M}/2}\to M$ is homotopic to the identity map as well, provided that
$m$ and $\varepsilon$ are sufficiently small.

By the compactness of $M$, there exists a positive constant $C_{M}$ such that
$\mathrm{dist}_{g}(p,q)\leq C_{M}\lVert p-q\rVert_{\mathbb{R}^{\tilde{n}}}$,
for every $p,q\in M$. We denote by $\mathrm{inj}(M_{\delta_{M}/2})$ the
_injectivity radius_ of $M_{\delta_{M}/2}$ in $M$, that is

$\mathrm{inj}(M^{\delta_{M}/2})\coloneqq\inf_{p\in
M^{\delta_{M}/2}}\sup\Big{\\{}R>0\text{ s.t. }\mathrm{exp}^{M}_{p}\colon
B^{M}_{g}(p,{R})\to M\mbox{ is a diffeomorphism}\Big{\\}}.$

Note that by this construction we have
$\mathrm{inj}(M^{\delta_{M}/2})\leq\delta_{M}/2$.

Recalling the definition of $r_{1}$ given in Remark 4.10, we set

$r^{*}=\frac{1}{2}\min\left\\{\frac{\delta_{M}}{C_{M}},\mathrm{inj}(M_{\delta_{M}/2}),r_{1}\right\\}\leq\frac{\delta_{M}}{4}.$

Let $m_{1}=m_{1}(M,g,W,\tau)$,
$m_{2}=m_{2}(M,g,r^{*},\mathrm{diam}_{\mathbb{R}^{\tilde{n}}}(M))$ and
$m_{3}=m_{3}(M,g,r^{*})$ be the positive constants given by Proposition 4.5,
Lemma 4.9 and Lemma 4.11, respectively. Let
$m^{*}=\min\\{m_{1},m_{2},m_{3}\\}>0$ and for every $m\in(0,m^{*})$ let
$\varepsilon^{*}=\min\\{\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}\\}$,
where $\varepsilon_{i}$ is defined by using the same results of $m_{i}$, for
$i=1,2,3$. Since $m<m_{1}$, for every $p\in M_{\delta_{M}/2}$ we have that
$\widetilde{\mathcal{P}}_{\varepsilon,m}(p)\in\overline{\mathcal{E}}_{\varepsilon,m}^{c}$,
with $c=\sigma\overline{I}_{M}(m)+\tau(m)$. Since $m<m_{3}$, this implies that
$\lVert\beta^{*}(\widetilde{\mathcal{P}}_{\varepsilon,m}(p))-p\rVert_{\mathbb{R}^{\tilde{n}}}\leq
r^{*}$ for every $p\in M_{\delta_{M}/2}$. Therefore, we have

$\mathrm{dist}_{g}\big{(}\mathcal{B}(\widetilde{\mathcal{P}}_{\varepsilon,m}(p)),p\big{)}\leq
C_{M}\lVert\pi_{M}(\beta^{*}(\widetilde{\mathcal{P}}_{\varepsilon,m}(p)))-p\rVert_{\mathbb{R}^{\tilde{n}}}\\\
\leq
C_{M}\big{(}\lVert\pi_{M}(\beta^{*}(\widetilde{\mathcal{P}}_{\varepsilon,m}(p)))-\beta^{*}(\widetilde{\mathcal{P}}_{\varepsilon,m}(p))\rVert_{\mathbb{R}^{\tilde{n}}}+\lVert\beta^{*}(\widetilde{\mathcal{P}}_{\varepsilon,m}(p))-p\rVert_{\mathbb{R}^{\tilde{n}}}\big{)}\\\
\leq 2C_{M}r^{*}\leq\frac{\delta_{M}}{2},$ (4.19)

so the map $F\colon[0,1]\times M_{\delta_{M}/2}\to M$, given by

$F(t,p)\coloneqq\mathrm{exp}^{M}_{p}\Big{(}t\big{(}\mathrm{exp}^{M}_{p}\big{)}^{-1}\big{(}\mathcal{B}(\widetilde{\mathcal{P}}_{\varepsilon,m}(p))\big{)}\Big{)},$

is well-defined, continuous and gives a homotopy equivalence between
$\mathcal{B}\circ\widetilde{\mathcal{P}}_{\varepsilon,m}$ and the identity map
in $M^{\delta_{M}/2}$. ∎

###### Proof of Theorem B.

As previously stated, this proof is based on Theorem C. Indeed, for every
$m,\varepsilon>0$ the functional $\overline{\mathcal{E}}_{\varepsilon,m}$ is
bounded below and it satisfies the Palais-Smale condition (see Lemma 2.1).
Moreover, let $m^{*}>0$ be given by Lemma 4.12 and for every $m\in(0,m^{*})$
let $\varepsilon^{*}>0$ be given by the same lemma. Setting
$c=\sigma\overline{I}_{M}(m)+\tau(m)$, where
$\tau\colon\mathbb{R}^{+}\to\mathbb{R}$ is defined in Proposition 4.5, we
establish that the photography map $\mathcal{P}_{\varepsilon,m}\colon
M\to\overline{\mathcal{E}}_{\varepsilon,m}^{c}$ (as defined in Definition 4.4)
and the barycenter map
$\mathcal{B}\colon\overline{\mathcal{E}}_{\varepsilon,m}^{c}\to M$ (defined as
$\pi_{M}\circ\beta^{*}$) are well-defined, for every
$\varepsilon\in(0,\varepsilon^{*})$. Furthermore, according to Lemma 4.12,
their composition $\mathcal{B}\circ\mathcal{P}_{\varepsilon,m}$ is homotopic
to the identity map. Then, all the conclusions of Theorem B can be derived by
applying Theorem C. ∎

##  5 Generic non-degeneracy

The conclusions of Theorems A and B are stronger if one can ensure that for
given $m$ and $\varepsilon$ all the solutions of (1.2) and (1.3) are
nondegenerate. In many situations, nondegeneracy of solutions is a generic
property, in the sense that it is obtained after some arbitrary small
perturbation of a key parameter of the problem. This is the case for
semilinear elliptic equations as the ones considered in this paper, meaning
that the stronger existence statements of Theorem A and B, based on Morse
inequalities, hold in generic situations. The purpose of this section is to
give some insight on how these genericity properties are obtained from the
available literature, which contains analogous results in similar settings.

###  5.1 Abstract transversality result

Generic nondegeneracy is usually obtained by applying Sard-Smale Theorem [53]
and abstract transversality results on infinite-dimensional Banach manfiolds,
see Quinn [51], Saut and Temam [52] and Uhlenbeck [56]. Following [52], we now
introduce some objects that will be fixed for the rest of this section and
recall some standard definitions. Let $X$, $Y$ and $Z$ be Banach spaces and
$U\subset X$, $V\subset Y$ open subsets. Let $F:U\times V\to Z$ be of class
$C^{k}$ for $k\geq 1$. We will denote by $DF(x_{0},y_{0})$ the (total)
differential of $F$ at the point $(x_{0},y_{0})\in U\times V$ and by
$D_{X}F(x_{0},y_{0})$, $D_{Y}F(x_{0},y_{0})$ the differentials with respect to
the $x$ and the $y$ components, respectively. Given $Z^{\prime}$ a Banach
space, $U^{\prime}\subset Z^{\prime}$ an open subset and $g:Z^{\prime}\to Z$
of class $C^{1}$, we say that $z_{0}\in Z$ is a regular value of $g$ if
$Dg(z_{0}^{\prime})$ (the differential of $g$ at $z_{0}^{\prime}$) is onto for
any $z_{0}^{\prime}\in U^{\prime}$ such that $g(z_{0}^{\prime})=z_{0}$. The
abstract result we use is the following:

######  Theorem 5.1 (Theorem 1.1 in [52]).

Assume that $z_{0}\in Z$ is a regular value of $F$ and that, moreover:

  1. 1.

For any $x\in U$ and $y\in V$, $D_{X}F(x,y):X\to Z$ is a Fredholm map of index
$l<k$.

  2. 2.

The set of $x\in U$ such that $F(x,y)=z_{0}$ for all $y$ in a compact subset
of $V$ is a relatively compact set of $U$.

Then, the set

$\mathcal{O}\coloneqq\\{y\in V:z_{0}\mbox{ is a regular value of }F(\cdot,y),$

is a dense open subset of $V$.

###  5.2 Genericity with respect to the Riemannian metric

Theorem 5.1 was used in [5] (cf. also [39]) in order to prove generic non-
degeneracy for solutions of (1.1) (i.e., critical points of the $C^{2}$
functionals $\mathcal{E}_{\varepsilon,m}$ and
$\overline{\mathcal{E}}_{\varepsilon,m}$) with respect to the Riemannian
metric. The analogous result holds in our setting:

######  Theorem 5.2.

Let $\mathrm{Met}^{\infty}(M)$ be the space of $C^{\infty}$ Riemannian metrics
on $M$ and $m\in{\mathbb{R}}$. For $\hat{g}\in\mathrm{Met}^{\infty}(M)$ and
$\varepsilon\in(0,+\infty)$, we denote by
$\mathcal{E}_{\varepsilon,m,\hat{g}}$ and
$\overline{\mathcal{E}}_{\varepsilon,m,\hat{g}}$ the functionals
$\mathcal{E}_{\varepsilon,m}$ and $\overline{\mathcal{E}}_{\varepsilon,m}$
with respect to the metric $\hat{g}$. Then, for any
$g_{0}\in\mathrm{Met}^{\infty}(M)$ the sets

$\displaystyle\left\\{\begin{aligned}
(\varepsilon,\hat{g})\in(0,+\infty)&\times\mathrm{Met}^{\infty}(M):\mbox{ any
critical point }\\\ &\mbox{ of }\mathcal{E}_{\varepsilon,m,\hat{g}}\mbox{ in
}H^{1}_{g_{0}}(M,{\mathbb{R}})\mbox{ is non-degenerate}\end{aligned}\right\\}$

and

$\displaystyle\left\\{\begin{aligned}
(\varepsilon,\hat{g})\in(0,+\infty)&\times\mathrm{Met}^{\infty}(M):\mbox{ any
critical point }\\\ &\mbox{ of
}\overline{\mathcal{E}_{\varepsilon,m,\hat{g}}}\mbox{ in
}H^{1}_{g_{0}}(M,{\mathbb{R}})\mbox{ is non-degenerate}\end{aligned}\right\\}$

are open dense subsets of $(0,+\infty)\times\mathrm{Met}^{\infty}(M)$.

The proof of Theorem 5.2 is obtained as an application of the abstract Theorem
5.1. The interested reader can find more details in [5, 39].

###  5.3 Genericity with respect to the domain

Notice that Theorem 5.2 has a drawback: In some applications, one might not
want to modify the metric of the problem. For instance, if $M$ is an Euclidean
domain $\Omega\subset{\mathbb{R}}^{n}$ then it would not make much sense to
replace the Euclidean metric on $\Omega$ by another one. It is therefore
reasonable to be intereseted in generic nondegeneracy with respect to a
different key parameter. It might be tempting to consider a genericity result
based on modifying the potential instead of the Riemannian metric. However,
Brunovský and Poláčik [19] produced a counterexample for semilinear elliptic
equations on the Euclidean ball with homogeneous Dirichlet boundary
conditions. Our setting is slightly different than theirs (in particular, more
restrictive) meaning that one could still hope to obtain genericity results
with respect to the potential. However, the investigation of this question
goes beyond the scope of this paper. Instead, we will present a genericity
result based on perturbation of the domains, obtained by adapting a result due
to Saut and Temam [52], see also the book by Henry [32]. This can be somehow
understood as a Euclidean counterpart of Theorem 5.2.

Following [32], let us give the precise statement of the genericity result.
Fix $\Omega\subset{\mathbb{R}}^{n}$ a bounded and smooth domain. For
$l\in\\{1,\ldots,+\infty\\}$, let $\mathrm{Diff}^{l}(\Omega)$ be the set of
maps in $\mathcal{C}^{l}(\Omega,{\mathbb{R}}^{n})$ such that $h\colon\Omega\to
h(\Omega)$ is a $C^{l}$-diffeomorphism. In particular,
$\mathrm{Id}\in\mathrm{Diff}^{l}(\Omega)$. For $l<+\infty$, let us endow
$\mathcal{C}^{l}(\Omega,{\mathbb{R}}^{n})$ with the usual sup norm, so it is a
Banach space. This induces the usual Whitney topology on
$\mathcal{C}^{\infty}(\Omega,{\mathbb{R}}^{n})$. Then, with these choices of
topologies, $\mathrm{Diff}^{l}(\Omega)$ is open in
$\mathcal{C}^{l}(\Omega,{\mathbb{R}}^{n})$. For
$h\in\mathrm{Diff}^{l}(\Omega)$ and $\varepsilon>0$, let
$\mathcal{E}_{\varepsilon,h}$ denote the energy functional with parameter
$\varepsilon$ in the domain $h(\Omega)$. For $m\in{\mathbb{R}}$, define
$\mathcal{E}_{\varepsilon,m,h}$ and $\overline{\mathcal{E}}_{\varepsilon,m,h}$
as the restriction of $\mathcal{E}_{\varepsilon,h}$ to $\mathcal{H}_{m}$ and
$\mathcal{H}_{m,0}$ respectively. The main result reads as follows:

######  Theorem 5.3.

The sets

$\big{\\{}(\varepsilon,h)\in(0,+\infty)\times\mathrm{Diff}^{\infty}(\Omega):\mbox{any
critical point of }\mathcal{E}_{\varepsilon,m,h}\mbox{ is
nondegenerate}\big{\\}}$

and

$\big{\\{}(\varepsilon,h)\in(0,+\infty)\times\mathrm{Diff}^{\infty}(\Omega):\mbox{any
critical point of }\overline{\mathcal{E}}_{\varepsilon,m,h}\mbox{ is
nondegenerate}\big{\\}}$

are dense in $(0,+\infty)\times\mathrm{Diff}^{\infty}(\Omega)$.

The proof of Theorem 5.3 is based on the transversality Theorem 5.1, and
follows by combining the ideas in [5] with those in [52, 32]. This result is
applied to $\mathrm{Diff}^{l}(\Omega,{\mathbb{R}}^{n})$, which is an open
subset of the Banach space $\mathcal{C}^{l}(\Omega,{\mathbb{R}}^{n})$. As a
consequence, one obtains that the sets

$\\{(\varepsilon,h)\in(0,+\infty)\times\mathrm{Diff}^{l}(\Omega):\mbox{Any
critical point of }\mathcal{E}_{\varepsilon,m,h}\mbox{ is nondegenerate}\\}$

and

$\\{(\varepsilon,h)\in(0,+\infty)\times\mathrm{Diff}^{l}(\Omega):\mbox{Any
critical point of }\overline{\mathcal{E}}_{\varepsilon,m,h}\mbox{ is
nondegenerate}\\}$

are open and dense in $(0,+\infty)\times\mathrm{Diff}^{\infty}(\Omega)$.
Theorem 5.3 then follows by Baire category Theorem, by taking the
intersections $\cap_{l\in{\mathbb{N}}^{*}}$ of the sets above. We skip the
details.

##  6 A partial result without the subcritical growth assumption

The sole purpose of the subcritical growth assumption (H3) is to ensure that
the variational problems under consideration are compact (more precisely, that
the functionals $\mathcal{E}_{\varepsilon,m}$ and
$\overline{\mathcal{E}}_{\varepsilon,m}$ satisfy the Palais-Smale condition).
However, as it was already observed in [14], it is possible to drop (H3) and
obtain weaker versions of Theorems A and B. More precisely, one has:

######  Theorem 6.1.

Assume that (H1) and (H2) hold. Then, there exists $m^{*}>0$ such that for all
$m\in(0,m^{*})$ there exist $\varepsilon_{m},c_{m}>0$ such that for any
$\varepsilon\in(0,\varepsilon_{m})$ the Neumann problem (1.2) has at least
$\mathrm{cat}(\partial M)$ solutions $u_{\varepsilon,m}$ with
$\mathcal{E}_{\varepsilon}(u_{\varepsilon,m})\leq c_{m}$. Moreover, if
$\varepsilon$ and $m$ as above are such that all critical points of
$\mathcal{E}_{\varepsilon,m}$ are non-degenerate, then (1.2) has at least
$\mathscr{P}_{1}(\partial M)$ solutions $u_{\varepsilon,m}$ with
$\mathcal{E}_{\varepsilon}(u_{\varepsilon,m})\leq c_{m}$.

######  Theorem 6.2.

Assume that (H1) and (H2) hold. Then, there exists $m^{*}>0$ such that for all
$m\in(0,m^{*})$ there exist $\varepsilon_{m},c_{m}>0$ such that for any
$\varepsilon\in(0,\varepsilon_{m})$ the Dirichlet problem (1.3) has at least
$\mathrm{cat}(M)$ solutions $u_{\varepsilon,m}$ with
$\mathcal{E}_{\varepsilon}(u_{\varepsilon,m})\leq c_{m}$. Moreover, if
$\varepsilon$ and $m$ as above are such that all critical points of
$\overline{\mathcal{E}}_{\varepsilon,m}$ are non-degenerate, then (1.3) has at
least $\mathscr{P}_{1}(M)$ solutions $u_{\varepsilon,m}$ with
$\mathcal{E}_{\varepsilon}(u_{\varepsilon,m})\leq c_{m}$.

The proofs of Theorems 6.1 and 6.2 work essentially like that of [14, Theorem
5.9]. However, the assumptions on the potential that we take in this paper are
slightly different than those in [14]. Therefore, we include the proofs of
Theorems 6.1 and 6.2 for completeness. The crucial ingredient is that for
small $\varepsilon$ one can obtain a priori bounds on the $L^{\infty}$ norm of
any solution $(u_{\varepsilon},\lambda_{\varepsilon})$ of the equation

$-\varepsilon\Delta
u_{\varepsilon}+\frac{1}{\varepsilon}\hat{W}^{\prime}(u_{\varepsilon})=\lambda_{\varepsilon}\mbox{
on }M.$ (6.1)

according to its energy (see Proposition 6.3 below). Such ingredient allows to
modify potentials close to infinity to turn them into potentials with
subcritical growth at infinity to which one may apply Theorems A and B. The a
priori bounds imply that the low energy solutions of the modified potential
are also solutions of the original potential. However, no such thing can be
said at this point regarding high energy solutions, hence the weaker
statements of Theorems 6.1 and 6.2.

######  Proposition 6.3.

Let $\hat{W}\in\mathcal{C}^{3}_{\mathrm{loc}}({\mathbb{R}},[0,+\infty])$ be a
double-well potential vanishing exactly on $\\{0,1\\}$ and satisfying (H1) and
(H2). There exists $\varepsilon_{ub}>0$ such that for any
$\varepsilon\in(0,\varepsilon_{\mathrm{ub}}]$ and any
$(u_{\varepsilon},\lambda_{\varepsilon})$ solution of (6.1) such that
$\mathcal{E}_{\varepsilon,\hat{W}}(u_{\varepsilon})<+\infty$ we have

$\lVert u_{\varepsilon}\rVert_{L^{\infty}(M,{\mathbb{R}})}\leq
C_{\mathrm{ub}}(\varepsilon_{ub},\mathcal{E}_{\varepsilon,\hat{W}}(u_{\varepsilon})),$

with
$C_{\mathrm{ub}}(\varepsilon_{ub},\mathcal{E}_{\varepsilon,\hat{W}}(u_{\varepsilon}))>0$
a constant depending on the quantities $\varepsilon_{ub}$ and
$\mathcal{E}_{\varepsilon,\hat{W}}(u_{\varepsilon})$ but independent on
$\varepsilon$.

The proof of Proposition 6.3 relies on the following result.

######  Lemma 6.4.

Let $\hat{W}\in\mathcal{C}^{3}_{\mathrm{loc}}({\mathbb{R}},[0,+\infty])$ be a
double-well potential vanishing exactly on $\\{0,1\\}$ and satisfying (H1) and
(H2). There exist $\varepsilon_{\mathrm{est}}>0$ and $C_{\mathrm{est}}>0$ such
that for any $\varepsilon\in(0,\varepsilon_{\mathrm{est}})$, $m\in[0,1]$ and
$(u_{\varepsilon},\lambda_{\varepsilon})$ a solution of (6.1) we have

$\lvert\lambda_{\varepsilon}\rvert\leq
C_{\mathrm{est}}\mathcal{E}_{\varepsilon,\hat{W}}(u_{\varepsilon}).$

Lemma 6.4 was proven by Chen in [22, Lemma 3.4] in the case of an Euclidean
domain of ${\mathbb{R}}^{n}$ and then extended to the setting of closed
manifolds in [14, Proposition 5.3]. In both cases, the proof is obtained by
combining standard elliptic estimates with mollification arguments. As the
arguments carry on directly to our setting, we skip the proof. We are now
ready to prove Proposition 6.3, which relies on standard arguments (see, for
instance, [14, Theorem 5.9]).

###### Proof of Proposition 6.3.

Assume that $\varepsilon\in(0,\varepsilon_{\mathrm{est}})$, with
$\varepsilon_{\mathrm{est}}$ as in Lemma 6.4. By (H2), we find
$u^{*}(\varepsilon_{\mathrm{est}},\mathcal{E}_{\varepsilon,\hat{W}}(u_{\varepsilon}))$
depending only on $\varepsilon_{\mathrm{est}}>0$ and
$\mathcal{E}_{\varepsilon,\hat{W}}(u_{\varepsilon})$ such that

$\frac{1}{\varepsilon}\hat{W}^{\prime}(u)>C_{\mathrm{est}}\mathcal{E}_{\varepsilon,\hat{W}}(u_{\varepsilon}),\qquad\text{
for all }u\geq
u^{*}(\varepsilon_{\mathrm{est}},\mathcal{E}_{\varepsilon,\hat{W}}(u_{\varepsilon}))$

and

$\frac{1}{\varepsilon}\hat{W}^{\prime}(u)<-C_{\mathrm{est}}\mathcal{E}_{\varepsilon,\hat{W}}(u_{\varepsilon}),\qquad\text{
for all }u\leq
u^{*}(\varepsilon_{\mathrm{est}},\\_{\varepsilon}(u_{\varepsilon})),$

with $C_{\mathrm{est}}>0$ as in Lemma 6.4. Let $x_{\mathrm{max}}\in M$ be a
maximum point for $u_{\varepsilon}$ and assume by contradiction that $\lVert
u_{\varepsilon}\rVert_{L^{\infty}(M,{\mathbb{R}})}>u^{*}(\varepsilon_{\mathrm{est}},\mathcal{E}_{\varepsilon,\hat{W}}(u_{\varepsilon}))$.
Then, using (6.1) and Lemma 6.4 it follows

$-\varepsilon\Delta
u_{\varepsilon}(x_{\mathrm{max}})=\lambda_{\varepsilon}-\frac{1}{\varepsilon}\hat{W}^{\prime}(u_{\varepsilon}(x_{\mathrm{max}}))<0,$

which gives the contradiction. Hence, one has that $\max_{M}u\leq
u^{*}(\varepsilon_{\mathrm{est}},\mathcal{E}_{\varepsilon,\hat{W}}(u_{\varepsilon}))$
and, in an analogous fashion, one finds that $\min_{M}u\geq
u^{*}(\varepsilon_{\mathrm{est}},\mathcal{E}_{\varepsilon,\hat{W}}(u_{\varepsilon}))$.
This establishes the result by choosing
$\varepsilon_{\mathrm{ub}}=\varepsilon_{\mathrm{est}}$ and
$C_{\mathrm{ub}}(\varepsilon_{ub},\mathcal{E}_{\varepsilon,\hat{W}}(u_{\varepsilon}))=u^{*}(\varepsilon_{\mathrm{est}},\mathcal{E}_{\varepsilon,\hat{W}}(u_{\varepsilon}))$.
∎

At this point, the proof of Theorem 6.1 can be completed as follows.

###### Proof of Theorem 6.1.

Let $\varepsilon\in(0,\varepsilon_{\mathrm{ub}})$, with
$\varepsilon_{\mathrm{ub}}$ as in Lemma 6.4 and
$C_{\mathrm{ub}}(\varepsilon_{\mathrm{ub}},1)>0$ the constant given by
Proposition 6.3. We can find $u^{*}\geq
C_{\mathrm{ub}}(\varepsilon_{\mathrm{ub}},1)$ and
$\hat{W}\in\mathcal{C}_{\mathrm{loc}}^{3}({\mathbb{R}},[0,+\infty))$ such that
$\hat{W}^{-1}(0)=\\{0,1\\}$ which satisfies (H1), (H2) and (H3) with
$\hat{W}(u)=W(u)$ for all $u\in[-u^{*},u^{*}]$. We now apply Theorem A to
$\hat{W}$. In particular, there exists $\hat{m}^{*}>0$ such that for all
$m\in(0,\hat{m}^{*})$ there exist $\hat{\varepsilon}_{m},\hat{c}_{m}>0$ such
that for all $\varepsilon\in(0,\hat{\varepsilon}_{m})$ we have that there
exist $\mathrm{cat}(\partial M)$ solutions of (1.2) (for $\hat{W}$)
$(\hat{u}_{\varepsilon,m},\hat{\lambda}_{\varepsilon,m})$ with
$\mathcal{E}_{\varepsilon,\hat{W}}(\hat{u}_{\varepsilon,m})\leq\hat{c}_{m}$.
Moreover $\hat{c}_{m}$ can be chosen such that $\hat{c}_{m}\to 0$ as $m\to 0$,
see the end of Section 3.5. Therefore, choose $m^{*}\leq\hat{m}^{*}$ such that
$\hat{c}_{m}\leq 1$ for all $m\in(0,m^{*})$. For all such $m$, let
$\varepsilon_{m}\coloneqq\frac{1}{2}\min\\{\hat{\varepsilon}_{m},\varepsilon_{\mathrm{est}}\\}>0$,
$\varepsilon\in(0,\varepsilon_{m})$ and $\hat{u}_{\varepsilon,m}$. Since
$\mathcal{E}_{\varepsilon}(\hat{u}_{\varepsilon,m})\leq 1$ whenever
$\hat{u}_{\varepsilon,m}$ is a solution as above, we can apply Proposition 6.3
and find that
$\lVert\hat{u}_{\varepsilon,m}\rVert_{L^{\infty}(M,{\mathbb{R}})}\leq
C_{\mathrm{ub}}(\varepsilon_{\mathrm{ub}},1)\leq u^{*}$ which implies that
$\hat{u}_{\varepsilon,m}$ is a solution of (1.2) for $W$ and hence the result.
∎

The proof of Theorem 6.2 works in the same way, so we skip it.

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