Ars Inveniendi Analytica (2023), Paper No. 1, 49 pp.

DOI 10.15781/e7f3-a487

ISSN: 2769-8505

00footnotetext: © Mark Allen, Dennis Kriventsov, and Robin
Neumayer00footnotetext:  Licensed under a Creative Commons Attribution License
(CC-BY).

Sharp Quantitative Faber-Krahn Inequalities and the

Alt-Caffarelli-Friedman
Monotonicity Formula

Mark Allen

Brigham Young University

Dennis Kriventsov

Rutgers University

Robin Neumayer

Carnegie Mellon University

Communicated by Guido De Philippis

Abstract. The objective of this paper is two-fold. First, we establish new
sharp quantitative estimates for Faber-Krahn inequalities on simply connected
space forms. In these spaces, geodesic balls uniquely minimize the first
eigenvalue of the Dirichlet Laplacian among all sets of a fixed volume. We
prove that the gap between the first eigenvalue of a given set $\Omega$ and
that of the ball quantitatively controls both the $L^{1}$ distance of this set
from a ball and the $L^{2}$ distance between the corresponding eigenfunctions:

$\lambda_{1}(\Omega)-\lambda_{1}(B)\gtrsim|\Omega\Delta
B|^{2}+\int|u_{\Omega}-u_{B}|^{2},$

where $B$ denotes the nearest geodesic ball to $\Omega$ with $|B|=|\Omega|$
and $u_{\Omega}$ denotes the first eigenfunction with suitable normalization.
On Euclidean space, this extends a result of Brasco-De Phillipis-Velichkov;
the eigenfunction control largely builds upon new regularity results for
minimizers of critically perturbed Alt-Cafarelli type functionals in our
companion paper. On the round sphere and hyperbolic space, the present results
are the first sharp quantitative results with respect to any distance; here
the local portion of the analysis is based on new implicit spectral analysis
techniques.
Second, we apply these sharp quantitative Faber-Krahn inequalities in order to
establish a quantitative form of the Alt-Caffarelli-Friedman (ACF)
monotonicity formula. A powerful tool in the study of free boundary problems,
the ACF monotonicity formula is nonincreasing with respect to its scaling
parameter for any pair of admissible subharmonic functions, and is constant if
and only if the pair comprises two linear functions truncated to complementary
half planes. We show that the energy drop in the ACF monotonicity formula from
one scale to the next controls how close a pair of admissible functions is
from a pair of complementary half-plane solutions. In particular, when the
square root of the energy drop summed over all scales is small, our result
implies the existence of tangents (unique blowups) of these functions.

Keywords. Faber-Krahn, ACF monotonicity formula, selection principle.

##  1\. Introduction

Let $(M,g)$ be a smooth Riemannian $n$-manifold, and let $\Omega$ be an open
bounded subset of $M$. The first Dirichlet eigenvalue, or principal frequency,
of $\Omega$ is defined by

(1.1)
$\lambda_{1}(\Omega)=\inf\left\\{\frac{\int_{\Omega}|{\nabla}u|^{2}}{\int_{\Omega}u^{2}}\
:\ u\in C^{\infty}_{0}(\Omega)\right\\}.$

Here and in the sequel,
$|{\nabla}u|^{2}=\langle\nabla_{g}u,\nabla_{g}u\rangle_{g}$ and integration is
with respect to the volume measure induced by the metric $g$. The infimum in
(1.1) is achieved, and a minimizer $u_{\Omega}$ satisfies the equation

(1.2) $\begin{cases}-\Delta u_{\Omega}=\lambda_{1}(\Omega)\,u_{\Omega}&\text{
in }\Omega\\\ u_{\Omega}=0&\text{ on }{\partial}\Omega,\end{cases}$

where $\Delta=\Delta_{g}$ is the Laplacian with respect to $g$. Such a
function $u_{\Omega}$ is called a first Dirichlet eigenfunction of $\Omega$
and $\lambda_{1}(\Omega)$ is the smallest number for which a nontrivial
solution to this eigenvalue problem exists. If $\Omega$ is connected, then the
first Dirichlet eigenfunction is unique up to constant multiples.

In Euclidean space, balls have the smallest principal frequency among subsets
$\Omega\subset{\varmathbb{R}}^{n}$ of a given volume. This fact, first
established by Faber [27] and Krahn [37] in the 1920s, readily follows from
the Polya-Szegö principle [44], which relies only on the coarea formula and
the isoperimetry of balls. On all simply connected space forms (Euclidean
space, hyperbolic space and the round sphere), geodesic balls are
isoperimetric sets for every volume. Consequently, on simply connected space
forms, the Polya-Szegö principle holds and geodesic balls minimize the
principal frequency among sets of a given volume. The resulting inequality is
known as the Faber-Krahn inequality: letting $B$ denote a geodesic ball and
$|\cdot|$ denote the volume measure induced by the metric on either Euclidean
space, hyperbolic space, or the round sphere, we have

$\lambda_{1}(\Omega)\geqslant\lambda_{1}(B)\qquad\text{ whenever
}|\Omega|=|B|.$

It is known, moreover, that equality is achieved if and only if $\Omega$ is a
geodesic ball, up to a set of capacity zero. On the round sphere, the Faber-
Krahn inequality was first established by Sperner in [45] and is often called
Sperner’s inequality. We refer the reader to [7, 10, 22] for a discussion of
spectral inequalities and symmetrization techniques in these settings.

Our first main theorem is a sharp quantitative stability result for the Faber-
Krahn inequality on simply connected space forms. Broadly speaking, a
quantitative stability result in this context states that

(1.3) $\lambda_{1}(\Omega)-\lambda_{1}(B)\geqslant c\text{
dist}(\Omega,B)^{\alpha}$

for some $\alpha>0$, where $B$ is a geodesic ball with $|\Omega|=|B|$ and
$\text{dist}(\Omega,B)$ is a suitable notion of distance between $\Omega$ and
the nearest ball $B$. In the statement below, we let $u_{\Omega}$ denote any
first eigenfunction of $\Omega$, i.e. solution of (1.2), that is normalized so
that $u_{\Omega}\geqslant 0$ and $\int u_{\Omega}^{2}=1$, and extended by $0$
to be defined on the entire space. We let $A\Delta B=(A\setminus
B)\cup(B\setminus A)$ denote the symmetric difference between sets.

######  Theorem 1.1 (Sharp quantitative stability for Faber-Krahn
inequalities).

Fix $n\geqslant 2$ and let $(M^{n},g)$ denote the round sphere, Euclidean
space, or hyperbolic space. For any $v>0$ (with $v<|S^{n}|$ in the case of the
round sphere), there exists a constant $c=c(n,v)$ such that the following
holds. For any open bounded set $\Omega\subset M$ with $|\Omega|=v,$ we have

(1.4) $\lambda_{1}(\Omega)-\lambda_{1}(B)\geqslant c\inf_{x\in
M}\left(\left|\Omega\Delta
B(x)\right|^{2}+\int_{M}\left|u_{\Omega}-u_{B(x)}\right|^{2}\right)\,.$

Here $B(x)$ is a geodesic ball centered at $x$ with radius uniquely chosen so
that $|B(x)|=v$, and $B$ denotes such a ball centered at any point.

######  Remark 1.2 (Optimality of the quadratic power).

Considering ellipsoidal perturbations of a geodesic ball (in normal
coordinates in the case of the sphere or hyperbolic space) shows that the
quadratic power $\alpha=2$ is optimal for both of the terms on the right-hand
side of (1.4), in the sense that the result is false if $2$ is replaced by any
$\alpha<2.$

On Euclidean space, quantitative stability for the Faber-Krahn inequality was
first considered by Hansen and Nadirashvili [32] and Melas [41]. Both of these
papers restrict their attention to convex sets or simply connected planar
sets, and consider “$L^{\infty}$” (Hausdorff) type set distances
$\text{dist}(\Omega,B)$. Among general open bounded sets, simple examples show
that even qualitative stability fails with respect to such a distance. For
this reason, the asymmetry,
$\text{dist}(\Omega,B)=\inf_{x\in{\varmathbb{R}}^{n}}|\Omega\Delta B(x)|$, is
a more natural notion of set distance to consider, and quantitative stability
with respect to the asymmetry was conjectured in [12] and [43]. Non-sharp
versions of this conjecture (i.e. with $\alpha>2$ in (1.3)) were established
in following years in [11] and [29]. All of these results are based on
applying quantitative forms of the isoperimetric inequality to level sets of
eigenfunctions.

The first sharp quantitative Faber-Krahn inequality on Euclidean space was
established by Brasco, De Philippis, and Velichkov [18], stating

(1.5) $\lambda_{1}(\Omega)-\lambda_{1}(B)\geqslant
c\inf_{x\in{\varmathbb{R}}^{n}}|\Omega\Delta B(x)|^{2}$

for all open bounded sets $\Omega$, where $|B(x)|=|\Omega|$. Their proof was
based on a selection principle approach introduced by Cicalese-Leonardi [23]
(see also [1]) using the regularity theory for Alt-Caffarelli type
functionals. The selection principle is also the basis of the proof of Theorem
1.1, and we discuss these proofs further in Section 1.1 below. Following the
work of [18], similar methods were used to establish sharp quantitative
stability for the $p$-Laplacian [30] as well as for the capacity [26] and
$p$-capacity [42] inequalities. We refer the reader to [17] for an overview on
spectral inequalities in quantitative form.

A shortfall of the asymmetry as a measure of the distance between sets is that
it only detects modifications up to sets of measure zero, while the eigenvalue
sees modifications of a set up to sets of zero capacity. For instance, if we
let $\Omega=B(0,1)\setminus\\{x_{1}=0\\}$ be a slit domain in
${\varmathbb{R}}^{2}$, then the left-hand side of (1.5) is strictly positive
while the right-hand side is zero. In contrast, the right-hand side of (1.4)
is strictly positive in this example (and more generally, whenever the left-
hand side is), and in this way, Theorem 1.1 nontrivially strengthens the
result of [18] even in Euclidean space. This issue is discussed in the survey
paper [17], and in particular Theorem 1.1 resolves [17, Open Problem 3].

On the sphere and hyperbolic space, significantly less progress has been made
toward quantitative Faber-Krahn inequalities. These spaces lack the scaling
invariance of Euclidean space, and certain computations that can be carried
out explicitly on Euclidean space cannot be done in these spaces. In fact, the
explicit forms of the first eigenvalue and eigenfunction of geodesic balls are
not known in closed form! Ávila established analogues of Melas’ aforementioned
results for convex subsets of $S^{2}$ and $H^{2}$ in [9], following
quantitative estimates for other spectral inequalities in [46]. Qualitative
stability results for the Faber-Krahn inequality and other spectral
inequalities on space forms is established in [8]. See also [13, 15] for
quantitative isoperimetric inequalities on space forms. To our knowledge,
Theorem 1.1 is the first (sharp or nonsharp) quantitative stability result for
Faber-Krahn inequalities on the round sphere and hyperbolic space that is
valid for all open bounded sets $\Omega$.

It has been understood in recent years that eigenvalue optimization problems
are closely linked to the study of free boundary problems. When optimizing
functionals involving (linear combinations of) higher eigenvalues, or of the
first eigenvalue with respect to additional constraints, one can no longer
expect to explicitly characterize minimizers. Instead, one hopes to show that
a minimizer exists in a weak class of sets, and then show that the boundary of
a minimizer is regular at most points. The paper [19] considered the
minimization of $\lambda_{1}(\Omega)$ among subsets $\Omega$ of some fixed
bounded open container. Here the authors established regularity properties of
minimizing sets by using the regularity theory of Alt and Caffarelli [5] for
the one-phase free boundary problem. More recently, minimization problems for
functionals that are linear combinations of Dirichlet eigenvalues have been
recast as vectorial free boundary problems in [38, 39, 40, 21], and regularity
of minimizers has been studied through this lens of free boundary regularity.

In these examples, regularity theory for free boundary problems is used as a
tool in the study of eigenvalue optimization problems. In the next main result
of this paper, the stream of tools will go in the opposite direction: we apply
the quantitative Faber-Krahn inequality of Theorem 1.1 on the sphere in order
to establish new quantitative estimates for the Alt-Caffarelli-Friedman
monotonicity formula, a powerful tool in the study of various types of free
boundary problems.

Fix $n\geqslant 2$ and let $B_{r}=B(0,r)\subset{\varmathbb{R}}^{n}$ denote the
ball of radius $r$ centered at the origin in Euclidean space. Let
$u_{1},u_{2}\in H^{1}(B_{2})$ be two nonnegative continuous functions
satisfying

(1.6) $\begin{cases}\Delta u_{i}\geqslant 0&\ \ \text{ in }\ \
\\{u_{i}>0\\}=:\Omega_{i}\\\ u_{1}\cdot u_{2}=0&\ \ \text{ in
}B_{2}.\end{cases}$

The Alt-Caffarelli-Friedman monotonicity formula (hereafter denoted ACF
formula) $J:(0,1)\to{\varmathbb{R}}$ is defined by

(1.7)
$J(r)=J[u_{1},u_{2}](r)=\frac{1}{r^{4}}\int_{B_{r}}|{\nabla}u_{1}|^{2}\int_{B_{r}}|{\nabla}u_{2}|^{2}\,$

when $n=2$ and

(1.8)
$J(r)=J[u_{1},u_{2}](r)=\frac{1}{r^{4}}\int_{B_{r}}\frac{|{\nabla}u_{1}|^{2}}{|x|^{n-2}}\int_{B_{r}}\frac{|{\nabla}u_{2}|^{2}}{|x|^{n-2}}$

for $n\geqslant 3$. Heuristically thinking, one can view $J(r)$ as the product
of the average Dirichlet energies of $u_{1},u_{2}$ on $B_{r}$. This quantity
has the important property of being nondecreasing for $r\in(0,1)$. Note that
this in particular guarantees the existence of the limit

$J(0^{+}):=\lim_{r\to 0}J(r).$

Furthermore, if $J(0^{+})$ is strictly positive, then $J(r)$ is constant with
respect to $r$ if and only if

$u_{1}=\beta_{1}(x\cdot\nu)^{+},\qquad\qquad u_{2}=\beta_{2}(x\cdot\nu)^{-},$

for some $\nu\in S^{n-1}$ and $\beta_{1},\beta_{2}>0$. Here $f^{+}$ and
$f^{-}$ respectively denote the positive and negative parts of a function $f$.
In other words, the ACF monotonicity formula is constant and positive if and
only if $u_{1},u_{2}$ are linear functions defined on complementary half
spaces $\Omega_{1}$ and $\Omega_{2}.$

Since its introduction in [6], the ACF monotonicity formula has proven useful
in both free boundary problems and harmonic analysis. One application of the
ACF monotonicity formula is the following. Suppose that a pair of functions
$u_{1},u_{2}$ satisfying (1.6) have $J(0^{+})>0$, and consider any sequence
$r_{k}\to 0$. There exists a subsequence (which we do not relabel), constants
$\beta_{1},\beta_{2}>0$ and a direction $\nu\in S^{n-1}$ such that

$\frac{u_{1}(r_{k}x)}{r_{k}}\to\beta_{1}(x\cdot\nu)^{+},\qquad\qquad\frac{u_{2}(r_{k}x)}{r_{k}}\to\beta_{2}(x\cdot\nu)^{-}.$

However, it is important to note that these blowup limits depend on the
sequence $r_{k}$ and does not imply that $u_{1}$ and $u_{2}$ or their
interfaces $\Omega_{1},\Omega_{2}$ have unique tangents at the origin. In
fact, this need not be the case! In [3] the first two authors gave an example
of two harmonic functions defined on complementary subsets of $B_{2}$, which
have a slowly spiraling interface, where different sequences $r_{k}\to 0$
result in different $\nu$ in the limit.

Toward understanding conditions that do guarantee the existence of unique
tangents for functions $u_{1}$ and $u_{2}$ as in (1.6), we establish the
following quantitative estimates for the Alt-Caffarelli-Friedman monotonicity
formula. Roughly speaking, this theorem says that if $J(1)-J(0^{+})$ is small,
then $u_{1}$ and $u_{2}$ are quantitatively close to linear functions
$\beta_{i}(x\cdot\nu)^{\pm}$. Note that if $J(0^{+})$ is strictly positive,
then up to rescaling we may assume that $J(0^{+})=1$.

######  Theorem 1.3 (Sharp quantitative stability for the ACF monotonicity
formula).

Fix $n\geqslant 2$ and let $\rho\in[0,1/2]$. There exists a constant
$C=C(n)>0$ such that the following holds. Suppose that $u_{1},u_{2}$ satisfy
(1.6). Then there exist constants $\beta_{1},\beta_{2}>0$ and $\nu\in S^{n-1}$
such that

$\int_{B_{1}\setminus
B_{\rho}}\left(u_{1}-\beta_{1}(x\cdot\nu)^{+}\right)^{2}+\left(u_{2}-\beta_{2}(x\cdot\nu)^{-}\right)^{2}\leqslant
C\log\left(\frac{J(1)}{J(\rho)}\right)\sum_{i=1}^{2}\|u_{i}\|_{L^{2}(B_{1})}^{2}.$

Furthermore, there exists a constant $\epsilon_{0}=\epsilon_{0}(n)>0$ such
that if the quotient $\log(J(1)/J(0^{+}))<\epsilon_{0}$, then $\beta_{i}$ and
$\nu$ may be chosen independently of $\rho\in[0,1/2]$.

This theorem is sharp, in the sense that the dependence on
$\log\frac{J(1)}{J(\rho)}$ on the right cannot be improved; this can be seen
from the examples constructed in [3, Section 2].

A direct consequence of Theorem 1.3 is the following corollary, which gives a
criterion for naturally scaled blow-ups of the functions $u_{i}$ at $0$ to be
unique. We believe this criterion is essentially sharp, for the same reasons
that Theorem 1.3 is sharp. This resolves a question raised in [3].

######  Corollary 1.4 (Uniqueness of blowups).

Fix $n\geqslant 2$. Suppose that $u_{1},u_{2}$ satisfy (1.6), assume
$J(0^{+})=1$, and let $\omega(r)=J(r)-1$. Assume in addition that

$\int_{0}^{1}\frac{\sqrt{\omega(r)}}{r}dr<\infty.$

Then there exists a number $\beta_{1}>0$ and a vector $\nu\in S^{n-1}$ such
that

$\lim_{r\searrow 0}\frac{1}{r^{n+2}}\int_{B_{r}}|u_{1}-\beta_{1}(\nu\cdot
x)_{+}|^{2}=0.$

###  1.1. Proofs of the main theorems

Let us now discuss some aspects of the proofs of our main theorems. We begin
by discussing Theorem 1.3, and particularly its connection to Theorem 1.1 in
the case of the round sphere. The proof of Theorem 1.3 is accomplished by
quantifying the following estimates when $J(1)-J(\rho)$ is small:

  1. (1)

For a large set of $r\in[\rho,1]$ and for $i=1,2$, the restriction of $u_{i}$
to ${\partial}B_{r}$ is close in an $L^{2}$ sense to the first Dirichlet
eigenfunction of the spherical Laplacian on the set $\\{u_{i}>0\\}\cap\partial
B_{r}$ .

  2. (2)

For $i=1,2,$ the function $u_{i}$ is close in an $L^{2}$ sense to a one-
homogeneous function in $B_{1}\setminus B_{\rho}$.

  3. (3)

For a large set of $r\in[\rho,1]$ and for $i=1,2$, the first Dirichlet
eigenvalue of the spherical Laplacian on the set $\\{u_{i}>0\\}\cap\partial
B_{r}$ is close to the first eigenvalue of a hemisphere in ${\partial}B_{r}$.

The one-homogeneous extension of the first eigenfunction of a hemisphere in
${\partial}B_{1}$ is a truncated linear function $(x\cdot\nu)^{+}$ for some
$\nu\in S^{n-1}$. Theorem 1.3 is established by combining the estimates above
to show that for $i=1,2$, the function $u_{i}$ is close to a constant multiple
$\beta_{i}>0$ of this linear function, for suitably chosen $\nu$.

In the case when $n=2$, the proof of Theorem 1.3 is significantly simpler than
in higher dimensions, fundamentally because the geometry of
$S^{1}(r)={\partial}B_{r}$ is sufficiently simple that much of the analysis
can be done in an explicit manner. In dimension $2$, the estimate (3) for the
the first eigenvalue of $\\{u_{i}>0\\}\cap\partial B_{r}$ immediately
determines the length of the longest connected curve of
$\\{u_{i}>0\\}\cap\partial B_{r}$ as well as the explicit first eigenfunction
on $\\{u_{i}>0\\}\cap\partial B_{r}$.

When $n>2$, on the other hand, the estimate (3) contains less immediate
information. Knowing the first eigenvalue on $\\{u_{i}>0\\}\cap\partial B_{r}$
does not determine the shape of $\\{u_{i}>0\\}\cap B_{r}$ nor the
eigenfunction on $\\{u_{i}>0\\}\cap B_{r}$. It is here that we make crucial
use of the sharp quantitive version of Sperner’s inequality established in
Theorem 1.1. Theorem 1.1 applied to $S^{n-1}={\partial}B_{1}$, in conjunction
with the estimate (3) allows us to deduce that the restriction of $u_{i}$ to
${\partial}B_{1}$ is quantitatively close to the first eigenfunction of a
spherical cap of the same volume. We emphasize that for this application, it
is essential that we have established the improved form of stability that
includes eigenfunction control in Theorem 1.1.

Let us now make some further remarks about Theorem 1.1 and its proof. We begin
with some comments about the statement and optimality of the theorem.

######  Remark 1.5 (Dependence of constants on the volume).

In Theorem 1.1, the constant $c$ in the quantitative estimates depend on the
volume. On Euclidean space, the scaling invariance can be used to state the
Faber-Krahn inequality without a volume constraint as
$\lambda_{1}(\Omega)|\Omega|^{2/n}\geqslant\lambda_{1}(B)|B|^{2/n}$ where
$B=B_{1}$, and similarly (1.4) can be restated in scale-invariant form as

$\lambda_{1}(\Omega)|\Omega|^{2/n}-\lambda_{1}(B)|B|^{2/n}\geqslant
c\left(\inf_{x\in{\varmathbb{R}}^{n}}\frac{|\Omega\Delta
B(x,r)|^{2}}{|\Omega|^{2}}+\inf_{x\in{\varmathbb{R}}^{n}}\int_{{\varmathbb{R}}^{n}}|u_{\Omega}-u_{B(x,r)}|^{2}\right)\,$

where $c$ is a constant depending on dimension only and $r$ is chosen so that
$|\Omega|=|B_{r}|$. On the sphere and hyperbolic space, the constant can
clearly be taken to be uniform for any compact subset of volumes. We make no
attempt to track the dependence of the constant $c=c(n,v)$ in (1.4) as $v\to
0$ or as $v\to|S^{n}|$ (in the case of the sphere) or $v\to\infty$ (in the
case of hyperbolic space), though as $v\to 0$ one expects that the constants
can be made uniform after the suitable rescaling.

######  Remark 1.6 (Optimality of the $L^{2}$ norm).

It is natural to wonder if the $L^{2}$ distance between eigenfunctions in
(1.4) can be strengthened to the $H^{1}$ distance. This is false, at least
with the quadratic power. Indeed, let $B$ be a ball of maximal overlap with
$\Omega$ and suppose by way of contradiction that there exists $c>0$ such that
$\lambda_{1}(\Omega)-\lambda_{1}(B)\geqslant
c\int|{\nabla}u_{\Omega}-{\nabla}u_{B}|^{2}$. On $B\setminus\Omega$, we have
${\nabla}u_{\Omega}=0$ and $|{\nabla}u_{B}|\approx 1.$ So,

$\int|{\nabla}u_{\Omega}-{\nabla}u_{B}|^{2}\geqslant\int_{B\setminus\Omega}|{\nabla}u_{B}|^{2}\geqslant
c\,|B\setminus\Omega|=\frac{c}{2}\,|B\Delta\Omega|.$

This would imply that the eigenvalue deficit
$\lambda_{1}(\Omega)-\lambda_{1}(B)$ controls the asymmetry linearly, which as
we have already discussed is false. It is not hard to show that qualitative
stability holds for the $H^{1}$ norm and we do not know if quantitative
stability may hold with a different modulus of continuity, e.g. for some
$\alpha>2$ in (1.3).

######  Remark 1.7 (Quotient spaces).

Naturally, the Polya-Szegö principle and thus the Faber-Krahn inequality hold
on quotients of Euclidean space, hyperbolic space, and the round sphere when
the volume of $\Omega$ is small. Theorem 1.1 also generalizes to quotient
spaces for the volume $v$ beneath a suitable threshold.

The proof of Theorem 1.1, like the proof of the quantitative Faber-Krahn
inequality of [18], is based on a selection principle argument. This global-
to-local reduction tool, introduced by Cicalese and Leonardi in [23] in the
context of the isoperimetric inequality, is perhaps the most robust method to
prove quantitative geometric inequalities and allows one to trade the task of
proving stability in the entire class of objects for the task of proving
stability among minimizers of some penalized functional, which typically enjoy
strong regularity properties.

Let us sketch the basic scheme of the selection principle argument. Suppose by
way of contradiction that the desired stability result (1.4) fails, and so
there exist a sequence of open bounded sets $\Omega_{j}$ of a fixed volume $v$
with $\lambda_{1}(\Omega_{j})\to\lambda_{1}(B)$ and yet

(1.9)
$\lambda_{1}(\Omega_{j})-\lambda_{1}(B)\leqslant\frac{\text{dist}(\Omega_{j},B)^{2}}{j},$

where we let $\text{dist}(\Omega,B)$ denote the distance on the right-hand
side of (1.4). Now, consider a functional of the form

(1.10)
$\mathcal{F}_{j}(\Omega)=\lambda_{1}(\Omega)+\tau\sqrt{c_{j}^{2}+(c_{j}-\text{dist}(\Omega,B)^{2})^{2}},$

where $c_{j}=\text{dist}(\Omega_{j},B)^{2}$ which is designed so that a
minimizer $\Omega_{j}^{\prime}$ of $\mathcal{F}_{j}$ among sets of a fixed
volume satisfies the same contradiction assumption (1.9). Theorem 1.1 will
then follow once we establish the following two main ingredients for a
selection principle:

  1. (1)

(Regularity) Minimizers $\Omega_{j}^{\prime}$ of the penalized functionals
$\mathcal{F}_{j}$ satisfy good $\varepsilon$-regularity properties. Combined
with the (easily shown) fact that the $\Omega_{j}^{\prime}$ converge to a ball
$B$ in a weak sense, this will guarantee that for $j$ sufficiently large, the
boundary of $\Omega_{j}^{\prime}$ is a small $C^{2,\alpha}$ perturbation of
the boundary of $B$.

  2. (2)

(Local stability) The quantitative stability estimate (1.4) holds for sets
whose boundaries are small $C^{2,\alpha}$ perturbations of the boundary of
$B$. This step is based on linear stability, i.e. strict positivity of the
second variation of $\lambda_{1}(B)$ in non-translational directions.

In the context of Theorem 1.1, significant challenges arise in both of these
two steps. Let us describe these challenges, and the new strategies introduced
to deal with them.

Step (1): For an energy like (1.10) in which $\lambda_{1}(\Omega)$ (or the
closely related torsional rigidity below) is the dominant term, the functional
$\mathcal{F}_{j}$ can be realized as a perturbation of the Alt-Caffarelli
functional [5]. However, in contrast to [18], the presence of $L^{2}$
eigenvalue penalization in the distance makes this perturbation critical and
leads to significant issues for regularity. Step (1) is carried out in our
companion paper [4], where we consider a selection principle associated to the
geometric inequality

$\lambda_{1}(\Omega)+\mathfrak{T}{\text{tor}}(\Omega)\geqslant\lambda_{1}(B)+\mathfrak{T}{\text{tor}}(B)$

for a small number $\mathfrak{T}>0$. Here ${\text{tor}}(\Omega)$ is the
(negative of the physical quantity) torsional rigidity of an open set
$\Omega\subset{\varmathbb{R}}^{n}$, defined and discussed further in Section
3.2. In Section 3.2, we will see how to connect this selection principle with
the one we need to prove Theorem 1.1, using the so-called Kohler-Jobin
inequality.

Step (2): In nearly all selection principle arguments, including the
quantitative Faber-Krahn inequality of [18] in Euclidean space, Step (2) can
be carried out using an explicit analysis of the second variation. On the
round sphere and hyperbolic space, these computations can no longer be carried
out explicitly. To overcome this challenge, we introduce a new technique,
based on the maximum principle, to perform an implicit spectral analysis. We
exploit the symmetry of the minimizers, which are balls, (thus indirectly
using their explicit form), and use an ODE maximum principle argument to
establish the desired spectral gap. To our knowledge, this is the first time
this method has been used in a quantitative stability estimate or in a
spectral analysis otherwise.

The remainder of the paper is organized in the following way. In Section 2 we
prove the quantitative estimates for the Alt-Caffarelli-Friedman monotonicity
formula of Theorem 1.3. We present the proof in the cases $n=2$ and
$n\geqslant 3$ separately because the two-dimensional case is simpler and
provides a basic outline of the proof for higher dimensions. Corollary 1.4 is
also proven in this section.

In Section 3, we prove Theorem 1.1. In the interest of clarity, we present the
proof in the specific case of the round sphere, which is also the essential
case in our application to Theorem 1.3. The proof carries over almost verbatim
to Euclidean space and hyperbolic space, and we therefore only remark on the
necessary modifications to the proof in these cases. As we have mentioned, the
regularity estimates of Step (1), as well as the details of the selection
prinicple, are shown in our companion paper [4]. Section 3 is thus largely
dedicated to proving Step (2) in Section 3.1, with the regularity results of
[4] recalled and the conclusion of Theorem 1.1 drawn in Section 3.2.

##  2\. Quantitative Stability for the Alt-Caffarelli-Friedman Monotonicity
Formula

This section is dedicated to the proof of Theorem 1.3 and Corollary 1.4. The
cases $n=2$ and $n\geqslant 3$ are carried out separately in Sections 2.1 and
2.2 respectively. Corollary 1.4 is shown in Section 2.3. We note that if
$\log(J(1)/J(\rho))\geqslant 1$, then Theorem 1.3 is true simply by choosing
$\beta_{i}=\|u\|_{L^{2}}(B_{1})$ and $\nu=e_{1}$ and choosing $C$ large enough
depending only on $n$. Therefore, throughout this section we assume that
$\log(J(1)/J(\rho))<1$. We begin by showing that it is sufficient to prove
Theorem 1.3 with the inequality

$\int_{B_{1}\setminus
B_{\rho}}\left(u_{1}-\beta_{1}(x\cdot\nu)^{+}\right)^{2}+\left(u_{2}-\beta_{2}(x\cdot\nu)^{-}\right)^{2}\leqslant
C\log\left(\frac{J(1)}{J(\rho)}\right)\sum_{i=1}^{2}\|u_{i}\|_{W^{1,2}(B_{1})}^{2}.$

Notice that in the above inequality the norm is $W^{1,2}$ rather than $L^{2}$.
This next Lemma will show how control the stronger $W^{1,2}$ norm with the
weaker $L^{2}$ norm.

######  Lemma 2.1.

Let $u_{1},u_{2}$ satisfy the assumptions of Theorem 1.3. There exists a
constant $C$ depending only on dimension $n$ such that

$\int_{B_{1}}|\nabla u_{i}|^{2}\leqslant
C\frac{J(1)}{J(1/2)}\int_{B_{1}}u_{i}^{2},$

for $i=1,2$.

###### Proof.

We note that multiplying either $u_{i}$ by a constant leaves $J(1)/J(1/2)$
invariant. Thus, we may assume without loss of generality that

$1=\int_{B_{1}}\frac{|\nabla u_{1}|^{2}}{|x|^{n-2}}=\int_{B_{1}}\frac{|\nabla
u_{2}|^{2}}{|x|^{n-2}}.$

Then for any $\rho\in[1/2,3/4]$ we have

$\displaystyle\int_{B_{1}}|\nabla u_{2}|^{2}$
$\displaystyle\leqslant\int_{B_{1}}\frac{|\nabla u_{2}|^{2}}{|x|^{n-2}}$
$\displaystyle=\left(\int_{B_{1}}\frac{|\nabla
u_{1}|^{2}}{|x|^{n-2}}\right)\left(\int_{B_{1}}\frac{|\nabla
u_{2}|^{2}}{|x|^{n-2}}\right)$ $\displaystyle\leqslant
2^{5}\left(\frac{J(1)}{J(1/2)}\right)\left(\int_{B_{1/2}}\frac{|\nabla
u_{1}|^{2}}{|x|^{n-2}}\right)\left(\int_{B_{1/2}}\frac{|\nabla
u_{2}|^{2}}{|x|^{n-2}}\right)$ $\displaystyle\leqslant
2^{5}\left(\frac{J(1)}{J(1/2)}\right)\left(\int_{B_{1}}\frac{|\nabla
u_{1}|^{2}}{|x|^{n-2}}\right)\left(\int_{B_{\rho}}\frac{|\nabla
u_{2}|^{2}}{|x|^{n-2}}\right)$
$\displaystyle=2^{5}\left(\frac{J(1)}{J(1/2)}\right)\left(\int_{B_{\rho}}\frac{|\nabla
u_{2}|^{2}}{|x|^{n-2}}\right)$

We now use the fact that $u_{2}\geqslant 0$, and $\Delta u_{2}\geqslant 0$ to
obtain (see [20]) that

$\int_{B_{\rho}}\frac{|\nabla u_{2}|^{2}}{|x|^{n-2}}\leqslant C\int_{\partial
B_{\rho}}|\nabla u_{2}|^{2}+u_{2}^{2}.$

Now there exists $\rho_{1}\in[1/2,3/4]$ such that

$\int_{\partial B_{\rho_{1}}}|\nabla u_{2}|^{2}\leqslant
4\int_{B_{3/4}}|\nabla u_{2}|^{2}.$

We now use that $u_{2}$ is subharmonic and the Caccioppoli inequality to
conclude that

$\int_{\partial B_{\rho_{1}}}|\nabla u_{2}|^{2}\leqslant
4\int_{B_{3/4}}|\nabla u_{2}|^{2}\leqslant C\int_{B_{1}}u_{2}^{2}.$

Using that $u_{2}$ is subharmonic we have that for any $\rho>\rho_{1}$ that

$\int_{\partial
B_{\rho_{1}}}u_{2}^{2}\leqslant\frac{\rho_{1}^{n-1}}{\rho^{n-1}}\int_{\partial
B_{\rho}}u_{2}^{2}\leqslant\int_{\partial B_{\rho}}u_{2}^{2}.$

Then

$\displaystyle\int_{\partial B_{\rho_{1}}}u_{2}^{2}$
$\displaystyle\leqslant\frac{1}{1-3/4}\int_{3/4}^{1}\int_{\partial
B_{\rho_{1}}}u_{2}^{2}\ d\rho\leqslant 4\int_{3/4}^{1}\int_{\partial
B_{\rho}}u_{2}^{2}\ d\rho$ $\displaystyle=4\int_{B_{1}\setminus
B_{3/4}}u_{2}^{2}\leqslant 4\int_{B_{1}}u_{2}^{2}.$

Thus we have shown that

$\int_{B_{1}}|\nabla u_{2}|^{2}\leqslant
C\left(\frac{J(1)}{J(1/2)}\right)\int_{B_{1}}u_{2}^{2}.$

The same argument applies to $u_{1}$. ∎

###  2.1. The two dimensional case

Throughout the section, we let $n=2$ and let
$u_{1},u_{2}:B_{1}\to{\varmathbb{R}}$ be two nonnegative functions satisfying
(1.6). Let $J=J[u_{1},u_{2}]:(0,1)\to{\varmathbb{R}}_{+}$ be the two
dimensional Alt-Caffarelli-Friedman monotonicity formula defined in (1.7).

To begin, we prove the monotonicity of the functional $J(r)$ following [6,
20], along the way introducing some notation and outlining the main ideas of
the proof of Theorem 1.3 when $n=2$. We directly compute

(2.1)
$\log(J(r))^{\prime}=\frac{\int_{{\partial}B_{r}}|{\nabla}u_{1}|^{2}}{\int_{B_{r}}|{\nabla}u_{1}|^{2}}+\frac{\int_{{\partial}B_{r}}|{\nabla}u_{2}|^{2}}{\int_{B_{r}}|{\nabla}u_{2}|^{2}}-\frac{4}{r}.$

For each fixed $r\in(0,1]$ and for $i=1,2$, we denote by
$\\{\lambda_{k,r}^{i}\\}$ and $\\{Y_{k,r}^{i}\\}$ the Dirichlet eigenvalues
and eigenfunctions of the Laplacian $-{\partial}_{\tau\tau}$ on the set
$\Omega_{r}^{i}=\\{u_{i}>0\\}\cap{\partial}B_{r}$, with the eigenfunctions
normalized so that $\|Y_{k,r}^{i}\|_{L^{2}(\Omega^{i}_{r})}=1$. Here and in
the sequel, we let ${\partial}_{\tau}$ denote the tangential derivative on
${\partial}B_{r},$ so that in polar coordinates $(r,\theta)$ for
${\varmathbb{R}}^{2}$, we have
${\partial}_{\tau}=\frac{1}{r}{\partial}_{\theta}$. The positivity set
$\Omega_{r}^{i}$ comprises a union of circular arcs in ${\partial}B_{r}$, and
so the the first eigenvalue and eigenfunction of this set are the (explicit)
first eigenvalue and eigenfunction of the longest circular arc. More
specifically, define $\theta_{i}(r)\in[0,2\pi]$ so that the length of the
longest circular arc in $\Omega_{r}^{i}$ is $r\theta_{i}(r)$. The first
eigenvalue and eigenfunction of $\Omega_{r}^{i}$ are given by (up to the
obvious rotation)

(2.2) $\lambda_{1,r}^{i}=\frac{\pi^{2}}{r^{2}\theta_{i}(r)^{2}},\qquad\qquad
Y_{1,r}^{i}=\sqrt{\frac{2}{r\theta_{i}(r)}}\,\sin\Big{(}r\sqrt{\lambda_{1,r}^{i}}\theta\Big{)}^{+}.$

Here we have written the eigenfunction in polar coordinates $(r,\theta)$ for
${\varmathbb{R}}^{2}$. It is worth noting that for a semicircle in
${\partial}B_{r},$ the first eigenfunction is $\lambda=1/r^{2}$ and the first
eigenfunction is $Y=\sqrt{2/\pi r}\sin(\theta)^{+},$ whose one-homogeneous
extension is a linear function restricted to a half plane. Observe that
$\theta_{1}(r)+\theta_{2}(r)\leqslant 2\pi$ because the supports of $u_{1}$
and $u_{2}$ are disjoint.

Now, we note that for $i=1,2$,

$\displaystyle\int_{\partial B_{r}}|\nabla u_{i}|^{2}=\int_{\partial
B_{r}}(\partial_{\tau}u_{i})^{2}+(\partial_{r}u_{i}^{2})$
$\displaystyle=\left(\int_{\partial
B_{r}}(\partial_{\tau}u_{i})^{2}-\lambda_{1,r}^{i}u_{i}^{2}\right)+\left(\int_{\partial
B_{r}}\lambda_{1,r}^{i}u_{i}^{2}-2\sqrt{\lambda_{1,r}^{i}}u_{i}\partial_{r}u_{i}\right)+\left(\int_{\partial
B_{r}}2\sqrt{\lambda_{1,r}^{i}}u_{i}\partial_{r}u_{i}\right).$

Using now the subharmonicity and nonnegativity of $u_{i}$ we have

$\int_{B_{r}}|\nabla u_{i}|^{2}\leqslant\int_{\partial
B_{r}}u_{i}{\partial}_{r}u_{i},$

so that dividing by $\int_{\partial B_{r}}|\nabla u_{i}|^{2}$, summing in $i$,
and subtracting $4/r$ we obtain from (2.1) and (2.2) that

(2.3)
$\log(J(r))^{\prime}\geqslant\delta_{A}(r)+\delta_{B}(r)+\delta_{C}(r)\,,$

where we define

$\displaystyle\delta_{A}(r)$ $\displaystyle=\sum_{i=1}^{2}\
\frac{1}{\int_{B_{r}}|{\nabla}u_{i}|^{2}}\left(\int_{{\partial}B_{r}}({\partial}_{\tau}u_{i})^{2}-\lambda_{1,r}^{i}\int_{{\partial}B_{r}}u_{i}^{2}\right)\,,$
$\displaystyle\delta_{B}(r)$ $\displaystyle=\sum_{i=1}^{2}\
\frac{1}{\int_{B_{r}}|{\nabla}u_{i}|^{2}}\left(\lambda_{1,r}^{i}\int_{{\partial}B_{r}}u_{i}^{2}+\int_{{\partial}B_{r}}({\partial}_{r}u_{i})^{2}-2\sqrt{\lambda_{1,r}^{i}}\int_{{\partial}B_{r}}u_{i}{\partial}_{r}u_{i}\right)\,,$
$\displaystyle\delta_{C}(r)$
$\displaystyle=\frac{2}{r}\left(\frac{\pi}{\theta_{1}(r)}+\frac{\pi}{\theta_{2}(r)}-2\right)\,.$

The ACF functional $J(r)$ is shown to be monotone in [6, 20] by showing the
nonnegativity of each of $\delta_{A}(r),\delta_{B}(r)$, and $\delta_{C}(r)$;
this is in particular a consequence of Lemma 2.2 below. The quantitative
monotonicity formula of Theorem 1.3 will be established by quantifying the
loss in each inequality $\delta_{(\cdot)}(r)\geqslant 0$ and integrating with
respect to $r$. Let us informally discuss the information contained in each
$\delta_{(\cdot)}(r)$.

The quantity $\delta_{A}(r)$ is nonnegative by the Poincaré inequality on
$\Omega_{r}^{i}$, and equals zero for a given $r\in(0,1]$ if and only if
$u_{i}$ restricted to ${\partial}B_{r}$ is equal to the first eigenfunction
$Y_{1,r}^{i}$ of $\Omega_{r}^{i}$. By quantifying the positivity of
$\delta_{A}(r)$ in (2.4) in Lemma 2.2 below, we see that $\delta_{A}(r)$
quantitatively controls the distance of $u_{i}$ to the eigenfunction on this
set.

The term in parentheses in the definition of $\delta_{B}(r)$ can be realized
as a square as in (2.5) in Lemma 2.2 below. This immediately shows that
$\delta_{B}(r)$ is nonnegative for all $r\in(0,1)$. Furthermore, it provides
quantitative control on how closely the restriction of $u_{i}$ to
${\partial}B_{r}$ behaves like a homogeneous function with homogeneity
$\sqrt{\lambda_{1,r}^{i}}/r$.

The quantity $\delta_{C}(r)$ can be bounded below by square functions; see
(2.6) in Lemma 2.2 below, which in particular shows that
$\delta_{C}(r)\geqslant 0$. The estimate (2.6) shows that the longest circular
arc in $\Omega_{r}^{i}$ (and thus $\Omega_{r}^{i}$ itself) is quantitatively
close to a semicircle in ${\partial}B_{r}$, and the corresponding first
eigenvalue $\lambda_{1,r}^{i}$ is quantitatively close to the first eigenvalue
$1/r^{2}$ of a semicircle in ${\partial}B_{r}$.

The proof of Theorem 1.3 roughly goes as follows. By integrating
$\delta_{B}(r)$ and $\delta_{C}(r)$ from $\rho$ to $1$, we show in Lemma 2.3
that each $u_{i}$ is quantitatively close to the one-homogeneous function
defined by $ru_{i}(1,\theta)$ in polar coordinates. Then, in Proposition 2.5,
we apply the estimate (2.4) for $\delta_{A}(1)$ to deduce that
$u_{i}(1,\theta)$ is almost equal to a multiple of first eigenfunction
$Y_{1,1}^{i}$ of $\Omega_{1}^{i}$, and applying the estimate (2.6) once again,
we see that $Y_{1,1}^{i}$ is quantitatively close to the first eigenfunction
of the semicircle. Finally, Theorem 1.3 follows from Proposition 2.5 once we
guarantee that the semicircles corresponding to $u_{1}$ and $u_{2}$ do not
have too much overlap.

The following lemma provides basic quantitative lower bounds for each of
$\delta_{A}(r),\delta_{B}(r)$, and $\delta_{C}(r)$ discussed above. Since each
of the right-hand sides of (2.4)-(2.6) are nonnegative, Lemmma 2.2 and (2.3)
in particular imply the monotonicity of $J(r)$.

######  Lemma 2.2.

For each $r\in(0,1]$ and $i=1,2$, let $Y_{1,r}^{i}$ be the eigenfunction
defined in (2.2) and let
$\beta_{1,r}^{i}=\int_{{\partial}B_{r}}u_{i}Y_{1,r}^{i}$. We have

(2.4) $\displaystyle\delta_{A}(r)$ $\displaystyle\geqslant\sum_{i=1}^{2}\
\frac{(\lambda_{2,r}^{i}-\lambda_{1,r}^{i})}{\int_{B_{r}}|{\nabla}u_{i}|^{2}}\int_{{\partial}B_{r}}\left(u_{i}-\beta_{1,r}^{i}Y_{1,r}^{i}\right)^{2},$
(2.5) $\displaystyle\delta_{B}(r)$ $\displaystyle=\sum_{i=1}^{2}\
\frac{1}{\int_{B_{r}}|{\nabla}u_{i}|^{2}}\int_{{\partial}B_{r}}\left(\sqrt{\lambda_{1,r}^{i}}u_{i}-{\partial}_{r}u_{i}\right)^{2},$
(2.6) $\displaystyle\delta_{C}(r)$
$\displaystyle\geqslant\sum_{i=1}^{2}\frac{2}{r}\frac{(\pi-\theta_{i}(r))^{2}}{\theta_{i}(r)(2\pi-\theta_{i}(r))}\geqslant\sum_{i=1}^{2}\frac{1}{\sqrt{\lambda_{1,r}^{i}}}\left(\sqrt{\lambda_{1,r}^{i}}-\frac{1}{r}\right)^{2}.$

###### Proof.

We fix $r\in(0,1]$, and for notational simplicity omit the subscripts $r$ in
the remainder of the proof. We first prove (2.4). For $i=1,2$, let
$\beta_{k}^{i}=\int_{{\partial}B_{r}}u_{i}Y_{k}^{i}$, so, expanding the term
in parentheses of $\delta_{A}(r)$ in this basis, we have

$\displaystyle\int_{{\partial}B_{r}}({\partial}_{\tau}u_{i})^{2}-\lambda_{1}^{i}\int_{{\partial}B_{r}}u_{i}^{2}$
$\displaystyle=\sum_{k=1}^{\infty}\lambda_{k}^{i}(\beta_{k}^{i})^{2}-\lambda_{1}^{i}\sum_{k=1}^{\infty}(\beta_{k}^{i})^{2}$
$\displaystyle\geqslant(\lambda_{2}^{i}-\lambda_{1}^{i})\sum_{k=2}^{\infty}(\beta_{k}^{i})^{2}=(\lambda_{2}^{i}-\lambda_{1}^{i})\int_{{\partial}B_{r}}(u_{i}-\beta_{1}^{i}Y_{1}^{i})^{2}.$

Dividing this expression by $\int_{B_{r}}|{\nabla}u_{i}|^{2}$ and summing from
$i=1$ to $2$ gives (2.4). Next, (2.5) is immediate by writing the integrand as
a square. Finally, to prove (2.6), recall that $\theta_{2}(r)\leqslant
2\pi-\theta_{1}(r)$, and so

$\displaystyle\delta_{C}(r)\geqslant\frac{2}{r}\left(\frac{\pi}{\theta_{1}(r)}+\frac{\pi}{2\pi-\theta_{1}(r)}-2\right)=\frac{2}{r}\left(\frac{2\pi^{2}}{\theta_{1}(r)(2\pi-\theta_{1}(r))}-2\right)=\frac{4}{r}\frac{(\pi-\theta_{1}(r))^{2}}{\theta_{1}(r)(2\pi-\theta_{1}(r))}.$

The analogous expression holds with $\theta_{2}$ in place of $\theta_{2}$.
Summing these expressions and dividing by $2$ gives the first estimate in
(2.6). For the second, use (2.2) to see

$\frac{4}{r}\frac{(\pi-\theta_{1}(r))^{2}}{\theta_{1}(r)(2\pi-\theta_{1}(r))}=\frac{1}{\sqrt{\lambda_{1}^{1}}}\frac{4\pi}{(2\pi-\theta_{1}(r))}\left(\sqrt{\lambda_{1}^{1}}-\frac{1}{r}\right)^{2}\geqslant\frac{2}{\sqrt{\lambda_{1}^{1}}}\left(\sqrt{\lambda_{1}^{1}}-\frac{1}{r}\right)^{2}.$

Again, the same expression holds with $\theta_{2}$ in place of $\theta_{2}$.
Summing these expressions and dividing by $2$ gives the second estimate in
(2.6). This concludes the proof of the lemma. ∎

The following lemma makes use of the estimates for $\delta_{B}(r)$ and
$\delta_{C}(r)$ in Lemma 2.2 and shows that $u_{1}$ and $u_{2}$ are
quantitatively close to being $1$-homogeneous.

######  Lemma 2.3.

Let $u_{1},u_{2}$ be as above. Then for a universal constant $C>0$ and any
$\rho\in(0,1)$,

$\int_{B_{1}\setminus B_{\rho}}[ru_{i}(1,\theta)-u_{i}(r,\theta)]^{2}\leqslant
C\log\left(\frac{J(1)}{J(\rho)}\right)\|u_{i}\|_{W^{1,2}(B_{1})}^{2}.$

###### Proof.

Fix $i\in\\{1,2\\}$. For notational simplicity, we omit the sub and
superscripts $i$ in this proof. We first note that

$\displaystyle\int_{B_{1}\setminus B_{\rho}}[ru(1,\theta)-u(r,\theta)]^{2}$
$\displaystyle\leqslant 2\int_{B_{1}\setminus
B_{\rho}}[ru(1,\theta)-A(r)u(1,\theta)]^{2}$
$\displaystyle\quad+2\int_{B_{1}\setminus
B_{\rho}}[A(r)u(1,\theta)-u(r,\theta)]^{2}=(I)+(II),$

where we define $A(r):=e^{-\int_{r}^{1}\sqrt{\lambda(t)}dt}$ and
$\lambda(t)=\lambda_{1,t}$. We bound $(II)$ in the following manner. Using
that

$u(1,\theta)A(r)-u(r,\theta)=\int_{r}^{1}[u_{r}(s,\theta)-\sqrt{\lambda(s)}u(s,\theta)]e^{-\int_{r}^{s}\sqrt{\lambda(t)}dt}\
ds$

as well as that $|e^{-\int_{r}^{s}\sqrt{\lambda(t)}dt}|\leqslant 1$ we have

(2.7)
$|u(1,\theta)A(r)-u(r,\theta)|^{2}\leqslant\int_{r}^{1}|u_{r}(s,\theta)-\sqrt{\lambda(s)}u(s,\theta)|^{2}\
ds.$

Now to continue with $(II)$ we have

$\displaystyle\int_{B_{1}\setminus B_{\rho}}[u(r,\theta)-A(r)u(1,\theta)]^{2}$
$\displaystyle=\int_{\rho}^{1}\int_{\partial
B_{1}}r[u(r,\theta)-A(r)u(1,\theta)]^{2}d\theta\ dr$
$\displaystyle\leqslant\int_{\rho}^{1}\int_{\partial
B_{1}}r\int_{r}^{1}|u_{r}(s,\theta)-\sqrt{\lambda(s)}u(s,\theta)|^{2}\ ds\
d\theta\ dr$ $\displaystyle\quad\text{ by }\eqref{e:sq1}$
$\displaystyle\leqslant\int_{\rho}^{1}\int_{\partial
B_{1}}\int_{r}^{1}s|u_{r}(s,\theta)-\sqrt{\lambda(s)}u(s,\theta)|^{2}\ ds\
d\theta\ dr$ $\displaystyle\quad\text{ since }r\leqslant s$
$\displaystyle\leqslant\int_{\rho}^{1}\int_{\partial
B_{1}}\int_{\rho}^{1}s|u_{r}(s,\theta)-\sqrt{\lambda(s)}u(s,\theta)|^{2}\ ds\
d\theta\ dr$ $\displaystyle\quad\text{ since }\rho\leqslant r$
$\displaystyle\leqslant\int_{\rho}^{1}\int_{\rho}^{1}\delta_{B}(s)\left(\int_{B_{s}}|\nabla
u|^{2}\right)\ ds\ dr$
$\displaystyle\leqslant\left(\int_{\rho}^{1}\delta_{B}(s)\
ds\right)\int_{B_{1}}|\nabla u|^{2}.$

In order to bound $(I)$ we first note that if $\sqrt{\lambda(s)}\geqslant
2/s$, then from (2.6) we have that $\sqrt{\lambda(s)}\leqslant
4\delta_{C}(s)$, so that $\max\\{\sqrt{\lambda(s)},2/s\\}\leqslant
4\delta_{C}(s)+2/s$. Now using that $e^{-x}$ is $1$-Lipschitz for $x\geqslant
0$ we have

$\displaystyle|A(r)-r|$
$\displaystyle=\left|e^{-\int_{r}^{1}\sqrt{\lambda(s)}\
ds}-e^{-\int_{r}^{1}1/s\ ds}\right|$
$\displaystyle\leqslant\left|\int_{r}^{1}\sqrt{\lambda(s)}-1/s\ ds\right|$
$\displaystyle\leqslant\left(\int_{r}^{1}\frac{(\sqrt{\lambda(s)}-1/s)^{2}}{\sqrt{\lambda(s)}}\
ds\right)^{1/2}\left(\int_{r}^{1}\sqrt{\lambda(s)}\ ds\right)^{1/2}$
$\displaystyle\leqslant\left(\int_{r}^{1}\delta_{C}(s)\
ds\right)^{1/2}\left(\int_{r}^{1}\max\\{\sqrt{\lambda(s)},2/s\\}\
ds\right)^{1/2}$ $\displaystyle\leqslant\left(\int_{r}^{1}\delta_{C}(s)\
ds\right)^{1/2}\left(\int_{r}^{1}4\delta_{C}(s)+2/s\ ds\right)^{1/2}$
$\displaystyle\leqslant\left(\int_{r}^{1}\delta_{C}(s)\
ds\right)^{1/2}\left(4\log(J(1)/J(r))-2\ln r\right)^{1/2}$
$\displaystyle\leqslant\left(\int_{r}^{1}\delta_{C}(s)\
ds\right)^{1/2}\left(4-2\ln r\right)^{1/2},$

so that

(2.8) $|A(r)-r|^{2}\leqslant\left(\int_{r}^{1}\delta_{C}(s)\right)(4-2\ln r).$

To continue with $(I)$, we use the trace inequality on $B_{1}$ and the fact
that $r\mapsto r(4-2\ln r)$ is bounded for $r\in(0,1]$ to find

$\displaystyle\int_{B_{1}\setminus
B_{\rho}}[ru(1,\theta)-A(r)u(1,\theta)]^{2}=\left(\int_{\partial
B_{1}}u^{2}(1,\theta)\ d\theta\right)\int_{\rho}^{1}r[r-A(r)]^{2}\ dr$
$\displaystyle\leqslant
C\|u\|_{W^{1,2}(B_{1})}^{2}\int_{\rho}^{1}r\left(\int_{r}^{1}\delta_{C}(s)\
ds\right)(4-2\ln r)\ dr\leqslant
C\|u\|_{W^{1,2}(B_{1})}^{2}\int_{\rho}^{1}\left(\int_{\rho}^{1}\delta_{C}(s)\
ds\right)\ dr$ $\displaystyle\leqslant
C\|u\|_{W^{1,2}(B_{1})}^{2}\int_{\rho}^{1}\delta_{C}(s)\ ds.$

Recalling (2.3) concludes the proof. ∎

This next lemma is a straightforward computation for sine functions that will
be used in the proofs of Proposition 2.5 and Theorem 1.3.

######  Lemma 2.4.

There exists a universal constant $C>0$ such that for any
$\hat{\theta}\in[0,2\pi)$ and any $\lambda>0$ with
$|1-\sqrt{\lambda}|\leqslant 1/2$,

(2.9)
$\displaystyle\int_{S^{1}}[\sin(\theta+\hat{\theta})^{+}-\sin(\theta)^{+}]^{2}$
$\displaystyle\leqslant C\hat{\theta}^{2}$ (2.10)
$\displaystyle\int_{S^{1}}[\sin(\sqrt{\lambda}\theta)^{+}-\sin(\theta)^{+}]^{2}$
$\displaystyle\leqslant C(1-\sqrt{\lambda})^{2}.$

###### Proof.

To estimate (2.9), from the bound $|\sin^{\prime}(\theta)|\leqslant 1$, one
has $(\sin(\theta+\hat{\theta})-\sin(\theta))^{2}\leqslant C\theta^{2}$. If
$\sin(\theta+\hat{\theta})>0$ and $\sin(\theta)\leqslant 0$, then
$(\sin(\theta+\hat{\theta}))^{2}\leqslant\hat{\theta}^{2}$. This is sufficient
to conclude (2.9).

To estimate (2.10) we break up the integral over $S^{1}$ into the regions
$(0,\pi)\cup(\pi,\bar{\theta})$ where we let
$\bar{\theta}=\pi/\sqrt{\lambda}$. If $\sqrt{\lambda}>1$, then it suffices to
only consider the interval $(0,\pi)$. We first have

$\displaystyle\int_{0}^{\pi}[\sin(\sqrt{\lambda}\theta)^{+}-\sin(\theta)^{+}]^{2}$
$\displaystyle\leqslant\int_{0}^{\pi}[\sin(\sqrt{\lambda}\theta)-\sin(\theta)]^{2}$
$\displaystyle=\pi-\frac{\sin((\sqrt{\lambda}-1)\pi)}{\sqrt{\lambda}-1}-\frac{\sin(2\sqrt{\lambda}\pi)}{4\sqrt{\lambda}}+\frac{\sin((\sqrt{\lambda}+1)\pi)}{\sqrt{\lambda}+1}$

which is a smooth function in $\sqrt{\lambda}$ and has a minimum of zero at
$\sqrt{\lambda}=1$, so there exists a universal constant $C$ such that

$\int_{0}^{\pi}[\sin(\sqrt{\lambda}\theta)^{+}-\sin(\theta)^{+}]^{2}\leqslant
C(1-\sqrt{\lambda})^{2}$

provided that $|\sqrt{\lambda}-1|\leqslant 1/2$. Furthermore, using a change
of variables and a Taylor expansion of $\sin^{2}(\cdot)$ at $\theta=0$, we
have

$\int_{\pi}^{\bar{\theta}}[\sin(\sqrt{\lambda}\theta)^{+}-\sin(\theta)^{+}]^{2}=\int_{\pi}^{\bar{\theta}}\sin^{2}(\sqrt{\lambda}\theta)=\int_{0}^{\bar{\theta}-\pi}\sin^{2}(\sqrt{\lambda}\theta)\leqslant
C(1-\sqrt{\lambda})^{3}$

for a universal constant $C$ when $|\sqrt{\lambda}-1|\leqslant 1/2$. We
conclude (2.10). ∎

The next proposition contains the core of the proof of Theorem 1.3 and shows
that $u_{1}$ and $u_{2}$ are each well-approximated by a linear function in a
quantitative sense. From Lemma 2.3, we know that $u_{1}$ and $u_{2}$ are each
well-approximated by one-homogeneous functions. In this proposition, we show
that these one-homogeneous functions can be well-approximated by linear
functions.

######  Proposition 2.5.

Assume $\log(J(1)/J(\rho))\leqslant 1$ with $\rho\in[0,1/2]$. For $i=1,2,$ let
$\hat{\beta}_{i}=\beta_{1,1}^{i}=\int_{{\partial}B_{1}}u_{i}Y_{1,1}^{i}$ and
let $\beta_{i}=\sqrt{2}\hat{\beta}_{i}/\sqrt{\pi}$. There exists
$\theta_{i}\in[0,2\pi]$ such that if we let $v_{i}$ be the linear function
defined in polar coordinates by $v_{i}=r\sin(\theta+\theta_{i})$, then

$\int_{B_{1}\setminus B_{\rho}}[u_{i}-\beta_{i}v_{i}^{+}]^{2}\leqslant
C\log\left(\frac{J(1)}{J(\rho)}\right)\|u_{i}\|_{W^{1,2}(B_{1})}^{2}.$

###### Proof.

Let

$\varepsilon=\log\left(\frac{J(1)}{J(\rho)}\right)=\int_{\rho}^{1}\log(J(r))^{\prime}\,dr\,.$

We note that it is sufficient to prove the result for $\varepsilon$ small.
Since the integrand is nonnegative, it follows from Chebyshev’s inequality
that $\log(J(r))^{\prime}\leqslant 10\varepsilon$ outside a set of measure
$\varepsilon/10$. In particular, $\log(J(r))^{\prime}\leqslant 10\varepsilon$
for some $r\in(1-\varepsilon/5,1]$. So, by scaling we may assume that

(2.11) $\log(J(1))^{\prime}\leqslant\epsilon.$

Applying the estimate (2.6) for $\delta_{C}(r)$ with $r=1$, we deduce that
there exists a universal constant $C$ such that

(2.12) $|\pi-\theta_{i}(1)|\leqslant C\sqrt{\epsilon}\quad\text{ and
}\quad|\sqrt{\lambda_{1,1}^{i}}-1|\leqslant
C\sqrt{\epsilon}.\quad\lambda_{2}-\lambda\geqslant c.$

Consequently, the other connected arcs have length less than
$C\sqrt{\epsilon}$, and it follows that
$\lambda_{2,1}^{i}=4\lambda_{1,1}^{i}$, so that

(2.13) $\lambda_{2,1}^{i}-\lambda_{1,1}^{i}\geqslant 1.$

Fix $i\in\\{1,2\\}$. For the remainder of the proof, we omit subscripts $i$
for notational simplicity. After a rotation, we assume that the longest arc on
which $u\cap{\partial}B_{1}$ is positive is $(0,\theta(1))$. After this
rotation, we let $v=r\sin(\theta)$. Here and in the remainder of the proof, we
let $\lambda=\lambda_{1,1},$ i.e. $\lambda$ is the first Dirichlet eigenvalue
of $\Omega_{1}$. Likewise, $\lambda_{2}$ denotes the second Dirichlet
eigenvalue of $\Omega_{1}$. We now apply the triangle inequality to split the
integral on the left-hand side into three pieces as follows:

(2.14) $\begin{split}\int_{B_{1}\setminus B_{\rho}}(u-\beta
v^{+})^{2}&\leqslant 4\int_{B_{1}\setminus B_{\rho}}(u-ru(1,\theta))^{2}\\\
&+4\int_{B_{1}\setminus B_{\rho}}(ru(1,\theta)-r\hat{\beta}Y_{1,1})^{2}\\\
&+4\int_{B_{1}\setminus B_{\rho}}(\hat{\beta}rY_{1,1}-\beta
r\sin(\theta)^{+})^{2}\,.\end{split}$

We apply Lemma 2.3 to the first term on the right-hand side of (2.14) to find
that for a universal constant $C$, we have

(2.15) $4\int_{B_{1}\setminus B_{\rho}}(u-ru(1,\theta))^{2}\leqslant
C\varepsilon\|u\|_{W^{1,2}(B_{1})}^{2}\,.$

Next, we use the estimate (2.4) for $\delta_{A}(r)$ with $r=1$ to bound the
second term on the right-hand side of (2.14). Indeed, by (2.4) and the the
uniform eigenvalue gap (2.12), we see that

(2.16) $\begin{split}4\int_{B_{1}\setminus
B_{\rho}}(ru(1,\theta)-\hat{\beta}rY_{1,1})^{2}&\leqslant
C\int_{{\partial}B_{1}}(u(1,\theta)-\hat{\beta}Y_{1,1})^{2}\\\ &\leqslant
C\log(J(1))^{\prime}\|\nabla u\|_{L^{2}(B_{1})}^{2}\leqslant
C\varepsilon\|\nabla u\|_{L^{2}(B_{1})}^{2}\,\end{split}$

for a universal constant $C$. In the final inequality we have made use of
(2.11).

Finally, the third term in (2.14) can be estimated using the eigenvalue
estimate in (2.12) and the estimate (2.10) from Lemma 2.4. Indeed, noting that
$\hat{\beta}\leqslant\|u\|_{L^{2}(\partial B_{1})}\leqslant
C\|u\|_{W^{1,2}(B_{1})}$, we conclude that

(2.17) $\begin{split}4\int_{B_{1}\setminus
B_{\rho}}(\hat{\beta}rY_{1,1}-&\beta
r\sin(\theta)^{+})^{2}\leqslant\hat{\beta}^{2}\int_{{\partial}B_{1}}\left[Y_{1,1}-\sqrt{\frac{2}{\pi}}\sin(\theta)^{+}\right]^{2}\\\
&\leqslant
C\hat{\beta}^{2}\int_{{\partial}B_{1}}\left[\sqrt{\frac{2}{\theta(1)}}\sin(\sqrt{\lambda}\theta)^{+}-\sqrt{\frac{2}{\pi}}\sin(\theta)^{+}\right]^{2}\\\
&\leqslant
C\hat{\beta}^{2}\int_{{\partial}B_{1}}\left[\sin(\sqrt{\lambda}\theta)^{+}-\sin(\theta)^{+}\right]^{2}\\\
&\quad+C\hat{\beta}^{2}\left(\frac{1}{\sqrt{\theta(1)}}-\frac{1}{\sqrt{\pi}}\right)^{2}\int_{{\partial}B_{1}}\left[\sin(\sqrt{\lambda}\theta)^{+}\right]^{2}\\\
&\leqslant
C\hat{\beta}^{2}(1-\sqrt{\lambda})^{2}+C\hat{\beta}^{2}(\pi-\theta(1))^{2}\leqslant
C\varepsilon\|u\|_{W^{1,2}(B_{1})}^{2}.\end{split}$

By combining (2.14), (2.15), (2.16) and (2.17), we conclude the proof of the
proposition. ∎

We are now ready to prove Theorem 1.3 in the case $n=2$.

###### Proof of Theorem 1.3 for $n=2$.

As noted in the beginning of the section, if $\log(J(1)/J(\rho))\geqslant 1$,
then we may choose $\beta_{i}=\|u\|_{L^{2}(B_{1})}$ and $\nu=e_{1}$, and the
result immediately follows for large enough $C$. If $\log(J(1)/J(\rho))<1$,
then we may choose for $i=1,2$, the constants $\beta_{1}^{i}$ and $\theta_{i}$
be as in Proposition 2.5. In order to conclude Theorem 1.3, the only issue is
that the functions $\sin(\theta+\theta_{1})^{+}$ and
$\sin(\theta+\theta_{2})^{+}$ may not have disjoint support. However, from
(2.12) we point out that the overlap may not exceed $C\sqrt{\epsilon}$.
Therefore, by Lemma 2.4, we may rotate both functions to have disjoint support
and thus obtain the result. ∎

###  2.2. The $n$-dimensional case

We now move to the proof of Theorem 1.3 in higher dimensions. Throughout this
section, let $n\geqslant 3$, let $u_{1},u_{2}$ be as in (1.6) and let
$J=J[u_{1},u_{2}]:(0,1]\to{\varmathbb{R}}$ be the $n$-dimensional Alt-
Caffarelli-Friedman monotonicity formula defined in (1.8). The proof of
Theorem 1.3 follows the same basic scheme as the proof in the two dimensional
setting, wherein we estimate the positivity of $\log(J(r))^{\prime}$ on each
sphere ${\partial}B_{r}$. Whereas in the two dimensional case the positivity
sets ${\partial}B_{r}\cap\\{u_{i}>0\\}$ were unions of circular arcs, in
higher dimensions the geometry of the positivity sets may be more complicated.
In particular, we can no longer hope to write the first eigenfunction and
eigenvalue in an explicit way. The key tool to overcome this difficulty is
Theorem 1.1 (specifically utilized in (2.33) in Proposition 2.11). In this
section only, we will use the notation ${\mathcal{B}}$ to denote an
$(n-1)$-dimensional spherical cap (i.e. geodesic ball in $S^{n-1}$), so as not
to be confused with the notation $B$ used for a ball in $n$-dimensional
Euclidean space.

To begin the $n$-dimensional setting, let us recall the proof from [6, 20]
that the ACF formula (1.8) is monotone. We directly compute

(2.18)
$\log(J(r))^{\prime}=\frac{r^{2-n}\int_{{\partial}B_{r}}|{\nabla}u_{1}|^{2}}{\int_{B_{r}}\frac{|{\nabla}u_{1}|^{2}}{|x|^{n-2}}}+\frac{r^{2-n}\int_{{\partial}B_{r}}|{\nabla}u_{2}|^{2}}{\int_{B_{r}}\frac{|{\nabla}u_{2}|^{2}}{|x|^{n-2}}}-\frac{4}{r}.$

For each $r\in(0,1)$ and $i=1,2$, set
$\Omega_{i}^{r}=\\{u_{i}>0\\}\cap{\partial}B_{r}$ . Let $\lambda_{1,r}^{i}$
and $\tilde{u}_{i,r}$ respectively denote the first Dirichlet eigenvalue and
eigenfunction $-\Delta_{\tau}$ on $\Omega_{i}^{r}$, where $\tilde{u}_{i,r}$ is
normalized so that $\|\tilde{u}_{i,r}\|_{L^{2}(\Omega_{i}^{r})}=1$. Here
$\Delta_{\tau}$ denotes the Laplacian of ${\partial}B_{r}$. For $n>2$, we may
no longer write $\lambda_{1,r}^{i}$ explicitly as we did in (2.2) in the two
dimensional case. For a hemisphere in ${\partial}B_{r}$, the first eigenvalue
is $\lambda_{1}=(n-1)/r^{2}$ with corresponding eigenfunction given by the
restriction a linear function on ${\varmathbb{R}}^{n}$ to this set. (By
symmmetry, one can see that the first eigenfunction of a hemisphere is the
restriction of the first spherical harmonic that vanishes on the boundary of
the hemisphere, which up to a rotation is the second spherical harmonic.)

An important quantity in this setting is the characteristic constant
$\alpha=\alpha_{i}(r)\in(0,\infty)$ of $\Omega_{i}^{r}$ defined by

(2.19) $\alpha_{i}(r)^{2}+(n-2)\alpha_{i}(r)-r^{2}\lambda_{1,r}^{i}=0,$

see [28]. The characteristic constant is defined in such a way so that the
$\alpha=\alpha_{i}(r)$ homogeneous extension $w(x)=r^{\alpha}\tilde{u}_{i,r}$
of $\tilde{u}_{i,r}$ is harmonic in the cone generated by $\Omega_{i}^{r}.$
Like the first Dirichlet eigenvalue, the characteristic constant is monotone
decreasing under set inclusion, and among sets of a fixed volume is minimized
by a spherical cap. A hemisphere in ${\partial}B_{r}$ has characteristic
constant $\alpha=1$.

Since $u_{i}$ is subharmonic and $u_{i}(0)=0$ we have [20] that

(2.20) $\int_{B_{r}}\frac{|{\nabla}u_{i}|^{2}}{|x|^{n-2}}\leqslant
r^{2-n}\int_{\partial
B_{r}}u_{i}(u_{i})_{r}+\frac{n-2}{2}\frac{u_{i}^{2}}{r}.$

From (2.19), we see that we can express $\lambda_{1,r}^{i}$ as
$\lambda_{1,r}^{i}=r^{-2}\alpha_{i}(r)^{2}+(n-2)r^{-2}\alpha_{i}(r)$. With
this in mind, we now write $|{\nabla}u_{i}|^{2}$ in the following way for any
point, where for a point $x\in{\partial}B_{r}$, we let ${\nabla}_{\tau}$
denote the projection of the gradient onto the tangent space of
${\partial}B_{r}$ at $x$.

$\displaystyle|{\nabla}u_{i}|^{2}=|{\nabla}_{\tau}u_{i}|^{2}+(\partial_{r}u_{i})^{2}$
$\displaystyle=|{\nabla}_{\tau}u_{i}|^{2}-\lambda_{1}^{i}u_{i}^{2}+\lambda_{1}^{i}u_{i}^{2}+(\partial_{r}u_{i})^{2}$
$\displaystyle=|{\nabla}_{\tau}u_{i}|^{2}-\lambda_{1}^{i}u^{2}+\frac{\alpha_{i}^{2}}{r^{2}}u_{i}^{2}+2\frac{\alpha_{i}}{r}\frac{n-2}{2r}u_{i}^{2}+(\partial_{r}u_{i})^{2}$
$\displaystyle=|{\nabla}_{\tau}u_{i}|^{2}-\lambda_{1}^{i}u^{2}$
$\displaystyle\quad+\frac{\alpha_{i}^{2}}{r^{2}}u_{i}^{2}-2\frac{\alpha_{i}u_{i}}{r}\partial_{r}u_{i}+(\partial_{r}u_{i})^{2}$
$\displaystyle\quad+2\frac{\alpha_{i}}{r}u_{i}\partial_{r}u_{i}+2\frac{\alpha_{i}}{r^{2}}\frac{n-2}{2}u_{i}^{2}.$

Using this expression in the numerator of (2.18), as well as the fact that
$|x|^{n-2}\leqslant r^{n-2}$ on $B_{r}$ and (2.20), we see that
$\log(J(r))^{\prime}\geqslant\delta_{A}(r)+\delta_{B}(r)+\delta_{C}(r)$ where

$\displaystyle\delta_{A}(r)$ $\displaystyle=\sum_{i=1}^{2}\
\frac{1}{\int_{B_{r}}|{\nabla}u_{i}|^{2}}\left(\int_{{\partial}B_{r}}({\nabla}_{\tau}u_{i})^{2}-\lambda_{1}^{i}\int_{{\partial}B_{r}}u_{i}^{2}\right)$
$\displaystyle\delta_{B}(r)$ $\displaystyle=\sum_{i=1}^{2}\
\frac{1}{\int_{B_{r}}|{\nabla}u_{i}|^{2}}\left(\int_{{\partial}B_{r}}\frac{\alpha_{i}(r)^{2}}{r^{2}}u_{i}^{2}+({\partial}_{r}u_{i})^{2}-2\frac{\alpha_{i}(r)}{r}u_{i}{\partial}_{r}u_{i}\right)$
$\displaystyle\delta_{C}(r)$
$\displaystyle=\frac{2}{r}\left(\alpha_{1}(r)+\alpha_{2}(r)-2\right)\,.$

In [6, 20], the ACF functional $J(r)$ is shown to be monotone by showing that
each of $\delta_{A}(r),\delta_{B}(r)$, and $\delta_{C}(r)$ is nonnegative,
which we will see is a consequence of Lemma 2.6 below. As in the two
dimensional case, the quantitative monotonicity formula of Theorem 1.3 will be
established by quantifying the loss in each inequality
$\delta_{(\cdot)}(r)\geqslant 0$ and integrating with respect to $r$. Let us
discuss the information captured by each $\delta_{(\cdot)}(r)$.

The quantity $\delta_{A}(r)$ is the direct analogue of $\delta_{A}(r)$ in the
two dimensional setting. It is nonnegative by the Poincaré inequality on
$\Omega_{i}^{r}$, and equals zero for a given $r\in(0,1]$ if and only if
$u_{i}$ restricted to ${\partial}B_{r}$ is a multiple of the first
eigenfunction $\tilde{u}_{i,r}$ of $\Omega_{i}^{r}$. As in the case $n=2$,
$\delta_{A}(r)$ quantitatively controls the distance of $u_{i}$ to the
eigenfunction on this set; see (2.21) in Lemma 2.6 below.

The term in parentheses in the definition of $\delta_{B}(r)$ can be realized
as a square as in (2.22) in Lemma 2.6 below, and so $\delta_{B}(r)$ is
nonnegative for all $r\in(0,1)$. It quantitatively controls how closely the
restriction of $u_{i}$ to ${\partial}B_{r}$ behaves like a homogeneous
function with homogeneity $\alpha_{i}(r)$.

The nonnegativity of $\delta_{C}(r)$ comes from the Friedland-Hayman
inequality [28]. In (2.23) of Lemma 2.6 below, we show that $\delta_{C}(r)$
provides quantitative control over how close the characteristic constant
$\alpha_{i}(r)$ of $\Omega_{i}^{r}$ is to the characteristic constant
$\alpha=1$ of a hemisphere in ${\partial}B_{r}$.

Let us outline the ingredients of the proof of Theorem 1.3 that make up the
rest of this section. By integrating $\delta_{B}(r)$ and $\delta_{C}(r)$ from
$\rho$ to $1$, we show in Lemma 2.7 that each $u_{i}$ is quantitatively close
to the one-homogeneous function defined by $ru_{i}(1,\theta)$ in polar
coordinates $(r,\theta)\in{\varmathbb{R}}_{+}\times S^{n-1}$ for
${\varmathbb{R}}^{n}$. Lemmas 2.8 and Lemma 2.9 provide quantitative and
qualitative information about the proximity of $\Omega_{i}^{1}$ to a spherical
cap when $\delta_{C}(1)$ is assumed to be small. Lemma 2.10 compares the
eigenfunctions of spherical caps of different radii. In Proposition 2.11, we
apply the estimate (2.21) for $\delta_{A}(1)$ to deduce that $u_{i}(1,\theta)$
is almost equal to a multiple of first eigenfunction $\tilde{u}_{1,1}^{i}$ of
$\Omega_{i}^{1}$, and applying Lemma 2.10 and Theorem 1.1, we see that
$\tilde{u}_{1,1}^{i}$ is quantitatively close to the first eigenfunction of a
hemisphere. Finally, Theorem 1.3 follows from Proposition 2.11 once we
guarantee that the hemispheres corresponding to $u_{1}$ and $u_{2}$ do not
have too much overlap.

The following lemma is the higher dimensional analogue of Lemma 2.2 and
contains the quantitative lower bounds for each of
$\delta_{A}(r),\delta_{B}(r)$, and $\delta_{C}(r)$ discussed above. Since each
of the right-hand sides of (2.21)-(2.23) are nonnegative, Lemmma 2.6 in
particular implies the monotonicity of $J(r)$.

######  Lemma 2.6.

For each $r\in(0,1]$ and $i=1,2$, let $\tilde{u}_{i,r}$ be the eigenfunction
of $\Omega_{i}^{r}$ as above and let
$\beta_{1,r}^{i}=\int_{{\partial}B_{r}}u_{i}\tilde{u}_{i,r}$. We have

(2.21) $\displaystyle\delta_{A}(r)$ $\displaystyle\geqslant\sum_{i=1}^{2}\
\frac{(\lambda_{2,r}^{i}-\lambda_{1,r}^{i})}{\int_{B_{r}}|{\nabla}u_{i}|^{2}}\int_{{\partial}B_{r}}\left(u_{i}-\beta_{1,r}^{i}\tilde{u}_{i,r}\right)^{2}$
(2.22) $\displaystyle\delta_{B}(r)$ $\displaystyle=\sum_{i=1}^{2}\
\frac{1}{\int_{B_{r}}|{\nabla}u_{i}|^{2}}\int_{{\partial}B_{r}}\left(\frac{\alpha_{i}(r)}{r}u_{i}-{\partial}_{r}u_{i}\right)^{2}$
(2.23) $\displaystyle\delta_{C}(r)$
$\displaystyle\geqslant\sum_{i=1}^{2}c(n)\frac{|\alpha_{i}(r)-1|^{2}}{r\max\\{\alpha_{i}(r),2\\}},$

with $c(n)$ a dimensional constant.

###### Proof.

The proof of (2.21) is exactly the same as that of (2.4) of Lemma 2.2, and as
in (2.5), we recognize the integrand in $\delta_{B}(r)$ above as a square to
see (2.22).

Observe that the inequality in (2.23) is straightforward when $\delta_{C}(r)$
is large. Therefore, it suffices to consider the case when $\delta_{C}(r)$ is
small. It also suffices to consider the case when $\Omega_{1}^{r}$ and
$\Omega_{2}^{r}$ are both spherical caps. Indeed, let $\bar{\alpha}_{1}$ and
$\bar{\alpha}_{2}$ denote the characteristic constants of disjoint spherical
caps of the same volume as $\Omega_{1}^{r}$ and $\Omega_{2}^{r}$ respectively.
Then by Sperner’s inequality (recall (2.19)), the corresponding
$\delta_{C}(r)$ can only decrease for by replacing $\Omega_{1}^{r}$ and
$\Omega_{2}^{r}$ by spherical caps, and also using the Friedland-Hayman
inequality, we see

(2.24)
$\delta_{C}(r)=\frac{1}{r}\left(|\alpha_{1}-\bar{\alpha}_{1}|+|\alpha_{2}-\bar{\alpha}_{2}|+\bar{\alpha}_{1}+\bar{\alpha}_{2}-2\right)\geqslant\frac{1}{r}\left(|\alpha_{1}-\bar{\alpha}_{1}|+|\alpha_{2}-\bar{\alpha}_{2}|\right).$

That is, the deficit $\delta_{C}(r)$ controls $|\alpha_{1}-\bar{\alpha}_{1}|$
linearly. Moreover, because the characteristic constant is monotone with
respect to set inclusion, we may assume without loss of generality that we
have two complementary caps.

So, define the function $\hat{\alpha}:(0,1)\to{\varmathbb{R}}$ by letting
$\hat{\alpha}(t)$ be the characteristic constant of the spherical cap of
radius $2\pi t$. Let
$\hat{\delta}(h)=\hat{\alpha}(1/2+h)+\hat{\alpha}(1/2-h)-2$, so that by the
Friedland-Hayman inequality we have $\hat{\delta}(h)\geqslant 0$.

We claim that $\hat{\delta}(h)\geqslant ch^{2}$ for a universal constant $c$.
It is known [20, 28, 33] that $\hat{\alpha}$ is convex with
$\hat{\alpha}^{\prime\prime}(1/2)=2c>0$, where $c$ is a universal constant,
and therefore there exists $h_{0}$ such that
$\hat{\alpha}^{\prime\prime}(1/2+h)\geqslant c$ for all $h$ with $|h|\leqslant
h_{0}$, so that $\hat{\alpha}$ is strictly convex in a neighborhood of $1/2$.
For the moment, we assume that $|h|\leqslant h_{0}$. So, a Taylor expansion
shows that

$\displaystyle\hat{\alpha}(1/2+h)$
$\displaystyle\geqslant\hat{\alpha}(1/2)+\hat{\alpha}^{\prime}(1/2)h+\frac{c}{2}h^{2}$

for any $h$ with $|h|\leqslant h_{0}$. Replacing $h$ by $-h$ and summing the
corresponding terms, we conclude that $\hat{\delta}(h)\geqslant ch^{2}$ for
all $|h|\leqslant h_{0}$. By the convexity of $\hat{\alpha}$, it follows that
$\hat{\delta}(h_{1})\geqslant\hat{\delta}(h_{0})$ for any $h_{1}\geqslant
h_{0}$. Therefore, by assuming that $\delta_{C}$ is small enough, we also have
necessarily that $|h|\leqslant|h_{0}|$.

Finally, since $\hat{\alpha}$ is a locally Lipschitz function, we have
$|\hat{\alpha}(1/2\pm h)-1|\leqslant C|h|$ from which it follows
$|\overline{\alpha}_{i}(r)-1|\leqslant h$. Then

$\displaystyle|\alpha_{i}(r)-1|^{2}$ $\displaystyle\leqslant
2|\alpha_{i}(r)-\overline{\alpha}_{i}(r)|^{2}+2|\overline{\alpha}_{i}-1|^{2}$
$\displaystyle\leqslant
2(\alpha_{1}+\alpha_{2}-2)^{2}+C(\alpha_{1}+\alpha_{2}-2)$
$\displaystyle\leqslant C(\alpha_{1}+\alpha_{2}-2),$

with the last inequality following since $\alpha_{1}+\alpha_{2}-2$ is assumed
to be small. By dividing by $r$ we arrive at (2.23). This concludes the proof.
∎

The following lemma is the higher dimensional analogue of Lemma 2.3. Using the
estimates for $\delta_{B}(r)$ and $\delta_{C}(r)$ in Lemma 2.6, this lemma
shows that $u_{1}$ and $u_{2}$ are quantitatively close to being
$1$-homogeneous.

######  Lemma 2.7.

Let $u_{1},u_{2}$ be as above. Then

$\int_{B_{1}\setminus B_{\rho}}[ru_{i}(1,\theta)-u_{i}(r,\theta)]^{2}\leqslant
C\log\left(\frac{J(1)}{J(\rho)}\right)\|u\|_{W^{1,2}(B_{1})}^{2}.$

###### Proof.

Fix $i\in\\{1,2\\}$. For notational simplicity, we omit the sub and
superscripts $i$ in this proof. The proof follows the proof of Lemma 2.3 using
Lemma 2.6 but with $A(r)$ now defined as
$A(r):=e^{-\int_{r}^{1}\frac{\alpha(t)}{t}\ dt}.$ ∎

In Lemma 2.6, we established a quantitative lower bound for $\delta_{C}(r)$ in
term of the characteristic constants of the regions $\Omega^{r}_{1}$ and
$\Omega_{2}^{r}$. In the following lemma, we prove some further quantitative
estimates for $\delta_{C}(1)$ under the assumption that $\delta_{C}(1)$ is
bounded above by one, which will be applied in the proof of Proposition 2.5.
Recall that we let ${\mathcal{B}}$ denote an $(n-1)$-dimensional spherical
cap.

######  Lemma 2.8.

Assume that $\delta_{C}(1)\leqslant 1$ and fix $i=1,2$. Let
$\bar{\mathcal{B}}_{i}\subset{\partial}B_{1}$ be a spherical cap such that
$|\bar{\mathcal{B}}_{i}|=|\Omega_{i}^{1}|$, and let $\bar{\lambda}^{i}$ denote
the first eigenvalue of $\bar{\mathcal{B}}_{i}$. There exists a constant
$C(n)$ such that

(2.25) $\displaystyle\delta_{C}(1)$ $\displaystyle\geqslant
C\left(\left|\lambda^{i}_{1,1}-\bar{\lambda}^{i}\right|+\left|\bar{\lambda}^{i}-(n-1)\right|^{2}\right)\,.$

###### Proof.

We fix $i$ and suppress the dependence on $i$ and $r=1$ in the remainder of
the proof. Let $\bar{\alpha}$ denote the characteristic constant of
$\bar{\mathcal{B}}$. Note that if $\delta_{C}(1)\leqslant 1$ then by
definition, $\alpha,\bar{\alpha}\leqslant 3$. We already showed in (2.24) in
the proof of Lemma 2.6 that $|\alpha-\bar{\alpha}|$ is linearly controlled by
$\delta_{C}(1)$. This combined with (2.19) shows the first estimate in (2.25):

(2.26)
$|\lambda-\bar{\lambda}|\leqslant(n-2)|\alpha-\bar{\alpha}|+|\alpha^{2}-\bar{\alpha}^{2}|\leqslant
C|\alpha-\bar{\alpha}|\leqslant C\delta_{C}(1).$

Similarly, it follows from the proof of Lemma 2.6 that $|\bar{\alpha}-1|$ is
quadradically controlled by $\delta_{C}(1)$, and using (2.19) in the same way,
we conclude the second estimate in (2.25) holds. ∎

The next lemma shows that for $\delta_{C}(1)$ sufficiently small, the sets
$\Omega_{1}^{1}$ and $\Omega_{2}^{1}$ enjoy a uniform spectral gap.

######  Lemma 2.9.

Fix $\eta>0$. There exists $\varepsilon_{0}=\varepsilon_{0}(\eta,n)$ such that
if $\delta_{C}(1)\leqslant\varepsilon_{0}$, then
$\lambda_{2}(\Omega_{i}^{1})-\lambda_{1}(\Omega_{i}^{1})\geqslant\eta$ for
$i=1,2$.

###### Proof.

We argue by contradiction. Suppose we may find a sequence of functions
$(u_{1,k},u_{2,k})$ satisfying (1.6) such that $\delta_{C,k}(1)\to 0$ and
$\Omega_{k}:=\\{u_{1,k}>0\\}\cap{\partial}B_{1}$ has
$|\lambda_{2}(\Omega_{k})-\lambda_{1}(\Omega_{k})|\to 0$.

By Lemma 2.8, we see that $|\lambda_{1}(\Omega_{k})-(n-1)|\to 0$ and
$|\lambda_{1}({\mathcal{B}}_{k})-(n-1)|\to 0$, where ${\mathcal{B}}_{k}$ is a
spherical cap with $|{\mathcal{B}}_{k}|=|\Omega_{k}|$. The first fact shows
that $|\lambda_{1}(\Omega_{k})-(n-1)|\to 0$, and the second ensures that
$|\Omega_{k}|\to|{\mathcal{B}}_{+}|$, where ${\mathcal{B}}_{+}\subset S^{n-1}$
denotes a hemisphere.

Now, for each $k,$ let $\tilde{u}_{1,k}$ and $\tilde{u}_{2,k}$ respectively
denote first and second eigenfunctions of $\Omega_{k}$, normalized to have
$L^{2}$ norm equal to one and extended by zero to be defined on the entire
sphere. So, there exist $\tilde{u}_{1,\infty},\tilde{u}_{2,\infty}\in
W^{1,2}(S^{n-1})$ such that after passing to a subsequence,

$\displaystyle\tilde{u}_{1,k}\to\tilde{u}_{1,\infty}$ $\displaystyle\ \text{
weakly in }W^{1,2}(S^{n-1}),\ \text{strongly in }L^{2}(S^{n-1}),\
\text{pointwise a.e.},$ $\displaystyle\tilde{u}_{2,k}\to\tilde{u}_{2,\infty}$
$\displaystyle\ \text{ weakly in }W^{1,2}(S^{n-1}),\ \text{strongly in
}L^{2}(S^{n-1}),\ \text{pointwise a.e.},$

and $\int\tilde{u}_{1,\infty}\cdot\tilde{u}_{2,\infty}=0$.

Let $\Omega=\\{\tilde{u}_{1,\infty}>0\\}$. By lower semicontinuity of the
norm, $\lambda_{1}(\Omega)\leqslant n-1$. By Fatou’s lemma,
$|\Omega|\leqslant\liminf|\\{\tilde{u}_{1,k}>0\\}|=\lim|\Omega_{k}|=|{\mathcal{B}}_{+}|$.
From Sperner’s inequality and the monotonicity of $\lambda_{1}$ with respect
to set inclusion, we see that up to a translation, $\Omega={\mathcal{B}}_{+}$
and $\tilde{u}_{1,\infty}$ is the first eigenfunction of ${\mathcal{B}}_{+}$.

Now, let $\Omega_{2}=\\{\tilde{u}_{2,\infty}>0\\}$. The same argument and the
uniqueness of the first eigenfunction show that
$\tilde{u}_{1,\infty}=\tilde{u}_{2,\infty}$, contradicting their
orthogonality. This completes the proof. ∎

######  Lemma 2.10.

Fix a compact interval $[r_{1},r_{2}]\subset(0,\pi)$, let $r_{1}\leqslant
r,\bar{r}\leqslant r_{2}$, and let
${\mathcal{B}},\bar{\mathcal{B}}\subset{\partial}B_{1}$ denote spherical caps
of radius $r,\bar{r}$ centered at the north pole. Let $\lambda$ and
$\bar{\lambda}$ denote the first eigenvalue of ${\mathcal{B}}$ and
$\bar{\mathcal{B}}$ respectively, and let $v$ and $\bar{v}$ denote the
corresponding first eigenfunctions extended by zero to be defined on the whole
sphere and normalized so that
$\|v\|_{L^{2}(S^{n-1})}=\|\bar{v}\|_{L^{2}(S^{n-1})}=1$. There exist constants
$C=C(r_{1},r_{2})$ and $\bar{c}=\bar{c}(r,\bar{r})$ such that

$\int_{S^{n-1}}|\bar{c}v-\bar{v}|^{2}\leqslant C(\lambda-\bar{\lambda})^{2}.$

###### Proof.

The functions $v$ and $\bar{v}$ are radially symmetric, and so with a slight
abuse of notation we let $v$, $\bar{v}$ denote the radial part of the
functions. These functions can be extended to $(r,\pi)$, $(\bar{r},\pi)$ to
satisfy

$\displaystyle
v^{\prime\prime}(\phi)+(n-2)\frac{\cos\phi}{\sin\phi}v^{\prime}(\phi)$
$\displaystyle=-\lambda v(\phi)\text{ for }\phi\in(0,\pi),$
$\displaystyle\bar{v}^{\prime\prime}(\phi)+(n-2)\frac{\cos\phi}{\sin\phi}\bar{v}^{\prime}(\phi)$
$\displaystyle=-\bar{\lambda}\bar{v}(\phi)\text{ for }\phi\in(0,\pi),$

with $v(r)=\bar{v}(\bar{r})=0$, $v^{\prime}(0)=\bar{v}^{\prime}(0)=0$ and
$v,\overline{v}\geqslant 0$ on $(0,r)$ and $(0,\overline{r})$ respectively.
Setting $f_{\lambda}(t)=\bar{c}\,v(\cos^{-1}(t))$, where $\bar{c}$ is a
constant to be chosen later in the proof, we see that $f_{\lambda}$ satisfies

$(1-t^{2})f_{\lambda}^{\prime\prime}(t)-(n-1)tf_{\lambda}^{\prime}(t)+\lambda
f_{\lambda}(t)=0,$

and likewise for $f_{\bar{\lambda}}(t)=\bar{v}(\cos^{-1}(t))$. The solutions
to the above equation are Legendre functions and are well known. Expanding
$\displaystyle v=\sum_{k=0}^{\infty}a_{k}(\lambda)(1-t)^{k}$ as a series
solution about $t=1$ we obtain the recursion formula

(2.27)
$a_{k+1}(\lambda)=a_{k}(\lambda)\frac{k^{2}+(n-2)k-\lambda}{(k+1)(2k+n-1)}.$

The series converges absolutely on the interval $(-1,3)$. We normalize by
choosing the zero order coefficient $\bar{a}_{0}$ for $f_{\bar{\lambda}}$ to
satisfy the constraint $\|\bar{v}\|_{L^{2}(S^{n-1})}=1$ and by choosing the
zero order coefficient $a_{0}$ of $f_{\lambda}$ so that $a_{0}=\bar{a}_{0}$,
in this way determining the constant $\bar{c}$ in the statement of the lemma.
Then

(2.28)
$a_{k+1}(\lambda)=\prod_{j=0}^{k}\frac{j^{2}+(n-2)j-\lambda}{(j+1)(2j+n-1)}.$

Taking the derivative of $\log a_{k+1}$ with respect to $\lambda$ we obtain

$|a_{k+1}^{\prime}(\lambda)|=|a_{k+1}(\lambda)|\sum_{j=0}^{k}\frac{1}{(j+1)(2j+n-1)}\leqslant
M|a_{k+1}(\lambda)|.$

Furthermore, from (2.27), we see that for any compact interval
$I\in(0,\infty)$, there is a constant $C=C(I)$ such that
$|a_{k+1}(\lambda)|\leqslant C/2^{k}$ for all $\lambda\in I$. Therefore,
$|a_{k+1}(\lambda)-a_{k+1}(\overline{\lambda})|\leqslant
C|\lambda-\overline{\lambda}|/2^{k}$ for any $\lambda,\bar{\lambda}\in I$.
Then summing over $k$ we obtain for any $t\in[0,t_{0}]$ with $t_{0}<\pi$ that

$|f_{\lambda}(t)-f_{\overline{\lambda}}(t)|\leqslant|\lambda-\overline{\lambda}|\sum_{k=0}^{\infty}\frac{C}{2^{k}}<C|\lambda-\overline{\lambda}|.$

The conclusion then follows after changing variables back to $\phi$ and
integrating over ${\partial}B_{1}$. ∎

The next proposition is the higher dimensional analogue of Proposition 2.5. It
shows that each of the $u_{i}$, $i=1,2$, is well approximated in $L^{2}$ by a
linear function.

######  Proposition 2.11.

There exist $\varepsilon_{0}=\varepsilon_{0}(n)$ and $C=C(n)$ such that the
following holds. Assume
$\log\left(\frac{J(1)}{J(\rho)}\right)\leqslant\varepsilon_{0}$ and
$0\leqslant\rho\leqslant 1/2$. Let $v$ be the first eigenfunction of the
spherical cap of radius $\pi/2$ in ${\partial}B_{1}$, extended by $0$ to be
defined on ${\partial}B_{1}$. For $i=1,2$ there exist
$\beta_{i}=\beta_{i}(u_{i})$ such that, up to a rotation, we have

$\int_{B_{1}\setminus B_{\rho}}\left(r\beta_{i}v-u_{i}\right)^{2}\leqslant
C\log\left(\frac{J(1)}{J(\rho)}\right)\|u_{i}\|_{W^{1,2}(B_{1})}^{2}.$

###### Proof.

Let $\varepsilon=\log(J(1)/J(\rho))$. Since
$\varepsilon=\int_{\rho}^{1}\log(J(r))^{\prime}\,dr$ and the integrand is
positive, it follows that $\log(J(r))^{\prime}\leqslant 10\varepsilon$ outside
a set of measure $\varepsilon/10$. In particular,
$\log(J(r))^{\prime}\leqslant 10\varepsilon$ for some
$r\in(1-\varepsilon/5,1]$. By scaling we assume that
$\log(J(r))^{\prime}\leqslant\varepsilon$ at $r=1$.

Fix $i\in\\{1,2\\}$. For the remainder of the proof, we omit subscripts $i$
for notational simplicity. Let $\Omega={\partial}B_{1}\cap\\{u>0\\}$ and let
$\tilde{u}$ denote the first eigenfunction of $\Omega$, normalized so that
$\|\tilde{u}\|_{L^{2}(\Omega)}=1$ and extended by zero to be defined
everywhere on ${\partial}B_{1}.$ Let $\bar{\mathcal{B}}\subset{\partial}B_{1}$
denote a spherical cap centered at the north pole such that
$|\bar{\mathcal{B}}|=|\Omega|$, and let $\bar{v}$ denote its first
eigenfunction with the normalization
$\|\bar{v}\|_{L^{2}(\bar{\mathcal{B}})}=1$ and extended by zero to be defined
on the whole ${\partial}B_{1}.$ Up to a rotation, we may assume that the
infimum over $x\in S^{n-1}$ in (1.4) of Theorem 1.1 is achieved at the north
pole. Lemmas 2.8 and 2.9 guarantee that for dimensional constants $c$ and $C$,

(2.29) $\left|\lambda_{1}-\lambda(\bar{\mathcal{B}})\right|\leqslant
C\varepsilon,\qquad\left|\lambda_{1}(\bar{\mathcal{B}})-(n-1)\right|\leqslant
C\sqrt{\varepsilon},\qquad\lambda_{2}-\lambda_{1}\geqslant c,$

Let $\hat{\beta}=\beta_{1}=\int_{{\partial}B_{1}}u\tilde{u}$ be as in (2.21),
and let $\bar{c}$ be as in Lemma 2.10 applied to $r=\pi/2$ and the radius
$\bar{r}$ of $\bar{\mathcal{B}}$. Let $\beta=\bar{c}\hat{\beta}$. We use the
triangle inequality to see that

(2.30) $\begin{split}\int_{B_{1}\setminus B_{\rho}}(u-\beta rv)^{2}&\leqslant
4\int_{B_{1}\setminus B_{\rho}}(u-ru(1,\theta))^{2}+4\int_{B_{1}\setminus
B_{\rho}}(ru(1,\theta)-\hat{\beta}r\tilde{u})^{2}\\\
&+4\hat{\beta}^{2}\int_{B_{1}\setminus
B_{\rho}}(r\tilde{u}-r\bar{v})^{2}+4\hat{\beta}^{2}\int_{B_{1}\setminus
B_{\rho}}(r\bar{v}-r\bar{c}v)^{2}.\end{split}$

We can estimate the first two terms on the right-hand side of (2.30) precisely
the way we estimated their two-dimensional analogues in the proof of
Proposition 2.5. Indeed, by Lemma 2.7, for the first term we have

(2.31) $4\int_{B_{1}\setminus B_{\rho}}(u-ru(1,\theta))^{2}\leqslant
C\varepsilon\|u\|_{W^{1,2}(B_{1})}^{2}.$

For the second term on the right-hand side of (2.30), we use the estimate
(2.21) for $\delta_{A}(1)\leqslant\varepsilon$ and the uniform eigenvalue gap
of (2.29) to see that for a dimensional constant $C$ we have

(2.32) $4\int_{B_{1}\setminus
B_{\rho}}(ru(1,\theta)-\hat{\beta}r\tilde{u})^{2}\leqslant
C\int_{{\partial}B_{1}}(u(1,\theta)-\hat{\beta}\tilde{u})^{2}\leqslant
C\varepsilon\|u\|_{W^{1,2}(B_{1})}^{2}\,.$

To estimate the third term on the right-hand side of (2.30), we call upon the
quantitative form of Sperner’s inequality established in Theorem 1.1:

(2.33) $\begin{split}4\int_{B_{1}\setminus
B_{\rho}}r^{2}(\tilde{u}-\bar{v})^{2}&\leqslant
C\hat{\beta}^{2}\int_{{\partial}B_{1}}(\tilde{u}-\bar{v})^{2}\leqslant
C\hat{\beta}^{2}\left(\lambda_{1}-\lambda_{1}(\bar{\mathcal{B}})\right)\leqslant
C\varepsilon\|u\|_{W^{1,2}(B_{1})}^{2},\end{split}$

where in the final inequality we recall (2.29) and that
$\hat{\beta}\leqslant\|u\|_{L^{2}({\partial}B_{1})}\leqslant
C\|u\|_{W^{1,2}(B_{1})}$.

Finally, we estimate the fourth term in (2.30). Applying Lemma 2.10 with
$r=\pi/2$ and $\bar{r}$ the radius of $\bar{\mathcal{B}}$ followed by (2.29),
we see that

(2.34) $\begin{split}4\hat{\beta}^{2}\int_{B_{1}\setminus
B_{\rho}}(r\bar{v}-r\bar{v})^{2}&\leqslant
C\hat{\beta}^{2}\int_{{\partial}B_{1}}(\bar{v}-cv)^{2}\\\ &\leqslant
C\hat{\beta}^{2}\left(\lambda_{1}(\bar{\mathcal{B}})-(n-1)\right)^{2}\leqslant
C\varepsilon\|u\|_{W^{1,2}(B_{1})}^{2}.\end{split}$

Combining (2.31)–(2.34), we conclude the proof of the proposition. ∎

We now prove Theorem 1.3 in the case $n\geqslant 3$.

###### Proof of Theorem 1.3 for $n\geqslant 3$.

Let $\epsilon_{0}$ be as in Proposition 2.11. If
$\log(J(1)/J(\rho)>\epsilon_{0}$, then we simply choose
$\beta_{i}=\|u_{i}\|_{L^{2}(B_{1})}$ and $\nu=e_{1}$, and the result follows
trivially for large enough $C$. Let now $\varepsilon=\log(J(1)/J(\rho)$ and
assume $\epsilon\leqslant\epsilon_{0}$. As in the proof of the two-dimensional
case, Theorem 1.3 almost follows from Proposition 2.11, but it remains to show
that the truncated linear functions obtained for each of $u_{1},u_{2}$ in
Proposition 2.11 can be taken to have disjoint supports. We also repeat the
following notation from Proposition 2.11, now emphasizing the dependence on
$i=1,2$: let $\Omega_{i}=\\{u_{i}>0\\}\cap{\partial}B_{1},$ let
$\bar{\mathcal{B}}_{i}\subset{\partial}B_{1}$ be the spherical cap of the same
volume as $\Omega^{i}$ with center achieving the infimum over $x\in S^{n-1}$
in (1.4) of Theorem 1.1, and ${\mathcal{B}}_{i}$ the hemisphere with the same
center. We first show that

(2.35) $|{\mathcal{B}}_{1}\Delta{\mathcal{B}}_{2}|\leqslant
C\varepsilon^{1/2}.$

Indeed, we repeatedly apply the triangle inequality to find

$|{\mathcal{B}}_{1}\Delta{\mathcal{B}}_{2}|\leqslant\left(\left|{\mathcal{B}}_{1}\Delta\bar{\mathcal{B}}_{1}\right|+\left|{\mathcal{B}}_{2}\Delta\bar{\mathcal{B}}_{2}\right|\right)+\left(\left|\bar{\mathcal{B}}_{1}\Delta\Omega_{1}\right|+\left|\bar{\mathcal{B}}_{2}\Delta\Omega_{2}\right|\right)+\left|\Omega_{1}\Delta\Omega_{2}\right|.$

Since the supports of $u_{1}$ and $u_{2}$ are disjoint,
$|\Omega_{1}\Delta\Omega_{2}|=0$. Next, we can apply Theorem 1.1 to
$\Omega_{1}$ and $\Omega_{2}$ and use (2.29) to see that

$\left|\bar{{\mathcal{B}}}_{1}\Delta\Omega_{1}\right|+\left|\bar{{\mathcal{B}}}_{2}\Delta\Omega_{2}\right|\leqslant
C\left(\lambda_{1}\left(\Omega_{1}\right)-\lambda_{1}(\bar{\mathcal{B}}_{2})\right)^{1/2}+\left(\lambda_{1}(\Omega_{2})-\lambda_{1}(\bar{\mathcal{B}}_{2})\right)^{1/2}\leqslant
C\varepsilon^{1/2}.$

Next, using (2.29), we see that

$\displaystyle\left|{\mathcal{B}}_{1}\Delta\bar{\mathcal{B}}_{1}\right|+\left|{\mathcal{B}}_{2}\Delta\bar{\mathcal{B}}_{2}\right|$
$\displaystyle\leqslant
C\left(\left|r_{1}-\frac{\pi}{2}\right|-\left|r_{2}-\frac{\pi}{2}\right|\right)$
$\displaystyle\leqslant
C\left(\left|\lambda_{1}(\bar{\mathcal{B}}_{1})-(n-1)\right|+\left|\lambda_{1}(\bar{{\mathcal{B}}}_{2})-(n-1)\right|\right)\leqslant
C\varepsilon^{1/2}.$

Together these estimates prove the claim (2.35).

Without loss of generality we may assume that ${\mathcal{B}}_{2}$ is centered
at the south pole. Let $x_{0}$ be the center of ${\mathcal{B}}_{1}$; from
(2.35) it follows that $d(x_{0},o)\leqslant C\varepsilon^{1/2}$ where $o$ is
the north pole and $d(\cdot,\cdot)$ denotes the distance on the sphere. Let
$w_{1}$ be the truncated linear function corresponding to $u_{1}$ provided by
Proposition 2.11 (so $w$ is defined in polar coordinates by $w(r,\theta)=\beta
rv(\theta)$) and let $\tilde{w}_{1}$ be its rotation so that
$\\{\tilde{w}_{1}>0\\}\cap{\partial}B_{1}$ is the hemisphere centered at the
north pole. Since $|w_{1}|\leqslant C\|u_{1}\|_{W^{1,2}(B_{1})}$, we see that

$\displaystyle\int_{B_{1}\setminus B\rho}|\tilde{w}_{1}-u_{1}|^{2}$
$\displaystyle\leqslant
C\varepsilon\|u\|_{W^{1,2}(B_{1})}^{2}+\int_{B_{1}\setminus
B\rho}|\tilde{w}_{1}-w_{1}|^{2}$ $\displaystyle\leqslant
C\varepsilon\|u\|_{W^{1,2}(B_{1})}^{2}+C\|u\|_{W^{1,2}(B_{1})}^{2}d(x_{0},o)^{2}\leqslant
C\|u\|_{W^{1,2}(B_{1})}^{2}\varepsilon.$

So, choosing $\tilde{w}_{1}$ and the truncated linear function $w_{2}$
corresponding to $u_{2}$ provided by Proposition 2.11 completes the proof of
the theorem. ∎

###  2.3. Uniqueness of blow-ups when the ACF formula decays fast

Here we sketch the proof of Corollary 1.4. The argument gives a precise
modulus for the convergence of the blow-ups as well.

###### Proof of Corollary 1.4..

We omit the subscripts $1,2$ in $u_{1},u_{2}$ below to simplify notation; any
statement involving $u$ or $\beta$ is valid for both $u=u_{1}$ and $u=u_{2}$,
up to replacing $(\nu\cdot x)^{+}$ by $(\nu\cdot x)^{-}$ if $i=2$.

For $k=0,1,2,\dots,$ consider the rescalings
$u^{k}(x)=\frac{u(2^{-k}x)}{2^{-k}}$. Recall that $\omega(r)=J(r)-J(0^{+})$.
We will show, by induction on $k$, that there are numbers $\beta^{k}>0$ and
unit vectors $\nu_{k}\in S^{n-1}$ such that

(2.36) $\int_{B_{1}\setminus B}\left|u^{k}-\beta^{k}(\nu_{k}\cdot
x)^{+}\right|^{2}\leqslant C_{1}\,\omega(2^{-k})$

and

(2.37)
$\left|\beta^{k}-\beta^{k-1}\right|^{2}+\left|\nu_{k}-\nu_{k-1}\right|^{2}\leqslant
C_{2}\,\omega(2^{-k-1}),$

where the constants $C_{1}$ and $C_{2}$ depend on $n$ and
$\|u_{i}\|_{W^{1,2}(B_{1})}$ but are independent of $k$. For each $k$, the
vector $\nu_{k}$ for $i=1$ will equal to $-\nu_{k}$ for $i=2$. For $k=0$ this
follows from a single application of Theorem 1.3.

Assume (2.36) and (2.37) hold for all $j\leqslant k-1$. Then we may apply
Theorem 1.3 (with $\rho=0$) to $u^{k}$, as
$J[u_{1}^{k},u_{2}^{k}](r)=J[u_{1},u_{2}](2^{-k}r)<1+\varepsilon$ by
assumption:

$\int_{B_{1}}|u^{k}-\beta^{k}(\nu_{k}\cdot x)^{+}|^{2}\leqslant
C\omega(2^{-k})\sum_{i=1}^{2}\|u^{k}_{i}\|^{2}_{W^{1,2}(B_{1})}.$

The main point now is to estimate $\|u^{k}\|_{W^{1,2}(B_{1})}$ in a manner
independent of $k$, and then show that $\beta^{k}$ is close to $\beta^{k-1}$
(and similarly for $\nu_{k}$).

We apply the Caccioppoli inequality to $u^{k}$ to give

$\|u^{k}\|_{W^{1,2}(B_{1})}\leqslant C\|u^{k}\|_{L^{2}(B_{2})},$

recalling they are subharmonic. From (2.36) with $k-1$, we have that

$\displaystyle\|u^{k}\|_{L^{2}(B_{2})}$
$\displaystyle=2^{n/2+1}\|u^{k-1}\|_{L^{2}(B_{1})}$ $\displaystyle\leqslant
2^{n/2+1}\|u^{k-1}-\beta^{k-1}(\nu_{k-1}\cdot
x)^{+}\|_{L^{2}(B_{1})}+C\beta_{i}^{k-1}$ $\displaystyle\leqslant
C\left[\sqrt{C_{1}\omega(2^{-k-1})}+\beta^{k-1}\right]$
$\displaystyle\leqslant C\left[\sqrt{C_{1}\varepsilon}+\beta^{k-1}\right].$

Now applying (2.37) for every $j\leqslant k-1$, we have

$\displaystyle\beta^{k-1}$
$\displaystyle\leqslant\beta^{0}+\sum_{j=1}^{k-1}\big{|}\beta^{j}-\beta^{j-1}\big{|}$
$\displaystyle\leqslant
C\|u\|_{W^{1,2}(B_{1})}+\sqrt{C_{2}}\sum_{j=1}^{k-1}\sqrt{\omega(2^{-j-1})}\leqslant
C\Big{(}1+\int_{0}^{2^{-k-1}}\frac{\sqrt{\omega(r)}}{r}\Big{)}\leqslant C.$

Choosing $C_{1}$ large enough and then $\varepsilon$ small in terms of $C_{1}$
gives

$\|u^{k}\|^{2}_{W^{1,2}(B_{1})}\leqslant
C[C_{1}\varepsilon+1]\leqslant\frac{C_{1}}{2}.$

This implies (2.36) for $k$.

To control $\beta^{k}$, we change variables:

$\int_{B_{1/2}}\left|u^{k-1}-\beta^{k}(\nu_{k}\cdot
x)^{+}\right|^{2}=2^{-n-2}\int_{B_{1}}\left|u^{k}-\beta^{k}(\nu_{k}\cdot
x)^{+}\right|^{2}\leqslant C\omega(2^{-k}).$

Then

$\int_{B_{1/2}}\left|\beta^{k}(\nu_{k}\cdot x)^{+}-\beta^{k-1}(\nu_{k-1}\cdot
x)^{+}\right|^{2}\leqslant C\omega(2^{-k-1})$

by combining with (2.36) with $k-1$. Direct evaluation leads to

$|\beta^{k}-\beta^{k-1}|^{2}+|\nu_{k}-\nu_{k-1}|^{2}\leqslant
C_{2}\omega(2^{-k-1}),$

and so (2.37) holds for $k$.

We are now in a position to conclude. From (2.37) the numbers $\beta^{k}$ and
vectors $\nu^{k}$ have

$\sum_{k=0}^{\infty}|\beta^{k}-\beta^{k-1}|+|\nu_{k}-\nu_{k-1}|\leqslant
C(\varepsilon+\int_{0}^{1}\frac{\sqrt{\omega(r)}}{r}dr)<\infty,$

so the sequences converge to some $\beta_{i},v$. Set

$\tilde{\omega}(r)=\omega(r)+\int_{0}^{2r}\frac{\sqrt{\omega(r)}}{r}dr,$

which a nondecreasing function tending to $0$ at $0$. Then from (2.36),

$\int_{B_{1}}|u^{k}-\beta(v\cdot x)^{+}|^{2}\leqslant
C\tilde{\omega}(2^{-k}),$

and so changing variables,

$2^{(n+2)k}\int_{B_{2^{-k}}}|u-\beta(\nu\cdot x)^{+}|^{2}\leqslant
C\tilde{\omega}(2^{-k}).$

This may be rewritten to give

$\frac{1}{r^{n+2}}\int_{B_{r}}|u-\beta(v\cdot x)^{+}|^{2}\leqslant
C\tilde{\omega}(2r)\rightarrow 0,$

concluding the proof. ∎

##  3\. Quantitative stability for the Faber-Krahn inequality

We now turn toward the proof of the quantitative Faber-Krahn inequality,
Theorem 1.1. As we discussed in the introduction, the proof is essentially
identical in the cases of the round sphere, Euclidean space, and hyperbolic
space. To keep the notation from becoming unreasonably heavy, we will
therefore present the proof in case of the round sphere, and include remarks
throughout the proof guiding the reader through any modifications needed to
generalize the proof to Euclidean or hyperbolic spaces. We give a few remarks
about the notation in this section. Since the ambient Euclidean space does not
play as prominent a role in this section, we will let $n$ denote the dimension
of the $n$-dimensional sphere and thus avoid using $n-1$ throughout the
section. Furthermore, in Section 2.1 it was necessary to distinguish between a
solid Euclidean ball and a spherical cap. In this section that is not
necessary, so we will simply use $B$ to denote the spherical cap. This
notation is also convenient for this section because - as already stated - the
proofs for Euclidean space and hyperbolic space are very similar.
Consequently, it is natural to use notation that easily adapts to the other
two settings: for the proof in Euclidean space $B$ would simply be a solid
Euclidean ball.

All three simply connected space forms share the property that balls centered
at any  point minimize the first eigenvalue. It it clear, then, that any
stability statement must involve the distance to a nearby ball to $\Omega$. In
(1.4), we chose to “mod out” by this symmetry by taking the infimum over all
$x$ in the space of the distance to balls $B(x)$ centered at $x$. It turns out
to be more convenient to prove the slightly stronger statement given below as
Theorem 3.1, in which we get rid of the translational invariance by choosing a
particular “set center” of a given set $\Omega$, defined in the following way.

Consider the standard embedding of the round sphere
$S^{n}(1)\subset{\varmathbb{R}}^{n+1}$ and define the set center $x_{\Omega}$
of $\Omega$ by setting

$x_{\Omega}=\frac{y_{\Omega}}{|y_{\Omega}|}$

whenever $0\neq
y_{\Omega}=\fint_{\Omega}y\,d{\rm{vol}}_{g}(y)\in{\varmathbb{R}}^{n+1}$. It is
worth noting that there are other possible ways that one may define a notion
of “barycenter” of a set $\Omega\subset S^{n}$. Another possible notion is
given in [14].

Throughout this section, given a set $\Omega\subset S^{n}$, we let
$u_{\Omega}$ denote its first eigenfunction extended by $0$ to be defined on
all of $S^{n}$ and normalized so that $\|u_{\Omega}\|_{L^{2}(\Omega)}=1$.

######  Theorem 3.1.

Fix $n\geqslant 1$ and $v\in(0,|S^{n}|)$. There exists a constant $c=c(n,v)$
such that the following holds. Let $\Omega\subset S^{n}$ with $|\Omega|=v$ be
a set for which $x_{\Omega}\in S^{n}$ is defined. Letting $B$ denote the
spherical cap centered at $x_{\Omega}$ with $|B|=v$, we have

(3.1) $\lambda_{1}(\Omega)-\lambda_{1}(B)\geqslant c\left(|\Omega\Delta
B|^{2}+\int_{S^{n}}|u_{\Omega}-u_{B}|^{2}\right)\,.$

######  Remark 3.2 (Sets with no barycenter).

The statement of Theorem 3.1 may look restrictive compared to Theorem 1.1.
However, if $\Omega$ is a set for which the barycenter is not defined, then
necessarily $\lambda_{1}(\Omega)-\lambda_{1}(B)\geqslant c$, and so (1.4)
holds trivially in this case. In particular, Theorem 3.1 implies Theorem 1.1
in the case of the round sphere.

######  Remark 3.3 (Set centers on Euclidean space and hyperbolic space).

On Euclidean space, an open set $\Omega\subset{\varmathbb{R}}^{n}$ has a
uniquely defined barycenter $x_{\Omega}=\fint_{\Omega}x\,dx$. For an open set
$\Omega\subset H^{n}$ in hyperbolic space, the set center
$x_{\Omega}=\text{argmin}_{x_{0}\in
H^{n}}\int_{\Omega}d(x,x_{0})^{2}\,d{\rm{vol}}(x)$ is well defined. The
analogues of Theorem 3.1 hold in each of these two spaces with $x_{\Omega}$
defined in this way; the proof will be identical up to the obvious
modifications. Once more, the analogues of Theorem 3.1 for Euclidean and
hyperbolic space imply Theorem 1.1 on these space forms.

We will work in spherical coordinates $(\theta,\phi)\in S^{n-1}\times(0,\pi)$
on $S^{n}$, in which the round metric takes the form
$g_{S^{n}}=d\phi^{2}+\sin^{2}(\phi)d\theta^{2}$, where $d\theta^{2}$ is used
to denote the round metric on $S^{n-1}$. A point $(\theta,\phi)\in S^{n}$ has
geodesic distance $\phi$ to the north pole $o\in S^{n}$. The Laplacian of a
function $f$ in spherical coordinates $(\theta,\phi)$ is given by

(3.2)
$\Delta_{S^{n}}f(\theta,\phi)=(\sin\phi)^{1-n}\frac{\partial}{\partial\phi}\left((\sin\phi)^{n-1}\frac{\partial
f}{\partial\phi}\right)+(\sin\phi)^{-2}\Delta_{S^{n-1}}f.$

######  Remark 3.4 (Coordinates on hyperbolic space and Euclidean space).

On hyperbolic space, we similarly use polar coordinates $(\theta,\phi)$, so
that the metric is expressed in these coordinates by
$g_{hyp}=d\phi^{2}+\sinh^{2}(\phi)d\theta^{2}$, where once more we let
$d\theta^{2}$ denote the round metric on $S^{n-1}$. A point $(\theta,\phi)\in
H^{n}$ has geodesic distance $\phi$ from the distinguished point from which
the polar coordinates are centered. The Laplacian of a function
$f:H^{n}\to{\varmathbb{R}}$ in these coordinates is given by

$\Delta_{H^{n}}f(\theta,\phi)=(\sinh\phi)^{1-n}\frac{\partial}{\partial\phi}\left((\sinh\phi)^{n-1}\frac{\partial
f}{\partial\phi}\right)+(\sinh\phi)^{-2}\Delta_{S^{n-1}}f.$

On Euclidean space we use standard polar coordinates $(\theta,\phi)$, where
$\phi$ is the distance to the origin and the Laplacian of a function
$f:{\varmathbb{R}}^{n}\to{\varmathbb{R}}$ is given by .

$\Delta_{{\varmathbb{R}}^{n}}f(\theta,\phi)=\phi^{1-n}\frac{\partial}{\partial\phi}\left(\phi^{n-1}\frac{\partial
f}{\partial\phi}\right)+\phi^{-2}\Delta_{S^{n-1}}f.$

Throughout this section, we will use the shorthand $B_{\rho}=B(o,\rho)$ to
denote the spherical cap of radius $\rho$ centered at the north pole. We will
frequently make the identification of the boundary of a geodesic ball,
${\partial}B(o,\rho)$, with the sphere $S^{n-1}$, noting that the induced
metric on ${\partial}B(o,\rho)$ is $\sin^{2}(\rho)d\theta^{2}$, where
$d\theta^{2}$ is the round metric on $S^{n-1}$.

We proceed in two steps: in Section 3.1, we establish Theorem 3.1 for sets
that are small perturbations of a spherical cap, and in Section 3.2, we call
upon results from our companion paper [4] to conclude Theorem 3.1 in the
general case.

###  3.1. Stability for nearly spherical sets

The main goal of this section is to establish Theorem 3.6 below, which is
Theorem 3.1 restricted to a class of sets in $S^{n}$ that are small
$C^{2,\alpha}$ perturbations of spherical caps.

The class of nearly spherical sets given in the following definition will be
our main objects of interest in this section.

######  Definition 3.5.

Given $\rho\in(0,\pi)$ and $\xi\in C^{2,\alpha}({\partial}B_{\rho})$ with
$\|\xi\|_{C^{2,\alpha}({\partial}B_{\rho})}\leqslant\varepsilon,$ we define
the following:

  1. (1)

A set $\Omega\subset S^{n}$ is parametrized by $\xi$ over $B_{\rho}$ if in
spherical coordinates ${\partial}\Omega$ takes the form

$\displaystyle{\partial}\Omega$
$\displaystyle=\\{(\theta,(1+\xi(\theta))\rho):(\theta,\rho)\in{\partial}B_{\rho}\\}.$

  2. (2)

A set $\Omega\subset S^{n}$ is a nearly spherical set if $\Omega$ is
parametrized by $\xi$ over $B_{\rho}$ such that $|\Omega|=|B_{\rho}|$ and the
barycenter $x_{\Omega}$ of $\Omega$ is the north pole $o$.

Note that the barycenter of a set parametrized over $B_{\rho}$ is always
defined provided that $\varepsilon$ is taken to be sufficiently small
depending on $n$ and $\rho$. The following theorem establishes quantitative
stability for Sperner’s inequality among nearly spherical sets.

######  Theorem 3.6 (Quantitative stability for nearly spherical sets).

Fix $n\geqslant 2$ and $\gamma\in(0,\pi/2)$. There exist postitive constants
$c$ and $\varepsilon$ depending only on $n$ and $\gamma$ such that the
following holds. Let $|\rho-\pi/2|<\gamma$ and let $\Omega$ be a nearly
spherical set parametrized by $\xi$ over $B_{\rho}$ with
$\|\xi\|_{C^{2,\alpha}({\partial}B_{\rho})}\leqslant\varepsilon$. Then

(3.3) $\lambda_{1}(\Omega)-\lambda_{1}(B_{\rho})\geqslant c\left(|\Omega\Delta
B_{\rho}|^{2}+\int_{S^{n}}|u_{\Omega}-u_{B_{\rho}}|^{2}\right)\,.$

The basic idea of the proof of Theorem 3.6 is the following. In Section 3.1.1,
we show that the deficit $\lambda_{1}(\Omega)-\lambda_{1}(B_{\rho})$ of a
nearly spherical set is equivalent to the second variation of
$\lambda_{1}(B_{\rho})$, up to an error that can be made arbitrarily small by
taking $\|\xi\|_{C^{2,\alpha}({\partial}B_{\rho})}$ to be small; see Theorem
3.7. Section 3.1.2 contains the core of the proof of Theorem 3.6: in Theorem
3.11 we establish a gap in the spectrum for the second variation operator. As
we discuss in the introduction, this spectral analysis is carried out through
an implicit argument based on the maximum principle. Theorem 3.11 is then
applied in Theorem 3.14 to show that the deficit
$\lambda_{1}(\Omega)-\lambda_{1}(B_{\rho})$ of a nearly spherical set controls
the $H^{1/2}$ norm of the parametrizing function $\xi$. Finally, in
Proposition 3.15 below of Section 3.1.3, we show that the right-hand side of
(1.4) is controlled linearly by $\|\xi\|_{H^{1/2}({\partial}B)}^{2}$.

Recall that the $H^{1/2}$ seminorm of a function
$\xi:{\partial}B_{\rho}\to{\varmathbb{R}}$ is defined by
$\|\xi\|_{\dot{H}^{1/2}({\partial}B_{\rho})}^{2}=\int_{B_{\rho}}|{\nabla}h_{\xi}|^{2},$
where $h_{\xi}$ is the harmonic extension of $\xi$, i.e. the unique solution
to

(3.4) $\begin{cases}\Delta h_{\xi}=0&\text{ in }B_{\rho}\\\ h_{\xi}=\xi&\text{
on }{\partial}B_{\rho}.\end{cases}$

The $H^{1/2}$ norm of a function $\xi:{\partial}B_{\rho}\to{\varmathbb{R}}$ is
defined by
$\|\xi\|_{H^{1/2}({\partial}B_{\rho})}^{2}=\|\xi\|_{L^{2}({\partial}B_{\rho})}^{2}+\|\xi\|_{\dot{H}^{1/2}({\partial}B_{\rho})}^{2}$.

####  3.1.1. The deficit and the second variation

The first eigenfunction $u_{B_{\rho}}$ of a spherical cap $B_{\rho}$ is
radially symmetric and decreasing. We let
$u_{\rho}(\phi)=u_{B_{\rho}}(\theta,\phi)$ and note that
$|{\nabla}u_{B_{\rho}}(\theta,\phi)|=-u_{\rho}^{\prime}(\phi)=|u_{\rho}^{\prime}(\phi)|.$
In the sequel, we will use the shorthand $|u_{\rho}^{\prime}|$ to refer to
number $|u_{\rho}^{\prime}(\rho)|>0$ and will let
$\lambda_{\rho}=\lambda_{1}(B_{\rho})$.

Given a function $\xi\in H^{1/2}({\partial}B_{\rho})$ with
$\int_{{\partial}B_{\rho}}\xi=0$, let $w_{\xi}:B_{\rho}\to{\varmathbb{R}}$ be
the solution to

(3.5) $\begin{cases}-\Delta w_{\xi}=\lambda_{\rho}w_{\xi}&\text{ in
}B_{\rho}\\\ \ \ \ \ \ w_{\xi}=|u_{\rho}^{\prime}|\xi&\text{ on
}{\partial}B_{\rho}\\\ \int_{B_{\rho}}w_{\xi}u_{B_{\rho}}=0.\end{cases}$

Such a solution exists and is unique by the Fredholm Alternative and is
continuous by [31]. For such a function $\xi$, the second variation
$\delta^{2}\lambda_{1}(B_{\rho})[\xi,\xi]$ of $\lambda_{1}$ at $B_{\rho}$ in
the direction $[\xi,\xi]$ is given by

(3.6)
$\delta^{2}\lambda_{1}(B_{\rho})[\xi,\xi]=2\int_{B_{\rho}}\left(|{\nabla}w_{\xi}|^{2}-\lambda_{\rho}w_{\xi}^{2}\right)\,dx+\mathcal{H}_{B_{\rho}}|u_{\rho}^{\prime}|^{2}\int_{{\partial}B_{\rho}}\xi^{2}\,dx;$

see [24, Lemma 2.8] or [18, Appendix A] for similar computations; note that on
${\varmathbb{R}}^{n}$ this expression matches [24, Lemma 2.8] since we take
$\xi$ with mean zero and since the ball has constant mean curvature. Here
$\mathcal{H}_{B_{\rho}}$ is the scalar mean curvature of ${\partial}B_{\rho}$
with the sign convention that $\mathcal{H}_{B_{\rho}}>0$ for $\rho<\pi/2$ and
$\mathcal{H}_{B_{\rho}}<0$ for $\rho\in(\pi/2,\pi).$ On Euclidean space and
hyperbolic space, the second variation takes the analogous form.

Let
$L_{\rho}:H^{1/2}({\partial}B_{\rho})\cap\\{\int_{{\partial}B_{\rho}}\xi=0\\}\to
L^{2}({\partial}B_{\rho})$ denote the Dirichlet-to-Neumann map corresponding
to the problem (3.5) translated by the zero order term
$\frac{1}{2}\mathcal{H}_{B_{\rho}}|u_{\rho}^{\prime}|^{2}$. More specifically,
given $\xi\in H^{1/2}({\partial}B_{\rho})$ with
$\int_{{\partial}B_{\rho}}\xi=0$, define

(3.7)
$L_{\rho}\xi={\partial}_{\nu}w_{\xi}+\frac{1}{2}\mathcal{H}_{B_{\rho}}|u_{\rho}^{\prime}|^{2}\xi,$

where $w_{\xi}$ is the unique solution to (3.5) and $\nu$ is the outer unit
normal to $B_{\rho}$. In this way, we can express the second variation on the
ball as

$\delta^{2}\lambda_{1}(B_{\rho})[\xi,\xi]=2\int_{{\partial}B_{\rho}}\xi
L_{\rho}\xi.$

The following theorem shows that the deficit in Sperner’s inequality for a
nearly spherical set is equal to the second variation, up to an error
depending on $\|\xi\|_{C^{2,\alpha}({\partial}B_{\rho})}$.

######  Theorem 3.7.

Fix $\gamma\in(0,\pi/2)$. There is a modulus of continuity $\omega$ depending
only on $n$ and $\gamma$ such that the following holds. Fix $\rho$ with
$|\pi/2-\rho|<\gamma$ and let $\Omega\subset S^{n}$ be a set parametrized by
$\xi$ over $B_{\rho}$ with $|\Omega|=|B_{\rho}|.$ Setting
$\hat{\xi}=\xi-\int_{{\partial}B_{\rho}}\xi,$ we have

(3.8)
$\lambda_{1}(\Omega)-\lambda_{1}(B_{\rho})=\frac{1}{2}\delta^{2}\lambda_{1}(B_{\rho})\big{[}\hat{\xi},\hat{\xi}\big{]}+\omega\left(\|\hat{\xi}\|_{C^{2,\alpha}({\partial}B_{\rho})}\right)\|\hat{\xi}\|_{H^{1/2}({\partial}B_{\rho})}^{2}.$

Theorem 3.7 was established in the Euclidean setting in [25, Theorem 1] (see
also [24, Theorem 1.4] for the $C^{2,\alpha}$ norm replaced by the $W^{2,p}$
norm for $p>n$). Adapting the proof to the spherical and hyperbolic settings
requires only technical modifications, which we omit.

######  Remark 3.8.

For a function $\xi\in H^{1/2}({\partial}B_{\rho})$ with nonzero mean, the
Fredholm Alternative implies that no solution exists to the problem (3.5)
(even without the orthogonality constraint
$\int_{B_{\rho}}w_{\xi}u_{B_{\rho}}=0$), and for this reason the second
variation is only defined for mean zero functions. However, for a nearly
spherical set $\Omega$, a Taylor expansion of the volume constraint shows that

$0=|\Omega|-|B_{\rho}|=\int_{{\partial}B_{\rho}}\xi+o(\|\xi\|_{L^{2}({\partial}B_{\rho})});$

see, for instance, [14]. This implies that
$\|\xi-\hat{\xi}\|_{L^{2}({\partial}B_{\rho})}=o(\|\xi\|_{L^{2}({\partial}B_{\rho})})$.
In particular,
$\|\xi\|_{H^{1/2}({\partial}B_{\rho})}=\|\hat{\xi}\|_{H^{1/2}({\partial}B_{\rho})}+o(\|\xi\|_{L^{2}({\partial}B_{\rho})})$
and

$\frac{1}{2}\|\xi\|_{C^{2,\alpha}({\partial}B_{\rho})}\leqslant\|\hat{\xi}\|_{C^{2,\alpha}({\partial}B_{\rho})}\leqslant
2\|\xi\|_{C^{2,\alpha}({\partial}B_{\rho})}.$

Therefore, the conclusion of Theorem 3.7 above may be equivalently written as

(3.9)
$\lambda_{1}(\Omega)-\lambda_{1}(B_{\rho})=\frac{1}{2}\delta^{2}\lambda_{1}(B_{\rho})\big{[}\hat{\xi},\hat{\xi}\big{]}+\omega\left(\|\xi\|_{C^{2,\alpha}({\partial}B_{\rho})}\right)\|\xi\|_{H^{1/2}({\partial}B_{\rho})}^{2}.$

####  3.1.2. The spectral gap

We study the spectrum of the shifted Dirichlet-to-Neumann operator $L_{\rho}$
introduced in (3.7) and establish a spectral gap in Theorem 3.11 below. In
contrast to the Euclidean case, the spectrum cannot be computed explicitly and
therefore the spectral analysis relies on implicit methods based on the
maximum principle.

We let $\\{Y_{i}\\}_{i=0}^{\infty}$ be the orthonormal basis for
$L^{2}({\partial}B_{\rho})$ of spherical harmonics, i.e. solutions to the
eigenvalue equation $\Delta_{S^{n-1}}Y_{i}=-\mu_{i}Y_{i}$, or equivalently
$\Delta_{{\partial}B_{\rho}}Y_{i}=-\sin^{-2}(\rho)\mu_{i}Y_{i}$, on
${\partial}B_{\rho}$ with each $Y_{i}$ normalized so that
$\int_{{\partial}B_{\rho}}Y_{i}^{2}=1.$ The eigenfunctions are ordered such
that $\mu_{i}\leqslant\mu_{i+1}$ for all $i$. Recall that

(3.10) $\begin{split}\mu_{0}&=0\\\ \mu_{1}&=\dots=\mu_{n}=(n-1)\\\
\mu_{n+1}&=2n.\end{split}$

The unique spherical harmonic corresponding to $\mu_{0}=0$ is constant, and
thus for each $i\geqslant 1$ we have

(3.11) $\int_{{\partial}B_{\rho}}Y_{i}=0.$

We let $w_{i}$ denote the solution of (3.5) with $\xi=Y_{i}$. That is,
$w_{i}\in H^{1}(B_{\rho})\cap C(B_{\rho})$ is the unique solution to

(3.12) $\begin{cases}-\Delta w_{i}=\lambda_{\rho}w_{i}&\text{ in }B_{\rho}\\\
\ \ \ \ w_{i}=|u_{\rho}^{\prime}|Y_{i}&\text{ on }{\partial}B_{\rho}\\\
\int_{B_{\rho}}w_{i}u_{B_{\rho}}=0.\end{cases}$

The following lemma shows that the solutions $w_{i}$ take a particular form
that will be crucial to our spectral analysis.

######  Lemma 3.9.

For each $i\geqslant 1$, the unique solution $w_{i}$ of (3.12) takes the form

(3.13) $w_{i}(\phi,\theta)=Y_{i}(\theta)g_{i}(\phi)$

in spherical coordinates, and $g_{i}$ satisfies the following three
properties:

$\displaystyle(1)\quad g_{i}(0)=0$ $\displaystyle(2)\quad
g_{i}(\rho)=|u_{\rho}^{\prime}|$ $\displaystyle(3)\quad g_{i}(\phi)>0\text{
for }0<\phi<\rho.$

###### Proof.

Fix $i\geqslant 1$ and the corresponding spherical harmonic $Y_{i}$ and
solution $w_{i}$ of (3.12). We omit the subscript $i$ in the remainder of this
proof for notational simplicity. Let us make the ansatz that $w$ takes the
form (3.13) in spherical coordinates. Then $w(\theta,\rho)=Y(\theta)g(\rho)$
on ${\partial}B_{\rho}$, so $w$ satisfies the boundary condition in (3.12)
provided $g(\rho)=|u_{\rho}^{\prime}|.$ Furthermore, thanks to the
orthogonality (3.11), we have

$\displaystyle\int_{B_{\rho}}w\,u_{B_{\rho}}$
$\displaystyle=\int_{0}^{\rho}\left(\int_{S^{n-1}}Y(\theta)\,d\mathcal{H}^{n-1}(\theta)\right)u_{\rho}(\phi)g(\phi)\,\sin(\phi)^{n-1}\,d\phi=0,$

so $w$ satisfies the orthogonality condition in (3.12). Finally, from the
expression (3.2) for the Laplacian of a function $f$ in spherical coordinates
$(\theta,\phi)$, we see that $g(\phi)$ satisfies the ordinary differential
equation

(3.14)
$\frac{d^{2}g}{d\phi^{2}}+(n-1)\frac{\cos\phi}{\sin\phi}\frac{dg}{d\phi}+\left(\lambda_{\rho}-\frac{\mu}{\sin^{2}\phi}\right)g=0.$

Under the change of variable $t=\cos\phi$, the solution $g$ is a solution to
the associated Legendre equation, which are well-known special functions.
Thus, up to choosing a constant multiple of $g$ so that the boundary condition
is satisfied, this function is the unique $H^{1}(B_{\rho})$ solution of (3.5).

Next we verify properties (1)–(3). Property (2) holds by our choice of
constant multiple of $g$. As a solution of (3.12), $w$ is nonconstant and
continuous, so it follows that $g(0)=0$ and thus $(1)$ holds. Finally, to show
(3), we suppose by way of contradiction that $g(\rho_{1})=0$ for some
$0<\rho_{1}<\rho$. Then we see that $w\in H^{1}_{0}(B_{\rho_{1}})$ solves

$\frac{\int_{B_{\rho_{1}}}|\nabla
w|^{2}}{\int_{B_{\rho_{1}}}w^{2}}=\lambda_{\rho}.$

Thus, $\lambda_{\rho_{1}}\leqslant\lambda_{\rho}$; however, since
$B_{\rho_{1}}\subset B_{\rho}$ we have that
$\lambda_{\rho}<\lambda_{\rho_{1}}$. This gives a contradiction and we see
that $g$ is nonvanishing in $(0,\rho)$. Together with (2) this proves (3). ∎

Using Lemma 3.9, we show in the following corollary that the spherical
harmonics for $i\geqslant 1$ are eigenfunctions of the Dirichlet-to-Neumann
operator (3.7).

######  Corollary 3.10.

Each spherical harmonic $Y_{i}$ for $i\geqslant 1$ is an eigenfunction of the
Dirichlet-to-Neumann operator $L_{\rho}$ defined in (3.7) with eigenvalue
$\eta_{i}$ given by

(3.15)
$\eta_{i}=g_{i}^{\prime}(\rho)+\frac{1}{2}\mathcal{H}_{B_{\rho}}|u_{\rho}^{\prime}|^{2},$

where $g_{i}$ is the function given in Lemma 3.9. If $\mu_{i}=\mu_{i+1}$, then
$\eta_{i}=\eta_{i+1}$. Furthermore, $\eta_{i}\geqslant 0$ for all
$i\in\varmathbb{N}$ and $\eta_{i}=0$ for $i=1,\dots,n$.

###### Proof.

Making use of Lemma 3.9, we compute that
$L_{\rho}Y_{i}=g_{i}^{\prime}(\rho)Y_{i}+\frac{1}{2}\mathcal{H}_{B_{\rho}}|u_{\rho}^{\prime}|^{2}Y_{i}$
where $g_{i}$ is the solution to (3.14) found in Lemma 3.9. In particular,
$Y_{i}$ is an eigenfunction of $L_{\rho}$ with eigenvalue $\eta_{i}$ as in
(3.15). If $\mu_{i}=\mu_{i+1}$, then we see from (3.14) that $g_{i}=g_{i+1}$
and thus $\eta_{i}=\eta_{i+1}$ by (3.15).

The next two claims follow from geometric considerations. Suppose by way of
contradiction that $\eta_{i}<0$ for some $i\geqslant 1$. For $t>0$ and small,
let $\Omega_{t}$ be the set parametrized by $\xi_{t}$ over $B_{\rho}$ where
$\xi_{t}=tY_{i}+s(t)Y_{0}$ and $s(t)$ is the function defined such that
$|\Omega_{t}|=|B_{\rho}|$. By Sperner’s inequality and Theorem 3.7,

$\displaystyle 0\leqslant\lambda_{1}(\Omega_{t})-\lambda_{1}(B_{\rho})$
$\displaystyle=t^{2}\eta_{i}+t^{3}\omega\left(\|Y_{i}\|_{C^{2,\alpha}({\partial}B_{\rho})}\right)\|Y_{i}\|_{H^{1/2}({\partial}B_{\rho})}^{2}\leqslant\frac{t^{2}}{2}\eta_{i}<0,$

where the penultimate inequality holds for $t$ sufficiently small depending on
$Y_{i}$. We reach a contradiction and conclude that $\eta_{i}\geqslant 0$ for
all $i\geqslant 1$.

We sketch the proof that $\eta_{i}=0$ for $i=1,\dots,n$. Note that
$\eta_{1}=\dots=\eta_{n-1}$ by (3.10) so it suffices to show that
$\eta_{1}=0$. Let $e_{1}\dots,e_{n+1}$ be an orthonormal basis for
${\varmathbb{R}}^{n+1}$. Consider the standard embedding of $S^{n}$ in
${\varmathbb{R}}^{n+1}$ such that the north pole is $e_{n+1}$. The spherical
harmonic $Y=Y_{1}$ is the restriction to ${\partial}B_{\rho}$ of the linear
function $x\cdot e_{1}$ on ${\varmathbb{R}}^{n+1}$.

For $t\in(-\varepsilon,\varepsilon)$, let $x_{t}=\exp(te_{1})$ and let
$\Omega_{t}=B(x_{t},\rho)$. For $t$ sufficiently small, there is a smooth and
uniquely defined one-parameter family of functions $\xi_{t}$ such that

${\partial}\Omega_{t}=\\{(\theta,(1+\xi_{t}(\theta))\rho)\ :\
(\theta,\rho)\in{\partial}B_{\rho}\\}$

in spherical coordinates. Moreover, for $t$ sufficiently small,
$\xi_{t}=\sum_{i=0}^{\infty}a_{i}(t)Y_{i}$ with

$a_{1}(t)=t/\rho,\qquad\text{ and
}\qquad\frac{d}{dt}\big{|}_{t=0}a_{i}(t)=\lim_{t\to 0}a_{i}(t)/t=0\text{ for
}i\neq 1.$

Because $\lambda_{1}(\Omega_{t})=\lambda_{1}(B_{\rho})$ for all $t$, we have,
in particular,

(3.16) $\frac{d^{2}}{dt^{2}}\lambda_{1}(\Omega_{t})=0$

for all $t$ sufficiently small. The expression for second variation at
$\Omega_{t}$ changes smoothly with respect to $t$. So, dividing (3.16) by
$t^{2}$ and letting $t\to 0$, we use the expression (3.6) to arrive at

$0=\lim_{t\to 0}\
\frac{1}{t^{2}}\left(\frac{d^{2}}{dt^{2}}\lambda_{1}(\Omega_{t})\right)=\lim_{t\to
0}\ \sum_{i=1}^{\infty}\eta_{i}\left(\frac{a_{i}(t)}{t}\right)^{2}=\eta_{1}.$

This concludes the proof of the corollary. ∎

Corollary 3.10 shows that $Y_{1},\dots,Y_{n}\in\ker L_{\rho}.$ The following
theorem is the key point in our spectral analysis: we establish a spectral gap
to show that the kernel of $L_{\rho}$ consists only of the span of
$\\{Y_{i}\\}_{i=1}^{n}$.

######  Theorem 3.11 (Spectral gap).

Fix $n\geqslant 2$ and $\rho\in(0,\pi)$. Let $\\{\eta_{i}\\}_{i=1}^{\infty}$
be the eigenvalues of $L_{\rho}$ given in (3.15). Then $\eta_{n+1}>\eta_{n}=0$
and so

$\eta_{i}\geqslant\eta_{n+1}>0\qquad\text{ for all }i\geqslant n+1.$

Furthermore, given any $\gamma\in(0,\pi/2)$, there exists $\bar{\eta}$
depending only on $n$ and $\gamma$ such that $\eta_{n+1}\geqslant\bar{\eta}$
for $|\rho-\pi/2|\leqslant\gamma$.

###### Proof of Theorem 3.11.

As noted in Corollary 3.10, if $\mu_{i}=\mu_{i+1}$, then
$\eta_{i}=\eta_{i+1}$. Let $w_{i}=Y_{i}(\theta)g_{i}(\phi)$ be the solution to
(3.5). We consider $h=g_{n}-g_{n+1}$. From properties (1) and (2) in Lemma
3.9, we have that $h(\rho)=h(0)=0$. Furthermore, from (3.14) and property (3)
of Lemma 3.9, we see that $h$ is a strict supersolution to the equation that
$g_{n}$ satisfies. That is,

(3.17)
$\frac{d^{2}h}{d\phi^{2}}+(n-1)\frac{\cos\phi}{\sin\phi}\frac{dh}{d\phi}+\left(\lambda_{\rho}-\frac{\mu_{n}}{\sin^{2}\phi}\right)h=\frac{\mu_{n}-\mu_{n+1}}{\sin^{2}\phi}g_{n-1}(\phi)<0.$

We will show that $h>0$ for $\phi\in(0,\rho)$ and $h^{\prime}(\rho)>0$. Notice
that in the case of the hemisphere $\rho=\pi/2$, we have $\lambda_{\rho}=n$
and $\mu_{n+1}=2n$ (recall (3.10)), hence the zero-order term
$(\lambda_{\pi/2}-\mu_{n+1}\sin^{-2}(\phi))\leqslant 0$ for all $\phi$.
Moreover, there exists $\epsilon$ depending on dimension $n$, such that for
$\rho\in(\pi/2-\epsilon,\pi)$, the zero-order term
$(\lambda_{\rho}-\mu_{n+1}\sin^{-2}(\phi))\leqslant 0$ for all $\phi.$ From
$\eqref{e:supersolt}$ we have $h_{\rho}$ is a strict supersolution, and from
the comparison principle we conclude that $h>0$, and so $g_{n}<g_{n+1}$ on
$(0,\rho)$. Furthermore, from the Hopf principle we conclude that
$h^{\prime}(\rho)<0$.

We will now show the same two properties for any $\rho\in(0,\pi)$. If
$h_{\rho}$ is the function associated with $\rho$, then define the set

$A:=\\{\rho\in(0,\pi)\mid h_{\rho}>0\text{ in }(0,\rho)\\}.$

We will show $A$ is both relatively open and closed in $(0,\pi)$. We have
shown already that $A\supset(\pi/2-\epsilon,\pi)$, and therefore $A$ is
nonempty. Since $h_{\rho}$ is the difference of two Legendre functions defined
on the whole interval $(0,\pi)$, then $h_{\rho}$ varies continuously with
$\rho$ in $C^{2}$ on $[0,\phi_{0}]$ for any $\phi_{0}<\pi$. Suppose
$\rho_{k}\in A$ and $\rho_{k}\to\rho\in(0,\pi)$. Then by uniform convergence
$h_{\rho}\geqslant 0$. If $h_{\rho}(\phi)=0$, then $\phi$ is a local minimum,
and so $h_{\rho}^{\prime}(\phi)=0$ and $h_{\rho}^{\prime\prime}(\phi)\geqslant
0$. But from (3.17) we have $h_{\rho}^{\prime\prime}(\phi)<0$, which is a
contradiction. Therefore, $h_{\rho}(\phi)>0$, and so $h_{\rho}\in A$. This
proves that $A$ is closed.

To show $A$ is also open, we will use the fact (shown below) that if
$h_{\rho}\geqslant 0$, then $h^{\prime}(\rho)<0$. Suppose by way of
contradiction that $\rho\in A$ and there exists a sequence $\rho_{k}\to\rho$
and $\phi_{k}\in(0,\rho_{k})$ such that $h_{\rho_{k}}(\phi_{k})\leqslant 0$.
Since $h_{\rho_{k}}$ converges uniformly to $h_{\rho}$, it follows that for a
subsequence, either $\phi_{k}\to\rho$ or $\phi_{k}\to 0$. If
$\phi_{k}\to\rho$, then from the uniform convergence, there also exists a
second sequence $\phi_{1,k}\to\rho$, such that $h_{\rho_{k}}(\phi_{1,k})=0$.
From the mean value theorem, there exists a third sequence
$\phi_{2,k}\to\rho$, such that $h^{\prime}_{\rho_{k}}(\phi_{2,k})=0$. From the
$C^{1}$ convergence up to the boundary point $\rho$, we would then have that
$h^{\prime}_{\rho}(\rho)=0$ which is a contradiction. If instead $\phi_{k}\to
0$, then from the uniform convergence, there exists a second sequence of local
minima $\phi_{1,k}\to 0$. But for $\phi$ close to zero, the zero-order term
$(\lambda_{\rho_{k}}-\mu_{n+1}\sin^{-2}(\phi))\leqslant 0$, so from the
comparison principle, we have a contradiction. We thus conclude that
$A=(0,\pi)$.

We now show the claim that $h^{\prime}(\rho)<0$. This claim is not only
necessary for the proof above, but we will also use it to show the eigenvalue
gap. We only need to consider the situation when
$(\lambda_{\rho}-\mu_{n+1}\sin^{-2}(\rho))>0$ and necessarily
$\rho\in(0,\pi/2-\epsilon)$. Under appropriate conditions one may still apply
the Hopf principle; however, in our one-dimensional case, we give the
following simpler argument.

Since $h\geqslant 0$ and the zero-order term is positive in
$(\rho-\delta,\rho)$ we have that

(3.18) $h^{\prime\prime}+(n-1)\cot\phi h^{\prime}=\zeta(\phi)\leqslant
0\quad\text{ for }\phi\in(\rho-\delta,\rho).$

Assume by way of contradiction that $h^{\prime}(\rho)=0$. Then applying the
explicit representation formula for first order linear equations to (3.18)
together with the initial condition $h^{\prime}(\rho)=0$, we have that

$h^{\prime}(\phi)=\frac{1}{p(\phi)}\int_{\rho}^{\phi}p(s)\zeta(s)\ ds\geqslant
0\quad(\text{since }\zeta\leqslant 0\text{ and }\phi<\rho),$

and where $p(s)>0$ is the integrating factor given by

$p(s)=\text{Exp}\left\\{\int_{\rho}^{s}(n-1)\cot t\ dt\right\\}.$

Then $h^{\prime}\geqslant 0$ in $(\rho-\delta,\rho)$. Since $h\geqslant 0$, we
must then have that $h^{\prime}\equiv 0$, so that $h\equiv 0$ on
$(\rho-\delta,\rho)$. But this is a contradiction since as already noted $h$
is a strict supersolution to the equation that $g_{n+1}$ satisfies. We must
then conclude that $h^{\prime}(\rho)<0$.

Consequently, we have $g_{n+1}^{\prime}(\rho)>g_{n}^{\prime}(\rho)$ and thus
$\eta_{n+1}>\eta_{n}$. In particular, $\eta_{i}\geqslant\eta_{n+1}>0$ for all
$i\geqslant n+1$. Finally, from (3.14), we see that $\eta_{n+1}$ depends
continuously on $\rho$, and so
$\eta_{n+1}=\eta_{n+1}(\rho)\geqslant\bar{\eta}$ for $\rho$ in any compact
subset $[\pi/2-\gamma,\pi/2+\gamma]$ of $(0,\pi)$. ∎

Theorem 3.11 will allow us to easily show that the second variation controls
the $L^{2}$ norm squared of any function $\xi\in H^{1/2}({\partial}B_{\rho})$
that is orthogonal to $Y_{0},\dots,Y_{n-1}$; see the proof of Corollary 3.13
below. In order to improve this $L^{2}$ control to $H^{1/2}$ control, we need
the following lemma.

######  Lemma 3.12.

Fix $n\geqslant 2$ and $\gamma\in(0,\pi/2)$. There exists a constant $C$
depending only on $n$ and $\gamma$ such that the following holds. Let
$|\pi/2-\rho|\leqslant\gamma$ and let $\xi\in H^{1/2}({\partial}B_{\rho})$
with $\int_{{\partial}B_{\rho}}\xi=0$. For the solution $w_{\xi}$ of (3.5), we
have
$\|w_{\xi}\|_{L^{2}(B_{\rho})}\leqslant\|\xi\|_{L^{2}({\partial}B_{\rho})}.$

###### Proof.

Let $w=w_{\xi}$ for notational simplicity. Let ${\varphi}$ be the unique
solution to the following auxiliary problem:

$\begin{cases}(-\Delta-\lambda_{\rho}){\varphi}=w&\qquad\text{ on }B_{\rho}\\\
{\varphi}=0&\qquad\text{ on }{\partial}{B_{\rho}}\\\
\int_{B_{\rho}}{\varphi}u_{B_{\rho}}=0.\end{cases}$

Such a solution exists and is unique by the Fredholm alternative. We multiply
the equation by $w$ and integrate by parts twice to find

$\displaystyle\int_{B_{\rho}}w^{2}$
$\displaystyle=\int_{B_{\rho}}w(-\Delta-\lambda_{\rho}){\varphi}$
$\displaystyle=-\int_{{\partial}B_{\rho}}{\partial}_{\nu}{\varphi}w+\int_{{\partial}B_{\rho}}{\partial}_{\nu}w{\varphi}+\int_{B_{\rho}}{\varphi}(-\Delta-\lambda_{\rho})w=-\int_{{\partial}B_{\rho}}{\partial}_{\nu}{\varphi}w.$

Then by Hölder’s inequality and the trace embedding (see [2, Theorem 5.22])
respectively, we see that

$\displaystyle\int_{B_{\rho}}w^{2}$
$\displaystyle\leqslant\|{\partial}_{\nu}{\varphi}\|_{L^{2}({\partial}B_{\rho})}|u^{\prime}_{\rho}|\|\xi\|_{L^{2}({\partial}B_{\rho})}$
$\displaystyle\leqslant
C\|{\nabla}{\varphi}\|_{H^{1}(B_{\rho})}|u^{\prime}_{\rho}|\|\xi\|_{L^{2}({\partial}B_{\rho})}\leqslant
C\|{\varphi}\|_{H^{2}(B_{\rho})}|u^{\prime}_{\rho}|\|\xi\|_{L^{2}({\partial}B_{\rho})},$

where the constant $C>0$ depends on $n$ and $\rho$, but is uniformly bounded
above for $|\pi/2-\rho|\leqslant\gamma.$ By standard elliptic estimates (see
[31, Theorem 9.13]), $\|{\varphi}\|_{H^{2}(B_{\rho})}\leqslant
C\|w\|_{L^{2}(B_{\rho})}$, where again $C>0$ is uniformly bounded above
depending on $n$ and $\gamma$. Since additionally $|u^{\prime}_{\rho}|$ is
bounded by a constant $C=C(n,\gamma)$, we find that

$\int_{B\rho}w^{2}\leqslant
C\|w\|_{L^{2}(B_{\rho})}\|\xi\|_{L^{2}({\partial}B_{\rho})}.$

Dividing through by $\|w\|_{L^{2}(B_{\rho})}$ establishes the claim. ∎

From Theorem 3.11 and Lemma 3.12, we can conclude that the second variation
controls $\|\xi\|_{H^{1/2}({\partial}B)}^{2}$ for any function $\xi\in
H^{1/2}({\partial}B_{\rho})$ that is orthogonal to $Y_{0},\dots,Y_{n-1}$:

######  Corollary 3.13.

Fix $n\geqslant 2$ and $\gamma\in(0,\pi/2)$. There is a positive constant
$\hat{\eta}$ depending only on $n$ and $\gamma$ such that the following holds.
Let $\rho\in(0,\pi)$ satisfy $|\pi/2-\rho|\leqslant\gamma.$ For any $\xi\in
H^{1/2}({\partial}B_{\rho})$ with $\int_{{\partial}B_{\rho}}\xi Y_{i}=0$ for
$i=0,\dots,n,$ we have

(3.19)
$\delta^{2}\lambda_{1}(B_{\rho})[\xi,\xi]\geqslant\hat{\eta}\|\xi\|_{H^{1/2}({\partial}B_{\rho})}^{2}.$

###### Proof.

We express $\xi=\sum_{i={n+1}}^{\infty}a_{i}Y_{i}$ in the basis of spherical
harmonics. By the linearity of $L_{\rho}$, we see that
$L_{\rho}\xi=\sum_{i={n+1}}^{\infty}\eta_{i}a_{i}Y_{i}.$ Thus, Theorem 3.11
shows that

(3.20)
$\frac{1}{2}\delta^{2}\lambda_{1}(B_{\rho})[\xi,\xi]=\sum_{i={n+1}}^{\infty}\eta_{i}a_{i}^{2}\geqslant\eta_{n}\sum_{i={n+1}}^{\infty}a_{i}^{2}=\bar{\eta}\|\xi\|_{L^{2}({\partial}B_{\rho})}^{2},$

where $\bar{\eta}$ depends only on $n$ and $\gamma$. In order to improve this
estimate to replace $\|\xi\|_{L^{2}({\partial}B_{\rho})}$ with
$\|\xi\|_{H^{1/2}({\partial}B_{\rho})}$, let $h_{\xi}$ denote the harmonic
extension of $|u_{\rho}^{\prime}|\xi$ as defined in (3.4) and note that

$|u_{\rho}^{\prime}|^{2}\|\xi\|_{\dot{H}^{1/2}(B_{\rho})}^{2}=\int_{B_{\rho}}|{\nabla}h_{\xi}|^{2}\leqslant\int_{B_{\rho}}|{\nabla}w_{\xi}|^{2}.$

Therefore, by first using the expression (3.6) for the second variation, and
then applying Lemma 3.12 to $w_{\xi}$, we have

$\displaystyle 2|u_{\rho}^{\prime}|^{2}\|\xi\|_{\dot{H}^{1/2}(B_{\rho})}^{2}$
$\displaystyle\leqslant 2\int_{B_{\rho}}|{\nabla}w_{\xi}|^{2}$
$\displaystyle\leqslant\delta^{2}\lambda_{1}(B_{\rho})[\xi,\xi]+|\mathcal{H}_{B_{\rho}}||u_{\rho}^{\prime}|^{2}\int_{{\partial}B_{\rho}}\xi^{2}+2\lambda_{\rho}\int_{B_{\rho}}w_{\xi}^{2}$
$\displaystyle\leqslant\delta^{2}\lambda_{1}(B_{\rho})[\xi,\xi]+C\int_{{\partial}B_{\rho}}\xi^{2},$

where $C=C(n,\gamma)$. Noting that $|u_{\rho}^{\prime}|>0$ depends
continuously on $\rho$ and thus $|u_{\rho}^{\prime}|\geqslant
c({n,\gamma})>0$, we apply the estimate (3.20) to the second term on the
right-hand side to conclude the proof. ∎

Finally, we can combine Corollary 3.13 with Theorem 3.7 to obtain the
following theorem.

######  Theorem 3.14.

Fix $n\geqslant 2$ and $\gamma\in(0,\pi/2)$. There are positive constants
$\varepsilon$ and $\hat{\eta}$ depending only on $n$ and $\gamma$ such that
the following holds. Let $|\pi/2-\rho|\leqslant\gamma$ and let $\Omega$ be a
nearly spherical set parametrized by $\xi$ over $B_{\rho}$ with
$\|\xi\|_{C^{2,\alpha}({\partial}B_{\rho})}\leqslant\varepsilon.$ Then

$\lambda_{1}(\Omega)-\lambda_{1}(B_{\rho})\geqslant\hat{\eta}\|\xi\|_{H^{1/2}({\partial}B_{\rho})}^{2}.$

###### Proof.

We express $\xi$ in the basis of spherical harmonics as
$\xi=\sum_{i=0}^{\infty}a_{i}Y_{i}$ where
$a_{i}=\int_{{\partial}B_{\rho}}Y_{i}\xi.$ The volume constraint and the set
center constraint respectively imply that

(3.21) $\displaystyle
a_{0}^{2}=o\left(\|\xi\|_{L^{2}({\partial}B_{\rho})}^{2}\right),\qquad
a_{1}^{2},\dots
a_{n}^{2}=o\left(\|\xi\|_{L^{2}({\partial}B_{\rho})}^{2}\right)$

with the $o(\cdot)$ term depending on $n$ and $\gamma$. Indeed, as we noted in
Remark 3.8, the first fact is a standard computation that can be found in [14,
Theorem 3.1]. To see the second, consider the standard embedding of
$S^{n}\subset{\varmathbb{R}}^{n+1}$ with the north pole $o=e_{n+1}$, and fix a
basis vector $e_{i}$ for $i=1,\dots,n$. In spherical coordinates we may write
$e_{i}=(1,\bar{\theta},\pi/2)$ for some $\bar{\theta}\in S^{n-1}$. So, from
the barycenter constraint $x_{\Omega}=e_{n+1}$, we have

$\displaystyle 0=e_{i}\cdot\int_{\Omega}x\,d{\rm{vol}}$
$\displaystyle=\int_{S^{n-1}}\int_{0}^{(1+\xi(\theta)\rho)}\left((\theta\cdot\bar{\theta})\sin\phi\right)\sin^{n-1}\phi\,d\phi\,d\theta$
$\displaystyle=\int_{S^{n-1}}(\theta\cdot\bar{\theta})\int_{0}^{(1+\xi(\theta)\rho)}\sin^{n}\phi\,d\phi\,d\theta$
$\displaystyle=\int_{S^{n-1}}(\theta\cdot\bar{\theta})\,F\left((1+\xi(\theta))\rho\right)\,d\theta,$

where the first inner product is the standard Euclidean inner product,
$F(t)=\int_{0}^{t}\sin(s)^{n}\,ds$ is a smooth hypergeometric function, and
$\theta\cdot\bar{\theta}$ indicates the inner product with respect to the
standard metric on $S^{n-1}$. A Taylor expansion of $F$ and Hölder’s
inequality thus show that

$\displaystyle 0$
$\displaystyle=\int_{S^{n-1}}(\theta\cdot\bar{\theta})\Big{(}F(\rho)+C_{\rho}\xi(\theta)+o\left(\|\xi\|_{L^{2}({\partial}B_{\rho})}\right)\Big{)}\,d\theta$
$\displaystyle=C_{\rho}\int_{S^{n-1}}(\theta\cdot\bar{\theta})\xi(\theta)\,d\theta+o\left(\|\xi\|_{L^{2}({\partial}B_{\rho})}\right).$

So, we see that for $i-1,\dots,n$, we have
$a_{i}=\int_{S^{n-1}}(\theta\cdot\bar{\theta})\xi(\theta))=o(\|\xi\|_{L^{2}(S^{n-1})})$
and conclude (3.21). Define the functions

$\hat{\xi}=\xi-
a_{0}Y_{0},\qquad\bar{\xi}=\hat{\xi}-\sum_{i=1}^{n}a_{i}Y_{i}\,.$

As a consequence of (3.21), we see that

$\displaystyle\|\xi\|_{H^{1/2}({\partial}(B_{\rho}))}$
$\displaystyle=\|\hat{\xi}\|_{H^{1/2}({\partial}B_{\rho})}+o\left(\|\xi\|_{L^{2}({\partial}B_{\rho})}^{2}\right)$
$\displaystyle=|\bar{\xi}\|_{H^{1/2}({\partial}B_{\rho})}+o\left(\|\xi\|_{L^{2}({\partial}B_{\rho})}^{2}\right)\,.$

By Theorem 3.7 and applying Corollary 3.13 to $\bar{\xi}$, we have

$\displaystyle\lambda_{1}(\Omega)-\lambda_{1}(B_{\rho})$
$\displaystyle\geqslant\frac{1}{2}\delta^{2}\lambda(B_{\rho})[\hat{\xi},\hat{\xi}]+o\left(\|\xi\|_{H^{1/2}({\partial}B_{\rho})}^{2}\right)$
$\displaystyle=\frac{1}{2}\delta^{2}\lambda(B_{\rho})[\bar{\xi},\bar{\xi}]+o\left(\|\xi\|_{H^{1/2}({\partial}B_{\rho})}^{2}\right)$
$\displaystyle=\frac{\hat{\eta}}{2}\|\bar{\xi}\|_{H^{1/2}({\partial}B_{\rho})}^{2}+o\left(\|\xi\|_{H^{1/2}({\partial}B_{\rho})}\right)$
$\displaystyle=\frac{\hat{\eta}}{2}\|\xi\|_{H^{1/2}({\partial}B_{\rho})}^{2}+o\left(\|\xi\|_{H^{1/2}({\partial}B_{\rho})}\right).$

Choosing $\varepsilon$ sufficiently small depending on $\bar{\eta}$ (and thus
on $\gamma$ and $n$), we may absorb the second term on the right-hand side to
conclude. ∎

####  3.1.3. The $H^{1/2}$ norm controls the distance

Thanks to Theorem 3.14, we will conclude the proof of Proposition 3.6 as soon
as we can show that the terms $|\Omega\Delta B_{\rho}|$ and
$\int_{S^{n}}|u_{\Omega}-u_{B_{\rho}}|$ are controlled linearly by
$\|\xi\|_{H^{1/2}(B)}.$ This is the content of the following proposition.

In the case of Euclidean space and hyperbolic space, the proof of this
proposition will be identical up to replacing the coarea factor
$\sin^{n-1}\phi$ with $\phi^{n-1}$ and $\sinh^{n-1}\phi$ respectively in
various integrals throughout the proof.

######  Proposition 3.15.

Fix $n\geqslant 2$ and $\gamma\in(0,\pi/2)$. There are positive constants
$\varepsilon$ and $C$ depending on $n$ and $\gamma$ such that the following
holds. Let $|\pi/2-\rho|<\gamma.$ Let $\Omega\subset S^{n}$ be a nearly
spherical set parameterized by $\xi$ with
$\|\xi\|_{C^{2,\alpha}({\partial}B_{\rho})}\leqslant\varepsilon$ and
$\lambda_{1}(\Omega)-\lambda_{1}(B_{\rho})\leqslant\varepsilon\lambda_{1}(B_{\rho}).$
Then

(3.22) $\displaystyle|\Omega\Delta
B_{\rho}|^{2}+\int_{S^{n}}|u_{\Omega}-u_{B_{\rho}}|^{2}$
$\displaystyle\leqslant C\|\xi\|^{2}_{H^{1/2}({\partial}B_{\rho})}.$

###### Proof.

The fact that $|\Omega\Delta B_{\rho}|\leqslant
C\|\xi\|_{L^{2}({\partial}B)}\leqslant C\|\xi\|_{H^{1/2}({\partial}B_{\rho})}$
is a standard computation; see [23, Proof of Theorem 4.3] or [18, Lemma 4.2].
Establishing $\int_{S^{n}}|u_{\Omega}-u_{B_{\rho}}|^{2}\leqslant
C\|\xi\|^{2}_{H^{1/2}({\partial}B_{\rho})}$ is more involved and will be
carried out in several steps.

Step 1. First, we define a diffeomorphism between $B_{\rho}$ and $\Omega$ and
express the equation solved by $u_{\Omega}$ pulled back by this
diffeomorphism. To this end, let $h:B_{\rho}\to{\varmathbb{R}}$ be the
harmonic extension of $\xi$ in $B_{\rho}$ defined in (3.4). Let
$\Psi:B_{\rho}\to S^{n}$ be the smooth map defined by

(3.23) $\Psi(x)=(1+h(x))\exp_{o}(x),$

where $\exp_{o}(x)$ is the exponential map at $o$. In normal coordinates
$\\{x^{\alpha}\\}$, the differential $d\Psi$ has coefficients

(3.24)
$\frac{{\partial}\Psi^{\alpha}}{{\partial}x^{i}}=(1+h(x))\delta_{i}^{\alpha}+\frac{{\partial}}{{\partial}x^{i}}h(x)x^{\alpha}.$

Since $h$ is harmonic and $|{\nabla}h|^{2}$ is subharmonic, the maximum
principle ensures that

$|h(x)|,|{\nabla}h(x)|\leqslant\|\xi\|_{C^{1}({\partial}B_{\rho})}\leqslant
C\|\xi\|_{C^{2,\alpha}({\partial}B_{\rho})}\leqslant C\varepsilon$

for all $x\in B_{\rho}$, where $C=C(n,p)$. The metric coefficients $g_{ij}$
are uniformly bounded uniformly bounded above, so
$|\frac{{\partial}}{{\partial}x^{i}}h(x)|\leqslant C\varepsilon$ as well. We
thus see that

(3.25)
$\left|\frac{{\partial}\Psi^{\alpha}}{{\partial}x^{i}}-\delta^{\alpha}_{i}\right|\leqslant
C\varepsilon,$

where $C$ is a constant depending only on $n$ and $\alpha$. In particular, the
differential $d\Psi$ is non-vanishing on $B_{\rho}$, so $\Psi$ is injective
and hence a diffeomorphism onto its image. Since $\Psi$ maps
${\partial}B_{\rho}$ to ${\partial}\Omega,$ it follows that $\Omega$ is the
image of $B_{\rho}$ under $\Psi$ and so $\Psi$ defines a smooth diffeomorphism
from $B_{\rho}$ to $\Omega.$ We let $\Phi:\Omega\to B_{\rho}$ denote the
inverse of $\Psi.$

Recall that $u_{\Omega}$ denotes the first Dirichlet eigenfunction of
$\Omega$, normalized so that $\int u_{\Omega}^{2}\,dx=1$ and extended by $0$
to be defined on all of $S^{n}$. We define
$\hat{u}_{\Omega}:B_{\rho}\to{\varmathbb{R}}$ to be the pullback
$\hat{u}_{\Omega}(x)=u_{\Omega}(\Psi(x))$ of $u_{\Omega}$ by $\Psi.$ Then
$\hat{u}_{\Omega}$ satisfies the equation

(3.26)
$\begin{cases}-\mathcal{L}\hat{u}_{\Omega}=m\lambda_{\Omega}\hat{u}_{\Omega}&\text{
in }B_{\rho}\\\ \hat{u}_{\Omega}=0&\text{ on }{\partial}B_{\rho}.\end{cases}$

Here and in the remainder of the proof we use the short-hand