# Linear Stability Implies Nonlinear Stability
for Faber-Krahn Type Inequalities

Mark Allen  Department of Mathematics, Brigham Young University, Provo, UT
<EMAIL_ADDRESS>,  Dennis Kriventsov  Department of Mathematics,
Rutgers University, Piscataway, NJ<EMAIL_ADDRESS>and  Robin Neumayer
Department of Mathematics, Northwestern University, Evanston, IL
<EMAIL_ADDRESS>
(Date: May 9, 2021)

###### Abstract.

For a domain $\Omega\subseteq{\mathbb{R}}^{n}$ and a small number
$\mathfrak{T}>0$, let

$\mathcal{E}_{0}(\Omega)=\lambda_{1}(\Omega)+\mathfrak{T}\mathrm{tor}(\Omega)=\inf_{u,w\in
H^{1}_{0}(\Omega)\setminus\\{0\\}}\frac{\int|\nabla u|^{2}}{\int
u^{2}}+\mathfrak{T}\int\frac{1}{2}|\nabla w|^{2}-w$

be a modification of the first Dirichlet eigenvalue of $\Omega$. It is well-
known that over all $\Omega$ with a given volume, the only sets attaining the
infimum of $\mathcal{E}_{0}$ are balls $B_{R}$; this is the Faber-Krahn
inequality. The main result of this paper is that, if for all $\Omega$ with
the same volume and barycenter as $B_{R}$ and whose boundaries are
parametrized as small $C^{2}$ normal graphs over $\partial B_{R}$ with bounded
$C^{2}$ norm,

$\int|u_{\Omega}-u_{B_{R}}|^{2}+|\Omega\triangle B_{R}|^{2}\leq
C[\mathcal{E}_{0}(\Omega)-\mathcal{E}_{0}(B_{R})]$

(i.e. the Faber-Krahn inequality is _linearly stable_), then the same is true
for _any_ $\Omega$ with the same volume and barycenter as $B_{R}$ without any
smoothness assumptions (i.e. it is _nonlinearly stable_). Here $u_{\Omega}$
stands for an $L^{2}$-normalized first Dirichlet eigenfunction of $\Omega$.
Related results are shown for Riemannian manifolds. The proof is based on a
detailed analysis of some critical perturbations of Bernoulli-type free
boundary problems. The topic of when linear stability is valid, as well as
some applications, are considered in a companion paper.

##  1\. Introduction

For a domain $\Omega\subseteq{\mathbb{R}}^{n}$, let

$\lambda_{1}(\Omega)=\inf_{u\in H^{1}_{0}(\Omega)}\frac{\int|\nabla
u|^{2}}{\int u^{2}}$

be the first Dirichlet eigenvalue, and let $u_{\Omega}$ be a nonnegative
function with $\int u_{\Omega}^{2}=1$ attaining the infimum. The Faber-Krahn
inequality asserts that

(1.1) $\lambda_{1}(\Omega)\geq\lambda_{1}(B_{R}),$

where $B_{R}$ is a ball with $|B_{R}|=|\Omega|$, and moreover this inequality
is strict unless $\Omega$ is a translate of $B_{R}$. In recent years, the
topic of _stability_ for this inequality has received considerable attention:
if $\lambda_{1}(\Omega)-\lambda_{1}(B_{R})$ is small, how closely must
$\Omega$ resemble $B_{R}$? For a survey of the recent literature on this
question, see [8]. In particular, _quantitative_ stability refers to
inequalities of the form

(1.2) $d^{\alpha}(B_{R}(x_{\Omega}),\Omega)\leq
C[\lambda_{1}(\Omega)-\lambda_{1}(B_{R})],$

where $d$ is some notion of distance between sets, $\alpha>0$ is a positive
number, and $x_{\Omega}=\fint_{\Omega}x$ is the barycenter. Here it is
desirable to find the strongest $d$, as well as the smallest number $\alpha$,
for which such an inequality may hold.

The best result of this type in the literature is due to Brasco, De Philippis,
and Velichkov [9], which establishes (1.2) with

(1.3) $d=d_{0}(E,E^{\prime})=|E\triangle E^{\prime}|$

and $\alpha=2$. This theorem is sharp, in the sense that (1.2) is false with
$d=d_{0}$ and $\alpha<2$. However, it does not fully resolve the issue of
quantitative stability, as it seems unlikely that $d_{0}$ is the “strongest”
distance for which (1.2) holds. To understand why, it is useful to separate
the stability question into two “regimes” where it may be studied: we say that
(1.1) is _linearly stable_ with respect to $d,\alpha$ if (1.2) holds for any
$\Omega$ whose boundary may be expressed as a $C^{2}$ normal graph over
$\partial B_{R}$, and we say (1.1) is _nonlinearly stable_ with respect to
$d,\alpha$ if (1.2) holds for any $\Omega$, unconditionally. For example, the
result of [9] is a nonlinear stability result.

It is not difficult to see that $d_{0}$ is not the strongest norm for which
_linear_ stability is valid. Indeed, if $\partial\Omega$ is given by a normal
graph $g$, then $d_{0}$ is comparable to the $L^{2}$ norm of $\xi$, while the
right-hand side of (1.2) is comparable to $\int_{\partial B_{R}}\xi L\xi$,
where $L$ is a first-order differential operator (this is implicit in the
proof of [9], and considered more carefully in the companion paper [3]). In
particular, Sobolev norms of $\xi$ up to $\|\xi\|_{H^{1/2}(\partial\Omega)}$
appear to be controllable. Unfortunately, this kind of observation does not
easily lead to improved nonlinear stability statements, as it is unclear how
to define a stronger Sobolev-type distance for arbitrary sets $\Omega$.

In the nonlinear context, $d_{0}$ has a different sort of drawback, which is
discussed in [8, Open Problem 3]: it is possible to modify $\Omega$ by
removing a set of measure $0$ in a way which strictly increases its eigenvalue
(consider, for example, $B_{1}$ vs. $B_{1}\setminus\\{te_{1}:t>0\\}$, a slit
domain in ${\mathbb{R}}^{2}$). Any change to $\Omega$ which modifies the local
capacity of its compliment should be reflected in $\lambda_{1}(\Omega)$. On
the other hand, $d_{0}$ is clearly insensitive to any such changes.

We propose a distance

$d_{1}(E,E^{\prime})=\sqrt{\int|u_{E}-u_{E^{\prime}}|^{2}}$

where $u_{E},u_{E^{\prime}}$ are the normalized first eigenfunctions as above
extended by zero to be defined on all of ${\mathbb{R}}^{n}$, partially in an
attempt to formulate nonlinear stability statements for (1.1) which address
these issues. This has advantages and disadvantages in comparison to $d_{0}$.
In the nonlinear context, it is sensitive to capacity-zero perturbations of
$\Omega$. It also turns out to be specifically relevant to some applications
to monotonicity formulas in free boundary theory, which we discuss in [3]. On
the other hand, $d_{1}$ is a kind of indirect measure of set difference and
may be difficult to use in practice; we also do not know whether
$d_{1}(E,B_{R})$ controls $d_{0}(E,B_{R})$ for all sets.

The results of this paper, in conjunction with our companion paper [3],
establish sharp quantitative stability for the Faber-Krahn inequality with
respect to the distance $d_{1}$. The result is proven on the round sphere and
hyperbolic space as well as in Euclidean space: we prove in [3, Theorem 1.1]
that (1.2) holds with $d=d_{1}+d_{2}$ and $\alpha=2$ on simply connected space
forms. The proof is based on a selection principle. This scheme, introduced by
Cicalese and Leonardi in [17] in the context of the isoperimetric inequality,
has been implemented in various settings in recent years (including [9]) to
establish quantitive forms of geometric inequalities. A selection principle
has two main steps:

  1. (1)

The reduction step to show that linear stability implies nonlinear stability,
based on regularity theory for a penalized functional.

  2. (2)

The local step to prove that linear stability holds.

This paper is dedicated to carrying out Step (1): if (1.1) is linearly stable
with $d=d_{1}$ and $\alpha=2$, then it is also nonlinearly stable. Step (2) on
simply connected space forms is established in [3]. Let us describe how the
classical selection principle argument for Step (1) might go in this context,
and where the significant challenges arise in this setting. Suppose by way of
contradiction that (1.2) fails. We may thus find a sequence of sets
$\Omega_{k}$ with $|\Omega_{k}|=|B_{R}|$ and
$\lambda_{1}(\Omega_{k})-\lambda_{1}(B_{R})\to 0$, while

(1.4) $c_{k}:=d^{2}_{1}(B_{R}(x_{\Omega_{k}}),\Omega_{k})\geq
k[\lambda_{1}(\Omega_{k})-\lambda_{1}(B_{R})]$

The goal, then, is to replace each $\Omega_{k}$ with a set $U_{k}$ that also
satisfies (1.4), but such that $\partial U_{k}$ a smooth normal graph over
$\partial B_{R}$, in this way contradicting the linear stability assumption.
Such a set is obtained by choosing $U_{k}$ to be a minimizer of a functional
that roughly takes the form

$\mathcal{F}_{\tau}^{0}(U)=\lambda_{1}(U)+\tau\sqrt{c_{k}^{2}+(c_{k}-d_{1}(U,B_{R}(x_{U}))^{2})^{2}}$

for a small parameter $\tau>0$ and establishing regularity estimates for such
minimizers. The second term in this functional forces
$d_{1}(U_{k},B_{R}(x_{U_{k}}))$ to be close to
$d_{1}(\Omega_{k},B_{R}(x_{\Omega_{k}}))$ so that (1.4) will be satisfied by
$U_{k}$, while the first term is hopefully regularizing the boundary of
$U_{k}$.

The Euler-Lagrange equation for minimizers of $\mathcal{F}_{\tau}^{0}$ leads
to a free boundary condition along $\partial U_{k}$ of Bernoulli type, similar
to the one studied in [5]. When working with $d_{0}$ or other weaker
distances, the second term in $\mathcal{F}_{\tau}^{0}$ is of lower order, and
can easily be seen to be a minor perturbation to the functional. In
particular, the regularity theory of Alt and Caffarelli [5] applies more or
less directly (at least in conjunction with more recent literature, such as
[20, 18]). In the case of $d_{1}$, however, this is no longer a sufficient
heuristic: the second term of $\mathcal{F}_{\tau}^{0}$ is now a _critical_ ,
or same-order, perturbation of $\lambda_{1}(U)$. The key point to establish
regularity becomes controlling the distance $d_{1}(U^{\prime},U)$ between a
minimizing set $U$ and various competitors $U^{\prime}$. This amounts to
needing estimates like

(1.5) $\int|u_{U}-u_{U^{\prime}}|\leq
C|\lambda_{1}(U)-\lambda_{1}(U^{\prime})|,$

at least for the competitors $U^{\prime}$ necessary to obtain estimates on
$U$. These competitors were essentially laid out in [5], and closely follow
minimal surface theory; they consist of $U\cup B_{r}(x)$, $U\setminus
B_{r}(x)$, and $\phi_{t}(U)$ for smooth diffeomorphisms $\phi_{t}$.

We do not know if (1.5) is valid for these competitors, and believe this is an
interesting problem. This means that we are unable establish the necessary
regularity theory for $\mathcal{F}_{\tau}^{0}$ to carry out Step (1) of the
selection principle as described above. In order to overcome this challenge,
we _modify the functional_ : let

(1.6)
$\mathcal{F}_{\tau}^{*}(U)=\lambda_{1}(\Omega)+\mathfrak{T}\mathrm{tor}(\Omega)+\tau\sqrt{c_{k}^{2}+(c_{k}-d_{1}(U,B_{R}(x_{U}))^{2})^{2}},$

where $\mathfrak{T}>0$ is a small parameter and

$\mathrm{tor}(\Omega)=\int\frac{1}{2}|\nabla w|^{2}-w$

is the torsional rigidity of $\Omega$. The torsional rigidity has a long
history of appearing when considering spectral optimization problems (see [11,
9]), and shares with $\lambda_{1}$ the ball as the unique volume-constrained
minimizer. The functional (1.6) corresponds to a vectorial  free boundary
problem. The crucial benefit of introducing this torsion term is that we can
establish the analogue of the key estimate (1.5) corresponding to this
functional:

(1.7) $\int|u_{U}-u_{U^{\prime}}|\leq
C\left(|\lambda_{1}(U)-\lambda_{1}(U^{\prime})|+|\mathrm{tor}(U)-\mathrm{tor}(U^{\prime})|\right)$

for all the relevant competitors $U^{\prime}$. This is shown in Proposition
3.9 and is the starting point toward establishing an existence and regularity
theory for $\mathcal{F}_{\tau}^{*}$ and its generalizations, which constitutes
the core analysis of this paper.

######  Theorem 1.1.

There exists a $\mathfrak{T}_{0}>0$ such that for each
$\mathfrak{T}<\mathfrak{T}_{0}$, there exists a $\tau_{0}(\mathfrak{T})>0$
such that if $\tau<\tau_{0}$:

  1. (1)

There exists a minimizer $U$ of $\mathcal{F}_{\tau}^{*}$ over the class of all
open sets $U$ with $|U|=|B_{R}|$.

  2. (2)

This $U$ has $C^{2,\alpha}$ boundary for any $\alpha<1$, and $\partial U$ may
be expressed as a normal graph $\xi$ over $\partial B_{R}$ with
$\|\xi\|_{C^{2}}=o_{\tau}(1)$ (i.e. for every $\varepsilon>0$, there is a
$\tau(\varepsilon)$ such that if $\tau<\tau(\varepsilon)$, then
$\|\xi\|_{C^{2,\alpha}}\leq\varepsilon$).

Beyond establishing the key estimate (1.7), further difficulties arise in
proving Theorem 1.1, which we discuss below. Theorem 1.1 holds for a more
general class of critical perturbations of the Alt-Caffarelli functional that
is introduced in Section 2.2. One generalization is that Theorem 1.1 holds for
the functional (1.6) with $d_{1}$ replaced by $\tilde{d}_{0}+d_{1}$ where
$\tilde{d}_{0}$ is an essentially equivalent regularization of $d_{0}$ in
(1.3) above.

Returning to the discussion of the selection principle, we use the functional
$\mathcal{F}_{\tau}^{*}$ to establish Step (1) of the selection principle
corresponding to the inequality
$\lambda_{1}(\Omega)+\mathfrak{T}\mathrm{tor}(\Omega)\geq\lambda_{1}(B_{R})+\mathfrak{T}\mathrm{tor}(B_{R})$
instead of (1.1). This linear stability-implies-nonlinear stability result
takes the following form.

######  Corollary 1.2.

Assume that, for a $\mathfrak{T}<\mathfrak{T}_{0}$ and $\varepsilon>0$, the
inequality

$d_{1}^{2}(B_{R}(x_{\Omega}),\Omega)+d_{0}^{2}(B_{R}(x_{\Omega}),\Omega)\leq
C[\lambda_{1}(\Omega)-\lambda_{1}(B_{R})+\mathfrak{T}(\mathrm{tor}(\Omega)-\mathrm{tor}(B_{R}))]$

holds for all $\Omega$ with $|\Omega|=|B_{R}|$ whose boundaries may be
expressed as a $C^{2,\alpha}$ normal graph over $\partial B_{R}$ with norm
bounded by $\varepsilon$. Then it also holds for all open sets $\Omega$ with
$|\Omega|=|B_{R}|$ (with a possibly larger constant).

Although it may appear that we have departed from the central aim of
establishing Step (1) of the selection principle for the Faber-Krahn
inequality, we show in the companion paper [3] that the statement of Corollary
1.2 actually implies that the same is valid with $\mathfrak{T}=0$ after
combining with the Kohler-Jobin inequality ([27, 28], see also [29, 7]), an
inequality that relates $\mathrm{tor}(\Omega)-\mathrm{tor}(B_{R})$ to
$\lambda_{1}(\Omega)-\lambda_{1}(B_{R})$. It is also shown in [3] that (1.1)
is linearly stable with $d=d_{1}+d_{2}$ and $\alpha=2$, which when combined
with Corollary 1.2 gives nonlinear stability as well.

While the preceding discussion was carried out in ${\mathbb{R}}^{n}$ to more
clearly illustrate the concepts in play, the results proved here also apply to
Riemannian manifolds, and are stated accordingly (see Theorem 2.13 and Theorem
2.14 for the general forms of Theorem 1.1 and Corollary 1.2 respectively). In
particular, Corollary 1.2 is also valid for any manifold for which balls are
the isoperimetric sets (i.e. they are the unique sets to attain the minimum in
the isoperimetric inequality), or with suitable modifications to the
statement, on any manifold on which minimizers to (1.1) are unique up to
isometry. The former condition is restrictive and only known for a handful of
manifolds (see [34] for a survey of the known results), but includes the round
sphere and hyperbolic space with standard metrics, which will be studied in
[3] in greater detail:

######  Remark 1.3.

Corollary 1.2 holds on hyperbolic space and the round sphere, with the
barycenter $x_{\Omega}$ replaced by the set centers defined in Examples 2.8
and 2.9 respectively.

The outline of our approach and the structure of the rest of the paper is as
follows. In Section 2 we give careful definitions of the quantities discussed
here on manifolds and set up notation for everything to follow. In Section
2.4, we discuss how to pass from Theorem 1.1 to Corollary 1.2 (and state it
more carefully and in greater generality). In Section 3 we establish basic
properties of the first eigenvalue and the torsional rigidity and discuss
Faber-Krahn inequalities on manifolds to the extent needed. The introduction
of the torsional rigidity is the first key idea in the paper, and in Section 3
we also prove (1.7). As previously discussed, this inequality will be
necessary to prove even basic estimates.

Typical approaches to Bernoulli free boundary problems involve first proving
existence of minimizers, usually via the direct method, and then proving
nondegeneracy and growth estimates at the free boundary. Due to the presence
of the eigenfunction penalization term, which is of the same order as the
other terms in the functional, a direct approach to the existence of
minimizers fails as the functional is not lower semicontinuous with respect to
topologies for which compactness is available. For this reason, we must
introduce a new approach to proving existence. In Section 4 we prove a priori
nondegeneracy estimates for the free boundary for inward minimizers. In
Section 5 we prove a priori Lipschitz growth estimates at the free boundary
for outward minimizers. In Section 6 we construct a minimizing sequence of
outward minimizers and utilize the Lipschitz estimate to prove the limit
$\Omega$ is a minimizer. This minimizer will then have both nondegeneracy and
Lipschitz estimates since it is also both an inward and outward minimizer. One
benefit of this approach is that the entire argument takes place in the class
of open sets; the a priori estimates allow us to avoid relaxing the problem to
the class of quasi-open sets as is common in the literature [12].

From here, we would like to follow the recent approaches of [14, 32] to
establish regularity for vectorial free boundary problems. The first step in
doing so is to apply the boundary Harnack principle in order to establish a
scalar form of the free boundary condition. Substantial difficulties arise
here due to the nonlinear term, because the contribution from $d_{1}$ to the
Euler-Lagrange equation along $\partial\Omega$ is of the same order as that
from $\lambda_{1}(\Omega)$ or $\mathrm{tor}(\Omega)$: it may be expressed as
$|\nabla v|^{2}$ for some function $v$ satisfying an elliptic PDE on $\Omega$.
In particular, there is no favorable sign appearing in this term (it is not
_elliptic_ , in some sense), and so it must be controlled by the other
contributions. The key tool toward controlling this term is a careful analysis
of the Green’s function $G_{\Omega}$ of $\Omega$, which we carry out in
Section 7. This involves utilizing an inhomogeneous boundary Harnack principle
recently shown in [4].

We begin Section 8 by considering more carefully how $d_{1}$ and $u_{\Omega}$
change under smooth deformations of domains. Applying the estimates from the
previous section we obtain several forms of the free boundary condition
satisfied along $\partial\Omega$ and arrive at a pointwise version. The final
fundamental point in the paper is to again apply the inhomogeneous boundary
Harnack principle to rewrite the free boundary condition in a form suitable to
known methods.

In Section 9 we apply recent free boundary theory to conclude that
$\partial\Omega$ is $C^{1,\alpha}$. In Section 10 we apply a higher order
inhomogeneous boundary Harnack principle [22] to prove Theorem 1.1 (that
$\partial\Omega$ may be parametrized as a $C^{2,\alpha}$ normal graph over the
sphere). Appendix A gives further examples of functionals to which our results
apply. Finally, Appendix B shows that domains supporting solutions to elliptic
PDE with linear growth away from $\partial U$ are nontangentially accessible,
a result which is known in the literature but, to the best of our knowledge,
not in the generality required here. This is a technical point relevant in
Section 7.

Acknowledgements. MA was partially supported by Simons Collaboration Grant ID
637757. During the course of this work, RN was partially supported by the
National Science Foundation under Grant No. DMS-1901427, as well as Grant No.
DMS-1502632 “RTG: Analysis on manifolds” at Northwestern University and Grant
No. DMS-1638352 at the Institute for Advanced Study. This project originated
from discussions at the 2018 PCMI Summer Session on Harmonic Analysis.

##  2\. Setup and definitions

Throughout this paper, we let $(M,g)$ be a complete, $n$-dimensional smooth
Riemannian manifold without boundary that is connected and oriented. We
further assume that $(M,g)$ has bounded geometry, that is,
$\text{inj}_{M}:=\inf_{x\in M}\text{inj}_{x}>0$ and $\sup_{x\in
M}|\text{Rm}|<\infty.$ All constants will depend on the metric $g$; it is
likely that these constants are uniform for Riemannian manifolds with
uniformly bounded geometry but we do not track this dependence.

Unless otherwise specified, all integrals are taken with respect to the volume
measure $m$ associated to $g$. We use the notation $|\Omega|$ for the volume
of a measurable set $\Omega\subseteq M$ and ${\mathcal{H}}^{n-1}$ for
$(n-1)$-dimensional Hausdorff measure. We let $d(x,y)$ denote the geodesic
distance between $x,y\in M$ and let

$B_{r}(x)=\\{y\in M\,:\,d(y,x)<r\\}$

denote a geodesic ball centered at $x$ of radius $r$. Given a set $\Omega$ and
a point $x\in M$, we let $d(x,\Omega)=\inf\\{d(x,y):y\in\Omega\\}.$ We let
$\nabla$ and $\operatorname*{div}$ denote the gradient and divergence with
respect to $g$ and let $\Delta u=\operatorname*{div}\nabla u$ be the Laplace-
Beltrami operator with respect to $g$.

Let $G$ denote the (possibly trivial) group of isometries of $(M,g)$. Several
global existence results in the sequel will hold for Riemannian manifolds such
that either $M$ is compact or $M/G$ is compact; examples of the latter to keep
in mind are Euclidean space, hyperbolic space, cylinders
$S^{n-k}\times{\mathbb{R}}^{k}$ equipped with the standard product metric, and
more generally products of space forms.

###  2.1. Base energy

Given a bounded open set $\Omega\subseteq M$, we may define the _torsional
rigidity_ of $\Omega$:

(2.1) $\mathrm{tor}(\Omega)=\inf_{u\in
H^{1}_{0}(\Omega)}\int_{\Omega}\frac{1}{2}|\nabla u|^{2}-u.$

We note that the definition of the torsional rigidity given here differs by a
sign from the typical definition. This is a negative quantity, and it is
straightforward to check (using the Sobolev inequality and the Lax-Milgram
theorem) that the infimum is finite and uniquely attained for any open bounded
$\Omega\neq M$ by a function $w_{\Omega}\geq 0$. We will refer to $w_{\Omega}$
as the _torsion function_. Note also that $\mathrm{tor}(\Omega)$ is decreasing
with respect to set inclusion.

The _first (Dirichlet) eigenvalue_ of $\Omega$ is defined as

(2.2) $\lambda_{1}(\Omega)=\inf_{u\in
H^{1}_{0}(\Omega)\setminus\\{0\\}}\frac{\int_{\Omega}|\nabla
u|^{2}}{\int_{\Omega}u^{2}}.$

This infimum is also attained by a nonnegative function $u_{\Omega}$, the
_first eigenfunction_ , and $\lambda_{1}(\Omega)$ is also a decreasing
function with respect to set inclusion. The first eigenfunction is unique up
to scalar multiples so long as $\lambda_{1}(\Omega)<\lambda_{2}(\Omega)$,
which in particular is true if $\Omega$ is connected. The notation
$u_{\Omega}$ will be used for the unique first eigenfunction with
$u_{\Omega}\geq 0$ and $\int u_{\Omega}^{2}=1$, whenever it exists and is
unique. We frequently will extend $u_{\Omega}$ and $w_{\Omega}$ by zero to be
defined on all of $M$, using the same notation to indicate the functions
$u_{\Omega},w_{\Omega}:M\to{\mathbb{R}}$.

In principle, we are interested in minimizing energy functionals of the type
(1.6) among sets $\Omega\subset M$ of a fixed volume $|\Omega|=v$. In
practice, it is more convenient to replace the volume constraint with a volume
penalization term. Following [1, 9], we define the volume penalization

(2.3) $f_{v,\eta}(t)=\begin{cases}\eta(t-v)&t\leq v\\\
\frac{1}{\eta}(t-v)&t>v\,.\end{cases}$

The idea of the volume penalization (2.3) is that by choosing the parameter
$\eta$ to be sufficiently small, a set $\Omega$ whose volume exceeds the
prescribed volume $v$ will have a large energy contribution coming from the
term $f_{v,\eta}(|\Omega|)$. We define the _base energy_ of $\Omega$ as

(2.4)
$\mathcal{E}(\Omega)=\lambda_{1}(\Omega)+\mathfrak{T}\mathrm{tor}(\Omega)+f_{v,\eta}(|\Omega|),$

where $\mathfrak{T}\in[0,1]$. The principal goal of this paper is to study
certain critical perturbations of the base energy introduced in the next
subsection. In order to do so, we will need to first consider the minimization
problem of the base energy itself. We discuss the problem of minimizing
$\mathcal{E}(\Omega)$ over sets briefly here and in more detail in Section 3.
Many of the existence and regularity results in later sections will apply to
this base energy problem as well. Let us begin with two observations regarding
the existence theory for this problem:

######  Remark 2.1 (A hard volume constraint is needed).

If we attempt to minimize the base energy $\mathcal{E}$ among all bounded open
subsets of $M$, then the energy may fail to be bounded from below and
minimizers may fail to exist or $\Omega=M$ may be a minimizer. For instance,
on Euclidean space, a simple scaling computation shows that
$\mathcal{E}(B_{R})\to-\infty$ as $R\to\infty$ for any fixed triple of
parameters $v,\eta,\mathfrak{T}>0$. Similarly, if $M$ has finite volume, then
$\mathcal{E}(M)=-\infty$ since $\mathrm{tor}(M)=-\infty$ while the other
quantities are finite.

To remedy this issue, we will introduce a “hard volume constraint” that is
much larger than $v$. Given the desired volume constraint $v\in(0,|M|)$, we
fix a larger number $v_{\text{max}}\in(v,|M|)$ and minimize with respect to
competitors $\Omega$ with $|\Omega|\leq v_{\text{max}}$. Ultimately this hard
constraint will never be saturated: in Proposition 9.5 we show that for small
values of $\eta$ (depending on $v$ and $v_{\text{max}}$) and $\mathfrak{T}$
(depending on $v,v_{\text{max}}$ and $\eta$), any minimizer over this class
actually satisfies the original desired volume constraint $|\Omega|=v$. In
particular, this means the volume-constrained problem (minimizing only over
$|\Omega|=v$) is equivalent to the volume-penalized problem where
$|\Omega|\leq v_{\text{max}}$.

######  Remark 2.2 (Compactness issues on non-compact manifolds).

On noncompact manifolds, a minimizing sequence $\Omega_{k}$ for the base
energy $\mathcal{E}$ can “drift to infinity” in the sense that
$d(x_{0},\Omega_{k})\rightarrow\infty$ for any reference point $x_{0}$. In
view of this issue, we first minimize over the class of $\Omega$ contained in
(smoothed out) balls of large radius $R$. Depending on the geometry of the
manifold at infinity, global minimizers may simply fail to exist when
$R\to\infty$. However, when $M/G$ is compact, the compactness issues are
solely caused by the symmetries of the metric and we may send the radius $R$
to infinity using concentration compactness techniques (see [31]). This will
be addressed in Section 3.4 for the base energy and then more completely in
Theorem 6.3.

Since geodesic balls need not have smooth boundary on an arbitrary Riemannian
manifold, it is more convenient from a technical standpoint to work with an
exhaustion $Q_{R}$ of smooth bounded open sets that play the role of smoothed-
out balls. More specifically, we let $Q_{R}$ be a fixed 1-parameter family of
open sets which have the following properties:

  1. (1)

$\bar{Q}_{R}\subseteq Q_{S}$ for any $S>R$

  2. (2)

$\operatorname*{diam}(Q_{R})\leq 2R$

  3. (3)

$\partial Q_{R}$ is either smooth (a finite union of disjoint $C^{2}$
codimension-$1$ submanifolds) or empty

  4. (4)

$M=\cup_{R}Q_{R}$.

In view of Remarks 2.1 and 2.2, we will fix parameters $R>0$ and
$v_{\text{max}}\in(0,|M|)$ and minimize the base energy (as well as the main
energy) over open sets in $Q_{R}$ of volume at most $v_{\text{max}}.$ We use
the notation

(2.5) $\mathcal{H}=\mathcal{H}_{R,v_{\text{max}}}=\\{\Omega\subseteq M\text{
open}:|\Omega|\leq v_{\text{max}},\Omega\subseteq Q_{R}\\}$

to denote this collection of sets. We will assume throughout the paper,
without further note, that $R$ is chosen to be large enough so that
$|Q_{R}|\geq v_{\text{max}}$. In this way, the container $Q_{R}$ does not
obstruct the hard volume constraint and the collection $\mathcal{H}$ is
nonempty. Moreover, we will always assume that $v_{\text{max}}<|M|$ when the
volume of $M$ is finite.

Given parameters $0<v<v_{\text{max}}<|M|,\eta>0$, $\mathfrak{T}\in(0,1]$ and
$R>0$, we consider the following minimization problem for the base energy:

(2.6)
$\mathcal{E}_{min}=\mathcal{E}_{min}(v,v_{\text{max}},\eta,\mathfrak{T},R)=\inf\\{\mathcal{E}(\Omega):\Omega\subseteq\mathcal{H}\\};$

we will drop the parameters unless ambiguous. The collection of minimizers for
this problem will be denoted by

(2.7)
$\mathcal{M}=\mathcal{M}(v,v_{\text{max}},\eta,\mathfrak{T},R)=\\{\Omega\subseteq\mathcal{H}:\mathcal{E}(\Omega)=\mathcal{E}_{min}\\},$

which we will show in Lemma 3.10 is always nonempty. While several of our
intermediate results will hold for larger parameter ranges, the reader should
typically think of first fixing the desired and hard volume constraints
$v,v_{\text{max}}$ and the radius $R$, then fixing the volume penalization
parameter $\eta$, and finally fixing the coefficient $\mathfrak{T}$ in front
of the torsional rigidity.

In the following example, we consider the minimization problem (2.6) on simply
connected space forms; generally speaking these are important motivating
examples to keep in mind throughout the paper. Since these spaces are compact
after modding out by the isometry group, we may send the parameter $R$ to
infinity and so the following example has no $R$ parameter.

######  Example 2.3.

Let $(M,g)$ be Euclidean space, hyperbolic space, or the round sphere. Fix
parameters $v,v_{\text{max}}$ with $0<v<v_{\text{max}}$, choosing
$v_{\text{max}}<|M|$ in the case of the round sphere. For
$\eta\leq\eta_{0}(v,v_{\text{max}})$ and
$\mathfrak{T}\leq\mathfrak{T}_{0}(v,v_{\text{max}},\eta)$, geodesic balls of
volume $v$ are the unique minimizers of the base energy $\mathcal{E}$ among
all open bounded subsets of $M$ with volume at most $v_{\text{max}}$.

For general Riemannian manifolds, we cannot hope to explicitly characterize
minimizers as in the example above. However, the following proposition
summarizes the basic facts that will be proven about minimizers of the base
energy in this paper. The proof follows from combining Lemma 3.10 and Theorem
6.3 (local and global existence respectively), Lemma 3.13 (connectedness), and
Lemma 9.5 (volume constraint satisfied).

######  Proposition 2.4 (Minimizers of the base energy).

Fix $R>0$ and $v\in(0,|M|)$. Fix any $v_{\text{max}}\in(v,|M|)$. There exists
$\eta_{0}=(R,v,v_{\text{max}})$ such that for $\eta\leq\eta_{0}$, there exists
$\mathfrak{T}_{0}=\mathfrak{T}_{0}(R,v,v_{\text{max}},\eta)$ such that the
following holds. If $\mathfrak{T}\leq\mathfrak{T}_{0}$, then
$\mathcal{E}_{min}>-\infty$ and a minimizer of the base energy $\mathcal{E}$
exists among sets in $\mathcal{H},$ i.e. the set $\mathcal{M}$ is nonempty.
Moreover, any minimizer in $\mathcal{M}$ is connected and has volume equal to
$v$.

If $M$ is compact or $M/G$ is compact, then constants above may be taken
independent of $R$ and the minimization of $\mathcal{E}$ may be taken among
all open bounded subsets of $M$ with volume at most $v_{\text{max}}.$

###  2.2. Main energy functional

Let us now introduce the class of energy functionals whose existence and
regularity theory constitute the heart of this paper and using which we can
carry out a selection principle as described in the introduction (in Section
2.4 below). Following the discussions in the introduction and in the previous
subsection, we are interested in functionals of the type (1.6), though as in
the previous subsection we relax the volume constraint. So, the general form
of the functionals we consider is

(2.8)
$\mathcal{F}_{\tau}(\Omega)=\mathcal{E}(\Omega)+\tau\mathfrak{h}(\Omega).$

Here $\mathfrak{h}(\Omega)$ is a functional mapping open bounded sets to
${\mathbb{R}}$ and $\tau<\tau_{0}$ is a parameter that will be chosen
sufficiently small depending on the parameters of the base energy
$\mathcal{E}$. The existence and regularity results of this paper will hold
whenever $\mathfrak{h}$ is an admissible nonlinearity as defined in Definition
2.11 below. Before defining this general class of admissible nonlinearities,
we motivate the definition with some important concrete examples that will be
used for our selection principle applications. In these examples, the relevant
nonlinearities take the form

(2.9) $\mathfrak{h}(\Omega)=\sqrt{c^{2}+(c-d_{*}(\Omega)^{2})^{2}}-c$

for a constant $c\in(0,1]$, where $d_{*}(\Omega)$ is a suitably defined
distance of $\Omega$ to the class of minimizers $\mathcal{M}$ of the base
energy. Note that (1.6) takes this form with
$d_{*}(\Omega)=d_{1}(\Omega,B_{R}(x_{\Omega}))$ on Euclidean space; the
subtraction of the constant $c$ in (2.9) is immaterial for the minimization
problem but conveniently normalizes the functional. We are interested in
generalizing this in two directions: (1) we want a distance $d_{*}$ that
measures the size of the symmetric difference as well as the eigenfunction
difference, and (2) we wish to suitably generalize this to certain Riemannian
manifolds.

In order to establish Step (1) of the selection principle (and ultimately
quantitative stability) in the form of Corollary 1.2 in which both the
eigenfunction distance $d_{1}$ and the asymmetry distance $d_{0}$ are
controlled (say on Euclidean space), a first approach might be to let
$d_{*}(\Omega)=d_{1}(\Omega,B_{R}(x_{\Omega}))+d_{0}(\Omega,B_{R}(x_{\Omega}))$
in (2.9), where $d_{0}$ is the symmetric difference defined in (1.3). The
issue here, already understood in [9], is that the distance $d_{0}$ is not
sufficiently smooth (as the integral of a characteristic function) and
consequently minimizers of such a functional will not be smooth.

To deal with this, we follow the approach of [9] and consider a smoothed out
version of the asymmetry term $d_{0}$. More specifically, consider a bounded,
open set $U$ with smooth boundary (say $\partial U$ is a disjoint union of
$C^{2}$ embedded submanifolds); in the relevant applications $U$ will be a
minimizer of the base energy $\mathcal{E}$, so in Euclidean space, hyperbolic
space, or the round sphere, take $U$ to be a geodesic ball. Take $f$ to be a
smooth nondecreasing function with $f(t)=t$ for $|t|\leq c_{0}$ and $f(t)$ is
locally constant for $|t|>2c_{0}$, where $c_{0}$ is selected so that
$d(x,\partial U)$ is a $C^{2}$ function on $\\{d(x,\partial U)<4c_{0}\\}$. One
can always find such a $c_{0}$; see [24, Lemma 14.16]). Now, let $\psi_{U}$ be
given by

$\psi_{U}(x)=\begin{cases}f(d(x,\partial U))&x\in U\\\ -f(d(x,\partial
U))&x\in M\setminus U.\end{cases}$

The essential point of the function $\psi_{U}$ is that
$\int_{\Omega}\psi_{U}-\int_{U}\psi_{U}$ is comparable to
$|U\triangle\Omega|^{2}$ for $\Omega$ whose boundary is a (small) normal graph
over $\partial U$, and controls $|U\triangle\Omega|^{2}$ in all cases. More
precisely, for any $\Omega$ we have

$\int_{\Omega}\psi_{U}-\int_{U}\psi_{U}=\int_{\Omega\triangle
U}|\psi_{U}|\geq\int_{\Omega\triangle U}\min\\{|d(x,\partial
U)|,c_{0}\\}dx\geq c|\Omega\triangle U|^{2},$

where the last step can be seen from the fact that the integral is minimized
by tubular neighborhoods of $U$, and then by changing variables and using the
smoothness of $\partial U$. On the other hand, if the boundary of $\Omega$ is
a $C^{2}$ normal graph $\xi$ over $\partial U$ with $|\xi|\leq c_{0}$, we may
change variables and integrate

$\int_{\Omega\triangle U}|\psi_{U}|\leq C\int_{\partial
U}\int_{0}^{|\xi(x)|}tdtd{\mathcal{H}}^{n-1}=C\int_{\partial
U}\int_{0}^{|\xi(x)|^{2}/2}1dtd{\mathcal{H}}^{n-1}\leq
C|U\triangle\Omega|^{2}.$

So, given any two bounded open sets $\Omega$ and $U$ with $U$ smooth as above,
we let

(2.10)
$d_{*}(\Omega,U)^{2}=\int_{\Omega}\psi_{U}-\int_{U}\psi_{U}+\int|u_{\Omega}-u_{U}|^{2}$

where $u_{\Omega}$ denotes the first eigenfunction of $\Omega$, normalized so
that $u\geq 0$ and $\|u_{\Omega}\|_{L^{2}}=1$ and extended by zero to be
defined on all of $M$. In each of the following examples, we will use
$d_{*}(\Omega,U)$ to define a distance $d_{*}(\Omega)$ to the collection
$\mathcal{M}$ of minimizers of the base energy and consider the functional
(2.8) with the nonlinearity (2.9) for this suitable definition of
$d_{*}(\Omega)$.

######  Remark 2.5 (Unique first eigenfunction).

Lemma 3.14 below guarantees that if
$\mathcal{E}(\Omega)\leq\mathcal{E}_{min}+\tau_{0}$ for $\tau_{0}$ is
sufficiently small, then $\Omega$ has a unique first eigenfunction and so
$d_{*}(\Omega,U)$ is well-defined for such a set. This means that by choosing
the parameter $\tau$ in (2.8) accordingly, $u_{\Omega}$ and $d_{*}(\Omega,U)$
will be well-defined for any minimizing sequence or minimizer of (2.8). As
such, we will generally only define $\mathfrak{h}$ for sets with
$\mathcal{E}(\Omega)\leq\mathcal{E}_{min}+\tau_{0}$. There is no loss of
generality, at least when considering minimizers, to define
$\mathfrak{h}(\Omega)=1$ if
$\mathcal{E}(\Omega)\geq\mathcal{E}_{min}+\tau_{0}$.

The simplest example is the case when only one minimizer of the base energy
exists.

######  Example 2.6 (Unique minimizer).

Suppose that the collection of minimizers $\mathcal{M}$ of the base energy
consists of a single set $U$, and $U$ has boundary of class $C^{2}$. Then the
distance $d_{*}(\Omega)$ of $\Omega$ to the collection of minimizers of the
base energy with $v=|\Omega|$ is defined by $d_{*}(\Omega)=d_{*}(\Omega,U).$

In the case when minimizers of the base energy are not unique, to define a
distance of a given set $\Omega$ to the collection of minimizers, we must
select the nearest minimizer to $\Omega$. On Euclidean space, where
translation invariance gives rise to nonuniqueness of minimizers, we do this
by choosing the ball with the same barycenter as $\Omega$.

######  Example 2.7 (Euclidean space).

Let $(M,g)$ be Euclidean space. Given a bounded open set $\Omega$, let $R$ be
the unique radius so that $|\Omega|=|B_{R}|$ and let
$x_{\Omega}=\fint_{\Omega}x\,dx$ be the barycenter of $\Omega$. Then the
distance $d_{*}(\Omega)$ of $\Omega$ to the collection of balls $B_{R}$ (i.e.
the minimizers of the base energy with $v=|\Omega|$) is defined by
$d_{*}(\Omega)=d_{*}(\Omega,B_{R}(x_{\Omega})).$

We saw in Example 2.3 that the situation on hyperbolic space and the round
sphere is similar to Euclidean space: the collection of minimizers of the base
energy comprises geodesic balls of suitable radius centered at any  point. In
these cases, we define an appropriate “set center” in analogy to the Euclidean
barycenter.

######  Example 2.8 (Hyperbolic space).

Let $(M,g)$ be hyperbolic space. Given a set bounded open set $\Omega\subset
M$, define the set center $x_{\Omega}$ of $\Omega$ by
$x_{\Omega}=\text{argmin}_{x}\int_{\Omega}d^{2}(x,y)\,dm(y)$, which is well-
defined. Then the distance $d_{*}(\Omega)$ of $\Omega$ to the collection of
balls $B_{R}$ (i.e. the minimizers of the base energy with $v=|\Omega|$) is
defined by $d_{*}(\Omega)=d_{*}(\Omega,B_{R}(x_{\Omega})).$

On the round sphere, we cannot define the notion of set center in analogy to
Example 2.8 as the minimum may not be uniquely achieved. We instead consider
the following.

######  Example 2.9 (The round sphere).

Let $(M,g)$ be the round sphere. Consider its standard embedding in
${\mathbb{R}}^{n+1}$. Given any open $\Omega\subseteq M$, let
$y_{\Omega}=\fint_{\Omega}yd{\mathcal{H}}^{n}(y),$ where
$y,y_{\Omega}\in{\mathbb{R}}^{n+1}$ and this is a (vector-valued) surface
integral. If $y_{\Omega}\neq 0$, then define the set center of $\Omega$ as
$x_{\Omega}=y_{\Omega}/|y_{\Omega}|$. The distance $d_{*}(\Omega)$ of $\Omega$
to the collection of balls $B_{R}$ (i.e. the minimizers of the base energy
with $v=|\Omega|$) is defined by
$d_{*}(\Omega)=d_{*}(\Omega,B_{R}(x_{\Omega})).$

In all three of the preceding examples, uniqueness of minimizers of the base
energy fails solely due to the isometries of the space, and modulo the
isometry group $G$, there is a unique minimizer $U$ with boundary of class
$C^{2}$. In more general situations of this type, we define a suitable notion
of set center, which is a mapping $\Omega\mapsto x_{\Omega}$ for which there
is a unique minimizer $U_{\Omega}$ of the base energy with the same set center
as $\Omega$. Because the definition is somewhat technical, we postpone it
until Definition A.2 in Appendix A. In this situation, we define
$d_{*}(\Omega)$ analogously to the examples above; the Euclidean barycenter
and the sets centers for hyperbolic space and the round sphere above are
examples of set centers in the sense of Definition A.2.

######  Example 2.10 (Minimizer is unique modulo isometries).

Assume that $\mathcal{M}$ consists of a single set $U$ up to the action of the
isometry group $G$. Also assume that it has smooth boundary, and that there
exists a set center $\Omega\mapsto x_{\Omega}$ adapted to $U$. We define

(2.11) $d_{*}(\Omega)=d_{*}(\Omega,U_{\Omega})$

where $U_{\Omega}$ the unique set in $\mathcal{M}$ with the same set center as
$\Omega$.

It may be tempting to give generic constructions of $d_{*}(\Omega)$,
independent of set centers, by solving minimization problems along the lines
of $U_{\Omega}=\text{argmin}\left\\{d_{*}(\Omega,U):U\in\mathcal{M}\right\\}$
and letting $d_{*}(\Omega)=d_{*}(\Omega,U_{\Omega}).$ However, such
variational problems have issues with uniqueness and regularity of the
solution map: these are closely related to the convexity and smoothness of the
“distance” functional being minimized, as well as the structure of the family
$\mathcal{M}$.

All of the examples given above fall into the general class of admissible
nonlinearities $\mathfrak{h}$ for which our main existence and regularity
results are valid. Below, $u_{\Omega}$ is used to denote the normalized,
nonnegative first eigenfunction of $\Omega$ (see (2.2)); recalling Remark 2.5,
this function exists and is unique when
$\mathcal{E}(\Omega)\leq\mathcal{E}_{min}+\tau_{0}$.

######  Definition 2.11 (Admissible nonlinearities).

Fix a (possibly trivial) closed subgroup $G_{0}\leq G$ of the isometry group
of $(M,g)$. A function $\mathfrak{h}$ mapping bounded open subsets of $M$ to
${\mathbb{R}}$ is said to be _an admissible nonlinearity_ with respect to
$G_{0}$ if the following properties hold.

  1. (N1)

$\mathfrak{h}\in[0,1]$.

  2. (N2)

$\mathfrak{h}$ is invariant under isometries in $G_{0}$.

  3. (N3)

If $\Omega,\Omega^{\prime}$ are bounded open sets with
$\mathcal{E}(\Omega)\leq\mathcal{E}_{min}+\tau_{0}$, then

$|\mathfrak{h}(\Omega)-\mathfrak{h}(\Omega^{\prime})|\leq|\Omega\triangle\Omega^{\prime}|+\int|u_{\Omega}-u_{\Omega^{\prime}}|.$

  4. (N4)

For any $x_{0}\in M$ and $r\leq r_{0}$, take a one-parameter family $\phi_{t}$
of diffeomorphisms with $\phi_{0}(x)=x$ and $|\partial_{t}\phi_{t}|\leq 1$
such that $|\\{x:\phi_{t}(x)\neq x\\}|\subseteq B_{r}(x_{0})$. Assume
$\phi_{t}(\Omega)$ is a set with
$\mathcal{E}(\phi_{t}(\Omega))\leq\mathcal{E}_{min}+\tau_{0}$ and
${\mathcal{H}}^{n-1}(\partial\Omega)<\infty$; then

$\limsup_{t\rightarrow
0}\frac{1}{|t|}\left|\mathfrak{h}(\phi_{t}(\Omega))-\mathfrak{h}(\Omega)-\int(u_{\phi_{t}(\Omega)}-u_{\Omega})a_{\Omega}-\left[\int_{\phi_{t}(\Omega)}b_{\Omega}-\int_{\Omega}b_{\Omega}\right]\right|\leq
C_{\mathfrak{h}}r^{n},$

where $a_{\Omega}$, $b_{\Omega}$ are continuous functions
$M\rightarrow{\mathbb{R}}$ with $|a_{\Omega}|,|b_{\Omega}|\leq 1$. Moreover,
$\|b_{\Omega}\|_{C^{1,1}(Q_{R})}\leq 1$ and
$\|a_{\Omega}\|_{C^{0,1}(\Omega)}\leq 1$.

Assumption 1 simply says that $\mathfrak{h}$ is nonnegative and uniformly
bounded on bounded open sets, and it is convenient to normalize any such
functional so that the upper bound is $1$. Assumption 2 should be understood
with Example 2.10 in mind; when the base energy $\mathcal{E}$ has a nonunique
minimizer due only to a subgroup of isometries $G_{0}$, then 2 is used to
compare a set $\Omega$ to the nearest minimizer of the base energy. Assumption
3 heuristically states that $\mathfrak{h}$ is a Lipschitz function
$\Omega\mapsto{\mathbb{R}}$, where the topology placed on the space of sets is
either the $L^{1}$ distance $|\Omega\triangle\Omega^{\prime}|$ or a distance
governed by the $L^{1}$ difference of their eigenfunctions. Assumption 4
should be viewed as a higher regularity assumption on $\mathfrak{h}$; it is
asking that $\mathfrak{h}$ is not only Lipschitz but also $C^{1}$, with
derivative represented by the functions $a_{\Omega}$ and $b_{\Omega}$.

Only the first three assumptions 1, 2, and 3 are needed for the existence
theory and core estimates up through Section 7. The last assumption 4 is used
to derive the Euler-Lagrange equation in Section 8 (indeed, the functions
$a_{\Omega}$ and $b_{\Omega}$ appear in the first variation) and for the
regularity of minimizers in Sections 9 and 10. In particular the extra
regularity of $a_{\Omega},b_{\Omega}$ is only needed in Section 10 to obtain
better smoothness of $\partial\Omega$ for minimizers.

In Appendix A, we verify that the examples given above are admissible
nonlinearities. Given an admissible nonlinearity, it is easy to construct
others via composition with functions of one variable:

######  Remark 2.12.

Given any admissible nonlinearity $\mathfrak{h}$, and any $C^{1}$ function
$\phi:[0,1]\rightarrow[0,1]$ with $|\phi^{\prime}|\leq 1$, the composition
$\phi\circ\mathfrak{h}$ is again an admissible nonlinearity. No monotonicity
or structure of $\phi$ is required here.

###  2.3. Minimizers of the main energy

As we did for the base energy, we will fix $R>0$ and $v_{\text{max}}$ and
minimize the main energy over the set
$\mathcal{H}=\mathcal{H}_{R,v_{\text{max}}}$ defined in (2.5). We say that a
bounded open set $\Omega$ is a _minimizer_ if $\Omega\in\mathcal{H}$ and for
any $\Omega^{\prime}\in\mathcal{H}$,

(2.12) $\mathcal{F}_{\tau}(\Omega)\leq\mathcal{F}_{\tau}(\Omega^{\prime}).$

We say that $\Omega$ is an _inward minimizer_ if (2.12) holds for any
$\Omega^{\prime}\in\mathcal{H}$ with $\Omega^{\prime}\subseteq\Omega$. We say
that $\Omega$ is an _outward minimizer_ if (2.12) holds for any
$\Omega^{\prime}\in\mathcal{H}$ with $\Omega^{\prime}\supset\Omega$.

The following theorem summarizes the main existence and regularity properties
for $\mathcal{F}_{\tau}$ that are established in this paper. The proof follows
by combining Theorem 6.2 and Theorem 6.3 (existence), Proposition 9.5 (volume
constraint), Theorem 8.1 (free boundary condition), Theorem 9.4
($C^{1,\alpha}$ regularity), and Corollary 10.5 (higher regularity).

We say that a collection $\mathcal{C}$ of bounded open subsets of $M$ is
_uniformly_ $C^{k}$ if there is a constant $C>2/\text{inj}_{M}$ such that for
every $U\in\mathcal{C}$ and every $x\in\partial U$, the set $\partial U\cap
B_{1/C}(x)$ may be expressed as a graph with $C^{k}$ norm at most $C$ over a
hyperplane in normal coordinates.

######  Theorem 2.13.

Fix $R>0$ and $v\in(0,|M|)$. Fix any $v_{\text{max}}\in(v,|M|).$ There exist
positive constants $\eta,\mathfrak{T}$ $\tau_{0}>0$ depending on
$R,v,v_{\text{max}}$ such that the following holds. Let $\mathfrak{h}$ be an
admissible nonlinearity with respect to the isometry group $G$ of $(M,g)$ and
for fixed $\tau<\tau_{0}$, consider the energy functional $\mathcal{F}_{\tau}$
defined in (2.8) with the parameters $v,\eta,$ and $\mathfrak{T}$ in the base
energy $\mathcal{E}$ of (2.8).

  1. (1)

There exists an open set $\Omega$ that minimizes $\mathcal{F}_{\tau}$ among
sets in $\mathcal{H}=\mathcal{H}_{R,v_{\text{max}}}$. Any such minimizer
satisfies the volume constraint $|\Omega|=v$, is a set of finite perimeter,
and has a unique first eigenfunction $u_{\Omega}$ up to scaling.

  2. (2)

There exist $C>0$ and $\alpha\in(0,1)$ depending only on $v,v_{\text{max}},R$,
a function $\rho\geq-C\tau$ with $\|\rho\|_{C^{0,\alpha}(\partial\Omega)}\leq
C$, and a constant $A_{0}\in[1/C,C]$ such that the normalized first
eigenfunction $u_{\Omega}$ satisfies the free boundary condition

$|\nabla u_{\Omega}(x)|^{2}(1+\rho(x))=A_{0}$

for ${\mathcal{H}}^{n-1}$-a.e. $x\in\partial^{*}\Omega\cap Q_{R}$.

  3. (3)

Assume further that the collection $\mathcal{M}$ of minimizers of the base
energy is uniformly $C^{4}$. For every $\alpha\in(0,1)$ and $r_{1}>0$ there is
a $\tau_{1}=\tau_{1}(v,v_{\text{max}},\eta,r_{1},\alpha)>0$ such that the
following holds. If $\tau<\tau_{1}$, then for any minimizer $\Omega$ of
$\mathcal{F}_{\tau}$, the set $\partial\Omega\cap\\{x\in Q_{R}:d(x,\partial
Q_{R})>r_{1}\\}$ may be parametrized as a $C^{2,\alpha}$ normal graph (with
$C^{2,\alpha}$ norm bounded by $r_{1}$) over $\partial U$ for some
$U\in\mathcal{M}$.

If $M/G$ is compact, then all constants may be taken to be independent of $R$,
the minimization of $\mathcal{F}_{\tau}$ may be taken among all bounded open
subsets of $M$ with volume at most $v_{\text{max}}$, and in (3) the entire
boundary of any minimizer $\Omega$ may be written as a $C^{2,\alpha}$ graph
over $\partial U$ for some $U\in\mathcal{M}$.

Theorem 2.13(1) implies that the minimization of $\mathcal{F}_{\tau}$ among
sets in $\mathcal{H}$ is equivalent to the minimization problem

$\inf\\{\mathcal{F}_{\tau}(\Omega):\Omega\in\mathcal{H}^{*},|\Omega|=v\\}$

where we let $\mathcal{H}^{*}$ denote either the collection of open bounded
subsets of $Q_{R}$ or, in the case that $M/G$ is compact, the collection of
open bounded subsets of $M$.

In order to better track the dependence of constants in various estimates in
the sections to follow, we define two quantities which measure growth
properties of the eigenfunction and torsion function of $\Omega$. Let $\Omega$
be an open subset of $Q_{R}$ with $\lambda_{1}(\Omega)<\lambda_{2}(\Omega)$,
and $u_{\Omega}$ the first eigenfunction (normalized so that $\int
u_{\Omega}^{2}=1$ and $u_{\Omega}\geq 0$ as usual), while $w_{\Omega}$ is the
torsion function. Then

(2.13)
$\text{UP}(\Omega)=\sup\left\\{\frac{u_{\Omega}(x)+\sqrt{\mathfrak{T}}w_{\Omega}(x)}{d(x,\partial\Omega)}:x\in\Omega,d(x,\partial\Omega)\in(0,1)\right\\}$

and

(2.14) $\text{DO}(\Omega)=\inf\left\\{\frac{1}{r}\sup_{x\in
B_{r}(y)}u_{\Omega}(x)+\sqrt{\mathfrak{T}}w_{\Omega}(x):y\in\bar{\Omega},r\in(0,1)\right\\}.$

In general, $\text{UP}(\Omega)\in(0,\infty]$ while
$\text{DO}(\Omega)\in[0,\infty]$, though we will show in Section 3.1 that
$\text{DO}(\Omega)<\infty$ by showing that $u_{\Omega}$ and $w_{\Omega}$ are
bounded. We also always have $\text{DO}(\Omega)\leq\text{UP}(\Omega)$. The
core nonlinear estimates for this type of variational problem, following [5],
are that outward minimizers have $\text{UP}(\Omega)\leq C<\infty$, while
inward minimizers have $\text{DO}(\Omega)\geq c>0$.

###  2.4. Selection Principle

In this section, we establish the main linear stability-implies-nonlinear
stability result of this paper, which is Theorem 2.14 below. This is a
generalization of Corollary 1.2. The proof is essentially self-contained
assuming Proposition 2.4 and Theorem 2.13, with the exception of some
continuity statements of Lemma 3.10 and Lemma 3.16. We operate under one of
the following set of assumptions:

Case 1: $M/G$ is compact.

Case 2:  $M$ is arbitrary and $R>0$ is fixed.

Let $\mathcal{H}^{*}=\\{\Omega\text{ open, bounded}\\}$ in Case 1 and
$\mathcal{H}^{*}=\\{\Omega\text{ open}:\Omega\subseteq Q_{R}\\}$ in Case 2.
Fix $0<v<v_{\text{max}}<|M|$, with $v_{\text{max}}<|Q_{R}|$ in Case 2. Let
$\eta=\eta(v,v_{\text{max}},R)$ be chosen small enough according to
Proposition 2.4 as well as Lemma 3.16 in Case 1 (Lemma 3.10 in Case 2). Then
choose $\mathfrak{T}_{*}=\mathfrak{T}_{*}(v,v_{\text{max}},R,\eta)>0$ small
enough according to Proposition 2.4. Fix any $\mathfrak{T}<\mathfrak{T}_{*}$.
All constants and parameters below depend on these fixed values.

We are interested in looking at the quantitative stability of minimizers of
the base energy $\mathcal{E}$ with a volume constraint:

(2.15)
$\inf\\{\lambda_{1}(\Omega)+\mathfrak{T}\mathrm{tor}(\Omega):\Omega\in\mathcal{H}^{*},|\Omega|=v\\},$

By Proposition 2.4, the infimum in (2.15) is equal to the infimum
$\mathcal{E}_{min}({v})$ of the unconstrained problem (2.6), and the two
minimization problems are equivalent in the sense that if a set
$\Omega\in\mathcal{H}^{*}$ attains the infimum in (2.6), then it has
$|\Omega|=v$ and it attains the infimum in (2.15), and conversely. Let
$\mathcal{M}^{*}$ denote the collection of minimizers of this variational
problem, which is nonempty by Proposition 2.4. In Theorem 2.14, we make the
following assumptions about $\mathcal{M}^{*}$:

In Case 1, assume that $\mathcal{M}^{*}$ may be represented as
$\\{e(U)\\}_{e\in G}$ for a set $U\in\mathcal{M}^{*}$, the boundary $\partial
U$ is smooth, and there exists a set center adapted to $U$ defined for sets in
$\\{\Omega:\mathcal{E}(\Omega)\leq\mathcal{E}_{min}(v)+\delta\\}$ for $\delta$
small enough. Recall that set centers are defined in Definition A.2 and that
examples to keep in mind are those on simply connected. space forms in
Examples 2.7–2.9.

In Case 2, assume that $\mathcal{M}^{*}$ contains only one element $U$ and
$U\subset\subset Q_{R}$ and that $\partial U$ is smooth.

More general situations, such as finitely many minimizers (modulo isometries
in Case 1) may be considered with suitable modifications to the arguments
here. Given an open bounded set $\Omega$, let $d_{*}(\Omega)$ be the distance
to the collection of minimizers as defined in Example 2.10. In Case 1, we let
$U_{\Omega}$ denote the unique minimizer with the same set center as $\Omega.$

######  Theorem 2.14.

For $M$ and $\mathcal{M}^{*}$ as above and $\varepsilon>0$ small, assume that
the inequality

(2.16)
$\lambda_{1}(\Omega)+\mathfrak{T}\mathrm{tor}(\Omega)-\mathcal{E}_{min}\geq
cd_{*}(\Omega)^{2}$

holds for all $\Omega\in\mathcal{H}^{*}$ with $|\Omega|=v$ and
$\partial\Omega$ expressible as a $C^{2}$ normal graph over $\partial
U_{\Omega}$ with $C^{2,\alpha}$ norm bounded by $\varepsilon$. Then, up to
possibly replacing $c$ with a smaller constant, (2.16) holds for all
$\Omega\in\mathcal{H}^{*}$ with $|\Omega|=v$.

###### Proof.

Suppose by way of contradiction that the theorem is false. We may find a
sequence of sets $\\{\Omega_{j}\\}$ with $\Omega_{j}\in\mathcal{H}^{*}$ and
$|\Omega_{j}|=v$ such that

(2.17)
$\mathcal{E}(\Omega_{j})=\lambda_{1}(\Omega_{j})+\mathfrak{T}\mathrm{tor}(\Omega_{j})\leq\mathcal{E}_{min}+\frac{d_{j}^{2}}{j}\,,$

where here and in the remainder of the proof, we let
$d_{j}=d_{*}(\Omega_{j})$. The quantity $d_{*}(\Omega)$ is bounded above
uniformly for any $\Omega\in\mathcal{H}^{*}$, so (2.17) implies that

$\lim_{j\rightarrow\infty}\mathcal{E}(\Omega_{j})=\mathcal{E}_{min}(v).$

Applying Lemma 3.16 in Case 1 or Lemma 3.10 in Case 2, we see that $d_{j}\to
0$, and so there exists $U\in\mathcal{M}^{*}$ such that, after passing to a
subsequence, $|\Omega_{j}\triangle e_{j}(U)|\to 0$ as $j\to\infty$ for some
$e_{j}\in G$.

For each $j\in\mathbb{N}$, consider the minimization problem

(2.18)
$\inf\\{\mathcal{F}_{\tau}(\Omega)=\mathcal{E}(\Omega)+\tau\mathfrak{h}(\Omega):\Omega\in\mathcal{H}^{*}\\},$

where we let

$\displaystyle\mathfrak{h}(\Omega)=\sqrt{d^{4}_{j}+\left(d_{*}(\Omega)^{2}-d_{j}^{2}\right)^{2}}-d_{j}^{2}$

where $\tau<\tau_{0}$ is a small parameter, and
$\tau_{0}=\tau_{0}(v,v_{\text{max}},R,\eta,\mathfrak{T})$ is chosen so that
Theorem 2.13 holds. By Theorem 2.13, a minimizer $V_{j}$ of this variational
problem exists and satisfies $|V_{j}|=v$. Let us take $\Omega_{j}$ as a
competitor in (2.18) in order to relate the deficits and distances of
$\Omega_{j}$ and $V_{j}$. We find

(2.19)
$\displaystyle\mathcal{E}(V_{j})+\tau\left(\sqrt{d_{j}^{4}+\left(d_{*}(V_{j})^{2}-d_{j}^{2}\right)^{2}}-d_{j}^{2}\right)$
$\displaystyle\leq\mathcal{E}(\Omega_{j})$ (2.20)
$\displaystyle\leq\mathcal{E}(U)+\frac{d_{j}^{2}}{j}.$

In (2.20) we have applied (2.17). Subtracting
$\mathcal{E}(U)=\mathcal{E}_{min}(v)$ throughout (2.19) and (2.20) (and
recalling that $\mathcal{E}_{min}(v)\leq\mathcal{E}(V_{j})$ by definition), we
directly see that

(2.21)
$\mathcal{E}(V_{j})-\mathcal{E}_{min}(v)\leq\mathcal{E}(\Omega_{j})-\mathcal{E}_{min}\to
0.$

Furthermore, (2.20) tells us that

$\sqrt{d_{j}^{4}+\left(d_{*}(V_{j})^{2}-d_{j}^{2}\right)^{2}}\leq\left(1+\frac{1}{\mathfrak{T}j}\right)d_{j}^{2}.$

After squaring both sides of this inequality, we find that
$(d_{*}(V_{j})^{2}-d_{j}^{2})^{2}\leq 3d_{j}^{4}/\tau j$. This guarantees,
first of all, that $V_{j}\not\in\mathcal{M}^{*}$ for $j$ sufficiently large.
Moreover, taking the square root of both sides, we find that for sufficiently
large $j$,

(2.22) $\frac{1}{2}d_{j}\leq d_{*}(V_{j}).$

So, combining the contradiction hypothesis (2.17) with the deficit and
distance comparisons (2.21) and (2.22), we see that

(2.23)
$\mathcal{E}(V_{j})-\mathcal{E}_{min}(v)\leq\mathcal{E}(\Omega_{j})-\mathcal{E}_{min}(v)\leq\frac{d_{j}^{2}}{j}\leq\frac{2d_{*}(V_{j})^{2}}{j}\,.$

By Theorem 2.13, $V_{j}$ may be represented as a $C^{2,\alpha}$ normal graph
over $\partial U_{j}$ for some $U_{j}\in\mathcal{M}$, with $C^{2,\alpha}$ norm
controlled by $o_{j}(1)$. Moreover, the set centers of $U_{j}$ and $V_{j}$
converge, since $d(x_{U_{j}},x_{V_{j}})\leq C|U_{j}\triangle V_{j}|=o_{j}(1),$
which immediately implies that $|U_{j}\triangle U_{V_{j}}|=o_{j}(1)$ from the
properties of set centers. Here $U_{V_{j}}$ is the unique minimizer with the
same set center as $V_{j}$. Using the smoothness of the minimizers $U$ and the
tubular neighborhood theorem, this means for $j$ large $\partial U_{j}$ (and
then also $\partial V_{j}$) may be written as a $C^{2,\alpha}$ normal graph
over $\partial U_{V_{j}}$ with small $C^{2,\alpha}$ norm. Applying the
hypothesis (2.16) to $V_{j}$, we see that

$cd_{*}(V_{j})^{2}\leq\mathcal{E}(V_{j})-\mathcal{E}_{min},$

which for $j$ large is a contradiction.This completes the proof. ∎

Recall from the discussion in Section 2.2 that $d_{*}(\Omega)^{2}$ is
comparable to

$\bar{d}_{*}(\Omega)^{2}:=|\Omega\triangle
U_{\Omega}|^{2}+\int|u_{\Omega}-u_{U_{\Omega}}|^{2}$

for $\Omega$ whose boundary is a (small) normal graph over $\partial U$, and
controls $\bar{d}_{*}(\Omega)^{2}$ in all cases. This leads to the following
corollary, which in particular proves Corollary 1.2.

######  Corollary 2.15.

Theorem 2.14 remains valid with $\bar{d}_{*}$ in place of $d_{*}$.

##  3\. Eigenfunctions, torsion functions, and the base energy

This section collects some useful facts about eigenfunctions and torsion
functions of bounded open subsets of $M$, as well some initial properties
about the base energy $\mathcal{E}$ and its optimizers. Section 3.1 is devoted
to general properties of eigenfunctions and torsion functions.

Section 3.2 contains Proposition 3.9, the key estimate that will allow us to
control the nonlinear perturbation in the main energy in the remainder of the
paper. This lemma estimates the difference of eigenfunctions on nested domains
in terms of their eigenvalue and torsional rigidity differences.
Fundamentally, this lemma highlights why we are able to establish existence
and regularity of minimizers of the main energy when the coefficient
$\mathfrak{T}$ in front of the torsional rigidity is positive (and suitably
small), but not when $\mathfrak{T}=0$ and the leading term in the base energy
is only the first eigenfunction.

In Section 3.3, we establish the existence and initial properties of
minimizers of the base energy in $Q_{R}$, while Section 3.4 extends these some
of these properties of minimizers of the base energy globally.

We recall the set $\mathcal{H}=\mathcal{H}_{R,v_{\text{max}}}$ defined in
(2.5). Recall that we always implicitly assume that $v_{\text{max}}$ and $R$
are chosen so that $v_{\text{max}}\leq|Q_{R}|$ and, when $|M|$ is finite, so
that $v_{\text{max}}<|M|$.

###  3.1. Eigenfunctions and torsion functions

Our first lemma collects some basic facts about eigenfunctions and torsion
functions. The proof is elementary and so we omit it.

######  Lemma 3.1.

Let $\Omega\subset M$ be a bounded open set. Then the following properties
hold.

  1. (1)

$w_{\Omega}$ is nonnegative and satisfies $\Delta w_{\Omega}\geq-1_{\Omega}$
on $M$ in the sense of distributions.

  2. (2)

$\Delta w_{\Omega}=-1$ on $\Omega$.

  3. (3)

$\int_{\Omega}|\nabla
w_{\Omega}|^{2}=\int_{\Omega}w_{\Omega}=-2\mathrm{tor}(\Omega)$.

  4. (4)

If $\lambda_{1}(\Omega)<\lambda_{2}(\Omega)$, the normalized eigenfunction
$u_{\Omega}$, up to replacing with $-u_{\Omega}$, is nonnegative and $\Delta
u_{\Omega}\geq-\lambda_{1}(\Omega)u_{\Omega}$ on $M$ in the sense of
distributions.

  5. (5)

If $u_{\Omega}$ is as in (4), $\Delta
u_{\Omega}=-\lambda_{1}(\Omega)u_{\Omega}$ on $\Omega$.

Next, we show that eigenfunctions of a domain $\Omega\subset M$ are bounded
uniformly in terms of an eigenvalue upper bound.

######  Lemma 3.2 (Eigenfunction upper bound).

For each $\lambda_{0}>0$, there is a constant $C_{1}=C_{1}(\lambda_{0})$ such
that the following holds. Let $\Omega$ be an open bounded set and fix
$\lambda\leq\lambda_{0}$. For any function $u\in H^{1}_{0}(\Omega)$ with
$\|u\|_{L^{2}}=1$ satisfying $\Delta|u|\geq-\lambda|u|$ on $\Omega$,

$|u|\leq C_{1}.$

In particular, this applies to any eigenfunction of $\Omega$.

###### Proof.

Let $K(x,y,t)$ be the heat kernel on $M$ (see [19]), and set
$C_{1}=e^{\lambda_{0}}\sup_{x\in M}\sqrt{K(x,x,2)}.$ Since $(M,g)$ has bounded
geometry, $C_{1}$ is finite. Extending $u$ by zero to be defined on all of
$M$, the function $e^{-t\lambda}|u(x)|$ is a subsolution to the heat equation
on $M$. Using the comparison principle,

$\displaystyle e^{-t\lambda}|u(x)|\leq\int K(x,y,t)|u(y)|dy$
$\displaystyle\leq\left(\int
K^{2}(x,y,t)dy\right)^{1/2}\|u\|_{L^{2}}=\sqrt{K(x,x,2t)}.$

Set $t=1$ to conclude. ∎

Next, the torsion function of an open bounded set is bounded in terms of its
torsional rigidity.

######  Lemma 3.3 (Torsion function upper bound).

For each $T>0,$ there exists a constant $C_{2}=C_{2}(T)$ such that the
following holds. Let $\Omega$ be an open bounded set with
$\mathrm{tor}(\Omega)\geq-T$.

$w_{\Omega}\leq C_{2}.$

###### Proof.

In local coordinates on a ball $B_{r}(x)$ around any point $x\in M$, we have
that $w_{\Omega}\geq 0$ and $\Delta w_{\Omega}\geq-1,$ so applying the local
maximum principle ([24], Theorem 8.17) gives

$\sup_{B_{r/2}(x)}w_{\Omega}^{2}\leq C(r,x)\bigg{[}\int_{B_{r}(x)}|\nabla
w_{\Omega}|^{2}+1\bigg{]}\leq C(r,x)[-\mathrm{tor}(\Omega)+1].$

In the second inequality we used Lemma 3.1(3). Since $(M,g)$ has bounded
geometry, the values of $r$ and the constant $C(r,x)=C$ may be taken uniformly
for all $x\in M$. We conclude by letting $C_{2}=C[T+1]$. ∎

The next lemma shows that the eigenfunction is controlled by the torsion
function.

######  Lemma 3.4 (Torsion function controls eigenfunction).

Let $\Omega$ be an open bounded set with $|\Omega|<|M|$ such that
$\lambda_{1}(\Omega)<\lambda_{2}(\Omega)$. Then there is a constant $C_{3}$
depending only on $C_{1}$ of Lemma 3.2 and $\lambda_{1}(\Omega)$ such that

$u_{\Omega}\leq C_{3}w_{\Omega}.$

###### Proof.

This follows directly by the maximum principle and Lemma 3.2: $-\Delta
u_{\Omega}\leq\lambda_{1}(\Omega)C_{1}=-\Delta\lambda_{1}(\Omega)C_{1}w_{\Omega}$
on $\Omega$, and both are in $H^{1}_{0}(\Omega)$, giving $u_{\Omega}\leq
C_{1}\lambda_{1}(\Omega)w_{\Omega}$ on $\Omega$. ∎

As a consequence of this lemma, if we write $\Delta
u_{\Omega}=-\lambda_{1}(\Omega)u_{\Omega}+\mu$ in the sense of distributions,
with $\mu$ a nonnegative measure, and similarly $\Delta
w_{\Omega}=-1_{\Omega}+\nu$ where $\nu$ is a nonnegative measure, then
$\mu\leq C\nu$. The opposite inequality $w_{\Omega}\leq Cu_{\Omega}$ is much
more subtle, and will be discussed for _minimizing_ $\Omega$ in Proposition
7.8.

We now establish two forms of a Poincaré inequality on the sets $Q_{R}$
defined at the beginning of Section 2; the key point here is that they depend
only on the parameters $R$ and $v_{\text{max}}$. We let
$\dot{H}^{1}_{0}(Q_{R})$ and $\dot{H}^{1}(Q_{R})$ denote the completions of
$C^{\infty}_{0}(Q_{R})$ and $C^{\infty}(Q_{R})$ respectively with respect to
the seminorm $\|\nabla u\|_{L^{2}(Q_{R})}.$

######  Lemma 3.5 (Poincaré Inequality).

For every $v_{\text{max}}<|M|$ and any $R>0$, the embedding

$X=\\{u\in\dot{H}^{1}_{0}(Q_{R}):|\\{|u|>0\\}|\leq
v_{\text{max}}\\}\rightarrow L^{p}(Q_{R})$

is continuous for $p\in[1,\frac{2n}{n-2}]$ and compact for $p<\frac{2n}{n-2}$.
If $v_{\text{max}}<|Q_{R}|$, then the same is true as well for
$X=\\{u\in\dot{H}^{1}(Q_{R}):|\\{|u|>0\\}|\leq v_{\text{max}}\\}\,.$

###### Proof.

It is well known (see for instance [25, Theorem 10.2] or [6, Theorems 2.30 and
2.34]) that on $Q_{R}$ (or more generally, any compact Riemannian manifold
with boundary), $H^{1}(Q_{R})$ embeds continuously into $L^{p}(Q_{R})$ for
$p\in[1,\frac{2n}{n-2}]$ and the embedding is compact for $p<\frac{2n}{n-2}$.

Suppose by way of contradiction that the second embedding stated in the lemma
fails for $p=2n/(n-2)$. Take a sequence of functions $\\{u_{k}\\}$ with
$\|u_{k}\|_{L^{p}(Q_{R})}=1$ and $|\\{u_{k}>0\\}|\leq v_{\text{max}}$ such
that $\|\nabla u_{k}\|_{L^{2}(Q_{R})}\to 0$. We see from Hölder’s inequality
that $\|u_{k}\|_{L^{2}(Q_{R})}\leq C(Q_{R})$. So, from the continuous
embedding $H^{1}(Q_{R})\hookrightarrow L^{p}(Q_{R})$ we find that $u_{k}\to c$
in $H^{1}(Q_{R})$, $L^{p}(Q_{R})$ and pointwise a.e. in $Q_{R}$ for some
constant $c\in\mathbb{R}$. On one hand, $\|c\|_{L^{p}(Q_{R})}=1$, while on the
other hand, Fatou’s lemma implies that $|\\{|c|>0\\}|\leq v_{\text{max}}$ and
so $c=0$. We reach a contradiction.

Now, for $p<2n/(n-2)$, the continuous embedding then follows by Hölder’s
inequality, while the compactness of the embedding follows from the compact
embedding of $H^{1}(Q_{R})$ into $L^{p}(Q_{R})$ and the continuous embedding
of $\\{u\in\dot{H}^{1}(Q_{R}):|\\{|u|>0\\}|\leq v_{\text{max}}\\}$ into
$L^{2}(Q_{R})$ that we have just established.

We prove the first embedding of the lemma analogously. If $M$ is compact, then
the argument above carries over directly. If $M$ is noncompact, we again argue
by way of contradiction and take a sequence of functions with
$\|u_{k}\|_{L^{p}(Q_{R})}=1$ and $\|\nabla u_{k}\|_{L^{2}(Q_{R})}\to 0$.
Extend each $u_{k}$ by zero to be defined on $Q_{2R}$. By Hölder’s inequality,
$\|u_{k}\|_{L^{2}(Q_{2R})}\leq C$ and so from the continuous embedding
$H^{1}(Q_{2R})\hookrightarrow L^{p}(Q_{2R})$ we see that $u_{k}\to c$ in
$H^{1}(Q_{2R}),L^{p}(Q_{2R})$ and pointwise a.e. in $Q_{2R}$. On one hand,
$\|c\|_{L^{p}(Q_{2R})}=1,$ while on the other hand, $u_{k}\equiv 0$ on
$Q_{2R}\setminus Q_{R}$ and so we see that $c=0$. We reach a contradiction.
Again, the continuous embedding for $p<2n/(n-2)$ then follows by Hölder’s
inequality, while the compactness of the embedding follows from the compact
embedding of $H^{1}_{0}(Q_{R})$ embeds continuously into $L^{p}(Q_{R})$ and
the continuous embedding of $\dot{H}^{1}_{0}(Q_{R})$ into $L^{2}(Q_{R})$ that
we have just established. ∎

As an immediate consequence of Lemma 3.5, we may bound the torsional rigidity
$\mathrm{tor}(\Omega)$ from below uniformly for all domains in
$\mathcal{H}=\mathcal{H}_{R,v_{\text{max}}}$, the class of admissible domains
defined in (2.5). Naturally, if $M$ has bounded diameter, then the lower bound
is independent of $R$.

######  Corollary 3.6 (Torsional rigidity lower bound).

Fix $R>0$ and $v_{\text{max}}<|M|$. There is a constant
$C(R,v_{\text{max}})<\infty$ such that for any $\Omega\in\mathcal{H}$,

$\mathrm{tor}(\Omega)\geq-C(R,v_{\text{max}}).$

###### Proof.

Fix $\Omega\in\mathcal{H}$ and let $w_{\Omega}$ be the corresponding torsion
function, so that $\mathrm{tor}(\Omega)=\int_{\Omega}\frac{1}{2}|\nabla
w_{\Omega}|^{2}-w_{\Omega}$. Then for any $\varepsilon>0$,

$\int|w_{\Omega}|\leq C(R,v_{\text{max}})\|\nabla
w_{\Omega}\|_{L^{2}}\leq\varepsilon\int|\nabla
w_{\Omega}|^{2}+C(\varepsilon,R,v_{\text{max}}),$

where the first inequality used Lemma 3.5 on $w_{\Omega}$ extended by $0$ and
the fact that $|\\{w_{\Omega}>0\\}|\leq|\Omega|\leq v_{\text{max}}$. Take
$\varepsilon=\frac{1}{4}$ and reabsorb to get

$\mathrm{tor}(\Omega)=\int_{\Omega}\frac{1}{2}|\nabla
w_{\Omega}|^{2}-w_{\Omega}\geq-C(R,v_{\text{max}})+\int_{\Omega}\frac{1}{4}|\nabla
w_{\Omega}|^{2}.$

This concludes the proof. ∎

######  Remark 3.7.

Notice that from Corollary 3.6, for any
$\Omega\in\mathcal{H}=\mathcal{H}_{R,v_{\text{max}}}$, the constant $C_{2}$ in
Lemma 3.3 depends only on $R$ and $v_{\text{max}}.$

The final lemma of this section is an elementary fact about the energy of the
difference between any normalized nonnegative function and the first
eigenfunction.

######  Lemma 3.8.

Let $\Omega$ be an open bounded set with
$\lambda_{1}(\Omega)+\alpha\leq\lambda_{2}(\Omega)$ for some $\alpha>0$, and
$v\in H^{1}_{0}(\Omega)$ with $\int v^{2}=1$ and $v\geq 0$. Then

$\int|\nabla(u_{\Omega}-v)|^{2}\leq\Big{(}1+\frac{2\lambda_{1}(\Omega)}{\alpha}\Big{)}\Big{[}\int|\nabla
v|^{2}-\lambda_{1}(\Omega)\Big{]}.$

###### Proof.

Let us use the shorthand $\lambda_{1}=\lambda_{1}(\Omega)$ and
$\lambda_{2}=\lambda_{2}(\Omega)$, and write $v=\beta u_{\Omega}+v_{\perp}$
where $\beta=\int u_{\Omega}v$. Because $\int u_{\Omega}v_{\perp}=0$ by
definition, we observe that

$\displaystyle\lambda_{2}(1-\beta^{2})=\lambda_{2}\int v_{\perp}^{2}$
$\displaystyle\leq\int|\nabla v_{\perp}|^{2}=\int|\nabla
v|^{2}-\lambda_{1}\beta^{2}=\int|\nabla
v|^{2}-\lambda_{1}+\lambda_{1}(1-\beta^{2}).$

In particular, $\alpha(1-\beta)^{2}\leq\int|\nabla v|^{2}-\lambda_{1}$ and
$\int|\nabla v_{\perp}|^{2}\leq(1+\lambda_{1}/\alpha)(\int|\nabla
v|^{2}-\lambda_{1})$. So, since $u_{\Omega}-v=(1-\beta)u_{\Omega}+v_{\perp}$
and $0\leq\beta\leq 1$, we see that

$\int|\nabla(u_{\Omega}-v)|^{2}=(1-\beta)^{2}\lambda_{1}+\int|\nabla
v_{\perp}|^{2}\leq\Big{(}1+\frac{2\lambda_{1}}{\alpha}\Big{)}\Big{(}\int|\nabla
v|^{2}-\lambda_{1}\Big{)}\,.$

This concludes the proof. ∎

###  3.2. The key estimate on the nonlinear perturbation

In order to study the existence and regularity of minimizers of the main
energy $\mathcal{F}_{\tau}$, we will require some sharp estimates on how an
admissible nonlinearity $\mathfrak{h}$ changes under a change in domain from
$\Omega$ to $\Omega^{\prime}$. From assumption 3 in Definition 2.11, this is
controlled by the sum of the symmetric difference
$|\Omega\triangle\Omega^{\prime}|$, which is easy to estimate, and the $L^{1}$
difference of eigenfunctions $\int|u_{\Omega}-u_{\Omega^{\prime}}|$, which is
not significantly more difficult to estimate. In fact, we do not know how to
control this quantity in terms of the eigenvalue difference
$|\lambda_{1}(\Omega^{\prime})-\lambda_{1}(\Omega)|$ alone, at least in the
case of outward perturbations. The following lemma does bound
$\int|u_{\Omega}-u_{\Omega^{\prime}}|$, but the difference
$|\mathrm{tor}(\Omega)-\mathrm{tor}(\Omega^{\prime})|$ appears on the right-
hand side of the estimate as well. The is the fundamental reason for
introducing the torsional rigidity term into the main energy functional
$\mathcal{F}_{\tau}$.

This lemma will be the crucial ingredient in proving the main estimates of
Sections 4 and 5 and establishing our main existence result in Section 6. At
least for the estimates, the specific form of the lemma, with _linear_
dependence on $|\mathrm{tor}(\Omega)-\mathrm{tor}(\Omega^{\prime})|$ and
$|\lambda_{1}(\Omega^{\prime})-\lambda_{1}(\Omega)|$, is essential; the (much)
simpler sublinear versions of this inequality are insufficient.

######  Proposition 3.9.

Fix $0<v_{\text{max}}<|M|$, $\alpha>0$ and $\lambda_{0}>0$. Let
$\Omega\subseteq\Omega^{\prime}$ be bounded open sets with
$|\Omega^{\prime}|\leq v_{\text{max}}$,
$\lambda_{2}(\Omega^{\prime})\geq\lambda_{1}(\Omega^{\prime})+\alpha$, and
$\lambda_{1}(\Omega)\leq\lambda_{0}$. Let $f:M\to{\mathbb{R}}$ be a bounded
function. Then there exists a constant
$C_{4}=C_{4}(\alpha,v_{\text{max}},\lambda_{0})$ such that

(3.1) $\Big{|}\int f(u_{\Omega^{\prime}}-u_{\Omega})\Big{|}\leq
C_{4}\|f\|_{L^{\infty}}\left(\mathrm{tor}(\Omega)-\mathrm{tor}(\Omega^{\prime})+\lambda_{1}(\Omega)-\lambda_{1}(\Omega^{\prime})\right).$

In the case that the first eigenvalue of $\Omega$ is not simple, then
Proposition 3.9 holds trivially for any first eigenfunction $u_{\Omega}$. To
see this, note that $\lambda_{1}(\Omega)\geq\lambda_{1}(\Omega^{\prime})$ and
$\lambda_{2}(\Omega)\geq\lambda_{2}(\Omega^{\prime})$ from
$\Omega\subseteq\Omega^{\prime}$. So, if $\lambda_{1}(\Omega)$ is not simple,
then $\lambda_{1}(\Omega)=\lambda_{2}(\Omega)$ and so
$\lambda_{1}(\Omega)-\lambda_{1}(\Omega^{\prime})\geq\alpha$, meaning the
estimate here holds automatically for _every_ normalized first eigenfunction
$u_{\Omega}$ (using Lemma 3.2). So, in the proof below, we will assume that
$\lambda_{1}(\Omega)<\lambda_{2}(\Omega)$, in which case $u_{\Omega}$ is
uniquely defined.

###### Proof.

We first consider the case of $\partial\Omega\cap\Omega^{\prime}$ smooth, but
all constants will be independent of the regularity of $\partial\Omega$. Let
$\beta=\int fu_{\Omega^{\prime}}$ and write $f=\beta
u_{\Omega^{\prime}}+f_{\perp}$. We estimate the contributions of $\beta
u_{\Omega^{\prime}}$ and $f_{\perp}$ to the left-hand side of (3.1)
separately. To estimate the contribution from the first term, use the
normalization of the eigenfunctions $u_{\Omega},u_{\Omega^{\prime}}$:

$\displaystyle\int\beta u_{\Omega^{\prime}}(u_{\Omega^{\prime}}-u_{\Omega})$
$\displaystyle=\beta\int
u_{\Omega^{\prime}}^{2}-u_{\Omega^{\prime}}u_{\Omega}=\frac{\beta}{2}\int
u_{\Omega^{\prime}}^{2}+u_{\Omega}^{2}-2u_{\Omega^{\prime}}u_{\Omega}=\frac{\beta}{2}\int|u_{\Omega}-u_{\Omega^{\prime}}|^{2}.$

Applying Lemma 3.8 and assuming without loss of generality that
$\alpha\leq\lambda_{1}(\Omega^{\prime})$, we have

$\int|u_{\Omega}-u_{\Omega^{\prime}}|^{2}\leq\frac{1}{\lambda_{1}(\Omega^{\prime})}\int|\nabla(u_{\Omega}-u_{\Omega^{\prime}})|^{2}\leq\frac{3}{\alpha}\left[\int|\nabla
u_{\Omega}|^{2}-\lambda_{1}(\Omega^{\prime})\right]=\frac{3}{\alpha}\left[\lambda_{1}(\Omega)-\lambda_{1}(\Omega^{\prime})\right]$

and therefore

(3.2) $\Big{|}\int\beta
u_{\Omega^{\prime}}(u_{\Omega^{\prime}}-u_{\Omega})\Big{|}\leq\frac{3}{\alpha}\|f\|_{L^{2}}\left[\lambda_{1}(\Omega)-\lambda_{1}(\Omega^{\prime})\right].$

Now, the remainder $f_{\perp}$ has
$\|f_{\perp}\|_{L^{2}}\leq\|f\|_{L^{2}}\leq\sqrt{v_{\text{max}}}\|f\|_{L^{\infty}}$
and $\int f_{\perp}u_{\Omega^{\prime}}=0$. This means we may solve for the
potential

(3.3) $\begin{cases}-\Delta q-\lambda_{1}(\Omega^{\prime})q=f_{\perp}&\text{
on }\Omega^{\prime}\\\ q=0&\text{ on }\partial\Omega^{\prime},\end{cases}$

which has a unique solution $q\in H^{1}_{0}(\Omega^{\prime})$ with $\int
qu_{\Omega^{\prime}}=0$, and enjoys the estimate
$\|q\|_{H^{1}_{0}(\Omega^{\prime})}\leq C(\alpha)\|f_{\perp}\|_{L^{2}}$ (this
follows from the Fredholm alternative). From Lemma 3.2 we have that
$\|u_{\Omega^{\prime}}\|_{L^{\infty}}\leq C$, so

$\|f_{\perp}\|_{L^{\infty}}\leq\|f\|_{L^{2}}\|u_{\Omega^{\prime}}\|_{L^{\infty}}+\|f\|_{L^{\infty}}\leq
C\|f\|_{L^{\infty}}.$

So, from (3.3), we see that $-\Delta q-\lambda_{1}(\Omega^{\prime})q\in
L^{\infty}$, we have from applying [24, Theorem 8.17] in charts that

$\|q\|_{L^{\infty}}\leq
C\big{[}\|f_{\perp}\|_{L^{\infty}}+\|q\|_{L^{2}}\big{]}\leq
C\|f\|_{L^{\infty}}.$

Now rewriting the PDE as $-\Delta q=\lambda_{1}(\Omega^{\prime})q+f_{\perp}\in
L^{\infty}$, applying the comparison principle with a multiple of the torsion
function $w_{\Omega^{\prime}}$ gives

$|q|\leq C\|f\|_{L^{\infty}}w_{\Omega^{\prime}}.$

This is the key fact about $q$.

We proceed in our estimate:

$\int f_{\perp}(u_{\Omega^{\prime}}-u_{\Omega})=\int
f_{\perp}u_{\Omega}=\int\left(-\Delta
q-\lambda_{1}(\Omega^{\prime})q\right)u_{\Omega}=\int q\left(-\Delta
u_{\Omega}-\lambda_{1}(\Omega^{\prime})u_{\Omega}\right),$

where the last equality holds _in the sense of distributions_. From Lemma 3.1,
on $\Omega^{\prime}$ we have $\Delta
u_{\Omega}=-\lambda_{1}(\Omega)u_{\Omega}+\mu$, where $\mu$ is a nonnegative
measure supported on $\partial\Omega\cap\Omega^{\prime}$. Continuing on,

$\int q\left(-\Delta
u_{\Omega}-\lambda_{1}(\Omega^{\prime})u_{\Omega}\right)=\int-
qd\mu+\left(\lambda_{1}(\Omega)-\lambda_{1}(\Omega^{\prime})\right)\int
qu_{\Omega}.$

The second term is controlled by

$\left(\lambda_{1}(\Omega)-\lambda_{1}(\Omega^{\prime})\right)\,\Big{|}\int
qu_{\Omega}\Big{|}\leq\left(\lambda_{1}(\Omega)-\lambda_{1}(\Omega^{\prime})\right)\|q\|_{L^{2}}\,\|u_{\Omega}\|_{L^{2}}\leq
C(\lambda_{1}(\Omega)-\lambda_{1}(\Omega^{\prime}))\|f\|_{L^{\infty}},$

so we need only focus on the first. Recall from Lemma 3.4 that writing $\Delta
w_{\Omega}=-1_{\Omega}+\nu$ on $\Omega^{\prime}$ for a nonnegative measure
$\nu$, then $\mu\leq C\nu$. As such,

$\Big{|}\int qd\mu\Big{|}\leq\int|q|d\mu\leq C\|f\|_{L^{\infty}}\int
w_{\Omega^{\prime}}d\nu=C\|f\|_{L^{\infty}}\int\left(w_{\Omega^{\prime}}+w_{\Omega}\right)d\nu,$

in the final identity using that $w_{\Omega}$ vanishes on the support of
$\nu$. Then using the equation for $w_{\Omega}$,

$\displaystyle\int\left(w_{\Omega^{\prime}}+w_{\Omega}\right)d\nu=\int(w_{\Omega^{\prime}}+w_{\Omega})(\Delta
w_{\Omega}+1_{\Omega})$
$\displaystyle\leq\int(w_{\Omega^{\prime}}+w_{\Omega})(\Delta
w_{\Omega}+1_{\Omega^{\prime}})$
$\displaystyle=\int(w_{\Omega^{\prime}}+w_{\Omega})(\Delta w_{\Omega}-\Delta
w_{\Omega^{\prime}})=\int|\nabla w_{\Omega^{\prime}}|^{2}-|\nabla
w_{\Omega}|^{2}.$

Recall that $\mathrm{tor}(\Omega)=-\frac{1}{2}\int|\nabla w_{\Omega}|^{2}$ and
similarly for $\Omega^{\prime}$, so we have shown that

$\Big{|}\int f_{\perp}(u_{\Omega^{\prime}}-u_{\Omega})\Big{|}\leq
C\|f\|_{L^{\infty}}[\lambda_{1}(\Omega)-\lambda_{1}(\Omega^{\prime})+\mathrm{tor}(\Omega)-\mathrm{tor}(\Omega^{\prime})].$

Together with (3.2) this gives the conclusion.

If $\partial\Omega\cap\Omega^{\prime}$ is not smooth, let
$\Omega_{t}=\\{u_{\Omega}>t\\}$, which has smooth boundary for almost every
$t$, and apply (3.1) to $\Omega_{t}$. We have that
$\lambda_{1}(\Omega_{t})\geq\lambda_{1}(\Omega)$ and
$\mathrm{tor}(\Omega_{t})\geq\mathrm{tor}(\Omega)$ from set inclusion, and
using $\frac{(u_{\Omega}-t)_{+}}{\|(u_{\Omega}-t)_{+}\|_{L^{2}}}\rightarrow
u_{\Omega}$ in $H^{1}$ as a competitor for $\lambda_{1}(\Omega_{t})$ gives
that $\lim_{t}\lambda_{1}(\Omega_{t})=\lambda_{1}(\Omega)$. On the other hand,
taking first eigenfunctions $u_{\Omega_{t}}\rightharpoonup u\in
H^{1}_{0}(\Omega)$ weakly and locally in $C^{k}$ topology for some $u$ along a
subsequence, and similarly $w_{\Omega_{k}}\rightharpoonup w$. Passing the PDEs
satisfied by these to the limit, we see that $-\Delta u=\lambda_{1}(\Omega)u$
and $-\Delta w=1$ on $\Omega$. This implies that $u_{\Omega}=u$ (recall we are
assuming that $\lambda_{1}(\Omega)$ is simple) and $w_{\Omega}=w$. In
particular, both sides of (3.1) pass to the limit. ∎

###  3.3. Base energy minimizers in $Q_{R}$

We now move toward minimizing the base energy $\mathcal{E}$ among subsets of a
fixed $Q_{R}$. In this section we establish existence, volume bounds and
connectedness properties of minimizers in the class
$\mathcal{H}=\mathcal{H}_{R,v_{\text{max}}}.$

We start with an existence and compactness theorem for minimizers. The
statement is formulated in a way to give information on $w_{\Omega}$ if and
only if $\mathfrak{T}>0$. We recall that the base energy $\mathcal{E}$, the
collection of sets over which we minimize $\mathcal{H}$, and the collection of
minimizers $\mathcal{M}$ are defined in (2.4), (2.5), and (2.7) respectively.
Recall from (2.6) that
$\mathcal{E}_{min}=\mathcal{E}_{min}(v,v_{\text{max}},\eta,\mathfrak{T},R)$
denotes the infimum of the base energy.

######  Lemma 3.10.

Fix $R>0$ and $v,v_{\text{max}}$ with $0<v<v_{\text{max}}$, as well as the
parameters $\eta>0$ and $\mathfrak{T}>0$ in the base energy. Then:

  1. (1)

$\mathcal{E}_{min}>-\infty$.

  2. (2)

Let $\Omega_{k}\in\mathcal{H}$ and $u_{k},w_{k}\in H^{1}_{0}(\Omega_{k})$ with
$\int u_{k}^{2}=1$ and

(3.4) $\lim_{k}\int|\nabla u_{k}|^{2}+\mathfrak{T}\int\frac{1}{2}|\nabla
w_{k}|^{2}-w_{k}+f_{v,\eta}(|\Omega_{k}|)=\mathcal{E}_{min}.$

Then there is a subsequence $\Omega_{k_{j}}$ and a set $\Omega\in\mathcal{H}$
such that

$\|u_{k_{j}}-u_{\Omega}\|_{H^{1}_{0}(Q_{R})}+\sqrt{\mathfrak{T}}\|w_{k_{j}}-w_{\Omega}\|_{H^{1}_{0}(Q_{R})}+|\Omega\triangle\Omega_{k}|\rightarrow
0,$

where $u_{\Omega}$ is a normalized first eigenfunction of $\Omega$ and
$w_{\Omega}$ is the torsion function of $\Omega$.

  3. (3)

$\mathcal{M}$ is nonempty, and for any $\Omega\in\mathcal{M}$, $u_{\Omega}$
and $\sqrt{\mathfrak{T}}w_{\Omega}$ are continuous functions (for any choice
of $u_{\Omega}$ normalized first eigenfunction).

###### Proof.

We first check that $\mathcal{E}_{min}>-\infty$. This amounts to showing that
$\mathrm{tor}(\Omega)$ is bounded from below uniformly in
$\Omega\in\mathcal{H}$ (the other two terms are bounded from below), so it
follows directly from Corollary 3.6.

Step 1: Basic compactness. Let $\Omega_{k}$ be a minimizing sequence as in
claim (2) of the lemma. As $\mathcal{E}_{min}$ is bounded from below, so are
each of the terms $\int|\nabla u_{k}|^{2},\mathfrak{T}\int\frac{1}{2}|\nabla
w_{k}|^{2}-w_{k},$ and $f_{v,\eta}(|\Omega_{k}|)$ individually. Because
$\mathcal{E}_{min}$ is also bounded from above, these three terms are bounded
above individually as well. This immediately gives that
$\|u_{k}\|_{H^{1}_{0}(Q_{R})}$ is uniformly bounded, while applying Lemma 3.5
as in the proof of Corollary 3.6 gives that

$C\geq\mathfrak{T}\int\frac{1}{2}|\nabla
w_{k}|^{2}-w_{k}\geq\mathfrak{T}\left[-C(R,v_{\text{max}})+\frac{1}{4}\int|\nabla
w_{k}|^{2}\right],$

giving $\sqrt{\mathfrak{T}}\|\nabla w_{k}\|_{L^{2}(Q_{R})}\leq C$ and so
$\sqrt{\mathfrak{T}}\|w_{k}\|_{H^{1}_{0}(Q_{R})}\leq C$ from Lemma 3.5 again.

Passing to subsequences, we have that $u_{k}\rightarrow u$ weakly in
$H^{1}_{0}(Q_{R})$, strongly in $L^{2}$, and a.e., and similarly for
$w_{k}\rightarrow w$ when $\mathfrak{T}>0$. This implies that
$1_{\\{|u|+\mathfrak{T}|w|>0\\}}\leq\liminf_{k}1_{\\{|u_{k}|+\mathfrak{T}|w_{k}|>0\\}}$,
so by Fatou’s lemma $|\\{|u|+\mathfrak{T}|w|>0\\}|\leq\liminf_{k}|\Omega_{k}|$
and

$f_{v,\eta}(|\\{|u|+\mathfrak{T}|w|>0\\}|)\leq\liminf_{k\rightarrow\infty}f_{v,\eta}(|\Omega_{k}|).$

Using the lower semicontinuity of the norm under weak convergence, we have
that

(3.5) $\int|\nabla u|^{2}+\mathfrak{T}\int\frac{1}{2}|\nabla
w|^{2}-w+f_{v,\eta}(|\\{|u|+\mathfrak{T}|w|>0\\}|)\leq\mathcal{E}_{min}$

and $\int u^{2}=1$. From Lemmas 3.2 and 3.3, we have $|u_{k}|\leq C_{1}$ and
$|w_{k}|\leq C_{2}$; these pass to the limit to give
$|u|+\sqrt{\mathfrak{T}}|w|\leq C$.

Step 2: Continuous representatives. We will now show that $u,w$ admit
continuous representatives; this will imply that the set
$\Omega:=\\{|u|+\mathfrak{T}|w|>0\\}$ is open (and hence in $\mathcal{H}$),
$u$ is a first eigenfunction of $\Omega$, $w=w_{\Omega}$, and the inequality
in (3.5) is an equality.

Step 2a.  To this end, we first claim that $u$ and $w$ have the following
“almost minimality” property. Fix a small ball $B_{2r}(x)$, $x\in Q_{R}$. We
claim that

(3.6) $\int|\nabla u|^{2}+\mathfrak{T}\int\frac{1}{2}|\nabla
w|^{2}-w\leq\frac{\int|\nabla a|^{2}}{\int
a^{2}}+\mathfrak{T}\int\frac{1}{2}|\nabla b|^{2}-b+Cr^{n}$

for any pair of functions $a,b$ with $a=u$ and $b=w$ on $B_{r}(x)^{c}$ and
such that $u-a$ and $w-b$ are in $H^{1}_{0}(B_{r}(x)\cap Q_{R})$.

Indeed, let $\psi$ be a smooth cutoff function which is $1$ on $B_{r}(x)$ and
supported on $B_{2r}(x)$. Let $\Omega_{k}^{\prime}=\Omega_{k}\cup B_{r}(x)$
and define the functions $a_{k},b_{k}\in H^{1}_{0}(\Omega_{k}^{\prime})$ by

$\displaystyle a_{k}$ $\displaystyle=\psi a+(1-\psi)u_{k},$ $\displaystyle
b_{k}$ $\displaystyle=\psi b+(1-\psi)w_{k}.$

Note that $a_{k}\to a$ and $b_{k}\to b$ in $L^{2}$. Moreover,
$f_{v,\eta}(|\Omega^{\prime}_{k}|)\leq f_{v,\eta}(|\Omega_{k}|)+Cr^{n}.$ So,
we may use $a_{k}/\|a_{k}\|_{L^{2}}$ as a competitor for
$u_{\Omega^{\prime}_{k}}$ and $b_{k}$ for $w_{\Omega^{\prime}_{k}}$ inside of
(2.2) and (2.1) to give

$\mathcal{E}_{min}\leq\mathcal{E}(\Omega^{\prime}_{k})\leq\frac{\int|\nabla
a_{k}|^{2}}{\int a_{k}^{2}}+\mathfrak{T}\int\frac{1}{2}|\nabla
b_{k}|^{2}-b_{k}+f_{v,\eta}(|\Omega_{k}|)+Cr^{n}.$

Recalling that $u_{k},w_{k}$ were chosen to satisfy (3.4), this gives that

(3.7) $0\leq\liminf_{k\rightarrow\infty}\frac{\int|\nabla a_{k}|^{2}}{\int
a_{k}^{2}}-\int|\nabla u_{k}|^{2}+\mathfrak{T}\left[\int\frac{1}{2}|\nabla
b_{k}|^{2}-b_{k}-\int\frac{1}{2}|\nabla w_{k}|^{2}-w_{k}\right]+Cr^{n},$

We first focus on the term in brackets. We have

$\displaystyle\int\frac{1}{2}$ $\displaystyle|\nabla
b_{k}|^{2}-\int\frac{1}{2}|\nabla w_{k}|^{2}=\frac{1}{2}\int|\nabla
w_{k}|^{2}\left[(1-\psi)^{2}-1\right]+\psi^{2}|\nabla
b|^{2}+2\psi(1-\psi)\nabla b\cdot\nabla w_{k}+o_{k}(1),$

where the $o_{k}(1)$ term contains all error terms containing $\nabla\psi$ and
can be seen to converge to $0$ using the weak convergence of $\nabla
w_{k}\rightarrow\nabla w$, the strong convergence of $w_{k}\to w$, and the
fact that $b=w$ on the support of $\nabla\psi$. Therefore, taking limits in
the term in brackets in (3.7), since $w=b$ when $1-\psi$ is nonzero, we have

$\displaystyle\limsup_{k\rightarrow\infty}$
$\displaystyle\int\frac{1}{2}|\nabla b_{k}|^{2}-b_{k}-\int\frac{1}{2}|\nabla
w_{k}|^{2}-w_{k}$ $\displaystyle\leq\frac{1}{2}\int|\nabla
w|^{2}\left[(1-\psi)^{2}-1\right]+\psi^{2}|\nabla b|^{2}+2\psi(1-\psi)|\nabla
w|^{2}+\int w-b$ $\displaystyle=\frac{1}{2}\int\psi^{2}[|\nabla b|^{2}-|\nabla
w|^{2}]+\int w-b$ $\displaystyle=\int\frac{1}{2}|\nabla
b|^{2}-b-\int\frac{1}{2}|\nabla w|^{2}-w,$

In the inequality we have used that $\int\phi|\nabla
w|^{2}\leq\liminf_{k\rightarrow\infty}\int\phi|\nabla w_{{}_{k}}|^{2}$ for
$\phi\geq 0$ from weak convergence (and setting $\phi=1-(1-\psi^{2})$). A
similar computation for the $a_{k}$ terms gives

$\limsup_{k\rightarrow\infty}\frac{\int|\nabla a_{k}|^{2}}{\int
a_{k}^{2}}-\int|\nabla u_{k}|^{2}\leq\frac{\int|\nabla a|^{2}}{\int
a^{2}}-\int|\nabla u|^{2}.$

Substituting these conclusions into (3.7), we arrive at (3.6), thus proving
the claim.

Step 2b. Now let $a$ be the harmonic replacement of $u$ on $B_{r}(x)\cap
Q_{R}$ (i.e. the minimizer of $\int_{B_{r}(x)\cap Q_{R}}|\nabla a|^{2}$ with
data $u$ on $\partial[B_{r}(x)\cap Q_{R}]$), and $b$ the harmonic replacement
of $w$. Then

$\int_{B_{r}(x)}|\nabla u|^{2}+\mathfrak{T}\frac{1}{2}|\nabla w|^{2}\leq
Cr^{n}+\int_{B_{r}(x)}|\nabla a|^{2}+\mathfrak{T}\frac{1}{2}|\nabla b|^{2},$

estimating the non-gradient terms using $|u|,|w|,|a|,|b|\leq C$ and absorbing
them into the $Cr^{n}$ term. Rewriting and using that $\int g(\nabla
a,\nabla(u-a))=0$ from the equation on $a$ (and similarly with $b$),

$\int_{B_{r}(x)}|\nabla(u-a)|^{2}+\mathfrak{T}\frac{1}{2}|\nabla(w-b)|^{2}\leq
Cr^{n}.$

This is valid for any $B_{2r}(x)$, and implies that $u,\sqrt{\mathfrak{T}}w$
are $C^{0,\alpha}(Q_{R})$ (see [15]).

Step 3: Conclusion. Finally, as $\Omega$ is open, $\Omega\in\mathcal{H}$ and
$\mathcal{E}_{min}\leq\mathcal{E}(\Omega)\leq\int|\nabla
u|^{2}+\int\frac{1}{2}|\nabla
w|^{2}-w+f_{v,\eta}(|\Omega|)\leq\mathcal{E}_{min}$. This implies that
$w=w_{\Omega}$, $u$ is a first eigenfunction of $\Omega$, and

$\lim_{k\rightarrow\infty}\int|\nabla u_{k}^{2}|=\int|\nabla
u|^{2}\qquad\lim_{k\rightarrow\infty}\int|\nabla w_{k}^{2}|=\int|\nabla
w|^{2}\qquad\lim_{k\rightarrow\infty}|\Omega_{k}|=|\Omega|.$

This gives the strong convergence of $u_{k}$ and $w_{k}$ in
$H^{1}_{0}(Q_{R})$. To see the convergence of $\Omega_{k}$, recall that
$1_{\Omega}\leq\liminf_{k}1_{\Omega_{k}}$; integrating over $\Omega$ and
applying Fatou’s lemma gives $|\Omega|\leq\liminf_{k}|\Omega\cap\Omega_{k}|$,
or $|\Omega\setminus\Omega_{k}|\rightarrow 0$. Together with
$|\Omega_{k}|\rightarrow|\Omega|$ this guarantees
$|\Omega_{k}\setminus\Omega|\rightarrow 0$ as well.

For (3), (1) and (2) immediately give that $\mathcal{M}$ is nonempty by
applying with $u_{k}=u_{\Omega_{k}}$, an eigenfunction of $\Omega_{k}$, and
$w_{k}=w_{\Omega_{k}}$, where $\Omega_{k}$ is any sequence in $\mathcal{H}$
with $\mathcal{E}(\Omega_{k})\rightarrow\mathcal{E}_{min}$. Choosing
$\Omega_{k}=\Omega$ and $u_{\Omega_{k}}=u_{\Omega}$ any first eigenfunction
and applying the continuity argument above gives that $u_{\Omega}$ and
$\sqrt{\mathfrak{T}}w_{\Omega}$ are continuous. ∎

Recall that the volume penalization term $f_{v,\eta}$ in the base energy does
not immediately guarantee that a minimizer $\Omega\in\mathcal{M}$ satisfies
the desired volume constraint $|\Omega|\leq v$. However, the following lemma
provides an initial upper bound on the volume of a minimizer, provided that
the parameter $\eta$ in the volume penalization term is taken to be
sufficiently small. We will eventually show in Proposition 9.5 that and
$\Omega\in\mathcal{M}$ actually satisfies the desired volume constrain
$|\Omega|=v$.

######  Lemma 3.11.

Fix $R>0$ and $v,v_{\text{max}}$ with $0<v<v_{\text{max}}$, and
$0<\mathfrak{T}\leq 1$. There exists an
$\eta_{0}=\eta_{0}(v,v_{\text{max}},R)>0$ so that for $\eta<\eta_{0}$, if
$\Omega\in\mathcal{M}$ then

$|\Omega|<\frac{v_{\text{max}}+v}{2}.$

###### Proof.

We have already verified in the proof of Lemma 3.10 that
$\mathcal{E}(\Omega)\geq-C+f_{v,\eta}(|\Omega|)$ for any
$\Omega\in\mathcal{H}$; this estimate is uniform for any $\mathfrak{T}\leq 1$.
By choosing some smooth $\Omega$ with precisely $|\Omega|=v$, we see that
$\mathcal{E}(\Omega)\leq C$ with $C$ independent of $\eta$ (and
$\mathfrak{T}$). We therefore have, for $\Omega\in\mathcal{M}$,

$\frac{|\Omega|-v}{\eta}\leq f_{v,\eta}(|\Omega|)\leq C.$

Choosing $\eta\leq\frac{v_{\text{max}}-v}{2C}$ gives the conclusion. ∎

######  Remark 3.12.

In the remainder of the paper, we will always assume, without further comment,
that the parameter in the base energy $\mathcal{E}$ and the main energy
$\mathcal{F}_{\tau}$ is taken such that $\eta\leq\eta_{0}$, where
$\eta_{0}=\eta_{0}(v,v_{\text{max}},R)$ is the constant obtained in Lemma
3.11.

In the next lemma, we show that the coefficient $\mathfrak{T}$ in front of the
torsional rigidity in the base energy can be taken to be sufficiently small to
guarantee that a minimizer $\Omega\in\mathcal{M}$ is connected and its first
eigenvalue is simple. Although the simplicity of the first eigenvalue follows
from the connectedness, we prove these two properties separately so that the
proof can be immediately generalized to prove Lemma 3.14 below.

######  Lemma 3.13.

Fix $R>0$, $v<v_{\text{max}}$, and $\eta>0$. There exists a
$\mathfrak{T}_{0}(v,v_{\text{max}},\eta,R)>0$ such that if
$\Omega\in\mathcal{M}$ with $\mathfrak{T}<\mathfrak{T}_{0}$, then $\Omega$ is
connected, $u_{\Omega}>0$ on $\Omega$ (up to changing sign), and

$\lambda_{2}(\Omega)>\lambda_{1}(\Omega).$

###### Proof.

Consider first the case of $\mathfrak{T}=0$. If
$\lambda_{2}(\Omega)=\lambda_{1}(\Omega)$, then there are two functions
$u_{1},u_{2}$ with $\int|\nabla u_{i}|^{2}=\lambda_{1}(\Omega)$, $\int
u_{i}^{2}=1$, and $\int u_{1}u_{2}=0$. We know from Lemma 3.10 that they are
both continuous. We may assume without loss of generality that $u_{i}\geq 0$
by replacing $u_{i}$ with $|u_{i}|$ in the Rayleigh quotient characterization
of eigenfunctions. So, the orthogonality shows that $\\{u_{1}>0\\}$ and
$\\{u_{2}>0\\}$ are disjoint open sets.

Now let $\Omega^{\prime}=\\{u_{1}>0\\}$, noting that
$\lambda_{1}(\Omega^{\prime})\leq\int|\nabla u_{1}|^{2}=\lambda_{1}(\Omega)$.
On the other hand, $|\Omega^{\prime}|<|\Omega|$, so
$f_{v,\eta}(|\Omega^{\prime}|)<f_{v,\eta}(|\Omega|)$ and
$\mathcal{E}(\Omega^{\prime})<\mathcal{E}(\Omega)=\mathcal{E}_{min}$. This is
a contradiction. If $\Omega$ is disconnected, then $u_{\Omega}=u_{E}$ where
$E$ is one of its connected components; the same argument shows that
$\mathcal{E}(E)<\mathcal{E}(\Omega)$.

To handle the case of $\mathfrak{T}>0$, argue by contradiction. Assume there
is a sequence of sets $\Omega_{k}\in\mathcal{M}_{\mathfrak{T}_{k}}$,
$\mathfrak{T}_{k}\rightarrow 0$, with
$\lambda_{2}(\Omega_{k})=\lambda_{1}(\Omega_{k})$ or $\Omega_{k}$
disconnected. Then

$\displaystyle\lambda_{1}(\Omega_{k})+\mathfrak{T}_{k}\mathrm{tor}(\Omega_{k})+f_{v,\eta}(\Omega_{k})$
$\displaystyle\leq\lambda_{1}(\Omega)+\mathfrak{T}_{k}\mathrm{tor}(\Omega)+f_{v,\eta}(\Omega)$
$\displaystyle\rightarrow\lambda_{1}(\Omega)+f_{v,\eta}(\Omega)$

for any $\Omega\in\mathcal{H}$. As $\mathrm{tor}(\Omega)\geq-C$ from Corollary
3.6, each of $\lambda_{1}(\Omega_{k}),\mathrm{tor}(\Omega_{k}),$ and
$f_{v,\eta}(\Omega_{k})$ are bounded uniformly in $k$, so the middle term
$\mathfrak{T}_{k}\mathrm{tor}(\Omega_{k})\rightarrow 0$. This means

$\displaystyle\mathcal{E}_{min}(\mathfrak{T}_{k})$
$\displaystyle=\lambda_{1}(\Omega_{k})+f_{v,\eta}(\Omega_{k})+o_{k}(1)$
$\displaystyle\leq\inf\\{\lambda_{1}(\Omega)+f_{v,\eta}(\Omega):\Omega\in\mathcal{H}\\}+o_{k}(1)=\mathcal{E}_{min}(0)+o_{k}(1).$

In particular,
$\mathcal{E}_{min}(\mathfrak{T}_{k})\rightarrow\mathcal{E}_{min}(0)$ and
$\lim_{k}\lambda_{1}(\Omega_{k})+f_{v,\eta}(\Omega_{k})=\mathcal{E}_{min}(0)$.
Fix first eigenfunctions $u_{\Omega_{k}}$ for $\Omega_{k}$; as above we may
take them nonnegative. Applying the compactness claim of Lemma 3.10, there is
an $\Omega\in\mathcal{M}_{0}$ with
$\|u_{\Omega_{k}}-u_{\Omega}\|_{H^{1}_{0}(Q_{R})}\rightarrow 0$ and
$|\Omega_{k}\triangle\Omega|\rightarrow 0$.

First, suppose that $\lambda_{2}(\Omega_{k})=\lambda_{1}(\Omega_{k})$ along
some subsequence. Let $v_{k}$ be functions in $H^{1}_{0}(\Omega_{k})$ with
$v_{k}\geq 0$, $\int v_{k}u_{\Omega_{k}}=0$, $\int v_{k}^{2}=1$, and
$\int|\nabla
v_{k}|^{2}=\lambda_{1}(\Omega_{k})\rightarrow\lambda_{1}(\Omega)$. We may
apply Lemma 3.10 to $v_{k}$ instead of $u_{\Omega_{k}}$ to obtain another set
$\Omega^{\prime}\in\mathcal{M}_{0}$ and a continuous first eigenfunction of
$\Omega^{\prime}$, $u_{\Omega^{\prime}}$, such that
$\|v_{k}-u_{\Omega^{\prime}}\|_{H^{1}_{0}(Q_{R})}\rightarrow 0$ and
$|\Omega_{k}\triangle\Omega^{\prime}|\rightarrow 0$. This immediately gives
that $|\Omega\triangle\Omega^{\prime}|=0$, and so $\Omega\cup\Omega^{\prime}$
is in $\mathcal{H}$,
$\lambda_{1}(\Omega^{\prime}\cup\Omega)\leq\lambda_{1}(\Omega)$, and
$|\Omega^{\prime}\cup\Omega|=|\Omega|$. In particular,
$\Omega\cup\Omega^{\prime}\in\mathcal{M}_{0}$ and
$\lambda_{1}(\Omega^{\prime}\cup\Omega)=\lambda_{1}(\Omega)$. This means both
$u_{\Omega^{\prime}}$ and $u_{\Omega}$ are first eigenfunctions of
$\Omega\cup\Omega^{\prime}$, and also

$\int u_{\Omega}u_{\Omega^{\prime}}=\lim_{k\rightarrow\infty}\int
u_{\Omega_{k}}v_{k}=0.$

Thus
$\lambda_{2}(\Omega\cup\Omega^{\prime})=\lambda_{1}(\Omega\cup\Omega^{\prime})$,
which is a contradiction.

Next, assume $\Omega_{k}$ is disconnected, and let $E^{i}_{k}$ be its
connected components. We claim that $|E^{i}_{k}|$ converges either to
$|\Omega|$ or to $0$. Indeed, let $u_{k}\geq 0$ be the first eigenfunction of
$\Omega_{k}$, and $E^{1}_{k}$ be the (unique) connected component of
$\Omega_{k}$ on which it is nonzero. We know that $u_{k}\rightarrow
u_{\Omega}$ a.e., which is strictly positive on $\Omega$; this gives that
$|E_{k}^{1}|\rightarrow|\Omega|$ while $|E_{k}^{i}|\rightarrow 0$ for any
other $i$.

It remains to check that for any $\Omega_{k}$, if $|E_{k}^{i}|$ is
sufficiently small, then it is empty. The basic idea is that the torsional
rigidity is additive on connected components, and a component of small volume
contributes too much torsional rigidity for a set to be minimizing. More
specifically, we consider the quantity

$a(t):=\inf\\{\mathrm{tor}(E):E\in\mathcal{H},|E|\leq t\\}.$

From Lemma 3.1(3), we have $\int|\nabla w_{E}|^{2}=\int
w_{E}=-2\mathrm{tor}(E)$ (using $w_{E}$ as a test function for itself), and
therefore

$-\mathrm{tor}(E)=\frac{1}{2}\int
w_{E}\leq|E|^{\frac{n+2}{2n}}\|w_{E}\|_{L^{\frac{2n}{n-2}}}\leq
C|E|^{\frac{n+2}{2n}}\|\nabla w_{E}\|_{L^{2}}\leq
C|E|^{\frac{n+2}{2n}}\sqrt{-\mathrm{tor}(E)},$

where the constant $C=C(R,v_{\text{max}})$ introduced in the second-to-last
inequality comes from the Poincaré inequality of Lemma 3.5. Dividing and
squaring, $\mathrm{tor}(E)\geq-C|E|^{\frac{n+2}{n}},$ so

(3.8) $a(t)\geq-Ct^{\frac{n+2}{n}}.$

Let $E_{k}^{1}$ be the large connected component of $\Omega_{k}$ and set
$|\Omega_{k}|=|E_{k}^{1}|+t_{k}$. We know that $t_{k}\rightarrow 0$ and assume
for contradiction that $t_{k}>0$. Since
$\lambda_{1}(\Omega_{k})=\lambda_{1}(E_{k}^{1})$, we take $E_{k}^{1}$ as a
competitor for $\Omega_{k}$ in the minimization of
$\mathcal{E}_{\mathfrak{T}_{k}}$ to find

$\mathfrak{T}_{k}\,\mathrm{tor}(\Omega_{k})+f_{v,\eta}(|\Omega_{k}|)\leq\mathfrak{T}_{k}\,\mathrm{tor}(E_{k}^{1})+f_{v,\eta}(|E_{k}^{1}|).$

Moreover, for disjoint $A,B$, $\mathrm{tor}(A\cup
B)=\mathrm{tor}(A)+\mathrm{tor}(B)$, which gives

$\mathfrak{T}_{k}\,\mathrm{tor}(\Omega_{k}\setminus
E_{k}^{1})+f_{v,\eta}(|\Omega_{k}|)\leq f_{v,\eta}(|E_{k}^{1}|).$

Consequently, we deduce that

$\mathfrak{T}_{k}a(t_{k})\leq
f_{v,\eta}(|E_{k}^{1}|)-f_{v,\eta}(|\Omega_{k}|)\leq-\eta t_{k},$

so applying (3.8) gives $\eta t_{k}\leq
C\mathfrak{T}_{k}t_{k}^{\frac{2+n}{n}}\leq Ct_{k}^{\frac{2+n}{n}}.$ For large
$k$, this is a contradiction. ∎

The same compactness argument can be applied to “approximate” minimizers
instead, except for the full connectedness conclusion. We omit the proof.

######  Lemma 3.14.

Fix $R>0,$ $0<v<v_{\text{max}}$, $\eta>0$ and $\varepsilon>0$. There exist
$\mathfrak{T}_{0}=\mathfrak{T}_{0}(v,v_{\text{max}},\eta,R)>0$ and
$\delta=\delta(v,v_{\text{max}},R,\eta,\varepsilon)>0$ such that if
$\mathfrak{T}<\mathfrak{T}_{0}$, $\Omega\in\mathcal{H}$, and

$\mathcal{E}(\Omega)\leq\mathcal{E}_{min}+\delta,$

then:

  1. (1)

There is an $\Omega^{\prime}\in\mathcal{M}$ with
$|\Omega^{\prime}\triangle\Omega|<\varepsilon$,
$\|u_{\Omega^{\prime}}-u_{\Omega}\|_{H^{1}}\leq\varepsilon$, and
$\sqrt{\mathfrak{T}}\|w_{\Omega^{\prime}}-w_{\Omega}\|_{H^{1}}\leq\varepsilon$.

  2. (2)

$\lambda_{2}(\Omega)>\lambda_{1}(\Omega)+c(v,v_{\text{max}},R)$.

  3. (3)

One connected component $E$ of $\Omega$ has $|E|\geq|\Omega|-\varepsilon$,
while every other (if any) has $|E|\leq\varepsilon$.

Note that under the assumptions of this lemma, it is simply false that
$\Omega$ must be connected: one may always add a ball of small measure to
$\Omega$, which perturbs $\mathcal{E}$ continuously.

###  3.4. Global minimizers of the base energy

In general, the constants in the previous section necessarily depend on $R$,
and there is no reason to expect, for instance, existence of global minimizers
on noncompact Riemannian manifolds with arbitrary (bounded geometry) behavior
at infinity. In this section, we turn our attention to Riemannian manifolds
$(M,g)$ where $M/G$ is compact. In this case, most of the above estimates
remain valid with constants independent of $R$, and minimizers of the base
energy $\mathcal{E}$ among all open bounded sets exist and have bounded
diameter. For the present section, we focus only on estimates which will be
relevant in later sections; in particular, the existence of minimizers is not
treated here (but does follow from Theorem 6.3 later).

Of course, in the case when $(M,g)$ is compact, then for $M\subset Q_{R}$ for
sufficiently large $R$ , and thus the constants in the previous section may be
taken independent of $R$ trivially. So, to simplify statements, in this
section we will assume that $(M,g)$ is noncompact but $(M/G,g)$ is compact.
Note in this case that $M$ has infinite volume.

For our first estimate, we show some initial bounds on the infimum of the
energy and on the torsion function of any open bounded set in terms of its
volume. Observe that $\mathcal{E}_{min}(R)$ is a nonincreasing function of
$R$, as the class $\mathcal{H}$ is increasing in $R$. Set
$\mathcal{E}_{min}(\infty)=\lim_{R\rightarrow\infty}\mathcal{E}_{min}(R)$.

######  Lemma 3.15.

Assume $M/G$ is compact and fix $v<v_{\text{max}}<\infty$. Then there exists
$C=C(v_{\text{max}})<\infty$ such that for any bounded open set $\Omega$ with
$|\Omega|\leq v_{\text{max}}$, we have $\mathcal{E}_{min}(\infty)\geq-C$ and
$w_{\Omega}\leq C$.

###### Proof.

First, we claim there exists a large $S\gg 0$ such that

$\bigcup_{e\in G}e(Q_{S})=M.$

If not, for every $k$ there is an $x_{k}\in M$ such that $e(x_{k})\notin
Q_{k}$ for any $e\in G$. There exists, however, a subsequence $x_{k_{j}}$ and
isometries $e_{j}$ with $e_{j}(x_{k_{j}})\rightarrow x\in M$. If $x\in Q_{J}$
for some $J$, then as $Q_{J}$ is open $e_{j}(x_{k_{j}})\in Q_{J}$ for $j$
large enough; this is impossible if $k_{j}\geq J$, so $x\notin Q_{J}$. This
contradicts $M=\cup_{R}Q_{R}$. Using the Vitali covering lemma and possibly
taking $S$ larger, there exists a countable collection $L\subseteq G$ such
that $\cup_{e\in L}e(Q_{S})=M$ and $\\{e(Q_{S})\\}_{e\in L}$ have finite
overlap. Up to possibly increasing $S$ depending on $v_{\text{max}},$ we also
assume that $|Q_{S}|>v_{\text{max}}$.

Now, fix any $\Omega$ open, bounded, with $|\Omega|\leq v_{\text{max}}$. For
any $e\in L$, we apply the Poincaré inequality of Lemma 3.5 to $e(Q_{S})$ to
obtain that

(3.9) $\|w_{\Omega}\|_{L^{2}(\Omega\cap e(Q_{S}))}\leq C\|\nabla
w_{\Omega}\|_{L^{2}(\Omega\cap e(Q_{S}))}$

for a constant $C=C(S,v_{\text{max}})=C(v_{\text{max}})$, using that
$|\Omega\cap e(Q_{S})|\leq v_{\text{max}}<|Q_{S}|$. Summing over all $e\in L$
and using the finite overlapping property gives
$\int_{\Omega}w_{\Omega}^{2}\leq C\int_{\Omega}|\nabla w_{\Omega}|^{2}.$ Then

$\int_{\Omega}w_{\Omega}\leq C\,|\Omega|^{1/2}\|\nabla
w_{\Omega}\|_{L^{2}(\Omega)}\leq C\,\|\nabla w_{\Omega}\|_{L^{2}(\Omega)},$

so $\mathrm{tor}(\Omega)=\int\frac{1}{2}|\nabla
w_{\Omega}|^{2}-w_{\Omega}\geq-C.$ On the other hand $\lambda_{1}(\Omega)>0$,
while $f_{v,\eta}(\Omega)\geq-v_{\text{max}}$, so
$\mathcal{E}(\Omega)\geq-C(v_{\text{max}})$. Taking the infimum over all
$\Omega$ gives the first conclusion.

To see that $w_{\Omega}\leq C$, we now have from (3.9), elliptic estimates
[24, Theorem 8.17], and a basic covering argument that

$\|w_{\Omega}\|_{L^{\infty}(\Omega\cap e(Q_{S}))}\leq
C[\|w_{\Omega}\|_{L^{2}(Q_{2S})}+1]\leq C.$

Applying to every $e$ gives the estimate. ∎

We now establish an analogue of Lemma 3.14 that is independent of $R$: that
any low energy set can be well approximated by a minimizer (on some $Q_{S}$
for $S$ uniform) and has one “large” connected component. We use a
concentration compactness argument to handle the loss of compactness coming
from the isometries of the space.

######  Lemma 3.16.

Assume $M/G$ is compact. Fix $v<v_{\text{max}}<\infty$, $\varepsilon>0$, and
$\eta>0$. There exist
$\mathfrak{T}_{0}=\mathfrak{T}_{0}(v,v_{\text{max}},\eta)>0$, and a
$\delta=\delta(v,v_{\text{max}},\eta,\varepsilon)>0$ such that if
$\mathfrak{T}<\mathfrak{T}_{0}$, then the following holds. For any open
bounded set $\Omega\subset M$ with $|\Omega|\leq v_{\text{max}}$ and
$\mathcal{E}(\Omega)\leq\mathcal{E}_{min}(\infty)+\delta$:

  1. (1)

There is an $S=S(v,v_{\text{max}},\eta,\varepsilon)$ and a
$U\in\mathcal{M}(S)$ (i.e. a minimizer of $\mathcal{E}$ over open
$\Omega\subseteq Q_{S}$ with $|\Omega|\leq v_{\text{max}}$) such that
$|U\triangle\Omega|<\varepsilon$,
$\|u_{U}-u_{\Omega}\|_{H^{1}}\leq\varepsilon$, and
$\sqrt{\mathfrak{T}}\|w_{U}-w_{\Omega}\|_{H^{1}}\leq\varepsilon$.

  2. (2)

$\lambda_{2}(\Omega)>\lambda_{1}(\Omega)+c(v,v_{\text{max}})$

  3. (3)

One connected component $E$ of $\Omega$ has $|E|\geq|\Omega|-\varepsilon$,
while every other (if any) has $|E|\leq\varepsilon$.

###### Proof.

Let $L\subseteq G$ and $S$ be as in the proof of Lemma 3.15. We use the
shorthand $\bar{Q}=Q_{S}$. Let $\\{\psi_{e}\\}_{e\in L}$ a partition of unity
subordinate to the cover of $M$ by $\\{e(\bar{Q})\\}_{e\in L}$ with
$\psi_{e}=\psi\circ e^{-1}\in C^{\infty}_{c}(\bar{Q})$ translates of each
other. Throughout this proof, $C$ and $c$ denote constants depending only on
$v_{\text{max}}$ (and, as usual, on $g$) whose values change from line to
line. Let $\delta\leq 1$ be a fixed number to be specified in the proof and
let $\Omega$ be as above. Applying Lemma 3.15, we know that

(3.10) $\int_{\Omega}|\nabla u_{\Omega}|^{2}+u_{\Omega}^{2}+|\nabla
w_{\Omega}|^{2}+w_{\Omega}^{2}\leq C.$

The first step of the proof is a concentration compactness argument that
allows us to replace $\Omega$ by a uniformly bounded set, losing only a small
amount of mass and increasing the base energy only a small amount.

Step 1: There exist $\mathfrak{T}_{0}$ and
$\bar{J}=\bar{J}(v_{\text{max}},\delta)\in\mathbb{N}$ such that we may choose
$x_{0}\in M$ and $j\leq\bar{J}$ depending on $\Omega$ such that

(3.11) $\begin{cases}\mathcal{E}(\Omega\cap
B_{8Sj}(x_{0}))\leq\mathcal{E}_{min}+2\delta\\\ |\Omega\setminus
B_{8Sj}(x_{0})|\leq 4\eta\delta.\end{cases}$

Because this step is rather involved, we divide it into several substeps.

Step 1a: Selection of $x_{0}$. We show that $\Omega$ has a (uniformly)
nontrivial amount of mass in $e_{0}(\bar{Q})$ for some $e_{0}\in L$, and
choose $x_{0}$ in this set. More specifically, for any $e\in L$, we apply
Lemma 3.5 to $\Omega\cap e(\bar{Q})$ to find

$\int_{e(\bar{Q})}u_{\Omega}^{2}\leq\left|\Omega\cap
e(\bar{Q})\right|^{\frac{2}{n}}\Big{(}\int_{e(\bar{Q})}u_{\Omega}^{\frac{2n}{n-2}}\Big{)}^{\frac{n-2}{n}}\leq
C\left|\Omega\cap e(\bar{Q})\right|^{\frac{2}{n}}\int_{e(\bar{Q})}|\nabla
u_{\Omega}|^{2}.$

Summing over $e\in L$, recalling the finite overlap property of $L$ and
(3.10), we have

$1=\int_{\Omega}u_{\Omega}^{2}\leq C\,\Big{(}\sup_{e\in L}|\Omega\cap
e(\bar{Q})|^{\frac{2}{n}}\Big{)}\int_{\Omega}|\nabla u_{\Omega}|^{2}\leq
C\,\Big{(}\sup_{e\in L}|\Omega\cap e(\bar{Q})|^{\frac{2}{n}}\Big{)}.$

Consequently, $|\Omega\cap e_{0}(\bar{Q})|\geq c$ for some $e_{0}$. We let
$x_{0}$ be any fixed point in $e_{0}(\bar{Q})$.

Step 1b: Selecting $j$ for each $J\in\mathbb{N}$. Next, we show how, for each
$J\in\mathbb{N}$, to suitably choose $j\leq J$ that will ultimately lead to
estimates of the form (3.11) but with a $C/J$ error; we will later choose $J$
depending on $\delta$ to absorb this term. For any $k\in\mathbb{N}$, we denote
by $L_{k}$ the finite collection of isometries such that $e(\bar{Q})$ lies in
the annulus $B_{16kS}(x_{0})\setminus B_{8kS}(x_{0})$. That is,

$L_{k}=\left\\{e\in L:e(\bar{Q})\cap\left[B_{8kS}(x_{0})\cup(M\setminus
B_{16kS}(x_{0}))\right]=\emptyset\right\\}.$

Since $\operatorname*{diam}\bar{Q}\leq 2S$ by assumption, the collections
$L_{k}$ are pairwise disjoint, and so from (3.10) we find that

$\sum_{k=1}^{\infty}\ \sum_{e\in L_{k}}\int_{e(\bar{Q})\cap\Omega}|\nabla
u_{\Omega}|^{2}+u_{\Omega}^{2}+|\nabla w_{\Omega}|^{2}+w_{\Omega}^{2}\leq C.$

Since this sum is finite, it follows that for every $J>0$, we may find some
$j\leq J$ such that

$\sum_{e\in L_{j}}\int_{e(\bar{Q})\cap\Omega}|\nabla
u_{\Omega}|^{2}+u_{\Omega}^{2}+|\nabla
w_{\Omega}|^{2}+w_{\Omega}^{2}\leq\frac{C}{J}.$

In other words, the functions $u_{\Omega}$ and $w_{\Omega}$ have most of their
mass and energy outside of the annulus corresponding to $L_{j}$. Let this
$j=j(J)$ be fixed.

Step 1c: Truncation and intermediate bounds. In this step, we consider
$J\in\mathbb{N}$ fixed and $j=j(J)$ as determined in the previous step. Let us
now define two smooth truncations of the eigenfunction $u_{\Omega}$, the
“inner part” $u_{I}$ in the ball $B_{8jS}$ and the “outer part” $u_{O}$ in the
complement of $B_{16Sj}$, by setting

$\displaystyle u_{I}=\sum_{e\in L_{I}}\psi_{e}u_{\Omega}$
$\displaystyle\qquad\qquad L_{I}=\left\\{e\in L\ :\ e(\bar{Q})\cap
B_{8jS}(x_{0})\neq\emptyset\right\\},$ $\displaystyle u_{O}=\sum_{e\in
L_{O}}\psi_{e}u_{\Omega}$ $\displaystyle\qquad\qquad L_{O}=\left\\{e\in L\ :\
e(\bar{Q})\cap(M\setminus B_{16jS}(x_{0}))\neq\emptyset\right\\}\,.$

Notice that the functions $u_{I},$ $u_{O}$ have disjoint support and the
remaining “annular part” of $u_{\Omega}$ is given by
$u_{\Omega}-u_{I}+u_{O}=\sum_{e\in L_{k}}\psi_{e}u_{\Omega}$. The truncations
$u_{I}$ and $u_{O}$ capture most of the energy of $u_{\Omega}$ in the
following sense:

$\displaystyle\left|\int|\nabla u_{\Omega}|^{2}-|\nabla u_{O}|^{2}-|\nabla
u_{I}|^{2}\right|=\Big{|}\int g(\nabla[u_{\Omega}-u_{O}$ $\displaystyle-
u_{I}],\nabla[u_{\Omega}+u_{O}+u_{I}])\Big{|}$ $\displaystyle\leq\sum_{e\in
L_{j}}\left(|\nabla\psi_{e}||u_{\Omega}|+\psi_{e}|\nabla
u_{\Omega}|\right)\cdot C|\nabla u_{\Omega}|$ $\displaystyle\leq C\sum_{e\in
L_{j}}\int_{e(\bar{Q})}|u_{\Omega}|^{2}+|\nabla
u_{\Omega}|^{2}\leq\frac{C}{J}.$

In particular, this together with the Poincaré inequality applied to $u_{O}$
provide the following upper bound for the energy of $u_{I}$:

(3.12) $\int|\nabla u_{I}|^{2}\leq\lambda_{1}(\Omega)-\int|\nabla
u_{O}|^{2}+\frac{C}{j}\leq\lambda_{1}(\Omega)\Big{(}1-\int
u_{O}^{2}\Big{)}+\frac{C}{J}.$

In a similar fashion, we find that

(3.13) $\int u_{\Omega}^{2}-u_{O}^{2}-u_{I}^{2}\leq\frac{C}{J}.$

The first main estimate of this sub-step is the following: there exists $C$
such that

(3.14) $\begin{split}\text{ if }\quad\int
u_{I}^{2}\geq\,\frac{1}{4}\qquad\text{ then }\quad\lambda_{1}\left(\Omega\cap
B_{8Sj}(x_{0})\right)&\leq\lambda_{1}(\Omega)+\frac{C}{J}\,,\\\ \text{ if
}\quad\int u_{O}^{2}\geq\,\frac{1}{4}\qquad\text{ then
}\quad\lambda_{1}\left(\Omega\setminus
B_{8Sj}(x_{0})\right)&\leq\lambda_{1}(\Omega)+\frac{C}{J}\,.\end{split}$

Note that (3.13) guarantees that one of the two alternatives in (3.14) must
hold provided $J$ is sufficiently large. If $\int u_{I}^{2}\geq 1/4$ then
(3.13) implies that $(1-\int u_{O}^{2})/\int u_{I}^{2}\leq 1+C/J$. So, if we
multiply (3.12) by $(1-\int u_{O}^{2})/\int u_{I}^{2}$ and divide both sides
by $1-\int u_{0}^{2}$, we arrive at

$\frac{\int|\nabla u_{I}|^{2}}{\int
u_{I}^{2}}\leq\lambda_{1}(\Omega)+\frac{C}{J}.$

Here we have also used the fact that $\lambda_{1}(\Omega)$ is uniformly
bounded; recall (3.10). So, the first estimate of (3.14) follows by taking
$u_{I}$ as a test function for $\lambda_{1}(\Omega\cap B_{8Sj}(x_{0}))$. The
proof of the second estimate of (3.14) is the same, using the analogous
estimate to (3.12) for $u_{O}$.

The second main estimate of this substep is the following:

(3.15) $\begin{split}\mathrm{tor}(\Omega\cap
B_{8Sj}(x_{0}))&\leq\mathrm{tor}(\Omega)+\frac{C}{J}+C|\Omega\setminus
B_{8Sj}(x_{0})|,\\\ \mathrm{tor}(\Omega\setminus
B_{8Sj}(x_{0}))&\leq\mathrm{tor}(\Omega)+\frac{C}{J}+C|\Omega\cap
B_{8Sj}(x_{0})|\end{split}$

The estimate (3.15) follows similarly to (3.14). Splitting the torsion
function $w_{\Omega}$ into an “inner part” $w_{I},$, and “outer part” $w_{O}$,
and an “annular part” $w_{\Omega}-w_{I}-w_{O}$ exactly as above leads to

$-\frac{1}{2}\int w_{I}+w_{O}\leq\mathrm{tor}(\Omega)+\frac{C}{J};$

we omit the details. This estimate is potentially much worse than the
eigenvalue one due to the presence of the $w_{O}$ term (the torsional rigidity
has poorer concentration-compactness behavior than the eigenvalue), but
recalling that $w_{\Omega}\leq C$ from Lemma 3.15, we find that

$\displaystyle\mathrm{tor}(\Omega\cap B_{8Sj}(x_{0}))\leq-\frac{1}{2}\int
w_{I}$ $\displaystyle\leq\mathrm{tor}(\Omega)+\frac{C}{J}+\frac{1}{2}\int
w_{O}$ $\displaystyle\leq\mathrm{tor}(\Omega)+\frac{C}{J}+C|\Omega\setminus
B_{8Sj}(x_{0})|,$

thus proving the first estimate in (3.15). The second estimate in (3.15) is
proven in the same way.

Step 1d: Energy comparison and selection of $\bar{J}$ and $\mathfrak{T}_{0}$.
Fix any $J\in\mathbb{N}$ and let $j=j(J)$ be as determined in Step 1b. Let us
first consider the case that $\int u_{I}^{2}\geq\frac{1}{4}.$ In this case, we
may use the first estimates in (3.14) and (3.15) to compare
$\mathcal{E}(\Omega)$ to $\mathcal{E}(\Omega\cap B_{8Sj}(x_{0}))$, which gives

$\displaystyle\mathcal{E}(\Omega)\leq\mathcal{E}_{min}+\delta$
$\displaystyle\leq\mathcal{E}(\Omega\cap B_{8Sj}(x_{0}))+\delta$
$\displaystyle\leq\lambda_{1}(\Omega)+\mathfrak{T}\mathrm{tor}(\Omega)+\frac{C}{J}+f_{v,\eta}(|\Omega\cap
B_{8Sj}(x_{0})|)+\delta+C\mathfrak{T}|\Omega\setminus B_{8Sj}(x_{0})|$
$\displaystyle\leq\mathcal{E}(\Omega)+\frac{C}{J}+f_{v,\eta}(|\Omega\cap
B_{8Sj}(x_{0})|)-f_{v,\eta}(|\Omega|)+\delta+C\mathfrak{T}|\Omega\setminus
B_{8Sj}(x_{0})|$
$\displaystyle\leq\mathcal{E}(\Omega)+\frac{C}{J}+\delta-[\eta-C\mathfrak{T}]|\Omega\setminus
B_{8Sj}(x_{0})|.$

At this point, we can fix our remaining parameters. Choosing
$\mathfrak{T}_{0}<\frac{\eta}{2C}$, the last term is negative and this leads
to

(3.16) $\begin{cases}\mathcal{E}(\Omega\cap
B_{8Sj}(x_{0}))\leq\mathcal{E}_{min}+\frac{C}{J}+\delta\leq 2\delta\\\
|\Omega\setminus B_{8Sj}(x_{0})|\leq 4\eta\delta.\end{cases}$

Finally, we choose $J=\bar{J}$ large enough in terms of $\delta$ so that
$\frac{C}{J}\leq\delta$, and (3.11) follows. This concludes the proof of Step
1 in the case when $\int u_{I}^{2}\geq\frac{1}{4}.$ holds. Now, assume instead
that $\int u_{O}^{2}\geq\frac{1}{4}.$ We may argue in the analogous fashion to
find that

(3.17) $\begin{cases}\mathcal{E}(\Omega\setminus
B_{8Sj}(x_{0}))\leq\mathcal{E}_{min}+\frac{C}{J}+\delta\leq 2\delta\\\
|\Omega\cap B_{8Sj}(x_{0})|\leq 4\eta\delta.\end{cases}$

This contradicts Step 1a for $\delta$ chosen sufficiently small. We conclude
that $\int u_{I}^{2}\geq\frac{1}{4}$ holds and thus have completed the proof
of Step 1.

Step 2: Conclusion. We are now in a position to prove the first conclusion of
the lemma. Let $\Omega^{\prime}=\Omega\cap B_{8Sj}(x_{0})$ be the truncation
of $\Omega$ obtained in Step 1. Let $S^{\prime}$ be chosen large enough that
$e_{0}^{-1}(B_{8SJ}(x_{0}))\subseteq Q_{S^{\prime}}$; note that this depends
only on $J$ and $S$, so only on $v_{\text{max}}$ and $\delta$. Applying Lemma
3.10 to $e_{0}^{-1}(\Omega^{\prime})$ on $Q_{S^{\prime}}$, there is a
$U\in\mathcal{M}(S^{\prime})$ with
$\|u_{U}-u_{I}\|_{H^{1}},\sqrt{\mathfrak{T}}\|w_{U}-w_{I}\|_{H^{1}}\leq\varepsilon/2$.
On the other hand, from the Caccioppoli inequality and letting
$L^{\prime}=\\{e\in L:e(Q_{S})\cap B_{8jS}(x_{0})=\emptyset\\}$,

$\int|\nabla(u_{\Omega}-u_{I})|^{2}\leq C\sum_{e\in
L^{\prime}}\int|\nabla\psi_{e}|^{2}|u_{\Omega}|^{2}+\psi_{e}^{2}|\nabla
u_{\Omega}|^{2}\leq C\int_{\Omega\setminus B_{8Sj}(x_{0})}|u_{\Omega}|^{2}\leq
C\delta,$

using that $|u_{\Omega}|\leq C$ and the volume estimate in (3.11) at the end.
Similarly, $\int|\nabla(w_{\Omega}-w_{I})|^{2}\leq C\delta$, and we obtain
conclusion (1) from the triangle inequality.

To prove (2), note that applying Lemma 3.14 gives that
$\lambda_{2}(\Omega^{\prime})\geq\lambda_{1}(\Omega)+c_{0}$ for
$c_{0}=c_{0}(S^{\prime},v,v_{\text{max}})$, and so it remains to show that
$\lambda_{2}(\Omega^{\prime})\leq\lambda_{2}(\Omega)+c_{0}/2$. To this end,
let $u_{2}$ be a second eigenfunction for $\Omega$. from Lemma 3.2 we have
that $|u_{2}|\leq C(\lambda_{1}(\Omega))\leq C$ unless
$\lambda_{2}(\Omega)>1+\lambda_{1}(\Omega)$ (in which case we would be done).
Then set $u_{2,I}=\sum_{e\in L_{I}}\psi_{e}u_{2}$ to be the “inner part,” i.e.
the analogue of $u_{I}$ for $u_{2}$. As
$\Delta|u_{2}|\geq-\lambda_{2}(\Omega)|u_{2}|$ in the sense of distributions,
the same Caccioppoli inequality argument gives that

$\int|\nabla(u_{2}-u_{2,I})|^{2}\leq C\int_{\Omega\setminus
B_{8Sj}(x_{0})}|u_{2}|^{2}\leq C\delta.$

Hence

$\frac{\int|\nabla u_{2,I}|^{2}}{\int
u_{2,I}^{2}}\leq\lambda_{2}(\Omega)+C\delta\qquad\text{and}\qquad\frac{|\int
u_{2,I}\,u_{I}|}{\|u_{2,I}\|_{L^{2}}\|u_{I}\|_{L^{2}}}\leq C\delta,$

implying that $\lambda_{2}(\Omega^{\prime})\leq\lambda_{2}(\Omega)+C\delta.$
Taking $\delta$ small enough completes the proof of (2).

To check the final conclusion (3) of the lemma, note that from Lemma 3.14, we
know that one connected component $A$ of $\Omega^{\prime}$ has
$|A|>|\Omega^{\prime}|-\varepsilon/2$, while every other has
$|A|\leq\varepsilon/2$ (choosing $\delta$ small enough in terms of
$\varepsilon$). As $|\Omega\setminus B_{8Sj}(x_{0})|\leq C\delta$, we may
conclude by choosing $C\delta<\varepsilon/2$. ∎

##  4\. The lower bound

In this section we consider inward minimizers of the main functional
$\mathcal{F}_{\tau}$ defined in (2.8). The main result of this section is
Theorem 4.1, which provides a linear lower bound for the growth of the
function $u_{\Omega}+\sqrt{\mathfrak{T}}w_{\Omega}$ away from the boundary of
an inward minimizer $\Omega$. The theorem is phrased in terms of a lower bound
on the quantity $\text{DO}(\Omega)$ defined in (2.14). One important
consequence of this theorem is a lower volume density estimate for inward
minimizers shown in Corollary 4.4. Throughout this section, we fix $R>0$,
$0<v<v_{\text{max}}$, and $0<\eta\leq\eta_{0}$ (recall Remark 3.12).

######  Theorem 4.1.

There are constants $\delta_{m},\mathfrak{T}_{m},c_{m}>0$ depending only on
$R,v,v_{\text{max}},$ and $\eta$ and a constant
$\tau_{m}(R,v,v_{\text{max}},\eta,\mathfrak{T})>0$ such that if we fix
$\mathfrak{T}<\mathfrak{T}_{m}$ and then $\tau<\tau_{m}$, the following holds.
Let $\Omega$ be an inward minimizer of $\mathcal{F}_{\tau}$ on $Q_{R}$
satisfying $\mathcal{E}(\Omega)\leq\mathcal{E}_{min}+\delta_{m}$. Then we have

$\text{DO}(\Omega)\geq c_{m}$

where $\text{DO}(\Omega)$ is defined in (2.14). If $M/G$ is compact, all
constants may be taken independent of $R$.

It is worth noting that Theorem 4.1 and Lemma 3.4 show that the torsion
function $w_{\Omega}$ grows at least linearly away from the boundary, but we
cannot immediately deduce an analogous lower bound for the growth of the
eigenfunction $u_{\Omega}$. It will require some delicate Green’s function
estimates to eventually show in Section 7 that, for minimizers, the first
eigenfunction also satisfies this type of linear lower bound.

The core arguments in the proof of Theorem 4.1 below are from David and Toro
[18]. We present the details to show how to apply the key estimate of
Proposition 3.9 in order to handle the nonlinear term $\mathfrak{h}$, as well
as to verify the dependence on all of the parameters. The proof will be
established by iteratively applying the following lemma.

######  Lemma 4.2.

There exist constants $\delta_{m},\mathfrak{T}_{m},c_{m}>0$ depending only on
$R,v,v_{\text{max}},$ and $\eta$ and a constant
$\tau_{m}(R,v,v_{\text{max}},\eta,\mathfrak{T})>0$ such that if we fix
$\mathfrak{T}<\mathfrak{T}_{m}$ and then $\tau<\tau_{m}$, the following holds.
Let $\Omega$ be an inward minimizer on $Q_{R}$ satisfying
$\mathcal{E}(\Omega)\leq\mathcal{E}_{min}+\delta_{m}$. For any $x\in Q_{R}$,
and $\sqrt{r}\leq c_{m}$, if

(4.1) $\sup_{B_{r}(x)}u_{\Omega}+\sqrt{\mathfrak{T}}w_{\Omega}\leq c_{m}r$

then

$\sup_{B_{r/2}(x)}u_{\Omega}+\sqrt{\mathfrak{T}}w_{\Omega}\leq\frac{1}{4}c_{m}r.$

If $M/G$ is compact, all constants may be taken independent of $R$.

###### Proof.

The basic idea of the proof is make use of the inward minimality property of
$\Omega$ using an energy competitor $\Omega^{\prime}$ obtained by removing a
small ball from $\Omega$.

Step 1: The competitor $\Omega^{\prime}$. Let
$c_{m}\leq\min\\{\text{inj}_{M},1\\}$ be a fixed number to be specified later
in the proof. For $r\leq c_{m}^{2}$ and $x\in Q_{R}$, we consider the
competitor $\Omega^{\prime}=\Omega\setminus B_{3r/4}(x).$ Choose $\delta_{m},$
and $\mathfrak{T}_{m}$ according to Lemma 3.14, so that $\Omega$ has a
spectral gap $\lambda_{2}(\Omega)>\lambda_{1}(\Omega)+c(R,v_{\text{max}},v)$
of a definite size. This allows us to apply the key estimate of Proposition
3.9 to $\Omega^{\prime}\subseteq\Omega$, which yields

$\|u_{\Omega^{\prime}}-u_{\Omega}\|_{L^{1}}\leq
C\big{[}\mathrm{tor}(\Omega^{\prime})-\mathrm{tor}(\Omega)+\lambda_{1}(\Omega^{\prime})-\lambda_{1}(\Omega)\big{]},$

where the constant depends only on $R,v_{\text{max}},v$. (Note that the
constant a priori also depends on an upper bound for
$\lambda_{1}(\Omega^{\prime})$; a basic cutoff function argument like the one
in step 3 below and the assumption on $\mathcal{E}(\Omega)$ show that
$\lambda_{1}(\Omega^{\prime})\leq C(R,v_{\text{max}},v).$) If $M/G$ is
compact, we may take the constants here independent of $R$ by using Lemma 3.16
in place of Lemma 3.14. This leads to

$|\mathfrak{h}(\Omega)-\mathfrak{h}(\Omega^{\prime})|\leq|\Omega\triangle\Omega^{\prime}|+\int|u_{\Omega^{\prime}}-u_{\Omega}|\leq|\Omega\setminus\Omega^{\prime}|+C\left[\mathrm{tor}(\Omega^{\prime})-\mathrm{tor}(\Omega)+\lambda_{1}(\Omega^{\prime})-\lambda_{1}(\Omega)\right]$

using property 3 of $\mathfrak{h}$. From the inward minimality of $\Omega$, we
have $\mathcal{F}_{\tau}(\Omega)\leq\mathcal{F}_{\tau}(\Omega^{\prime})$ and
so

$\displaystyle 0$
$\displaystyle\leq\mathcal{F}_{\tau}(\Omega^{\prime})-\mathcal{F}_{\tau}(\Omega)$
$\displaystyle\leq\lambda_{1}(\Omega^{\prime})-\lambda_{1}(\Omega)+\mathfrak{T}[\mathrm{tor}(\Omega^{\prime})-\mathrm{tor}(\Omega)]+f_{v,\eta}(\Omega^{\prime})-f_{v,\eta}(\Omega)$
$\displaystyle\qquad\qquad+\tau\Big{(}\left|\Omega\setminus\Omega^{\prime}\right|+C\big{[}\mathrm{tor}(\Omega^{\prime})-\mathrm{tor}(\Omega)+\lambda_{1}(\Omega^{\prime})-\lambda_{1}(\Omega)\big{]}\Big{)}$
$\displaystyle\leq
2\big{[}\mathfrak{T}(\mathrm{tor}(\Omega^{\prime})-\mathrm{tor}(\Omega))+\lambda_{1}(\Omega^{\prime})-\lambda_{1}(\Omega)\big{]}+(\tau-\eta)|\Omega\setminus\Omega^{\prime}|,$

where the final inequality holds provided $\tau_{m}$ is chosen small enough in
terms of $C$ and $\mathfrak{T}$. So long as $\tau<\eta/2$, we may rewrite this
as

(4.2) $|\Omega\cap B_{3r/4}(x)|=|\Omega\setminus\Omega^{\prime}|\leq
C\big{[}\mathfrak{T}(\mathrm{tor}(\Omega^{\prime})-\mathrm{tor}(\Omega))+\lambda_{1}(\Omega^{\prime})-\lambda_{1}(\Omega)\big{]}.$

Step 2: Estimates for the eigenfunction and torsion function.  From Lemma 3.1
as well as the bounds from Lemmas 3.2 and 3.3, we have that $\Delta
u_{\Omega},\Delta w_{\Omega}\geq-C$ on $M$. Fix $\phi$ a cutoff function which
is $1$ on $B_{\frac{7r}{8}}(x)$, compactly supported on $B_{1}$, and has
$|\nabla\phi|\leq C/r$. Applying the Caccioppoli inequality to $w_{\Omega}$
(i.e. use $\phi^{2}w_{\Omega}$ as a test function for $w_{\Omega}$ and
rearrange terms) and the assumption (4.1),

$\mathfrak{T}\int\phi^{2}|\nabla
w_{\Omega}|^{2}\leq\mathfrak{T}C\int\left(|\nabla\phi|^{2}w_{\Omega}^{2}+\phi^{2}w_{\Omega}\right)\leq
C[r^{n-2}c_{m}^{2}r^{2}+r^{n}\sqrt{\mathfrak{T}}c_{m}r]\leq Cr^{n}c_{m}^{2}.$

In the final inequality we have used that $r\leq\sqrt{r}\leq c_{m}$. We may
estimate $u_{\Omega}$ in a similar manner:

$\int\eta^{2}|\nabla u_{\Omega}|^{2}\leq
C\int\left(|\nabla\eta|^{2}u_{\Omega}^{2}+\eta^{2}u_{\Omega}\right)\leq
Cr^{n}c_{m}^{2}.$

All together, this implies

$\int_{B_{7r/8}(x)}|\nabla u_{\Omega}|^{2}+\mathfrak{T}|\nabla
w_{\Omega}|^{2}\leq
Cr^{n}c_{m}^{2},\qquad\int_{B_{7r/8}(x)}u_{\Omega}^{2}+\mathfrak{T}w_{\Omega}^{2}\leq
Cr^{n+2}c_{m}^{2}.$

Step 3: Estimates for the eigenvalue and torsional rigidity of
$\Omega^{\prime}$. Now take another cutoff function $\phi_{1}$ which is $1$ on
$M\setminus B_{7r/8}(x)$, vanishes on $B_{3r/4}(x)$, and has
$|\nabla\phi_{1}|\leq C/r$. We use $\phi_{1}u_{\Omega}$ as a competitor for
$\lambda_{1}(\Omega^{\prime})$ to find
$\lambda_{1}(\Omega^{\prime})\leq{\int|\nabla(\phi_{1}u_{\Omega})|^{2}}/{\int(\phi_{1}u_{\Omega})^{2}}.$
Estimating the denominator,

$\int\phi_{1}^{2}u_{\Omega}^{2}\geq\int u_{\Omega}^{2}-\int_{\Omega\setminus
B_{7r/8}(x)}u_{\Omega}^{2}\geq 1-Cr^{n}c_{m}^{2}.$

For the numerator,

$\int|\nabla(\phi_{1}u_{\Omega})|^{2}\leq\int|\nabla
u_{\Omega}|^{2}+\int|\nabla
u_{\Omega}|^{2}\phi^{2}+|\nabla\phi|^{2}u_{\Omega}^{2}\leq\lambda_{1}(\Omega)+Cr^{n}c_{m}^{2},$

so

$\lambda_{1}(\Omega^{\prime})\leq\lambda_{1}(\Omega)+Cr^{n}c_{m}^{2}.$

An identical estimate on $w_{\Omega}$ gives

$\mathfrak{T}\mathrm{tor}(\Omega^{\prime})\leq\mathfrak{T}\mathrm{tor}(\Omega)+Cr^{n}c_{m}^{2}.$

Step 4: Conclusion.  Substituting the estimates of Step 3 into (4.2), we
arrive at

(4.3) $|\Omega\cap B_{3r/4}(x)|\leq Cr^{n}c_{m}^{2}.$

We now apply the local maximum principle for subsolutions, [24, Theorem 8.17],
to $u_{\Omega}+\sqrt{\mathfrak{T}}w_{\Omega}$:

$\displaystyle\sup_{B_{r/2}(x)}u_{\Omega}+\sqrt{\mathfrak{T}}w_{\Omega}$
$\displaystyle\leq
C\left[r^{-n/2}\|u_{\Omega}+\sqrt{\mathfrak{T}}w_{\Omega}\|_{L^{2}(B_{3r/4}(x))}+r^{2}\right]$
$\displaystyle\leq C\left[c_{m}rr^{-n/2}|\Omega\cap
B_{3r/4}(x)|^{1/2}+rc_{m}^{2}\right]\leq Crc_{m}^{2},$

where we used (4.3) and that $r\leq c_{m}^{2}$ by assumption. Finally, we
choose $c_{m}$ small enough so that $Cc_{m}\leq\frac{1}{4}$ to conclude the
proof. ∎

Theorem 4.1 now follows by iteratively applying Lemma 4.2:

###### Proof of Theorem 4.1.

Set all constants as in Lemma 4.2. Take any $y\in\bar{\Omega}$ and any
$r<r_{0}:=c_{m}^{2}/16$. We claim that

$\sup_{B_{r}(y)}u_{\Omega}+\sqrt{\mathfrak{T}}w_{\Omega}\geq c_{m}r.$

Indeed, if this is not the case, we apply Lemma 4.2 to obtain

$\sup_{B_{r/2}(y)}u_{\Omega}+\sqrt{\mathfrak{T}}w_{\Omega}\leq\frac{c_{m}}{2}\frac{r}{2}.$

Then take any $x\in B_{r/4}(y)$:

$\sup_{B_{r/4}(x)}u_{\Omega}+\sqrt{\mathfrak{T}}w_{\Omega}\leq
c_{m}\frac{r}{4},$

so repeatedly applying Lemma 4.2 gives

$\sup_{B_{r/4\cdot
2^{-k}}(x)}u_{\Omega}+\sqrt{\mathfrak{T}}w_{\Omega}\leq\frac{c_{m}}{2}\frac{r}{4}2^{-k}.$

In particular, $x$ is a Lebesgue point of $u_{\Omega}$, $w_{\Omega}$ and
$u_{\Omega}(x)=w_{\Omega}(x)=0$. This is true of all points in $B_{r/4}(y)$,
so $u_{\Omega}\equiv w_{\Omega}\equiv 0$ there. Using $\Omega\setminus
B_{r/4}(x)$ as a competitor for $\Omega$ implies that $|\Omega\cap
B_{r/4}|=0$, contradicting that $y\in\bar{\Omega}$ and $\Omega$ open.

For $r\in[r_{0},1]$, we simply use
$\sup_{B_{r}(y)}u_{\Omega}+\sqrt{\mathfrak{T}}w_{\Omega}\geq c_{m}r_{0}\geq
c_{m}r_{0}r.$ This gives $\text{DO}(\Omega)\geq c_{m}r_{0}$. ∎

Note that while for an absolute minimizer (should one exist)
$\mathcal{E}(\Omega)\leq\mathcal{E}_{min}+\delta$ follows from $\tau$ being
small (see Section 6), there is no reason for that to be the case for an
inward minimizer in general.

######  Remark 4.3.

The careful reader may observe that it is possible to modify the argument in
this section to give $\sup_{B_{r}(y)}w_{\Omega}\geq c_{m}r$ as long as
$\tau<\tau_{m}(R,v,v_{\text{max}})$ independent of $\mathfrak{T}$ (by using
Lemma 3.4 to estimate $u_{\Omega}\leq Cw_{\Omega}$). Estimates in this spirit
were used by Bucur [11] for other problems. We do not pursue this point
further here because the opposite estimate on $\text{UP}(\Omega)$, discussed
in the next section, cannot be altered in this way without a version of
Proposition 3.9 which omits the torsional rigidity terms on the right-hand
side.

The following corollary will help us bound the diameter of minimizers when
taking $R\rightarrow\infty$. We will discuss much sharper estimates in Section
7.1, using the upper bound $\text{UP}(\Omega)$ as well.

######  Corollary 4.4.

Let $\Omega$ be as in Theorem 4.1. Then for any $x\in\Omega$ and
$r<c_{m}^{4}$,

$|\Omega\cap B_{r}(x)|\geq c(r,v,v_{\text{max}},\eta,R).$

If $M/G$ is compact, $c$ may be taken independent of $R$.

###### Proof.

As in the proof of Lemma 4.2, apply the local maximum principle on $B_{r}(x)$
to obtain

$\displaystyle\sup_{B_{r/2}(x)}u_{\Omega}^{2}+\mathfrak{T}w_{\Omega}^{2}\leq
Cr^{-n}\int_{B_{r}(x)\cap\Omega}(u_{\Omega}^{2}+\mathfrak{T}w_{\Omega}^{2})+Cr^{4}$
$\displaystyle\leq C|\Omega\cap
B_{r}(x)|r^{-n}\sup_{\Omega}(u_{\Omega}^{2}+\mathfrak{T}w_{\Omega}^{2})+Cr^{4}$
$\displaystyle\leq C|\Omega\cap B_{r}(x)|r^{-n}+Cr^{4},$

with the last step using Lemmas 3.2 and 3.3. The quantities
$|\mathrm{tor}(\Omega)|$ and $\lambda_{1}(\Omega)$ are controlled by
$\mathcal{E}(\Omega)$, which is bounded from the assumption
$\mathcal{E}_{min}\leq\mathcal{E}(\Omega)\leq\mathcal{E}_{min}+\delta$ and
Lemma 3.10 or 3.15. The constant $C$ can be taken to depend only on
$M,v,v_{\text{max}},\eta$ if $M/G$ compact, and otherwise also depends on $R$.
Then applying Theorem 4.1 to the left-hand side, $c_{m}^{2}r^{2}/2\leq