# A strict maximum principle
for nonlocal minimal surfaces

Serena Dipierro(1) and Ovidiu Savin(2) and Enrico Valdinoci(1)

###### Abstract.

In the setting of fractional minimal surfaces, we prove that if two nonlocal
minimal sets are one included in the other and share a common boundary point,
then they must necessarily coincide.

This strict maximum principle is not obvious, since the surfaces may touch at
an irregular point, therefore a suitable blow-up analysis must be combined
with a bespoke regularity theory to obtain this result.

For the classical case, an analogous result was proved by Leon Simon. Our
proof also relies on a Harnack Inequality for nonlocal minimal surfaces that
has been recently introduced by Xavier Cabré and Matteo Cozzi and which can be
seen as a fractional counterpart of a classical result by Enrico Bombieri and
Enrico Giusti.

In our setting, an additional difficulty comes from the analysis of the
corresponding nonlocal integral equation on a hypersurface, which presents a
remainder whose sign and fine properties need to be carefully addressed.

OS was supported by the NSF grant DMS-2055617. EV was supported by the
Australian Laureate Fellowship FL190100081.

(1) – Department of Mathematics and Statistics

University of Western Australia

35 Stirling Highway, WA6009 Crawley (Australia)

(2) – Department of Mathematics

Columbia University

2990 Broadway, NY 10027 New York (USA)

E-mail addresses<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

##  1\. Introduction

The maximum principle is a fundamental tool to study geometric properties of
minimal hypersurfaces. Roughly speaking, it says that if two sets with smooth
boundaries are included one into the other and touch at some point, then the
mean curvature of the inner set at the contact point is larger than that of
the outer set.

The strict maximum principle aims at imposing a separation between surfaces
with the same mean curvature. While this property is rather straightforward
for smooth hypersurfaces, it becomes much more delicate in the presence of
singularities. For classical minimal surfaces, such a strong maximum principle
has been established by Leon Simon in the celebrated article [MR906394], also
relying on a Harnack Inequality for minimal surfaces which was put forth by
Enrico Bombieri and Enrico Giusti in [MR308945].

Related results were put forth in [MR482508]. See also [MR1017330] for a
strict maximum principle for stationary integer rectifiable varifolds in which
one of them is smooth and [MR1402732] for stationary hypersurfaces under
suitable assumptions and relying on an “extrinsic” Harnack inequality.

The goal of this article is to establish a strict maximum principle in the
framework of nonlocal minimal surfaces. This result can be seen as a
fractional counterpart of the main result in [MR906394] and also leverages a
recent Harnack Inequality for nonlocal minimal surfaces introduced by Xavier
Cabré and Matteo Cozzi in [MR3934589].

The setting that we consider is that of nonlocal minimal surfaces, as
introduced in [MR2675483]. Namely, given a (bounded and Lipschitz) domain
$\Omega\subset\mathbb{R}^{n}$ and a (measurable) set
$E\subseteq\mathbb{R}^{n}$, we say that $E$ is $s$-minimal in $\Omega$ if,
whenever a (measurable) set $F$ coincides with $E$ outside $\Omega$, it holds
that

$\operatorname{Per}_{s}(E,\Omega)\leqslant\operatorname{Per}_{s}(F,\Omega),$

where

$\displaystyle\operatorname{Per}_{s}(E,\Omega)$ $\displaystyle:=$
$\displaystyle\iint_{(E\cap\Omega)\times(E^{c}\cap\Omega)}\frac{dx\,dy}{|x-y|^{n+s}}$
$\displaystyle\qquad+\iint_{(E\cap\Omega)\times(E^{c}\cap\Omega^{c})}\frac{dx\,dy}{|x-y|^{n+s}}+\iint_{(E\cap\Omega^{c})\times(E^{c}\cap\Omega)}\frac{dx\,dy}{|x-y|^{n+s}}.$

Here above and in what follows, $s\in(0,1)$ is a fractional parameter and the
superscript “c” denotes the complement set in $\mathbb{R}^{n}$.

Nonlocal minimal surfaces are a fascinating, and very difficult topic of
investigation. While their regularity is known in the plane (see [MR3090533])
and up to dimension $7$ when $s$ is close to $1$ (see [MR3107529]), the full
regularity theory of these objects is not well-understood. Though no examples
of singular sets are presently known, it is commonly believed that nonlocal
minimal surfaces do develop singularities, therefore, for the validity of a
strict maximum principle, in general it does not suffice to take into
consideration only regular points.

In this setting, our main result is the following:

######  Theorem 1.1.

Let $r>0$ and $p\in\mathbb{R}^{n}$. Let $E_{1}$, $E_{2}$ be $s$-minimal sets
in $B_{r}(p)$, with $E_{1}\subseteq E_{2}$. Assume that $p\in(\partial
E_{1})\cap(\partial E_{2})$.

Then, $E_{1}=E_{2}$.

We recall that a better understanding of nonlocal minimal surfaces is
important also in connection with hybrid heat equations (see [MR2564467]),
geometric motions (see [MR2487027, MR3401008, MR3713894, MR3778164, MR3951024,
MR4104832, MR4175821] ), nonlocal soap bubble problems (see [MR3836150,
MR3881478]), capillarity problems (see [MR3717439, MR4404780]), long-range
phase coexistence models (see [MR2948285, MR4581189]), etc.

Interestingly, the nonlocal perimeter functional interpolate the classical
perimeter as $s\nearrow 1$ (see [MR1945278, MR1942130, MR2782803, MR2765717])
with a suitably weighted Lebesgue measure as $s\searrow 0$ (see [MR1940355,
MR3007726]). In this spirit, on the long run, it is expected that nonlocal
minimal surfaces can serve as auxiliary tools to better understand classical
minimal surfaces as well (see [MR4116635, HARDY, case]) and can provide new
perspectives on classical geometric objects by bringing forth tools
alternative to, and different than, differential geometry (see [INSPIRED]).

For other types of nonlocal minimal surfaces in which the interaction kernel
is of integrable type, see [MR3930619].

The rest of this paper focuses on the proof of Theorem 1.1. Roughly speaking
is that, by a blow-up and dimensional reduction argument, one reduces to the
case in which the two $s$-minimal sets under consideration touch at a point
which exhibit the same limit cone for both surfaces.

One then writes a geometric equation coming from the vanishing of the nonlocal
mean curvature of one of these surfaces and a suitable translation of the
other one and then scales to have a normalized picture in which the separation
between these two sheets is of order one (assuming, for a contradiction, that
they do not coincide to start with).

Some caveat is in place, since one has to switch from the notion of solution
in the smooth or viscosity sense to the one in the weak sense: for this, some
energy and capacity estimates will be required.

In this scenario, after a limit procedure, one aims at applying a suitable
Harnack Inequality to obtain a contradiction between a linear separation close
to the origin and the assumption that the two surfaces share the same111As
usual in geometric measure theory (see e.g. [MR906394]), here and in what
follows by the “same tangent cone” we mean that any blow-up sequence possesses
a subsequence along which the two dilated surfaces share the same tangent cone
in the limit. That is, the two surfaces have the same tangent cone in the
“strong sense”, namely along each convergent subsequence and not only along
one particular convergent subsequence. tangent cone.

However, a number of difficulties emerge, since the equations under
consideration present some complicated kernels, are not defined everywhere,
and produce a remainder which needs to be specifically analyzed (in
particular, we will need to check that this remainder is bounded and has a
sign, to be able to infer uniform Hölder estimates on the normalized
oscillation, pass them to the limit, obtain a limit inequality, and apply to
it an appropriate version of the Harnack Inequality).

Interestingly, the regularity theory utilized in this process is twofold:
first we use Hölder estimates for a nonlocal equation which is not symmetric
to perform a compactness argument on a sequence of normalized solutions which
encode the distance between the original surfaces and thus obtain a positive
supersolution on the limit cone for a fractional Jacobi operator, then we
apply to this setting the geometric Harnack Inequality to obtain the desired
conclusion.

A slightly more detailed and technical sketch of the proof will be presented
in the forthcoming Section 2: after this, we focus on the rigorous arguments
needed to make the actual proof work. More specifically, we will present in
Section 3 a suitable geometric equation which will play a major role in our
analysis. The interplay between viscosity and weak solutions of this equation
will be discussed in Section 4, through some capacity and cut-off arguments
(some technical estimates being deferred to Appendix A).

The geometric Hölder estimates needed to pass to the limit the rescaled
configuration in which the sheets are separated by an order one will be
presented in Section 5 (actually, these estimates are of general flavor and
can find application in other geometric problems as well).

To apply these Hölder estimates one will also need a uniform bound on
solutions of fractional equations in a geometric setting and this argument is
contained in Section 6. In turn, in our case, this uniform bound will be a
consequence of an integral bound, which is presented in Section 7.

The proof of Theorem 1.1 is thus completed in Section 8 (the details about the
good set of regular boundary points utilized in the proof being contained in
Appendix B).

##  2\. The plan of the proof

The proof will combine the blow-up methods introduced in [MR906394] with the
fine analysis of the nonlocal setting needed in our context. The idea of the
proof is to consider blow-up sequences of the minimal surfaces which approach
the same limit cone at the origin and take a normal parameterization of these
surfaces away from the singular set. We then look at the difference between
these parameterizations, say $w$ (or, more precisely, $w_{k}$, since it
depends on the step). The inclusion of the two sets guarantees that $w$ has a
sign, say $w\geqslant 0$ (and, by contradiction, we can suppose that $w>0$ at
some point).

The function $w$ may behave very badly in the vicinity of the singular set,
thus we perform our analysis in the complement of a neighborhood of the
singular set (we will shrink this neighborhood at the end of the proof,
relying on the regularity theory of the nonlocal minimal surfaces). Moreover,
since the convergence of $w=w_{k}$ is only local, we restrict our attention to
a given ball $B_{R}$ (we will send $R\to+\infty$ in the end).

Roughly speaking, the gist is to choose a point $x_{\star}$ (in $B_{R}$ and
away from a given small neighborhood of the singular set) and a sequence of
sets so that

(2.1) $2w_{k}(x_{\star})\geqslant t^{-1}w_{k}(tx_{\star})$

for all $t\in(0,1]$: it is indeed possible to fulfill this inequality (or a
slight modification of it) since the two minimal surfaces share the same
tangent cone (hence their difference is “sublinear” at the origin, see
Appendix B for full details about this technical construction). We can also
normalize $w_{k}$ at $x_{\star}$ to be $1$, i.e. look at the normalized
function $v_{k}:=\frac{w_{k}}{w_{k}(x_{\star})}$.

In this way, we see that $w=w_{k}$ (and therefore $v=v_{k}$) satisfies a
suitable nonlocal equation of geometric type (in view of Theorem 3.4) and we
show that $v$ remains locally bounded (owing to Lemma 6.1) and therefore
Hölder continuous in compact sets that avoid the singular set (thanks to Lemma
5.1).

We can thereby pick a convergent subsequence to a limit function $v_{\infty}$
which is a positive supersolution to a suitable Jacobi operator on the cone
(first in the viscosity sense and then in the distributional sense, due to
Lemma 4.1). From this, we will apply the weak Harnack Inequality of
[MR3934589] and deduce that $v_{\infty}$ is locally bounded below by a
universal constant and we contradict in this way the inequality in (2.1)
(normalized and passed to the limit).

The details to carry over this plan are very technical and several
complications arise from the nonstandard form of the integro-differential
equations involved (such as the domain of integration not being flat and the
kernels involved not being symmetric) and by the fact of dealing with non-
standard domains of definitions (such as large balls with neighbourhoods of
singular sets removed). These important technicalities occupy the rest of this
paper.

##  3\. A geometric equation

Now we consider the case in which two $s$-minimal sets, one included into the
other, can be written locally as a graph in terms of another smooth
hypersurface (concretely, in our case, the regular part of their common limit
cone, though the arguments presented are of general flavor).

In this scenario, we obtain a geometric inequality for the difference between
the parameterizations of the nonlocal minimal surfaces, which is reminiscent
of the Jacobi field for nonlocal minimal surfaces (see e.g. [MR3798717,
MR3951024]). The advantage of this approach in general is that these Jacobi
fields leverage additional cancellations with respect to the two individual
equations for the nonlocal mean curvature of the two surfaces. The specialty
of our setting is however that the two nonlocal minimal surfaces are nice
graphs with respect to the limit cone only away from the possible
singularities of the cone (hence, in our application, we will have to consider
graphs converging to the limit cone away from its singularities and carefully
estimate the error terms).

Actually, the methods that we develop work in a greater generality, which is
also useful to appreciate the geometric structures arising in the limit
construction. Thus, we split our calculations in different regions of the
space (namely, near a regular point in Lemma 3.1, in an intermediate ring in
Lemma 3.2, and far away in Lemma 3.3) and we collect these calculations into
the general form given by Theorem 3.4.

This type of results are influenced by the classical theory of minimal
surfaces but present significant differences with respect to the classical
case. For instance, in the local setting, one can apply directly the theory of
Jacobi fields and obtain a linear equation describing the normal displacement
of a surface approaching a limit cone (see equation (7) in [MR906394]).
Instead, in the nonlocal setting, remote interactions highly complicate the
structure of the equations and a fine theory of cancellations is needed to
obtain the convergence of the desired approximation and effective estimates on
the remainder terms.

In the forthcoming calculations, we use the notation

$\widetilde{\chi}_{E}(x):=\chi_{\mathbb{R}^{n}\setminus
E}(x)-\chi_{E}(x)=\begin{cases}1&{\mbox{ if }}x\in\mathbb{R}^{n}\setminus
E,\\\ -1&{\mbox{ if }}x\in E.\end{cases}$

Figure 1. Projections along the tangent hyperplane.

Also, here below we will take into consideration suitable kernels defined on a
hypersurface, specifically, a certain regular subset of $\partial E_{1}$, with
$E_{1}\subseteq\mathbb{R}^{n}$: in this situation, given $x_{0}$ on this
hypersurface and a vector $y$ (with small norm) in the linear tangent
hyperplane at $x_{0}$ we will denote by $y^{+}_{x_{0}}$ the point on the
hypersurface whose projection onto the affine tangent hyperplane at $x_{0}$
coincides with $x_{0}+y$ and by $y^{-}_{x_{0}}$ the point on the hypersurface
whose projection onto the affine tangent hyperplane at $x_{0}$ coincides with
$x_{0}-y$, see Figure 1.

######  Lemma 3.1.

Let $E_{1}\subseteq E_{2}\subseteq\mathbb{R}^{n}$.

Suppose that there exist $\overline{X}\in\mathbb{R}^{n}$ and $\delta>0$ such
that $(\partial E_{1})\cap B_{\delta}(\overline{X})$ is a hypersurface of
class $C^{2}$. Let $\nu$ be the unit external normal to $E_{1}$ in
$B_{\delta}(\overline{X})$.

Suppose that

$(E_{2}\setminus E_{1})\cap
B_{\delta}(\overline{X})=\Big{\\{}x+t\nu(x),\;{\mbox{with }}x\in\partial
E_{1},\;t\in\big{[}0,w(x)\big{)}\Big{\\}}\cap B_{\delta}(\overline{X}),$

for some function $w$ of class $C^{2}$ with values in $[0,\delta/4]$.

Given $x_{0}\in(\partial E_{1})\cap B_{\delta/8}(\overline{X})$, let

$\widetilde{E}_{1}:=E_{1}+w(x_{0})\nu(x_{0})\qquad{\mbox{and}}\qquad
X_{0}:=x_{0}+w(x_{0})\nu(x_{0}).$

Then, if $\delta$ and $\|w\|_{C^{2}(B_{\delta}(\overline{X}))}$ are small
enough,

(3.1)
$\begin{split}&\frac{1}{2}\int_{B_{\delta/4}(\overline{X})}\frac{\widetilde{\chi}_{E_{2}}(X)-\widetilde{\chi}_{\widetilde{E}_{1}}(X)}{|X-X_{0}|^{n+s}}\,dX\\\
&\qquad=\int_{(\partial E_{1})\cap
B_{\delta/4}(\overline{X})}\big{(}w(x_{0})-w(x)\big{)}\,K_{1}(x,x_{0})\,d{\mathcal{H}}^{n-1}_{x}\\\
&\qquad\qquad-w(x_{0})\int_{(\partial E_{1})\cap
B_{\delta/4}(\overline{X})}\big{(}1-\nu(x_{0})\cdot\nu(x)\big{)}\,K_{2}(x,x_{0})\,d{\mathcal{H}}^{n-1}_{x}+O(w^{2}(x_{0})),\end{split}$

where $K_{j}:({(\partial E_{1})\cap
B_{\delta/4}(\overline{X})})^{2}\to[0,+\infty]$, with $j\in\\{1,2\\}$,
satisfies, for a suitable $C\geqslant 1$, that

(3.2) $\begin{split}&{\mbox{for all\leavevmode\nobreak\ $x\neq x_{0}$, we have
that }}\;\frac{1}{C|x-x_{0}|^{n+s}}\leqslant
K_{j}(x,x_{0})\leqslant\frac{C}{|x-x_{0}|^{n+s}},\\\ &{\mbox{and, for
all\leavevmode\nobreak\ $y\in B_{\delta}\setminus\\{0\\}$,
}}K_{j}(y^{+}_{x_{0}},x_{0})-K_{j}(y^{-}_{x_{0}},x_{0})=O(|y|^{1-n-s}).\end{split}$

The “big $O$” terms here above depend only on the curvatures of $\partial
E_{1}$ in $B_{9\delta/10}(\overline{X})$, on
$\|w\|_{C^{2}(B_{\delta}(\overline{X}))}$, on $n$ and $s$.

Also, the kernel $K_{j}$ approaches, up to a normalizing constant, the kernel
$|x-x_{0}|^{-n-s}$ when $\|w\|_{C^{2}(B_{\delta}(\overline{X}))}$ tends to
zero.

###### Proof.

Given $X\in B_{\delta}(\overline{X})$, we write $X=x+t\nu(x)$, with
$x\in\partial E_{1}$ and $t\in\mathbb{R}$. If $\delta>0$ is small enough, the
map linking $X$ to $(x,t)$ is a diffeomorphism and the intersection with
$\partial E_{1}$ occurs only for $t=0$.

Thus, we denote the corresponding geometric Jacobian determinant by $J(x,t)$
(set to zero when $x+t\nu(x)\not\in B_{\delta/4}(\overline{X})$) and we have
that

(3.3)
$\begin{split}&\int_{B_{\delta/4}(\overline{X})}\frac{\widetilde{\chi}_{E_{2}}(X)-\widetilde{\chi}_{\widetilde{E}_{1}}(X)}{|X-X_{0}|^{n+s}}\,dX\\\
&\qquad=\iint_{((\partial E_{1})\cap
B_{\delta/4}(\overline{X}))\times\mathbb{R}}\frac{\widetilde{\chi}_{E_{2}}(x+t\nu(x))-\widetilde{\chi}_{\widetilde{E}_{1}}(x+t\nu(x))}{|x-x_{0}+t\nu(x)-w(x_{0})\nu(x_{0})|^{n+s}}\,J(x,t)\,d{\mathcal{H}}^{n-1}_{x}\,dt.\end{split}$

Now, given $x\in(\partial E_{1})\cap B_{\delta/4}(\overline{X})$, we have that

$\widetilde{\chi}_{E_{2}}(x+t\nu(x))=\begin{cases}1&{\mbox{ if }}t\geqslant
w(x),\\\ -1&{\mbox{ if }}t<w(x).\end{cases}$

Also, $\widetilde{\chi}_{\widetilde{E}_{1}}(x+t\nu(x))=-1$ if and only if
$x+t\nu(x)\in\widetilde{E}_{1}$, and so if and only if
$x(t):=x+t\nu(x)-w(x_{0})\nu(x_{0})\in E_{1}$.

Now, the signed distance of $x(t)$ to the tangent hyperplane of $E_{1}$ at $x$
is equal to

$\displaystyle d(x):=(x(t)-x)\cdot\nu(x)=t-w(x_{0})\nu(x_{0})\cdot\nu(x).$

Moreover, the projection of $x(t)-x$ onto the tangent plane is

$\displaystyle(x(t)-x)-\big{(}(x(t)-x)\cdot\nu(x)\big{)}\nu(x)=t\nu(x)-w(x_{0})\nu(x_{0})-\big{(}t-w(x_{0})\nu(x_{0})\cdot\nu(x)\big{)}\nu(x)$
$\displaystyle\qquad=-w(x_{0})\nu(x_{0})+w(x_{0})\nu(x_{0})\cdot\nu(x)\nu(x)=w(x_{0})\Big{(}\nu(x_{0})\cdot\nu(x)\nu(x)-\nu(x_{0})\Big{)},$

which has length equal to

$w(x_{0})\Big{|}\nu(x_{0})\cdot\nu(x)\nu(x)-\nu(x_{0})\Big{|}=w(x_{0})\sqrt{1-(\nu(x_{0})\cdot\nu(x))^{2}}.$

Since $\partial E_{1}$ detaches at most quadratically from its tangent
hyperplane, we have that

$\widetilde{\chi}_{\widetilde{E}_{1}}(x+t\nu(x))=\begin{cases}1&{\mbox{ if
}}t\geqslant\widetilde{w}(x),\\\ -1&{\mbox{ if
}}t<\widetilde{w}(x),\end{cases}$

where

$\widetilde{w}(x):=w(x_{0})\nu(x_{0})\cdot\nu(x)+O\big{(}w^{2}(x_{0})|\nu(x)-\nu(x_{0})|^{2}\big{)}.$

From these observations, we infer that, for all $x\in(\partial E_{1})\cap
B_{\delta/4}(\overline{X})$,

$\displaystyle\int_{\mathbb{R}}\frac{\widetilde{\chi}_{E_{2}}(x+t\nu(x))}{|x-x_{0}+t\nu(x)-w(x_{0})\nu(x_{0})|^{n+s}}\,J(x,t)\,dt=h_{1}(w(x))-h_{2}(w(x))$
and
$\displaystyle\int_{\mathbb{R}}\frac{\widetilde{\chi}_{\widetilde{E}_{1}}(x+t\nu(x))}{|x-x_{0}+t\nu(x)-w(x_{0})\nu(x_{0})|^{n+s}}\,J(x,t)\,dt=h_{1}(\widetilde{w}(x))-h_{2}(\widetilde{w}(x)),$

where

(3.4)
$\begin{split}&h_{1}(\xi):=\int_{\xi}^{+\infty}\frac{J(x,t)\,dt}{|x-x_{0}+t\nu(x)-w(x_{0})\nu(x_{0})|^{n+s}}\\\
{\mbox{and
}}\;&h_{2}(\xi):=\int_{-\infty}^{\xi}\frac{J(x,t)\,dt}{|x-x_{0}+t\nu(x)-w(x_{0})\nu(x_{0})|^{n+s}}.\end{split}$

We insert this information into (3.3) and we find that

(3.5)
$\begin{split}&\int_{B_{\delta/4}(\overline{X})}\frac{\widetilde{\chi}_{E_{2}}(X)-\widetilde{\chi}_{\widetilde{E}_{1}}(X)}{|X-X_{0}|^{n+s}}\,dX\\\
&\qquad=\int_{(\partial E_{1})\cap
B_{\delta/4}(\overline{X})}\Big{(}h_{1}(w(x))-h_{1}(\widetilde{w}(x))-h_{2}(w(x))+h_{2}(\widetilde{w}(x))\Big{)}\,d{\mathcal{H}}^{n-1}_{x}\\\
&\qquad=\int_{(\partial E_{1})\cap
B_{\delta/4}(\overline{X})}\Big{(}h(w(x))-h(\widetilde{w}(x))\Big{)}\,d{\mathcal{H}}^{n-1}_{x},\end{split}$

where

(3.6) $h:=h_{1}-h_{2}.$

We also observe that

(3.7)
$-h^{\prime}(\xi)=-h_{1}^{\prime}(\xi)+h_{2}^{\prime}(\xi)=\frac{2J(x,\xi)}{|x-x_{0}+\xi\nu(x)-w(x_{0})\nu(x_{0})|^{n+s}}.$

Therefore, letting

$\displaystyle A(x_{0}):={\rm Id}+w(x_{0})I\\!I(x_{0})$ and
$\displaystyle\sigma(x):=A(x_{0})(x-x_{0})=\big{(}{\rm
Id}+w(x_{0})I\\!I(x_{0})\big{)}(x-x_{0}),$

where $I\\!I$ denotes the second fundamental form along $\partial E_{1}$, and
using the change of variable $\theta:=\frac{\xi-w(x_{0})}{|\sigma(x)|}$, we
have that

(3.8) $\begin{split}&\frac{h(a+w(x_{0}))-h(b+w(x_{0}))}{2}\\\
=\,&-\frac{1}{2}\int_{a+w(x_{0})}^{b+w(x_{0})}h^{\prime}(\xi)\,d\xi\\\
=\,&\int_{a+w(x_{0})}^{b+w(x_{0})}\frac{J(x,\xi)\,d\xi}{|x-x_{0}+\xi\nu(x)-w(x_{0})\nu(x_{0})|^{n+s}}\\\
=\,&\int_{a/|\sigma(x)|}^{b/|\sigma(x)|}\frac{|\sigma(x)|\,J\big{(}x,w(x_{0})+\theta|\sigma(x)|\big{)}\,d\theta}{\Big{|}x-x_{0}+\big{(}w(x_{0})+\theta|\sigma(x)|\big{)}\nu(x)-w(x_{0})\nu(x_{0})\Big{|}^{n+s}}.\end{split}$

We also remark that, as $|x-x_{0}|\to 0$, the vector $x-x_{0}$ becomes tangent
to $\partial E_{1}$, therefore

$(x-x_{0})\cdot\nu(x)=O(|x-x_{0}|^{2}).$

Moreover,

$\nu(x)-\nu(x_{0})=I\\!I(x_{0})(x-x_{0})+O(|x-x_{0}|^{2}).$

For this reason,

$\displaystyle\big{(}w(x_{0})+\theta|\sigma(x)|\big{)}\nu(x)-w(x_{0})\nu(x_{0})$
$\displaystyle=$
$\displaystyle\big{(}w(x_{0})+\theta|\sigma(x)|\big{)}\big{(}\nu(x_{0})+I\\!I(x_{0})(x-x_{0})+O(|x-x_{0}|^{2})\big{)}-w(x_{0})\nu(x_{0})$
$\displaystyle=$ $\displaystyle
w(x_{0})I\\!I(x_{0})(x-x_{0})+\theta|\sigma(x)|\nu(x_{0})+O(|x-x_{0}|^{2}).$

As a result,

$\displaystyle\Big{|}x-x_{0}+\big{(}w(x_{0})+\theta|\sigma(x)|\big{)}\nu(x)-w(x_{0})\nu(x_{0})\Big{|}^{2}$
$\displaystyle=$ $\displaystyle\Big{|}\big{(}{\rm
Id}+w(x_{0})I\\!I(x_{0})\big{)}(x-x_{0})+\theta|\sigma(x)|\nu(x_{0})+O(|x-x_{0}|^{2})\Big{|}^{2}$
$\displaystyle=$
$\displaystyle\Big{|}\sigma(x)+\theta|\sigma(x)|\nu(x_{0})+O(|x-x_{0}|^{2})\Big{|}^{2}$
$\displaystyle=$
$\displaystyle(1+\theta^{2})|\sigma(x)|^{2}+2\theta|\sigma(x)|\sigma(x)\cdot\nu(x_{0})+O(|x-x_{0}|^{3}).$

Actually, since

$\displaystyle\sigma(x)\cdot\nu(x_{0})=\big{(}w(x_{0})I\\!I(x_{0})(x-x_{0})\big{)}\cdot\nu(x_{0})+O(|x-x_{0}|^{2})$
$\displaystyle\qquad=w(x_{0})\big{(}\nu(x)-\nu(x_{0})\big{)}\cdot\nu(x_{0})+O(|x-x_{0}|^{2})$
$\displaystyle\qquad=w(x_{0})\big{(}\nu(x)\cdot\nu(x_{0})-1\big{)}+O(|x-x_{0}|^{2})$
$\displaystyle\qquad=-\frac{w(x_{0})}{2}\big{|}\nu(x)-\nu(x_{0})\big{|}^{2}+O(|x-x_{0}|^{2})$
$\displaystyle\qquad=O(|x-x_{0}|^{2}),$

we find that

$\displaystyle\Big{|}x-x_{0}+\big{(}w(x_{0})+\theta|\sigma(x)|\big{)}\nu(x)-w(x_{0})\nu(x_{0})\Big{|}^{2}=(1+\theta^{2})|\sigma(x)|^{2}+O(|x-x_{0}|^{3}).$

This gives that

$\displaystyle\Big{|}x-x_{0}+\big{(}w(x_{0})+\theta|\sigma(x)|\big{)}\nu(x)-w(x_{0})\nu(x_{0})\Big{|}^{-(n+s)}$
$\displaystyle=$
$\displaystyle\Big{(}(1+\theta^{2})|\sigma(x)|^{2}+O(|x-x_{0}|^{3})\Big{)}^{-\frac{n+s}{2}}$
$\displaystyle=$
$\displaystyle(1+\theta^{2})^{-\frac{n+s}{2}}|\sigma(x)|^{-(n+s)}\Big{(}1+O(|x-x_{0}|)\Big{)}^{-\frac{n+s}{2}}$
$\displaystyle=$
$\displaystyle(1+\theta^{2})^{-\frac{n+s}{2}}|\sigma(x)|^{-(n+s)}\Big{(}1+O(|x-x_{0}|)\Big{)}$

and consequently, in view of (3.8),

(3.9) $\begin{split}&\frac{h(a+w(x_{0}))-h(b+w(x_{0}))}{2}\\\
&\qquad=\int_{a/|\sigma(x)|}^{b/|\sigma(x)|}\frac{|\sigma(x)|\,J\big{(}x,w(x_{0})+\theta|\sigma(x)|\big{)}\,\big{(}1+O(|x-x_{0}|)\big{)}}{(1+\theta^{2})^{\frac{n+s}{2}}|\sigma(x)|^{n+s}}\,d\theta.\end{split}$

Now we take

$a:=w(x)-w(x_{0})\qquad{\mbox{and}}\qquad b:=\widetilde{w}(x)-w(x_{0}).$

We observe that

$|w(x)-w(x_{0})|=O\big{(}\|\nabla
w\|_{L^{\infty}(B_{\delta}(\overline{X}))}|x-x_{0}|\big{)}$

and

$\displaystyle|\widetilde{w}(x)-w(x_{0})|$ $\displaystyle\leqslant$
$\displaystyle
w(x_{0})\big{|}\nu(x_{0})\cdot\nu(x)-1\big{|}+O\big{(}w^{2}(x_{0})|x-x_{0}|^{2}\big{)}$
$\displaystyle\leqslant$ $\displaystyle O\big{(}w(x_{0})|x-x_{0}|^{2}\big{)}.$

This says that, in our setting,

(3.10) $\frac{a}{|\sigma(x)|}=O\big{(}\|\nabla
w\|_{L^{\infty}(B_{\delta}(\overline{X}))}\big{)}\qquad{\mbox{and}}\qquad\frac{b}{|\sigma(x)|}=O\big{(}w(x_{0})|x-x_{0}|\big{)}.$

Therefore, with reference to the integral in (3.9), in this setting we have
that $\theta=O\big{(}\|w\|_{C^{1}(B_{\delta}(\overline{X}))}\big{)}$ and
therefore

$J\big{(}x,w(x_{0})+\theta|\sigma(x)|\big{)}=J\big{(}x,w(x_{0})\big{)}+O\big{(}|x-x_{0}|\big{)}=J\big{(}x_{0},w(x_{0})\big{)}+O\big{(}|x-x_{0}|\big{)},$

with the latter quantity depending on the curvatures of $\partial E_{1}$ in
$B_{9\delta/10}(\overline{X})$, on $\|w\|_{C^{2}(B_{\delta}(\overline{X}))}$,
on $n$ and $s$.

Gathering this and (3.9) we conclude that

$\begin{split}&\frac{h(a+w(x_{0}))-h(b+w(x_{0}))}{2}\\\
&\qquad=\frac{|\sigma(x)|\,\big{(}J(x_{0},w(x_{0}))+O(|x-x_{0}|)\big{)}}{|\sigma(x)|^{n+s}}\,\int_{a/|\sigma(x)|}^{b/|\sigma(x)|}\frac{d\theta}{(1+\theta^{2})^{\frac{n+s}{2}}}\\\
&\qquad=\frac{J(x_{0},w(x_{0}))+O(|x-x_{0}|)}{|\sigma(x)|^{n+s}}\,\left(b\Phi\left(\frac{b}{|\sigma(x)|}\right)-a\Phi\left(\frac{a}{|\sigma(x)|}\right)\right),\end{split}$

where

$\Phi(\alpha):=\frac{1}{\alpha}\int_{0}^{\alpha}\frac{d\theta}{(1+\theta^{2})^{\frac{n+s}{2}}}.$

Now we let

(3.11)
${\mathcal{A}}_{1}(x,x_{0}):=\Phi\left(\frac{w(x)-w(x_{0})}{|\sigma(x)|}\right)\qquad{\mbox{and}}\qquad{\mathcal{A}}_{2}(x,x_{0}):=\Phi\left(\frac{\widetilde{w}(x)-w(x_{0})}{|\sigma(x)|}\right)$

and we see that

$\displaystyle\frac{h(w(x))-h(\widetilde{w}(x))}{2}$ $\displaystyle=$
$\displaystyle\frac{J(x_{0},w(x_{0}))+O(|x-x_{0}|)}{|\sigma(x)|^{n+s}}\,\Big{(}(\widetilde{w}(x)-w(x_{0})){\mathcal{A}}_{2}(x,x_{0})-(w(x)-w(x_{0})){\mathcal{A}}_{1}(x,x_{0})\Big{)}$
$\displaystyle=$
$\displaystyle\frac{J(x_{0},w(x_{0}))+O(|x-x_{0}|)}{|\sigma(x)|^{n+s}}$
$\displaystyle\times\Big{[}\Big{(}w(x_{0})(\nu(x_{0})\cdot\nu(x)-1)+O(w^{2}(x_{0})|x-x_{0}|^{2})\Big{)}{\mathcal{A}}_{2}(x,x_{0})-(w(x)-w(x_{0})){\mathcal{A}}_{1}(x,x_{0})\Big{]}$
$\displaystyle=$
$\displaystyle\Big{(}w(x_{0})(\nu(x_{0})\cdot\nu(x)-1)+O(w^{2}(x_{0})|x-x_{0}|^{2})\Big{)}K_{2}(x,x_{0})-(w(x)-w(x_{0}))K_{1}(x,x_{0}),$

where

$K_{j}(x,x_{0}):=\frac{\big{(}J(x_{0},w(x_{0}))+O(|x-x_{0}|)\big{)}\,{\mathcal{A}}_{j}(x,x_{0})}{|\sigma(x)|^{n+s}}.$

By inserting this into (3.5), we have thereby obtained the desired result in
(3.1), but we need to check (3.2) in order to ensure the necessary
cancellations to make sense of the integrals involved.

To this end, we observe that the Jacobian $J(x_{0},t)$ approaches $1$ as $t\to
0$. Moreover,

$\Phi(\alpha)\leqslant\frac{1}{\alpha}\int_{0}^{\alpha}\,d\theta=1$

and, if $\alpha\in(-1,1)$,

$\Phi(\alpha)\geqslant\frac{1}{\alpha}\int_{0}^{\alpha}\frac{d\theta}{2^{\frac{n+s}{2}}}=\frac{1}{2^{\frac{n+s}{2}}}.$

This, (3.10) and (3.11) entail that, in the range of interest, also
${\mathcal{A}}_{j}(x,x_{0})\in\left[\frac{1}{2^{\frac{n+s}{2}}},1\right]$.

Besides, for small $w(x_{0})$, we have that
$|\sigma(x)|\in\left[\frac{|x-x_{0}|}{2},2|x-x_{0}|\right]$ and these
considerations establish the first claim in (3.2).

Additionally,

$|\sigma(x_{0}\pm y)|=|\pm A(x_{0})y|=|A(x_{0})y|.$

Furthermore,

$\Phi(\alpha)=\frac{1}{\alpha}\int_{0}^{\alpha}\left(1-{\frac{n+s}{2}}\,\theta^{2}+O(\theta^{4})\right)\,d\theta=1-{\frac{n+s}{6}}\,\alpha^{2}+O(\alpha^{4}),$

giving that, if $\alpha\in(-1,1)$, then $|\Phi^{\prime}(\alpha)|\leqslant C$
and accordingly

$|\Phi(\alpha)-\Phi(\beta)|\leqslant C|\alpha-\beta|.$

We also have that

$w(y^{+}_{x_{0}})-w(y^{-}_{x_{0}})=O(|y|^{2})$

and

$\widetilde{w}(y_{x_{0}}^{\pm})-w(x_{0})=w(x_{0})\big{(}\nu(x_{0})\cdot\nu(y_{x_{0}}^{\pm})-1\big{)}+O(|y|^{2})=O(|y|^{2}).$

As a consequence,

$\displaystyle\Big{|}{\mathcal{A}}_{1}(y^{+}_{x_{0}},x_{0})-{\mathcal{A}}_{1}(y^{-}_{x_{0}},x_{0})\Big{|}$
$\displaystyle\qquad=\left|\Phi\left(\frac{w(y^{+}_{x_{0}})-w(x_{0})}{|A(x_{0})y|}\right)-\Phi\left(\frac{w(y^{+}_{x_{0}})-w(x_{0})}{|A(x_{0})y|}\right)\right|$
$\displaystyle\qquad\leqslant
C\left|\frac{w(y^{+}_{x_{0}})-w(y^{-}_{x_{0}})}{|A(x_{0})y|}\right|\leqslant
C|y|$

and

$\displaystyle\Big{|}{\mathcal{A}}_{2}(y^{+}_{x_{0}},x_{0})-{\mathcal{A}}_{2}(y^{-}_{x_{0}},x_{0})\Big{|}$
$\displaystyle\qquad=\left|\Phi\left(\frac{\widetilde{w}(y^{+}_{x_{0}})-w(x_{0})}{|A(x_{0})y|}\right)-\Phi\left(\frac{\widetilde{w}(y^{-}_{x_{0}})-w(x_{0})}{|A(x_{0})y|}\right)\right|\leqslant
C|y|.$

The proof of (3.2) is thereby complete.

The fact that $K_{1}$ and $K_{2}$ can be seen as perturbations of the kernel
$|x-x_{0}|^{-n-s}$ when $\|w\|_{C^{2}(B_{\delta}(\overline{X}))}$ tends to
zero follows since in this asymptotics $\sigma(x)$ approaches $|x-x_{0}|$,
${\mathcal{A}}_{1}$ and ${\mathcal{A}}_{2}$ approach $\Phi(0)=1$ and the
Jacobian term approaches $J(x,0)=1$. ∎

######  Lemma 3.2.

Let $\delta\in(0,1)$, $R>1+2\delta$, $\overline{X}\in\mathbb{R}^{n}$ and
$E_{1}\subseteq E_{2}\subseteq\mathbb{R}^{n}$.

Assume that there exists ${\mathcal{G}}\subseteq B_{2R}(\overline{X})$ such
that $(\partial E_{1})\cap{\mathcal{G}}$ is a hypersurface of class $C^{2}$
with unit external normal $\nu$.

Suppose that

$(E_{2}\setminus E_{1})\cap{\mathcal{G}}=\Big{\\{}x+t\nu(x),\;{\mbox{with
}}x\in\partial E_{1},\;t\in\big{[}0,w(x)\big{)}\Big{\\}}\cap{\mathcal{G}},$

for some function $w$ of class $C^{2}$ with values in $[0,\delta/4]$.

Assume also that

(3.12) ${\mbox{the Hausdorff distance between\leavevmode\nobreak\ $\partial
E_{1}$ and\leavevmode\nobreak\ $\partial E_{2}$ in\leavevmode\nobreak\
$B_{2R}(\overline{X})$ is less than }}\frac{\delta}{20}.$

Let $x_{0}\in(\partial E_{1})\cap B_{\delta/8}(\overline{X})$,
$\nu_{0}\in\partial B_{1}$,

$\widetilde{E}_{1}:=E_{1}+w(x_{0})\nu_{0}\qquad{\mbox{and}}\qquad
X_{0}:=x_{0}+w(x_{0})\nu_{0}.$

Let also ${\mathcal{G}}^{\prime}\Subset{\mathcal{G}}$ and assume that

(3.13)
$B_{2R}(\overline{X})\setminus{\mathcal{G}}^{\prime}\subseteq\bigcup_{j\in\mathbb{N}}B_{r_{j}}(p_{j}),$

for some balls $\\{B_{r_{j}}(p_{j})\\}_{j\in\mathbb{N}}$, with

(3.14) $\eta:=\sum_{j\in\mathbb{N}}r_{j}^{n-1}<1.$

Then, if $\delta$ is small enough, and $\eta$ and
$\|w\|_{L^{\infty}(B_{R}(\overline{X})\cap{\mathcal{G}})}$ are small compared
to $\delta$ and $1/R$,

$\displaystyle\frac{1}{2}\int_{B_{R}(\overline{X})\setminus
B_{\delta/4}(\overline{X})}\frac{\widetilde{\chi}_{E_{2}}(X)-\widetilde{\chi}_{\widetilde{E}_{1}}(X)}{|X-X_{0}|^{n+s}}\,dX$
$\displaystyle\qquad=\int_{((\partial
E_{1})\cap{\mathcal{G}}^{\prime})\setminus
B_{\delta/4}(\overline{X})}\big{(}w(x_{0})-w(x)\big{)}\,K_{1}(x,x_{0})\,d{\mathcal{H}}^{n-1}_{x}$
$\displaystyle\qquad\qquad\qquad-w(x_{0})\int_{((\partial
E_{1})\cap{\mathcal{G}}^{\prime})\setminus
B_{\delta/4}(\overline{X})}\big{(}1-\nu_{0}\cdot\nu(x)\big{)}\,K_{2}(x,x_{0})\,d{\mathcal{H}}^{n-1}_{x}$
$\displaystyle\qquad\qquad\qquad+\frac{1}{2}\int_{(B_{R}(\overline{X})\setminus
B_{\delta/4}(\overline{X}))\setminus{\mathcal{G}}^{\prime}}\frac{\widetilde{\chi}_{E_{2}}(X)-\widetilde{\chi}_{E_{1}}(X)}{|X-X_{0}|^{n+s}}\,dX+O\left(w(x_{0})\eta\right),$

where the “big $O$” term here above depends only on $\delta$, $R$, $n$ and
$s$.

In addition, $K_{j}:({((\partial E_{1})\cap{\mathcal{G}}^{\prime})\setminus
B_{\delta/4}(\overline{X})})^{2}\to[0,+\infty]$, with $j\in\\{1,2\\}$,
satisfies (3.2) and approaches, up to a normalizing constant, the kernel
$|x-x_{0}|^{-n-s}$ when $\|w\|_{C^{2}(B_{R}(\overline{X})\cap{\mathcal{G}})}$
tends to zero.

###### Proof.

As in the proof of Lemma 3.1, given $X\in(B_{R}(\overline{X})\setminus
B_{\delta/4}(\overline{X}))\cap{\mathcal{G}}^{\prime}$, we write
$X=x+t\nu(x)$, with $x\in\partial E_{1}$ and $t\in\mathbb{R}$.

By (3.12), we see that

${\mbox{if\leavevmode\nobreak\ $|t|\geqslant$}}\frac{\delta}{10},{\mbox{ then
}}\widetilde{\chi}_{E_{2}}(x+t\nu(x))-\widetilde{\chi}_{\widetilde{E}_{1}}(x+t\nu(x))=0.$

In this setting, if $\delta>0$ is small enough,

(3.15) $\begin{split}&\int_{(B_{R}(\overline{X})\setminus
B_{\delta/4}(\overline{X}))\cap{\mathcal{G}}^{\prime}}\frac{\widetilde{\chi}_{E_{2}}(X)-\widetilde{\chi}_{\widetilde{E}_{1}}(X)}{|X-X_{0}|^{n+s}}\,dX\\\
=\;&\iint_{((\partial E_{1})\cap({B_{R}(\overline{X})\setminus
B_{\delta/4}(\overline{X})})\cap{\mathcal{G}}^{\prime})\times\mathbb{R}}\frac{\widetilde{\chi}_{E_{2}}(x+t\nu(x))-\widetilde{\chi}_{\widetilde{E}_{1}}(x+t\nu(x))}{|x-x_{0}+t\nu(x)-w(x_{0})\nu_{0}|^{n+s}}\,J(x,t)\,d{\mathcal{H}}^{n-1}_{x}\,dt,\end{split}$

where $J(x,t)$ denotes the geometric Jacobian determinant (set to zero when
$x+t\nu(x)\not\in B_{R}(\overline{X})$).

Given $x\in(\partial E_{1})\cap{\mathcal{G}}^{\prime}$, we have that

(3.16) $\widetilde{\chi}_{E_{2}}(x+t\nu(x))=\begin{cases}1&{\mbox{ if
}}t\geqslant w(x),\\\ -1&{\mbox{ if }}t<w(x).\end{cases}$

Moreover, $\widetilde{\chi}_{\widetilde{E}_{1}}(x+t\nu(x))=-1$ if and only if
$x+t\nu(x)\in\widetilde{E}_{1}$, and so if and only if
$x(t):=x+t\nu(x)-w(x_{0})\nu_{0}\in E_{1}$.

Now, the signed distance of $x(t)$ to the tangent hyperplane of $E_{1}$ at $x$
is equal to

$\displaystyle d(x):=(x(t)-x)\cdot\nu(x)=t-w(x_{0})\nu_{0}\cdot\nu(x).$

Moreover, the projection of $x(t)-x$ onto the tangent plane is

$\displaystyle(x(t)-x)-\big{(}(x(t)-x)\cdot\nu(x)\big{)}\nu(x)=t\nu(x)-w(x_{0})\nu_{0}-\big{(}t-w(x_{0})\nu_{0}\cdot\nu(x)\big{)}\nu(x)$
$\displaystyle\qquad\qquad\qquad=w(x_{0})\nu_{0}\cdot\nu(x)\nu(x)-w(x_{0})\nu_{0}$

which has length equal to

$\ell(x):=\sqrt{w^{2}(x_{0})-(w(x_{0})\nu_{0}\cdot\nu(x))^{2}}.$

We stress that $\ell(x)\leqslant w(x_{0})$, hence if $w(x_{0})$ is small
enough then $B_{\ell(x)}(x)$ lies in ${\mathcal{G}}$ and accordingly $\partial
E_{1}$ detaches at most quadratically from its tangent hyperplane in this
region. This gives that

(3.17) $\widetilde{\chi}_{\widetilde{E}_{1}}(x+t\nu(x))=\begin{cases}1&{\mbox{
if }}t\geqslant\widetilde{w}(x),\\\ -1&{\mbox{ if
}}t<\widetilde{w}(x),\end{cases}$

where

$\widetilde{w}(x):=w(x_{0})\nu_{0}\cdot\nu(x)+O\big{(}w^{2}(x_{0})\big{)}.$

By using (3.16) and (3.17), and in the notation of (3.4) and (3.6) (with
$\nu(x_{0})$ replaced by $\nu_{0}$ here), we deduce that, if $x\in(\partial
E_{1})\cap({B_{R}(\overline{X})\setminus
B_{\delta/4}(\overline{X})})\cap{\mathcal{G}}^{\prime}$, then

$\displaystyle\int_{\mathbb{R}}\frac{\widetilde{\chi}_{E_{2}}(x+t\nu(x))-\widetilde{\chi}_{\widetilde{E}_{1}}(x+t\nu(x))}{|x-x_{0}+t\nu(x)-w(x_{0})\nu_{0}|^{n+s}}\,J(x,t)\,dt=h(w(x))-h(\widetilde{w}(x))$

and therefore

(3.18) $\begin{split}&\iint_{((\partial
E_{1})\cap({B_{R}(\overline{X})\setminus
B_{\delta/4}(\overline{X})}\cap{\mathcal{G}}^{\prime})\times\mathbb{R}}\frac{\widetilde{\chi}_{E_{2}}(x+t\nu(x))-\widetilde{\chi}_{\widetilde{E}_{1}}(x+t\nu(x))}{|x-x_{0}+t\nu(x)-w(x_{0})\nu_{0}|^{n+s}}\,J(x,t)\,d{\mathcal{H}}^{n-1}_{x}\,dt\\\
&\qquad=\int_{(\partial E_{1})\cap({B_{R}(\overline{X})\setminus
B_{\delta/4}(\overline{X})})\cap{\mathcal{G}}^{\prime}}\Big{(}h(w(x))-h(\widetilde{w}(x))\Big{)}\,d{\mathcal{H}}^{n-1}_{x}.\end{split}$

Thus, we define

$K(x,x_{0}):=-\frac{h(w(x))-h(\widetilde{w}(x))}{2(w(x)-\widetilde{w}(x))}$

and we remark that this kernel satisfies (3.2). Indeed, we can take here
$K_{1}=K_{2}=K$ and, since in our setting
$|x-x_{0}|\geqslant\frac{\delta}{100}$, we only need to check the second claim
in (3.2).

And this claim holds true, because, by (3.7),

$\displaystyle-\frac{h(w(x))-h(\widetilde{w}(x))}{w(x)-\widetilde{w}(x)}=-\int_{0}^{1}h^{\prime}(tw(x)+(1-t)\widetilde{w}(x))\,dt$
$\displaystyle\qquad=\int_{0}^{1}\frac{2J(x,tw(x)+(1-t)\widetilde{w}(x))}{|x-x_{0}+(tw(x)+(1-t)\widetilde{w}(x))\nu(x)-w(x_{0})\nu_{0}|^{n+s}}\,dt,$

which, when $|x-x_{0}|\geqslant\frac{\delta}{100}$ is bounded from above and
below by $\frac{1}{|x-x_{0}|^{n+s}}$, up to multiplicative constants.

We also deduce from (3.18) that

(3.19) $\begin{split}&\frac{1}{2}\int_{(B_{R}(\overline{X})\setminus
B_{\delta/4}(\overline{X}))\cap{\mathcal{G}}^{\prime}}\frac{\widetilde{\chi}_{E_{2}}(X)-\widetilde{\chi}_{\widetilde{E}_{1}}(X)}{|X-X_{0}|^{n+s}}\,dX\\\
&\qquad=\int_{(\partial E_{1})\cap({B_{R}(\overline{X})\setminus
B_{\delta/4}(\overline{X})})\cap{\mathcal{G}}^{\prime}}\big{(}\widetilde{w}(x)-w(x)\big{)}\,K(x,x_{0})\,d{\mathcal{H}}^{n-1}_{x}.\end{split}$

Now we deal with the term

$\displaystyle\int_{(B_{R}(\overline{X})\setminus
B_{\delta/4}(\overline{X}))\setminus{\mathcal{G}}^{\prime}}\frac{\widetilde{\chi}_{E_{2}}(X)-\widetilde{\chi}_{\widetilde{E}_{1}}(X)}{|X-X_{0}|^{n+s}}\,dX$
$\displaystyle=$ $\displaystyle\int_{(B_{R}(\overline{X})\setminus
B_{\delta/4}(\overline{X}))\setminus{\mathcal{G}}^{\prime}}\frac{\widetilde{\chi}_{E_{2}}(X)-\widetilde{\chi}_{E_{1}}(X)}{|X-X_{0}|^{n+s}}\,dX+\int_{(B_{R}(\overline{X})\setminus
B_{\delta/4}(\overline{X}))\setminus{\mathcal{G}}^{\prime}}\frac{\widetilde{\chi}_{E_{1}}(X)-\widetilde{\chi}_{\widetilde{E}_{1}}(X)}{|X-X_{0}|^{n+s}}\,dX.$

Specifically, to estimate the latter term, we define $F_{1}$ as the symmetric
difference between $E_{1}$ and $\widetilde{E}_{1}$ and we point out that, by
(3.13),

(3.20) $\begin{split}\left|\int_{(B_{R}(\overline{X})\setminus
B_{\delta/4}(\overline{X}))\setminus{\mathcal{G}}^{\prime}}\frac{\widetilde{\chi}_{E_{1}}(X)-\widetilde{\chi}_{\widetilde{E}_{1}}(X)}{|X-X_{0}|^{n+s}}\,dX\right|&\leqslant
2\int_{((B_{R}(\overline{X})\setminus
B_{\delta/4}(\overline{X}))\setminus{\mathcal{G}}^{\prime})\cap
F_{1}}\frac{dX}{|X-X_{0}|^{n+s}}\\\
&\leqslant\frac{C}{\delta^{n+s}}\sum_{j\in\mathbb{N}}\big{|}B_{r_{j}}(p_{j})\cap
F_{1}\big{|}.\end{split}$

To complete this estimate, it is useful to observe that, for all $r>0$, all
$\tau\in\mathbb{R}^{n}$ and all measurable sets $L$, we have that

$\displaystyle\Big{|}B_{r}\cap\big{(}(L+\tau)\setminus
L\big{)}\Big{|}=\left|\int_{B_{r}\cap(L+\tau)}\,dx-\int_{B_{r}\cap
L}\,dx\right|=\left|\int_{(B_{r}-\tau)\cap L}\,dx-\int_{B_{r}\cap
L}\,dx\right|$ $\displaystyle\qquad=\Big{|}L\cap\big{(}(B_{r}-\tau)\setminus
B_{r}\big{)}\Big{|}\leqslant\big{|}(B_{r}-\tau)\setminus
B_{r}\big{|}\leqslant\big{|}B_{r+\tau}\setminus B_{r}\big{|}\leqslant
Cr^{n-1}\tau.$

Plugging this information into (3.20) and exploiting (3.14), we conclude that

$\left|\int_{(B_{R}(\overline{X})\setminus
B_{\delta/4}(\overline{X}))\setminus{\mathcal{G}}^{\prime}}\frac{\widetilde{\chi}_{E_{1}}(X)-\widetilde{\chi}_{\widetilde{E}_{1}}(X)}{|X-X_{0}|^{n+s}}\,dX\right|\leqslant\frac{Cw(x_{0})}{\delta^{n+s}}\sum_{j\in\mathbb{N}}r_{j}^{n-1}\leqslant\frac{C\eta
w(x_{0})}{\delta^{n+s}}.$

The desired result follows from this inequality and (3.19) (the asymptotics of
the kernel being similar to those discussed in Lemma 3.1). ∎

######  Lemma 3.3.

Let $\delta\in(0,1)$, $R>1+2\delta$, $\overline{X}\in\mathbb{R}^{n}$ and
$E_{1}\subseteq E_{2}\subseteq\mathbb{R}^{n}$.

Let $x_{0}\in B_{\delta/8}(\overline{X})$, $\nu_{0}\in\partial B_{1}$,
$w_{0}\in[0,\delta/4]$

$\widetilde{E}_{1}:=E_{1}+w_{0}\nu_{0}\qquad{\mbox{and}}\qquad
X_{0}:=x_{0}+w_{0}\nu_{0}.$

Then,

$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{n}\setminus
B_{R}(\overline{X})}\frac{\widetilde{\chi}_{E_{2}}(X)-\widetilde{\chi}_{\widetilde{E}_{1}}(X)}{|X-X_{0}|^{n+s}}\,dX=\frac{1}{2}\int_{\mathbb{R}^{n}\setminus
B_{R}(\overline{X})}\frac{\widetilde{\chi}_{E_{2}}(X)-\widetilde{\chi}_{E_{1}}(X)}{|X-X_{0}|^{n+s}}\,dX+O\left(\frac{w_{0}}{R^{1+s}}\right),$

where the “big $O$” term here above depends only on $n$ and $s$.

###### Proof.

We observe that

$\displaystyle\left|\int_{\mathbb{R}^{n}\setminus
B_{R}(\overline{X})}\frac{\widetilde{\chi}_{E_{1}}(X)-\widetilde{\chi}_{\widetilde{E}_{1}}(X)}{|X-X_{0}|^{n+s}}\,dX\right|$
$\displaystyle\qquad=\left|\int_{\mathbb{R}^{n}\setminus
B_{R}(\overline{X})}\frac{\widetilde{\chi}_{E_{1}}(X)}{|X-X_{0}|^{n+s}}\,dX-\int_{\mathbb{R}^{n}\setminus
B_{R}(\overline{X}-w_{0}\nu_{0})}\frac{\widetilde{\chi}_{E_{1}}(X)}{|X+w_{0}\nu_{0}-X_{0}|^{n+s}}\,dX\right|$
$\displaystyle\qquad\leqslant\left|\int_{\mathbb{R}^{n}\setminus
B_{R}(\overline{X})}\frac{\widetilde{\chi}_{E_{1}}(X)}{|X-X_{0}|^{n+s}}\,dX-\int_{\mathbb{R}^{n}\setminus
B_{R}(\overline{X}-w_{0}\nu_{0})}\frac{\widetilde{\chi}_{E_{1}}(X)}{|X-X_{0}|^{n+s}}\,dX\right|+\frac{Cw_{0}}{R^{1+s}}$
$\displaystyle\qquad\leqslant\frac{Cw_{0}}{R^{1+s}},$

up to renaming $C$, as usual, from line to line. ∎

Now we recall the notion of nonlocal mean curvature of a set $E$ at a point
$X\in\partial E$, namely

$H^{s}_{E}(X):=\int_{\mathbb{R}^{n}}\frac{\widetilde{\chi}_{E}(Y)}{|X-Y|^{n+s}}\,dY.$

Putting together Lemmata 3.1, 3.2 and 3.3, we conclude that:

######  Theorem 3.4.

Let $E_{1}\subseteq E_{2}\subseteq\mathbb{R}^{n}$. Let $\delta\in(0,1)$,
$R>1+2\delta$ and $\overline{X}\in\mathbb{R}^{n}$.

Suppose that $(\partial E_{1})\cap{\mathcal{G}}$ is a hypersurface of class
$C^{2}$, for some ${\mathcal{G}}\subseteq\mathbb{R}^{n}$ such that
${\mathcal{G}}\Supset B_{2\delta}(\overline{X})$, and let $\nu$ be the unit
external normal to $E_{1}$ in ${\mathcal{G}}$.

Suppose that

$(E_{2}\setminus E_{1})\cap{\mathcal{G}}=\Big{\\{}x+t\nu(x),\;{\mbox{with
}}x\in\partial E_{1},\;t\in\big{[}0,w(x)\big{)}\Big{\\}}\cap{\mathcal{G}},$

for some function $w$ of class $C^{2}$ with values in $[0,\delta/4]$.

Assume also that the Hausdorff distance between $\partial E_{1}$ and $\partial
E_{2}$ in $B_{2R}(\overline{X})$ is less than $\delta/20$.

Given $x_{0}\in(\partial E_{1})\cap B_{\delta/8}(\overline{X})$, let

$\widetilde{E}_{1}:=E_{1}+w(x_{0})\nu(x_{0})\qquad{\mbox{and}}\qquad
X_{0}:=x_{0}+w(x_{0})\nu(x_{0}).$

Let also ${\mathcal{G}}^{\prime}\Subset{\mathcal{G}}$ with
${\mathcal{G}}^{\prime}\Supset B_{2\delta}(\overline{X})$ and assume that

$B_{2R}(\overline{X})\setminus{\mathcal{G}}^{\prime}\subseteq\bigcup_{j\in\mathbb{N}}B_{r_{j}}(p_{j}),$

for some balls $\\{B_{r_{j}}(p_{j})\\}_{j\in\mathbb{N}}$, with

(3.21) $\eta:=\sum_{j\in\mathbb{N}}r_{j}^{n-1}<1.$

Then, if $\delta$ is small enough, and $\eta$ and
$\|w\|_{C^{2}(B_{R}(\overline{X})\cap{\mathcal{G}})}$ are small compared to
$\delta$ and $1/R$,

(3.22)
$\begin{split}&\frac{H^{s}_{E_{2}}(X_{0})-H^{s}_{\widetilde{E}_{1}}(X_{0})}{2}=\int_{(\partial
E_{1})\cap{\mathcal{G}}^{\prime}}\big{(}w(x_{0})-w(x)\big{)}\,K_{1}(x,x_{0})\,d{\mathcal{H}}^{n-1}_{x}\\\
&\qquad\qquad-w(x_{0})\int_{(\partial
E_{1})\cap{\mathcal{G}}^{\prime}}\big{(}1-\nu(x_{0})\cdot\nu(x)\big{)}\,K_{2}(x,x_{0})\,d{\mathcal{H}}^{n-1}_{x}\\\
&\qquad\qquad+\frac{1}{2}\int_{\mathbb{R}^{n}\setminus{\mathcal{G}}^{\prime}}\frac{\widetilde{\chi}_{E_{2}}(X)-\widetilde{\chi}_{E_{1}}(X)}{|X-X_{0}|^{n+s}}\,dX+O\left(w(x_{0})\eta\right)+O\left(\frac{w(x_{0})}{R^{1+s}}\right),\end{split}$

where $K_{j}:((\partial E_{1})\cap{\mathcal{G}}^{\prime})^{2}\to[0,+\infty]$,
with $j\in\\{1,2\\}$, satisfies, for a suitable $C\geqslant 1$, that

(3.23) $\begin{split}&{\mbox{for all\leavevmode\nobreak\ $x\neq x_{0}$, we
have that }}\;\frac{1}{C|x-x_{0}|^{n+s}}\leqslant
K_{j}(x,x_{0})\leqslant\frac{C}{|x-x_{0}|^{n+s}},\\\ &{\mbox{and, for
all\leavevmode\nobreak\ $y\in B_{\delta}\setminus\\{0\\}$,
}}K_{j}(y_{x_{0}}^{+},x_{0})-K_{j}(y_{x_{0}}^{-},x_{0})=O(|y|^{1-n-s}).\end{split}$

All the “big $O$” terms here above depend only on the curvatures of $\partial
E_{1}$ in $B_{9\delta/10}(\overline{X})$, on
$\|w\|_{C^{2}(B_{R}(\overline{X})\cap{\mathcal{G}})}$, on $\delta$, $R$, $n$
and $s$, except for the last one in (3.22), which depends only on $n$ and $s$.

Also, the kernel $K_{j}$ approaches, up to a normalizing constant, the kernel
$|x-x_{0}|^{-n-s}$ when $\|w\|_{C^{2}(B_{R}(\overline{X})\cap{\mathcal{G}})}$
tends to zero.

##  4\. Cut-off arguments

Having obtained a geometric equation for a nonnegative function in Theorem
3.4, our objective would be to apply a Harnack Inequality to it (actually, to
a suitable limit of it, and this will indeed be implemented in the forthcoming
Section 8). This is however rather delicate, since in principle, in our
setting, the equation will only be valid in a pointwise sense along the
regular part of a limit $s$-minimal cone, but we do not have any information
about the equation along the singular points of this cone.

Also, the equation obtained needs to be set into a distributional framework,
which, again, is not for free since the quantities involved may explode at
singular points, not allowing for a formulation in a suitable energy space.

These difficulties can be overcome by an appropriate cut-off argument, based
on a fine covering of the singular points (if any) of the limit cone. Namely,
the regularity theory of $s$-minimal cones (see [MR3090533, MR3331523]) will
allow us to confine these singular points within a small region, which can be
covered by suitable balls that produce a negligible energy term.

Similar difficulties arose in [MR3934589, Section 6] and this issue was solved
there in Lemma 6.3 via a bespoke capacity argument. We provide here a self-
contained approach. The technical result that we use goes as follows:

######  Lemma 4.1.

Let ${\mathcal{R}}\subset\mathbb{R}^{n}$ and
$S:=\overline{{\mathcal{R}}}\setminus{\mathcal{R}}$. Assume that
${\mathcal{R}}$ is locally a $C^{2}$-smooth hypersurface and that $S$ is
closed and has Hausdorff dimension $d\leqslant n-3$.

Assume also that, for every $r>0$ and $p\in\mathbb{R}^{n}$,

(4.1) ${\mathcal{H}}^{n-1}({\mathcal{R}}\cap B_{r}(p))\leqslant Cr^{n-1},$

for some $C>0$.

Let $u:{\mathcal{R}}\to[0,+\infty)$ be such that, for every
$x\in{\mathcal{R}}$,

(4.2)
$\int_{\mathcal{R}}\frac{u(x)-u(y)}{|x-y|^{n+s}}\,d{\mathcal{H}}^{n-1}_{y}\geqslant
0.$

Given $Q>0$, let $u_{Q}:=\min\\{u,Q\\}$.

Then, for every $\zeta\in W^{1,\infty}(\mathbb{R}^{n},[0,+\infty))$ with
support of $\zeta\big{|}_{\mathcal{R}}$ contained in ${\mathcal{R}}$, we have
that

(4.3)
$\iint_{{\mathcal{R}}\times{\mathcal{R}}}\frac{(u_{Q}(x)-u_{Q}(y))(\zeta(x)-\zeta(y))}{|x-y|^{n+s}}\,d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}\geqslant
0.$

Moreover, for every $R_{0}>0$,

(4.4) $\iint_{({\mathcal{R}}\cap
B_{R_{0}})\times{\mathcal{R}}}\frac{(u_{Q}(x)-u_{Q}(y))^{2}}{|x-y|^{n+s}}\,d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}<+\infty$

and, for every $\phi\in C^{\infty}_{0}(\mathbb{R}^{n},[0,+\infty))$,

(4.5)
$\iint_{{\mathcal{R}}\times{\mathcal{R}}}\frac{(u_{Q}(x)-u_{Q}(y))(\phi(x)-\phi(y))}{|x-y|^{n+s}}\,d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}\geqslant
0.$

For the reader’s facility, the rather technical proof of Lemma 4.1 will be
given in Appendix A.

##  5\. Hölder estimates for fractional operators in a geometric setting

Here we present a Hölder regularity result in a geometric setting. Namely,
differently from the cases already treated in the literature, the equation
considered here is defined by an integral on a portion of a hypersurface and
the kernel is not necessarily symmetric, but only symmetric up to a suitable
remainder.

######  Lemma 5.1.

Let ${\mathcal{R}}$ be a portion of a $C^{2}$ hypersurface in $B_{2}$ that is
$C^{2}$-diffeomorphic to $B_{2}\cap\\{x_{n}=0\\}$. Let $\nu$ be the unit
normal vector field of ${\mathcal{R}}$ and $f\in L^{\infty}(B_{2})$.

Let $K:{\mathcal{R}}\times{\mathcal{R}}\to[0,+\infty]$ be such that

(5.1) $\begin{split}&{\mbox{for all\leavevmode\nobreak\ $x\neq x_{0}$, we have
that }}\;\frac{1}{C|x-x_{0}|^{n+s}}\leqslant
K(x,x_{0})\leqslant\frac{C}{|x-x_{0}|^{n+s}},\\\ &{\mbox{and, for
all\leavevmode\nobreak\ $y\in B_{\delta}\setminus\\{0\\}$,
}}|K(y_{x_{0}}^{+},x_{0})-K(y_{x_{0}}^{-},x_{0})|\leqslant
C|y|^{1-n-s},\end{split}$

for some $C\geqslant 1$.

Assume that, for all $x_{0}\in{{\mathcal{R}}}\cap B_{1}$, the function $v\in
L^{\infty}({\mathcal{R}})$ is a solution of

$\int_{{\mathcal{R}}}\big{(}v(x_{0})-v(x)\big{)}\,K(x,x_{0})\,d{\mathcal{H}}^{n-1}_{x}=f(x_{0}).$

Then, $v$ is Hölder continuous in ${{\mathcal{R}}}\cap B_{1/2}$. More
precisely,

(5.2) $\|v\|_{C^{\alpha}({\mathcal{R}}\cap B_{1/2})}\leqslant
C_{0}\Big{(}\|f\|_{L^{\infty}(B_{2})}+\|v\|_{L^{\infty}({\mathcal{R}})}\Big{)},$

where $\alpha\in(0,1)$ and $C_{0}>0$ depend only on $n$, $s$, the regularity
parameters of ${\mathcal{R}}$ and the structural constant $C$ of the kernel
$K$ in (5.1).

###### Proof.

We let $u:=v\chi_{B_{3/4}}$ and we see that, for all
$x_{0}\in{{\mathcal{R}}}\cap B_{2/3}$,

$\displaystyle\int_{{\mathcal{R}}}\big{(}u(x_{0})-u(x)\big{)}\,K(x,x_{0})\,d{\mathcal{H}}^{n-1}_{x}=\int_{{\mathcal{R}}}\big{(}v(x_{0})-v(x)\chi_{B_{3/4}}(x)\big{)}\,K(x,x_{0})\,d{\mathcal{H}}^{n-1}_{x}$
$\displaystyle\qquad=f(x_{0})+\int_{{\mathcal{R}}\setminus
B_{3/4}}v(x)\,K(x,x_{0})\,d{\mathcal{H}}^{n-1}_{x}=:g(x_{0})$

and

$\displaystyle|g(x_{0})|\leqslant\|f\|_{L^{\infty}(B_{2})}+C\|v\|_{L^{\infty}({\mathcal{R}})}\int_{{\mathcal{R}}\setminus
B_{3/4}}\frac{d{\mathcal{H}}^{n-1}_{x}}{|x-x_{0}|^{n+s}}$
$\displaystyle\qquad\qquad\qquad\leqslant
C\Big{(}\|f\|_{L^{\infty}(B_{2})}+{\mathcal{H}}^{n-1}({\mathcal{R}})\|v\|_{L^{\infty}({\mathcal{R}})}\Big{)}=:C_{\star},$

up to renaming $C$ from line to line.

We now take $B_{2}^{\prime}:=B_{2}\cap\\{x_{n}=0\\}$ and a
$C^{2}$-diffeomorphism $\phi:{\mathcal{R}}\to B^{\prime}_{2}$. We can also
adapt the diffeomorphism so that $\phi({\mathcal{R}}\cap B_{2/3})$ contains
$B^{\prime}_{3/4}$. We define $U(y):=u(\phi^{-1}(y))$ and we observe that, for
all $y_{0}=\phi(x_{0})\in B^{\prime}_{3/4}$,

(5.3)
$\begin{split}&G(y_{0}):=g(\phi^{-1}(y_{0}))=\int_{{\mathcal{R}}}\big{(}u(\phi^{-1}(y_{0}))-u(x)\big{)}\,K(x,\phi^{-1}(y_{0}))\,d{\mathcal{H}}^{n-1}_{x}\\\
&\qquad=\int_{B^{\prime}_{2}}\big{(}U(y_{0})-U(y)\big{)}\,K_{*}(y,y_{0})\,d{\mathcal{H}}^{n-1}_{y},\end{split}$

where

(5.4) $K_{*}(y,y_{0}):=K(\phi^{-1}(y),\phi^{-1}(y_{0}))\,J(y),$

for a suitable Jacobian function $J$, which is bounded and bounded away from
zero.

Thus, by (5.1), we have that $K_{*}(y,y_{0})$ is bounded from above and below,
up to constants, by $|\phi^{-1}(y)-\phi^{-1}(y_{0})|^{-(n+s)}$, which in turn
is comparable to ${|y-y_{0}|^{-(n+s)}}$, since $\phi$ is a diffeomorphism.

Additionally, if

$L(y_{0},z):=K_{*}(y_{0}+z,y_{0})-K_{*}(y_{0}-z,y_{0}),$

we have that

(5.5)
$\begin{split}|L(y_{0},z)|&\leqslant|K(\phi^{-1}(y_{0}+z),\phi^{-1}(y_{0}))-K(\phi^{-1}(y_{0}-z),\phi^{-1}(y_{0}))|\,J(y_{0}+z)\\\
&\qquad\qquad\qquad+|K(\phi^{-1}(y_{0}-z),\phi^{-1}(y_{0}))|\,|J(y_{0}+z)-J(y_{0}-z)|\\\
&\leqslant
C|K(\phi^{-1}(y_{0}+z),\phi^{-1}(y_{0}))-K(\phi^{-1}(y_{0}-z),\phi^{-1}(y_{0}))|\\\
&\qquad\qquad\qquad+C|z|^{-n-s}\,|J(y_{0}+z)-J(y_{0}-z)|\\\ &\leqslant
C|z|^{1-n-s},\end{split}$

up to renaming $C$ line after line.

Actually, up to renaming $G$ by an additional bounded function, if we extend
$U$ by zero outside $B^{\prime}_{2}$ we can rewrite (5.3) in the form

(5.6)
$\int_{\mathbb{R}^{n-1}}\big{(}U(y_{0})-U(y)\big{)}\,K_{*}(y,y_{0})\,d{\mathcal{H}}^{n-1}_{y}=G(y_{0}),$

for all $y_{0}\in B_{3/4}^{\prime}$.

We can thereby apply the regularity theory for integro-differential equations
(see e.g. [MR2244602, Theorem 5.4 and Remark 4.4], or [MR2995098, Theorem
3.1], or [KW, Theorem 1.1(ii)]) and obtain the desired Hölder estimate for
$U$, and therefore for $u$, and then for $v$. ∎

##  6\. An upper bound in a geometric setting

In the proof of our main result, as it will be presented in Section 8, a
somewhat delicate issue comes from the possibility of extracting a convergent
sequence from the renormalized distance between minimal sheets. Roughly
speaking, the plan would be to remove an arbitrarily small neighborhood of the
singular set and focus on an arbitrarily large ball, obtain estimates in this
domain which are uniform with respect to the sequences under consideration
(with constants possibly depending on the neighborhood of the singular set and
on the radius of the large ball), pass the sequence to the limit, and only at
the end of the argument shrink the neighborhood of the singular set and invade
the space by larger and larger balls.

In this setting, Lemma 5.1 is instrumental to provide the necessary
compactness. However, to use this result, one needs a bound in $L^{\infty}$,
as dictated by the right-hand side of (5.2).

Such a bound does not come completely for free, not even in the situation in
which the distance between minimal sheets is normalized to be $1$ at some
point, due to the possible divergence of this function at the singular set and
the correspondingly large values that this function could, in principle,
attain in the domain under consideration. To get around such a difficulty, we
present here an $L^{\infty}$ bound in terms of an integral estimate (as made
precise by (6.3) here below). This integral control will be then checked in
our specific case via a sliding method (as made precise by the forthcoming
Lemma 7.1).

This strategy will thus allow us to pass to the limit locally away from the
singular set: summarizing, one needs to just establish an integral bound (that
will come from Lemma 7.1), to deduce from it a bound in $L^{\infty}$ (coming
from the next result), and thus obtain a bound in $C^{\alpha}$ (coming from
Lemma 5.1), which in turns provides the desired compactness property (by the
Arzelà-Ascoli Theorem).

######  Lemma 6.1.

Let ${\mathcal{R}}$ be a portion of a $C^{2}$ hypersurface in $B_{2}$ that is
$C^{2}$-diffeomorphic to $B_{2}\cap\\{x_{n}=0\\}$. Let $\nu$ the unit normal
vector field of ${\mathcal{R}}$.

Let $K:{\mathcal{R}}\times{\mathcal{R}}\to[0,+\infty]$ be such that

(6.1) $\begin{split}&{\mbox{for all\leavevmode\nobreak\ $x\neq x_{0}$, we have
that }}\;\frac{1}{C|x-x_{0}|^{n+s}}\leqslant
K(x,x_{0})\leqslant\frac{C}{|x-x_{0}|^{n+s}},\\\ &{\mbox{and, for
all\leavevmode\nobreak\ $y\in B_{\delta}\setminus\\{0\\}$,
}}|K(y_{x_{0}}^{+},x_{0})-K(y_{x_{0}}^{-},x_{0})|\leqslant
C|y|^{1-n-s},\end{split}$

for some $C\geqslant 1$.

Assume that, for all $x_{0}\in{{\mathcal{R}}}\cap B_{1}$, the function $v$
satisfies

(6.2)
$\begin{split}&\int_{{{\mathcal{R}}}}\big{(}v(x_{0})-v(x)\big{)}\,K(x,x_{0})\,d{\mathcal{H}}^{n-1}_{x}-a(x_{0})v(x_{0})\leqslant
M,\end{split}$

for some $M\geqslant 0$ and $a\in L^{\infty}(\mathbb{R}^{n})$, and that

(6.3) $\int_{{\mathcal{R}}}v^{+}(x)\,d{\mathcal{H}}^{n-1}_{x}\leqslant
M^{\prime},$

for some $M^{\prime}\geqslant 0$.

Then, $v$ is bounded from above in ${\mathcal{R}}\cap B_{1/2}$, with

$\sup_{{\mathcal{R}}\cap B_{1/2}}v\leqslant C_{0}\left(1+M+M^{\prime}\right),$

with $C_{0}>0$ depending only on $n$, $s$,
$\|a\|_{L^{\infty}(\mathbb{R}^{n})}$, the regularity parameters of
${\mathcal{R}}$ and the structural constant $C$ of the kernel $K$ in (6.1).

###### Proof.

The argument presented here extends to the geometric framework and for more
general kernels a method of proof utilized in [MR3661864, Lemma 5.2].
Specifically, recalling the notation $B^{\prime}_{r}:=B_{r}\cap\\{x_{n}=0\\}$,
as in (5.4) we consider a $C^{2}$-diffeomorphism $\phi:{\mathcal{R}}\to
B^{\prime}_{2}$, define $V(y):=v(\phi^{-1}(y))$ and look at the corresponding
integral equation driven by the kernel $K_{*}$

We look at the function $B^{\prime}_{1}\ni
y\mapsto\Psi(y):=\Theta\,(1-|y|^{2})^{-n}$, with $\Theta>0$ suitably large,
and we slide the graph of $\Psi$ by above in $B_{1}^{\prime}$ till we touch
the graph of $V$. That is, we find $t\in\mathbb{R}$ such that
$V\leqslant\Psi+t$ in $B_{1}^{\prime}$ with equality holding at some $q\in
B_{1}^{\prime}$. We can assume that $t\geqslant 0$ (otherwise,
$v\leqslant\Psi+t\leqslant\Psi$ and this would give the desired bound).

Let also $d:=1-|q|$. Let $r\in\left(0,\frac{d}{2}\right)$ to be taken suitably
small. We consider the tangent plane of the barrier $\Psi+t$ at $q$, namely
the linear function

$\ell(y):=\nabla\Psi(q)\cdot(y-q).$

It would be desirable to freely subtract tangent planes in the operators, but
this is not possible in our framework, due to the lack of symmetry of the
kernel, therefore we need to take care of an additional remainder. Namely,

$\displaystyle\left|2\int_{B^{\prime}_{r}(q)}\ell(y)\,K_{*}(y,q)\,d{\mathcal{H}}^{n-1}_{y}\right|=\left|2\nabla\Psi(q)\cdot\int_{B_{r}^{\prime}(q)}(y-q)\,K_{*}(y,q)\,d{\mathcal{H}}^{n-1}_{y}\right|$
$\displaystyle\qquad=\left|\nabla\Psi(q)\cdot\int_{B_{r}^{\prime}}z\big{(}K_{*}(q+z,q)-K_{*}(q-z,q)\big{)}\,d{\mathcal{H}}^{n-1}_{z}\right|.$

Now, as observed in (5.5), it follows from (6.1) that

$\displaystyle|K_{*}(q+z,q)-K_{*}(q-z,q)|\leqslant C|z|^{1-n-s}$

As a result,

$\begin{split}&\left|2\int_{B^{\prime}_{r}(q)}\ell(y)\,K_{*}(y,q)\,d{\mathcal{H}}^{n-1}_{y}\right|\leqslant|\nabla\Psi(q)|\,\int_{B_{r}^{\prime}}|z|\,\big{|}K_{*}(q+z,q)-K_{*}(q-z,q)\big{|}\,d{\mathcal{H}}^{n-1}_{z}\\\
&\qquad\qquad\leqslant
C|\nabla\Psi(q)|\,\int_{B_{r}^{\prime}}|z|^{2-n-s}\,d{\mathcal{H}}^{n-1}_{z}\leqslant
C|\nabla\Psi(q)|\,r^{1-s}.\end{split}$

Consequently, if $q:=\phi(p)$ and
${\mathcal{R}}_{r}(p):=\phi^{-1}(B_{r}^{\prime}(q))$,

(6.4)
$\begin{split}&\int_{{\mathcal{R}}_{r}(p)}\big{(}v(p)-v(x)\big{)}\,K(x,p)\,d{\mathcal{H}}^{n-1}_{x}\\\
&\qquad=\int_{B_{r}^{\prime}(q)}\big{(}V(q)-V(y)+\ell(y)\big{)}\,K_{*}(y,q)\,d{\mathcal{H}}^{n-1}_{y}-\int_{B^{\prime}_{r}(q)}\ell(y)\,K_{*}(y,q)\,d{\mathcal{H}}^{n-1}_{y}\\\
&\qquad\geqslant-\int_{B^{\prime}_{r}(q)}\big{(}V(y)-V(q)-\ell(y)\big{)}\,K_{*}(y,q)\,d{\mathcal{H}}^{n-1}_{x}-C|\nabla\Psi(q)|\,r^{1-s}.\end{split}$

Now we use the notation of positive and negative parts of a function, namely
$g=g_{+}-g_{-}$, where $g_{+}:=\max\\{g,0\\}$ and $g_{-}:=\max\\{-g,0\\}$, to
see that, if $y\in B^{\prime}_{r}(q)$,

$|y|\leqslant|q|+|y-q|\leqslant 1-d+r\leqslant 1-\frac{d}{2}$

and, as a consequence,

$\displaystyle\big{(}V(y)-V(q)-\ell(y)\big{)}_{+}$ $\displaystyle\leqslant$
$\displaystyle\max\\{\Psi(y)-\Psi(q)-\ell(y),0\\}$ $\displaystyle\leqslant$
$\displaystyle\frac{C|y-q|^{2}}{d^{n+2}}.$

For this reason,

(6.5)
$\begin{split}&\int_{B^{\prime}_{r}(q)}\big{(}V(y)-V(q)-\ell(y)\big{)}_{+}\,K_{*}(y,q)\,d{\mathcal{H}}^{n-1}_{y}\leqslant\frac{C}{d^{n+2}}\int_{B^{\prime}_{r}(q)}|y-q|^{2}\,K_{*}(y,q)\,d{\mathcal{H}}^{n-1}_{y}\\\
&\qquad\qquad\qquad\leqslant\frac{C}{d^{n+2}}\int_{0}^{r}\frac{d\rho}{\rho^{s}}=\frac{Cr^{1-s}}{d^{n+2}}.\end{split}$

Moreover,

$\displaystyle\big{(}V(y)-V(q)-\ell(y)\big{)}_{-}$ $\displaystyle\geqslant$
$\displaystyle V(q)-V(y)+\ell(y)$ $\displaystyle\geqslant$
$\displaystyle\Psi(q)+t-V^{+}(y)+\ell(y)$

and therefore

$\begin{split}&\int_{B^{\prime}_{r}(q)}\big{(}V(y)-V(q)-\ell(y)\big{)}_{-}\,K_{*}(y,q)\,d{\mathcal{H}}^{n-1}_{y}\\\
&\qquad\geqslant\frac{1}{C}\int_{B_{r}^{\prime}(q)}\big{(}V(y)-V(q)-\ell(y)\big{)}_{-}\,\frac{d{\mathcal{H}}^{n-1}_{y}}{|y-q|^{n+s}}\\\
&\qquad\geqslant\frac{1}{Cr^{n+s}}\int_{B_{r}^{\prime}(q)}\big{(}V(y)-V(q)-\ell(y)\big{)}_{-}\,d{\mathcal{H}}^{n-1}_{y}\\\
&\qquad\geqslant\frac{1}{Cr^{n+s}}\int_{B_{r}^{\prime}(q)}\big{(}\Psi(q)+t-V^{+}(y)+\ell(y)\big{)}\,d{\mathcal{H}}^{n-1}_{y}\\\
&\qquad\geqslant\frac{1}{Cr^{n+s}}\left(\big{(}\Psi(q)+t\big{)}r^{n-1}-\int_{B_{r}^{\prime}(q)}V^{+}(y)\,d{\mathcal{H}}^{n-1}_{y}\right)-\frac{C|\nabla\Psi(q)|}{r^{s}}\\\
&\qquad\geqslant\frac{\Psi(q)+t}{Cr^{1+s}}-\frac{C}{r^{n+s}}\int_{{\mathcal{R}}_{r}(p)}v^{+}(x)\,d{\mathcal{H}}^{n-1}_{x}-\frac{C|\nabla\Psi(q)|}{r^{s}}.\end{split}$

Note that

$\Psi(q)=\Theta\,(1-|q|^{2})^{-n}=\Theta\,(1+|q|)^{-n}(1-|q|)^{-n}\geqslant
2^{-n}\Theta d^{-n}$

and

$|\nabla\Psi(q)|=\frac{2n\Theta|q|}{(1-|q|^{2})^{n+1}}\leqslant\frac{2n\Theta}{d^{n+1}},$

giving that

$\begin{split}&\int_{B_{r}^{\prime}(q)}\big{(}V(y)-V(q)-\ell(y)\big{)}_{-}\,K_{*}(y,q)\,d{\mathcal{H}}^{n-1}_{y}\\\
&\qquad\geqslant\frac{\Theta}{Cd^{n}r^{1+s}}+\frac{t}{Cr^{1+s}}-\frac{C}{r^{n+s}}\int_{{\mathcal{R}}_{r}(p)}v^{+}(x)\,d{\mathcal{H}}^{n-1}_{x}-\frac{C\Theta}{d^{n+1}r^{s}}.\end{split}$

Notice that

$\displaystyle\frac{\Theta}{Cd^{n}r^{1+s}}-\frac{C\Theta}{d^{n+1}r^{s}}=\frac{\Theta}{Cd^{n}r^{s}}\left(\frac{1}{r}-\frac{C^{2}}{d}\right)>\frac{\Theta}{Cd^{n}r^{1+s}}$

up to renaming $C$, if $r$ is smaller than a small constant times $d$, and
therefore

$\begin{split}&\int_{B_{r}^{\prime}(q)}\big{(}V(y)-V(q)-\ell(y)\big{)}_{-}\,K_{*}(y,q)\,d{\mathcal{H}}^{n-1}_{y}\\\
&\qquad\geqslant\frac{\Theta}{Cd^{n}r^{1+s}}+\frac{t}{Cr^{1+s}}-\frac{C}{r^{n+s}}\int_{{\mathcal{R}}_{r}(p)}v^{+}(x)\,d{\mathcal{H}}^{n-1}_{x}.\end{split}$

Combining this estimate and (6.5), we infer that

$\displaystyle\int_{B_{r}^{\prime}(q)}\big{(}V(y)-V(q)-\ell(y)\big{)}\,K_{*}(y,q)\,d{\mathcal{H}}^{n-1}_{y}$
$\displaystyle\qquad\leqslant-\frac{\Theta}{Cd^{n}r^{1+s}}-\frac{t}{Cr^{1+s}}+\frac{C}{r^{n+s}}\int_{{\mathcal{R}}_{r}(p)}v^{+}(x)\,d{\mathcal{H}}^{n-1}_{x}+\frac{Cr^{1-s}}{d^{n+2}}.$

From this and (6.4) we conclude that

(6.6)
$\begin{split}&\int_{{\mathcal{R}}_{r}(p)}\big{(}v(p)-v(x)\big{)}\,K(x,p)\,d{\mathcal{H}}^{n-1}_{x}\\\
&\qquad\geqslant\frac{\Theta}{Cd^{n}r^{1+s}}+\frac{t}{Cr^{1+s}}-\frac{C}{r^{n+s}}\int_{{\mathcal{R}}_{r}(p)}v^{+}(x)\,d{\mathcal{H}}^{n-1}_{x}-\frac{Cr^{1-s}}{d^{n+2}},\end{split}$

where, as customary, we have renamed constants line after line.

Also,

$v(p)=V(q)=\Psi(q)+t\geqslant\Psi(q)\geqslant 0$

and therefore

$\displaystyle\int_{{\mathcal{R}}\setminus{\mathcal{R}}_{r}(p)}\big{(}v(p)-v(x)\big{)}\,K(x,p)\,d{\mathcal{H}}^{n-1}_{x}\geqslant-C\int_{{\mathcal{R}}\setminus{\mathcal{R}}_{r}(p)}v^{+}(x)\,\frac{d{\mathcal{H}}^{n-1}_{x}}{|x-p|^{n+s}}$
$\displaystyle\qquad\geqslant-\frac{C}{r^{n+s}}\int_{{\mathcal{R}}\setminus{\mathcal{R}}_{r}(p)}v^{+}(x)\,d{\mathcal{H}}^{n-1}_{x}\geqslant-\frac{C}{r^{n+s}}\int_{{\mathcal{R}}}v^{+}(x)\,d{\mathcal{H}}^{n-1}_{x}.$

This, (6.2) and (6.6) yield that, if $\Theta$ is large as specified above,

$\displaystyle M$ $\displaystyle\geqslant$
$\displaystyle\int_{{{\mathcal{R}}}}\big{(}v(p)-v(x)\big{)}\,K(x,p)\,d{\mathcal{H}}^{n-1}_{x}-a(p)v(p)$
$\displaystyle\geqslant$
$\displaystyle\frac{\Theta}{Cd^{n}r^{1+s}}+\frac{t}{Cr^{1+s}}-\frac{C}{r^{n+s}}\int_{{\mathcal{R}}}v^{+}(x)\,d{\mathcal{H}}^{n-1}_{x}-\frac{Cr^{1-s}}{d^{n+2}}-a(p)(\Psi(q)+t)$
$\displaystyle\geqslant$
$\displaystyle\frac{\Theta}{Cd^{n}r^{1+s}}+\frac{t}{Cr^{1+s}}-\frac{C}{r^{n+s}}\int_{{\mathcal{R}}}v^{+}(x)\,d{\mathcal{H}}^{n-1}_{x}-\frac{Cr^{1-s}}{d^{n+2}}-a(p)\Psi(q)$
$\displaystyle\geqslant$
$\displaystyle\frac{\Theta}{Cd^{n}r^{1+s}}+\frac{t}{Cr^{1+s}}-\frac{C}{r^{n+s}}\int_{{\mathcal{R}}}v^{+}(x)\,d{\mathcal{H}}^{n-1}_{x}-\frac{Cr^{1-s}}{d^{n+2}}-\frac{C\Theta}{d^{n}}.$

Notice that for $r$ small enough, the latter term can be reabsorbed. In
particular, taking $r=\alpha d$, for a small $\alpha\in(0,1)$, we find that

$\displaystyle M$ $\displaystyle\geqslant$
$\displaystyle\frac{\Theta}{Cd^{n}r^{1+s}}+\frac{t}{Cr^{1+s}}-\frac{C}{r^{n+s}}\int_{{\mathcal{R}}}v^{+}(x)\,d{\mathcal{H}}^{n-1}_{x}-\frac{Cr^{1-s}}{d^{n+2}}$
$\displaystyle=$
$\displaystyle\frac{\Theta}{C\alpha^{1+s}d^{n+1+s}}+\frac{t}{C\alpha^{1+s}d^{1+s}}-\frac{C}{\alpha^{n+s}d^{n+s}}\int_{{\mathcal{R}}}v^{+}(x)\,d{\mathcal{H}}^{n-1}_{x}-\frac{C\alpha^{1-s}}{d^{n+1+s}}$
$\displaystyle\geqslant$
$\displaystyle\frac{t}{C\alpha^{1+s}d^{1+s}}+\frac{\Theta}{\alpha^{1+s}d^{n+1+s}}\left[\frac{1}{C}-\frac{Cd}{\alpha^{n-1}\Theta}\int_{{\mathcal{R}}}v^{+}(x)\,d{\mathcal{H}}^{n-1}_{x}-\frac{C\alpha^{2}}{\Theta}\right].$

The smallness of $\alpha$ having played its role, we omit it from the notation
from now on, absorbing it into the constants. In particular, if

$\Theta:=C\left(1+\int_{{\mathcal{R}}}v^{+}(x)\,d{\mathcal{H}}^{n-1}_{x}\right),$

with $C$ large enough, we have that

$\displaystyle
M\geqslant\frac{t}{Cd^{1+s}}+\frac{\Theta}{Cd^{n+1+s}}\geqslant\frac{t}{Cd^{1+s}}.$

All in all, we have found that, if $\xi\in{\mathcal{R}}\cap B_{1/2}$,

$\displaystyle v(\xi)=V(\phi(\xi))$ $\displaystyle\leqslant$
$\displaystyle\Psi(\phi(\xi))+t$ $\displaystyle\leqslant$ $\displaystyle
C\left(1+M+\int_{{\mathcal{R}}}v^{+}(x)\,d{\mathcal{H}}^{n-1}_{x}\right)(1-|\phi(\xi)|^{2})^{-n}$
$\displaystyle\leqslant$ $\displaystyle
C\left(1+M+\int_{{\mathcal{R}}}v^{+}(x)\,d{\mathcal{H}}^{n-1}_{x}\right),$

up to renaming $C$. ∎

##  7\. An integral bound in a geometric setting

In retrospect, the Hölder regularity theory established in Lemma 5.1 relied on
a specific hypothesis, namely an $L^{\infty}$ bound, which in turn was reduced
to an integral bound by means of Lemma 6.1.

However, in our setting, integral bounds do not come completely for free. As
already mentioned, the complication arises from the fact that we need to apply
this regularity theory to a normalized parameterization between minimal
sheets: thus, in view of the possible divergence of normal parameterizations
at singular points, $L^{\infty}$ bounds, and even $L^{1}$ bounds, may be
nontrivial. The next result fills this gap and ensures an $L^{1}$ bound,
which, in our application, will lead to an $L^{\infty}$ bound (via Lemma 6.1)
and thus to a Hölder estimate (owing to Lemma 5.1), from which one will deduce
suitable compactness properties of the normalized distance of minimal sheets.

######  Lemma 7.1.

Let ${\mathcal{R}}$ be a portion of a $C^{2}$ hypersurface in $B_{2}$ that is
$C^{2}$-diffeomorphic to $B_{2}\cap\\{x_{n}=0\\}$. Let $\nu$ be the unit
normal vector field of ${\mathcal{R}}$.

Let $K:{\mathcal{R}}\times{\mathcal{R}}\to[0,+\infty]$ be such that

(7.1) $\begin{split}&{\mbox{for all\leavevmode\nobreak\ $x\neq x_{0}$, we have
that }}\;\frac{1}{C|x-x_{0}|^{n+s}}\leqslant
K(x,x_{0})\leqslant\frac{C}{|x-x_{0}|^{n+s}},\\\ &{\mbox{and, for
all\leavevmode\nobreak\ $y\in B_{\delta}\setminus\\{0\\}$,
}}|K(y_{x_{0}}^{+},x_{0})-K(y_{x_{0}}^{-},x_{0})|\leqslant
C|y|^{1-n-s},\end{split}$

for some $C\geqslant 1$.

Assume that, for all $x_{0}\in{{\mathcal{R}}}\cap B_{1}$, the function $v$
satisfies

(7.2)
$\begin{split}&\int_{\mathcal{R}}\big{(}v(x_{0})-v(x)\big{)}\,K(x,x_{0})\,d{\mathcal{H}}^{n-1}_{x}-a(x_{0})v(x_{0})\in[-M_{0},M_{0}],\end{split}$

for some $M_{0}>0$ and $a\in L^{\infty}(\mathbb{R}^{n},[0,+\infty))$.

Assume also that $v$ is nonnegative and that $v(x_{\star})=1$, for some
$x_{\star}\in{{\mathcal{R}}}\cap B_{1}$.

Then,

$\int_{{\mathcal{R}}}v(x)\,d{\mathcal{H}}^{n-1}_{x}\leqslant C_{0},$

with $C_{0}>0$ depending only on $n$, $s$, $M_{0}$,
$\|a\|_{L^{\infty}(\mathbb{R}^{n})}$, the regularity parameters of
${\mathcal{R}}$ and the structural constant $C$ of the kernel $K$ in (7.1).

###### Proof.

We straighten ${\mathcal{R}}$ by a diffeomorphism $\phi:{\mathcal{R}}\to
B^{\prime}_{2}:=B_{2}\cap\\{x_{n}=0\\}$ and set $V(y):=v(\phi^{-1}(y))$. We
let $y_{\star}:=\phi(x_{\star})$ and slide the parabola $-M|y-y_{\star}|^{2}$
from below till we touch the graph of $V$ at some point $y_{\sharp}$. Notice
that

$M|y_{\sharp}-y_{\star}|^{2}\leqslant
V(y_{\sharp})+M|y_{\sharp}-y_{\star}|^{2}\leqslant
V(y_{\star})=v(x_{\star})=1.$

Thus, the parameter $M>0$ is fixed (once and for all) sufficiently large such
that the touching point $y_{\sharp}$ lies in $B_{3/2}^{\prime}$.

We remark that

$P(y):=V(y_{\sharp})+M|y_{\sharp}-y_{\star}|^{2}-M|y-y_{\star}|^{2}\leqslant
V(y)$

and in particular $P(y_{\sharp})=V(y_{\sharp})\geqslant 0$.

Hence, if $\widetilde{V}:=V-P$, we have that $\widetilde{V}(y_{\sharp})=0$ and
thus, recalling the notation in (5.4),

$\displaystyle\int_{B_{2}^{\prime}}\widetilde{V}(y)\,K_{*}(y,y_{\sharp})\,d{\mathcal{H}}^{n-1}_{y}=\int_{B_{2}^{\prime}}\big{(}\widetilde{V}(y)-\widetilde{V}(y_{\sharp})\big{)}\,K_{*}(y,y_{\sharp})\,d{\mathcal{H}}^{n-1}_{y}$
$\displaystyle\qquad=\int_{B_{2}^{\prime}}\big{(}V(y)-V(y_{\sharp})\big{)}\,K_{*}(y,y_{\sharp})\,d{\mathcal{H}}^{n-1}_{y}-\int_{B_{2}^{\prime}}\big{(}P(y)-P(y_{\sharp})\big{)}\,K_{*}(y,y_{\sharp})\,d{\mathcal{H}}^{n-1}_{y}$
$\displaystyle\qquad\leqslant\int_{B_{2}^{\prime}}\big{(}V(y)-V(y_{\sharp})\big{)}\,K_{*}(y,y_{\sharp})\,d{\mathcal{H}}^{n-1}_{y}+C_{0}$
$\displaystyle\qquad=\int_{\mathcal{R}}\big{(}v(x)-v(x_{\sharp})\big{)}\,K(x,x_{\sharp})\,d{\mathcal{H}}^{n-1}_{x}+C_{0},$

with $C_{0}$ bounded in terms of $n$, $s$, the regularity parameters of
${\mathcal{R}}$ and the structural constant $C$ of the kernel $K$ in (7.1).

From this observation and (7.2), we infer that

$\displaystyle\int_{B_{2}^{\prime}}\widetilde{V}(y)\,K_{*}(y,y_{\sharp})\,d{\mathcal{H}}^{n-1}_{y}\leqslant
M_{0}-a(x_{\sharp})v(x_{\sharp})+C_{0}\leqslant C_{0},$

with $C_{0}$ now depending also on $M_{0}$.

Since $\widetilde{V}\geqslant 0$, we thereby conclude that

$\displaystyle\int_{B_{1/100}^{\prime}(y_{\sharp})}\widetilde{V}(y)\,K_{*}(y,y_{\sharp})\,d{\mathcal{H}}^{n-1}_{y}\leqslant
C_{0}.$

We point out that, if $y\in B_{1/100}^{\prime}(y_{\sharp})$,

$\displaystyle
K_{*}(y,y_{\sharp})\geqslant\frac{1}{C|y-y_{\sharp}|^{n+s}}\geqslant\frac{100^{n+s}}{C},$

and thus

$\int_{B_{1/100}^{\prime}(y_{\sharp})}\widetilde{V}(y)\,d{\mathcal{H}}^{n-1}_{y}\leqslant\frac{C}{100^{n+s}}\int_{B_{1/100}^{\prime}(y_{\sharp})}\widetilde{V}(y)\,K_{*}(y,y_{\sharp})\,d{\mathcal{H}}^{n-1}_{y}\leqslant
C_{0}.$

As a result, up to keeping renaming $C_{0}$,

(7.3)
$\begin{split}&\int_{B_{1/100}^{\prime}(y_{\sharp})}V(y)\,d{\mathcal{H}}^{n-1}_{y}\leqslant\int_{B_{1/100}^{\prime}(y_{\sharp})}P(y)\,d{\mathcal{H}}^{n-1}_{y}+C_{0}\leqslant
C_{0}.\end{split}$

Furthermore,

$\displaystyle\int_{B_{1/100}^{\prime}(y_{\sharp})}\big{(}V(y_{\sharp})-V(y)\big{)}\,K_{*}(y,y_{\sharp})\,d{\mathcal{H}}^{n-1}_{y}$
$\displaystyle\qquad\leqslant\int_{B_{1/100}^{\prime}(y_{\sharp})}\big{(}P(y_{\sharp})-P(y)\big{)}\,K_{*}(y,y_{\sharp})\,d{\mathcal{H}}^{n-1}_{y}\leqslant
C_{0}$

and therefore one deduces from (7.2) that

$\displaystyle C_{0}+\int_{{B^{\prime}_{2}\setminus
B_{1/100}^{\prime}(y_{\sharp})}}\big{(}V(y_{\sharp})-V(y)\big{)}\,K_{*}(y,y_{\sharp})\,d{\mathcal{H}}^{n-1}_{y}$
$\displaystyle\qquad\geqslant\int_{B_{2}^{\prime}}\big{(}V(y_{\sharp})-V(y)\big{)}\,K_{*}(y,y_{\sharp})\,d{\mathcal{H}}^{n-1}_{y}$
$\displaystyle\qquad=\int_{\mathcal{R}}\big{(}v(x_{\sharp})-v(x)\big{)}\,K(x,x_{\sharp})\,d{\mathcal{H}}^{n-1}_{x}$
$\displaystyle\qquad\geqslant a(x_{\sharp})v(x_{\sharp})-C_{0}$
$\displaystyle\qquad\geqslant-C_{0}.$

This gives that

$\displaystyle\int_{B_{2}^{\prime}\setminus
B_{1/100}^{\prime}(y_{\sharp})}V(y)\,K_{*}(y,y_{\sharp})\,d{\mathcal{H}}^{n-1}_{y}$
$\displaystyle\qquad\leqslant V(y_{\sharp})\int_{B_{2}^{\prime}\setminus
B_{1/100}^{\prime}(y_{\sharp})}\,K_{*}(y,y_{\sharp})\,d{\mathcal{H}}^{n-1}_{y}+C_{0}$
$\displaystyle\qquad=C_{0}\big{(}V(y_{\sharp})+1\big{)}$
$\displaystyle\qquad\leqslant
C_{0}\big{(}V(y_{\star})-M|y_{\sharp}-y_{\star}|^{2}+1\big{)}$
$\displaystyle\qquad\leqslant C_{0},$

up to keeping renaming $C_{0}$, and consequently

$\int_{B_{2}^{\prime}\setminus
B_{1/100}^{\prime}(y_{\sharp})}V(y)\,d{\mathcal{H}}^{n-1}_{y}\leqslant C_{0}.$

This and (7.3) yield the desired result. ∎

We now present a useful variant222Actually, not only the proofs of Lemmata 7.1
and 7.2 are similar, but one could state just one single, albeit more
complicated, result to condensate Lemmata 7.1 and 7.2 into a single statement.
For the sake of simplicity, however, we prefer to keep the two statements
separate. of Lemma 7.1. Its utility in our context is that the geometric
equation that we deal with will present an integral term coming from the
nonlocal mean curvature of two minimal sets that we cannot reabsorb into
“smooth” objects (due to the fact that the domain of integration is a “bad
set”, say ${\mathcal{B}}$, containing far away points and points close to the
singular set, for which any regularity information is missing). The next
result however will provide a uniform control of such additional term.

######  Lemma 7.2.

Let ${\mathcal{R}}$ be a portion of a $C^{2}$ hypersurface in $B_{2}$ that is
$C^{2}$-diffeomorphic to $B_{2}\cap\\{x_{n}=0\\}$. Let $\nu$ be the unit
normal vector field of ${\mathcal{R}}$.

Let $K:{\mathcal{R}}\times{\mathcal{R}}\to[0,+\infty]$ be such that

(7.4) $\begin{split}&{\mbox{for all\leavevmode\nobreak\ $x\neq x_{0}$, we have
that }}\;\frac{1}{C|x-x_{0}|^{n+s}}\leqslant
K(x,x_{0})\leqslant\frac{C}{|x-x_{0}|^{n+s}},\\\ &{\mbox{and, for
all\leavevmode\nobreak\ $y\in B_{\delta}\setminus\\{0\\}$,
}}|K(y_{x_{0}}^{+},x_{0})-K(y_{x_{0}}^{-},x_{0})|\leqslant
C|y|^{1-n-s},\end{split}$

for some $C\geqslant 1$.

Assume that, for all $\widetilde{x}\in{{\mathcal{R}}}\cap B_{3/2}$, the
nonnegative function $w$ satisfies

(7.5)
$\begin{split}&\int_{\mathcal{R}}\big{(}w(\widetilde{x})-w(x)\big{)}\,K(x,\widetilde{x})\,d{\mathcal{H}}^{n-1}_{x}-\int_{{\mathcal{B}}}\frac{\Phi(X)}{|X-\widetilde{X}|^{n+s}}\,dX\geqslant-\mu,\end{split}$

where ${\mathcal{B}}$ is some measurable set of
$\mathbb{R}^{n}\setminus{\mathcal{R}}$, $\Phi\in
L^{\infty}(\mathbb{R}^{n},[0,1])$, $\mu\geqslant 0$, and
$\widetilde{X}:=\widetilde{x}+w(\widetilde{x})\nu(\widetilde{x})$.

Assume that there exists a point $x_{0}$ in ${\mathcal{R}}\cap B_{1}$ whose
distance from ${\mathcal{B}}$ is bounded from below by some
$r_{0}>\widetilde{C}\|w\|_{L^{\infty}({{\mathcal{R}}}\cap B_{2})}$, with
$\widetilde{C}>1$.

Then, there exists $C_{0}>0$, depending only on $n$, $s$, $r_{0}$, the
regularity parameters of ${\mathcal{R}}$ and the structural constant $C$ of
the kernel in (7.4), such that, if $\widetilde{C}\geqslant C_{0}$,

(7.6) $\int_{\mathcal{B}}\frac{\Phi(X)}{|X-X_{0}|^{n+s}}\,dX\leqslant
C_{0}\,\big{(}\|w\|_{L^{\infty}({{\mathcal{R}}}\cap B_{2})}+\mu\big{)},$

where $X_{0}:=x_{0}+w(x_{0})\nu(x_{0})$.

###### Proof.

We straighten ${\mathcal{R}}$ by a diffeomorphism $\phi:{\mathcal{R}}\to
B^{\prime}_{2}:=B_{2}\cap\\{x_{n}=0\\}$ and set $W(y):=w(\phi^{-1}(y))$. We
can also suppose for simplicity that ${\mathcal{R}}\cap B_{1}$ is mapped by
$\phi$ in $B_{1}^{\prime}$ and that ${\mathcal{R}}\cap B_{3/2}$ is mapped by
$\phi$ in $B_{3/2}^{\prime}$. Let $x_{0}\in{\mathcal{R}}\cap B_{1}$ with
$B_{r_{0}}(x_{0})\cap{\mathcal{B}}=\varnothing$. Let also $y_{0}:=\phi(x_{0})$
and notice that $y_{0}\in B_{1}^{\prime}$.

Given $M>0$, we slide the parabola $P(y):=-M|y-y_{0}|^{2}$ from below till we
touch $W$ at some point $\widehat{y}$. The touching condition between the slid
parabola and the graph of $W$ entails that

(7.7) $W(\widehat{y})+M|\widehat{y}-y_{0}|^{2}-M|y-y_{0}|^{2}\leqslant W(y).$

Here we choose

(7.8) $M:=C_{*}^{2}\,\|w\|_{L^{\infty}({{\mathcal{R}}}\cap
B_{2})}\max\left\\{1,\frac{1}{r_{0}^{2}}\right\\},$

with $C_{*}$ to be chosen sufficiently large. In this way, by evaluating (7.7)
at $y:=y_{0}$,

$\displaystyle|\widehat{y}-y_{0}|\leqslant\sqrt{\frac{W(y_{0})-W(\widehat{y})}{M}}\leqslant\sqrt{\frac{W(y_{0})}{M}}=\sqrt{\frac{w(x_{0})}{M}}$
$\displaystyle\qquad\leqslant\sqrt{\frac{1}{C_{*}^{2}\max\left\\{1,\frac{1}{r_{0}^{2}}\right\\}}}=\frac{1}{C_{*}}\min\\{1,r_{0}\\}.$

In particular, if $C_{*}$ is large enough, we have that $\widehat{y}\in
B_{3/2}^{\prime}$ and $|\widehat{y}-y_{0}|<\frac{r_{0}}{10}$.

In this way, the point $\widehat{x}:=\phi^{-1}(\widehat{y})$ belongs to
${\mathcal{R}}\cap B_{3/2}$. We can therefore utilize (7.5) at $\widehat{x}$,
finding that

(7.9)
$\begin{split}&\int_{\mathcal{B}}\frac{\Phi(X)}{|X-\widehat{X}|^{n+s}}\,dX\leqslant\int_{\mathcal{R}}\big{(}w(\widehat{x})-w(x)\big{)}\,K(x,\widehat{x})\,d{\mathcal{H}}^{n-1}_{x}+\mu\\\
&\qquad=\int_{B_{2}^{\prime}}\big{(}W(\widehat{y})-W(y)\big{)}\,K_{*}(y,\widehat{y})\,d{\mathcal{H}}^{n-1}_{y}+\mu,\end{split}$

where $\widehat{X}:=\widehat{x}+w(\widehat{x})\nu(\widehat{x})$ and we used
the notation in (5.4).

Notice that, on the one hand,

$\displaystyle|X-\widehat{X}|\leqslant|X-X_{0}|+|X_{0}-\widehat{X}|\leqslant|X-X_{0}|+|x_{0}-\widehat{x}|+2\|w\|_{L^{\infty}({{\mathcal{R}}}\cap
B_{2})}$
$\displaystyle\qquad\qquad\qquad\leqslant|X-X_{0}|+\frac{r_{0}}{10}+2\|w\|_{L^{\infty}({{\mathcal{R}}}\cap
B_{2})}\leqslant|X-X_{0}|+\frac{3r_{0}}{10}.$

On the other hand, if $X\in{\mathcal{B}}$,

$|X-X_{0}|\geqslant|X-x_{0}|-\|w\|_{L^{\infty}({{\mathcal{R}}}\cap
B_{2})}\geqslant r_{0}-\|w\|_{L^{\infty}({{\mathcal{R}}}\cap
B_{2})}\geqslant\frac{9r_{0}}{10}.$

By combining these observations, we find that

$|X-\widehat{X}|\leqslant|X-X_{0}|+\frac{3}{10}\cdot\frac{10}{9}|X-X_{0}|\leqslant
C_{0}|X-X_{0}|,$

for some $C_{0}>1$.

This and (7.9), up to renaming $C_{0}$, give that

$\begin{split}&\frac{1}{C_{0}}\int_{\mathcal{B}}\frac{\Phi(X)}{|X-X_{0}|^{n+s}}\,dX\leqslant\int_{B_{2}^{\prime}}\big{(}W(\widehat{y})-W(y)\big{)}\,K_{*}(y,\widehat{y})\,d{\mathcal{H}}^{n-1}_{y}+\mu.\end{split}$

As a consequence, by (7.7),

$\begin{split}\frac{1}{C_{0}}\int_{\mathcal{B}}\frac{\Phi(X)}{|X-X_{0}|^{n+s}}\,dX\leqslant\;&M\int_{B_{2}^{\prime}}\big{(}|y-y_{0}|^{2}-|\widehat{y}-y_{0}|^{2}\big{)}\,K_{*}(y,\widehat{y})\,d{\mathcal{H}}^{n-1}_{y}+\mu\\\
\leqslant\;&C_{0}\big{(}M+\mu\big{)}.\end{split}$

The desired result thus follows in view of (7.8). ∎

##  8\. Completion of the proof of Theorem 1.1

We can now complete the proof of Theorem 1.1 by relying on the work carried
out so far and on the Harnack Inequality by Cabré and Cozzi in [MR3934589].

###### Proof of Theorem 1.1.

Suppose, by contradiction, that $E_{1}\neq E_{2}$. Without loss of generality,
we may suppose that, in the notation recalled in footnote 1,

(8.1) $E_{1}$ and $E_{2}$ share the same tangent cone at the origin.

This is a standard procedure in geometric measure theory, based on blow-up,
dimensional reduction, and regularity theory. We recall the details for the
facility of the reader.

Indeed, if (8.1) does not hold, we have a converging blow-up sequence for
$E_{1}$ approaching a cone ${\mathcal{C}}_{1}$ with the corresponding blow-up
sequence for $E_{2}$ approaching a cone
${\mathcal{C}}_{2}\neq{\mathcal{C}}_{1}$. We take a rotation
${\mathcal{R}}_{1}$ of ${\mathcal{C}}_{1}$ such that
$E_{1}^{(1)}:={\mathcal{R}}_{1}{\mathcal{C}}_{1}\subseteq{\mathcal{C}}_{2}=:E_{2}^{(1)}$
and there exists $p^{(1)}\in(\partial E_{1}^{(1)})\cap(\partial
E_{2}^{(1)})\cap(\partial B_{1})$.

We then consider a blow-up of $E_{1}^{(1)}$ and $E_{2}^{(1)}$ at $p^{(1)}$,
which produces two new $s$-minimal cones in $\mathbb{R}^{n}$, which will be
denoted by ${\mathcal{C}}_{1}^{(1)}$ and ${\mathcal{C}}_{2}^{(2)}$, with
${\mathcal{C}}_{1}^{(1)}\subseteq{\mathcal{C}}_{2}^{(2)}$.

Now, two cases can occur. If
${\mathcal{C}}_{1}^{(1)}={\mathcal{C}}_{2}^{(2)}$, it suffices to replace $p$,
$E_{1}$, $E_{2}$, ${\mathcal{C}}_{1}$ and ${\mathcal{C}}_{2}$ respectively
with $p^{(1)}$, $E_{1}^{(1)}$, $E_{2}^{(1)}$, ${\mathcal{C}}_{1}^{(1)}$ and
${\mathcal{C}}_{2}^{(1)}$: in this way we have obtained (8.1) in this new
configuration.

If instead ${\mathcal{C}}_{1}^{(1)}\neq{\mathcal{C}}_{2}^{(1)}$, we observe
that both these cones are cylinders over $\mathbb{R}^{n-1}$. In this way, by
the dimensional reduction (see [MR2675483]), we have found two $s$-minimal
cones in $\mathbb{R}^{n-1}$, which we denote by $\widetilde{E}_{1}^{(2)}$ and
$\widetilde{E}_{2}^{(2)}$, such that
$\widetilde{E}_{1}^{(2)}\subseteq\widetilde{E}_{2}^{(2)}$ and
$\widetilde{E}_{1}^{(2)}\neq\widetilde{E}_{2}^{(2)}$ (and, up to an isometry,
${\mathcal{C}}_{j}^{(1)}=\widetilde{E}_{j}^{(2)}\times\mathbb{R}$ for
$j\in\\{1,2\\}$).

We thus repeat the previous algorithm, taking a rotation ${\mathcal{R}}_{2}$
in $\mathbb{R}^{n-1}$, such that
$E_{1}^{(2)}:={\mathcal{R}}_{2}\widetilde{E}_{1}^{(2)}\subseteq\widetilde{E}_{2}^{(1)}=:E_{2}^{(1)}$
and there exists $p^{(2)}\in(\partial E_{1}^{(2)})\cap(\partial
E_{2}^{(2)})\cap(\partial B_{1})$. Then, a blow-up at $p_{2}$ produces two
$s$-minimal cones ${\mathcal{C}}_{1}^{(2)}$ and ${\mathcal{C}}_{2}^{(2)}$,
which either coincide (whence we replace $p$, $E_{1}$, $E_{2}$,
${\mathcal{C}}_{1}$ and ${\mathcal{C}}_{2}$ respectively with $p^{(2)}$,
$E_{1}^{(2)}$, $E_{2}^{(2)}$, ${\mathcal{C}}_{1}^{(2)}$ and
${\mathcal{C}}_{2}^{(2)}$) or not, in which case we apply again the
dimensional reduction (reducing now to $\mathbb{R}^{n-2}$) and proceed.

This algorithm will stop at a certain dimension, producing the same
$s$-minimal cones after a blow-up, since, by the regularity of $s$-minimal
cones in dimension 2 (see [MR3090533]), we know that halfspaces are the only
nontrivial minimal cones in $\mathbb{R}^{2}$. The proof of (8.1) is thereby
complete.

Hence, from now on, we will denote by ${\mathcal{C}}$ the common tangent cone
of $E_{1}$ and $E_{2}$ along a blow-up sequence that we are now going to
specify. To this end, we write $E_{2}$ in normal coordinates with respect to
$E_{1}$ at its regular points (this is possible up to an initial dilation),
that is suppose that, at a set of regular points, $E_{2}\setminus E_{1}$ has
the form $x+t\nu(x)$, with $x\in\partial E_{1}$ and
$t\in\big{[}0,w_{0}(x)\big{)}$, for a suitable function $w_{0}>0$. Then, by
Corollary B.2, we find a set ${\mathcal{M}}_{0}$ of regular points for $E_{1}$
and an infinitesimal sequence of points $z_{\star,k}\in{\mathcal{M}}_{0}$ such
that

(8.2) $\frac{w_{0}(x)}{|x|}\leqslant\frac{2w_{0}(z_{\star,k})}{|z_{\star,k}|}$

for all $x\in{\mathcal{M}}_{0}$ with $|x|\leqslant|z_{\star,k}|/2$. In
addition, by (B.1),

(8.3) the distance of any point $x\in{\mathcal{M}}_{0}$ from the singular set
is at least $\theta_{0}|x|$,

with $\theta_{0}>0$.

What is more, by (B.2), we also know that, for all $\rho\in(0,\rho_{0}]$,

(8.4) ${\mathcal{M}}_{0}\cap(\partial B_{\rho})\neq\varnothing.$

Hence, we choose

(8.5) $r_{k}:=|z_{\star,k}|$

and consider the corresponding blow-up sequences for $j\in\\{1,2\\}$ defined
by

(8.6) $E_{j,k}:=\frac{E_{j}-p}{r_{k}}.$

We now apply Theorem 3.4. More specifically, the sets $E_{1}$ and $E_{2}$ in
the statement of Theorem 3.4 are here the sets $E_{1,k}$ and $E_{2,k}$, as
defined in (8.6), with $k$ large enough.

The gist of the construction is that we cover the singular set of
${\mathcal{C}}$ by a set of small balls (the fact that the singular set has
dimension at most $n-3$, due to [MR3090533, MR3331523] guarantees that these
balls can be chosen to satisfy (3.21) for $\eta$ as small as we wish).

Since, for $k$ large enough, $E_{1,k}$ and $E_{2,k}$ locally lie in a small
neighborhood of ${\mathcal{C}}$, we have that both $E_{1,k}$ and $E_{2,k}$ are
smooth away from the above covering of the singular set of ${\mathcal{C}}$,
thanks to the improvement of flatness of nonlocal minimal surfaces put forth
in [MR2675483, Corollary 4.4 and Theorem 6.1] (in this way, the convergence of
${E_{1,k}}$ and ${E_{2,k}}$ to the limit cone ${\mathcal{C}}$ occurs in the
$C^{1,\alpha}$ sense away from any small neighborhood of the singular set of
${\mathcal{C}}$, and actually in the $C^{k}$ sense for any $k\geqslant 2$,
thanks to the bootstrap regularity in [MR3331523]).

This allows us to look at a “large” set333Note that this set ${\mathcal{G}}$
possibly contains ${\mathcal{M}}_{0}$. ${\mathcal{G}}$ containing regular
points as in Theorem 3.4 (and at a small shrinkage of it, namely
${\mathcal{G}}^{\prime}$) such that all the singular points of
$\partial{\mathcal{C}}$, $\partial E_{1}$ and $\partial E_{2}$ in a large ball
$B_{R}$ lie outside ${\mathcal{G}}$ (also, ${\mathcal{G}}$ covers all $B_{R}$,
up to a negligible covering of balls, as specified in (3.21)).

Accordingly, for $j\in\\{1,2\\}$ and $k$ sufficiently large, one can
parameterize ${E_{j,k}}$ as a graph of class $C^{2}$ in the normal direction
of ${\mathcal{C}}$ away from a small neighborhood of the singular set.

It is however technically simpler to recenter this parameterization on the
“middle surface”, namely to parameterize $E_{2}=E_{2,k}$ in terms of
$E_{1}=E_{1,k}$, and this corresponds to the normal parameterization $w=w_{k}$
in Theorem 3.4. We stress that since both $E_{1,k}$ and $E_{2,k}$ converge to
${\mathcal{C}}$ locally in the $C^{2}$-sense away from the singular points of
$\partial{\mathcal{C}}$, we also know that the $C^{2}$-norm of $w=w_{k}$ is as
small as we like, provided that $k$ is chosen large enough (possibly in
dependence also of the covering of the singular set, which is now fixed once
and for all).

We can therefore utilize Theorem 3.4 in this context and conclude that the
normal parameterization $w=w_{k}$ of $E_{2}=E_{2,k}$ with respect to
$E_{1}=E_{1,k}$ satisfies the equation in (3.22), with suitable integrable
kernels, as defined in (3.23). More specifically, since $E_{1}$ and $E_{2}$
are $s$-minimal sets and therefore $H^{s}_{E_{1}}=H^{s}_{E_{2}}$ at every
regular boundary point (see [MR2675483, Theorem 5.1]), we obtain from Theorem
3.4 that

(8.7) $\begin{split}&0=\int_{(\partial
E_{1})\cap{\mathcal{G}}^{\prime}}\big{(}w(x_{0})-w(x)\big{)}\,K_{1}(x,x_{0})\,d{\mathcal{H}}^{n-1}_{x}\\\
&\qquad\qquad-w(x_{0})\int_{(\partial
E_{1})\cap{\mathcal{G}}^{\prime}}\big{(}1-\nu(x_{0})\cdot\nu(x)\big{)}\,K_{2}(x,x_{0})\,d{\mathcal{H}}^{n-1}_{x}\\\
&\qquad\qquad+\frac{1}{2}\int_{\mathbb{R}^{n}\setminus{\mathcal{G}}^{\prime}}\frac{\widetilde{\chi}_{E_{2}}(X)-\widetilde{\chi}_{E_{1}}(X)}{|X-X_{0}|^{n+s}}\,dX+O\left(w(x_{0})\eta\right)+O\left(\frac{w(x_{0})}{R^{1+s}}\right),\end{split}$

where $\nu=\nu_{k}$ is the outer unit normal of $\partial E_{1}=\partial
E_{1,k}$ at its regular points and $X_{0}:=x_{0}+w(x_{0})\nu(x_{0})$.

Strictly speaking, the function $w$ does not need to be defined everywhere in
$\mathbb{R}^{n}$ for the validity of (8.7), since such an equation does not
consider values of $w$ outside certain sets, but we will implicitly suppose
that $w$ is defined everywhere for definiteness.

We now take ${\mathcal{G}}^{\prime\prime}\Subset{\mathcal{G}}^{\prime}$ and
$x_{0}\in(\partial E_{1})\cap B_{R/2}\cap{\mathcal{G}}^{\prime\prime}$. In
this setting, owing to Lemma 7.2 and to the fact that $E_{1}\subseteq E_{2}$,
we see that, when the size of $w$ is small,

(8.8)
$0\leqslant\int_{\mathbb{R}^{n}\setminus{\mathcal{G}}^{\prime}}\frac{\widetilde{\chi}_{E_{1}}(X)-\widetilde{\chi}_{E_{2}}(X)}{|X-X_{0}|^{n+s}}\,dX=O(\|w\|_{L^{\infty}(B_{R}\cap{\mathcal{G}})}).$

Now, since $E_{1}\not=E_{2}$, we have that $w$ does not vanish identically,
and in fact $w>0$ on the regular part of $\partial E_{1}$ (otherwise, we could
compute the nonlocal mean curvature of $E_{1}$ at that point in the smooth
sense, as well as the one of $E_{2}$ in the viscosity sense and get a
contradiction). Therefore we can normalize $w$ to take value $1$ at a suitable
point. Roughly speaking, it would be desirable to pick this point to reach the
supremum of $\frac{w(x)}{|x|}$, so to obtain a “linear” separation about the
minimal sheets at the origin and thus contradict (8.1): however, this choice
of the normalizing point may be impossible, since the supremum of
$\frac{w(x)}{|x|}$ might well be on the singular set (where $w$ is not even
defined) or dangerously close to it. This is the reason for which we have
chosen an appropriate sequence of blow-up radii $r_{k}$ in (8.2) and (8.5).

In this setting, we define
$x_{\star,k}:=\frac{z_{\star,k}}{|z_{\star,k}|}=\frac{z_{\star,k}}{r_{k}}$
and, in light of (8.3), we stress that the distance of $x_{\star,k}$ from the
singular set is at least $\theta_{0}$. This ensures that, up to a subsequence,
$x_{\star,k}$ converges444Let us stress that the fact that the limit point
$x_{\star,\infty}$ remains bounded and bounded away from the singular set is
essential to apply the Harnack Inequality in [MR3934589]: this is the reason
for which we can well allow intermediate constants to depend on the radius of
the large ball that we are considering, on the set of small balls chosen to
cover the singular set, as well as on the regularity of the $s$-minimal sheets
away of this cover, since we will pass $k\to+\infty$ before removing this
large ball and this small cover, but we will employ the normalization at the
limit point $x_{\star,\infty}$ to bound from below the last term in (8.13). to
a regular point $x_{\star,\infty}$ of the limit cone as $k\to+\infty$.

Furthermore, by (8.4), given $\delta\in\left(0,\frac{1}{2}\right]$, to be
chosen conveniently small in what follows, we can pick a point
$y_{k}\in{\mathcal{M}}_{0}\cap(\partial B_{\delta r_{k}})$. In this way, by
(8.3), we also have that the distance of $y_{\star,k}:=\frac{y_{k}}{r_{k}}$
from the singular set is at least $\delta\theta_{0}$ and therefore, up to a
subsequence, $y_{\star,k}$ converges to a regular point $y_{\star,\infty}$ of
the limit cone as $k\to+\infty$.

We also recall (8.2) and write that

$\displaystyle\frac{w_{0}(y_{k})}{\delta
r_{k}}=\frac{w_{0}(y_{k})}{|y_{k}|}\leqslant\frac{2w_{0}(z_{\star,k})}{|z_{\star,k}|}=\frac{2w_{0}(z_{\star,k})}{r_{k}}$

and accordingly

(8.9) $w_{k}(y_{\star,k})\leqslant 2\delta w_{k}(x_{\star,k}).$

We now define

$v(x)=v_{k}(x):=\frac{w_{k}(x)}{w(x_{\star,k})}$

and we divide (8.7) by $w(x_{\star,k})=w_{k}(x_{\star,k})$, recalling also
(8.8), to find that, for all $x_{0}\in(\partial E_{1})\cap
B_{R/2}\cap{\mathcal{G}}^{\prime\prime}$,

(8.10) $\begin{split}&0=\int_{(\partial
E_{1})\cap{\mathcal{G}}^{\prime}}\big{(}v(x_{0})-v(x)\big{)}\,K_{1}(x,x_{0})\,d{\mathcal{H}}^{n-1}_{x}\\\
&\qquad\qquad-v(x_{0})\int_{(\partial
E_{1})\cap{\mathcal{G}}^{\prime}}\big{(}1-\nu(x_{0})\cdot\nu(x)\big{)}\,K_{2}(x,x_{0})\,d{\mathcal{H}}^{n-1}_{x}\\\
&\qquad\qquad-\psi(x_{0})+O\left(\eta\right)+O\left(\frac{1}{R^{1+s}}\right),\end{split}$

where $0\leqslant\psi=O(1)$.

Also, if
${\mathcal{G}}^{\prime\prime\prime}\Subset{\mathcal{G}}^{\prime\prime}$, we
have that, in $(\partial E_{1})\cap
B_{R/4}\cap{\mathcal{G}}^{\prime\prime\prime}$,

(8.11) $0\leqslant v\leqslant O(1).$

Indeed, we first observe that $v\geqslant 0$, since $w\geqslant 0$. The upper
bound in (8.11) follows from the previously developed regularity theory.
Specifically, one first employs Lemma 7.1 and finds that

$\int_{(\partial
E_{1})\cap{\mathcal{G}}^{\prime}}v(x)\,d{\mathcal{H}}^{n-1}_{x}\leqslant
O(1).$

This inequality provides the uniform bound corresponding to (6.3) in the
present setting. Now, the upper bound in (8.11) is a consequence of Lemma 6.1.
The proof of (8.11) is thereby complete.

Accordingly, by the Hölder estimates in Lemma 5.1, we deduce that the
$C^{\alpha}$-norm of $v$ is bounded locally uniformly in $k$ (and we stress
that, since $w=w_{k}$, also $v=v_{k}$).

Therefore, up to a subsequence, $v_{k}$ converges locally uniformly in
$B_{R/4}\cap{\mathcal{G}}^{\prime\prime\prime}$ to a function $v_{\infty}$. In
view of (8.10), $v_{\infty}$ satisfies, for all $x_{0}\in(\partial E_{1})\cap
B_{R/4}\cap{\mathcal{G}}^{\prime\prime\prime}$,

$\begin{split}&0\leqslant\int_{(\partial{\mathcal{C}})\cap{\mathcal{G}}^{\prime}}\frac{v_{\infty}(x_{0})-v_{\infty}(x)}{|x-x_{0}|^{n+s}}\,d{\mathcal{H}}^{n-1}_{x}\\\
&\qquad\qquad-
v_{\infty}(x_{0})\int_{(\partial{\mathcal{C}})\cap{\mathcal{G}}^{\prime}}\frac{1-\nu(x_{0})\cdot\nu(x)}{{|x-x_{0}|^{n+s}}}\,d{\mathcal{H}}^{n-1}_{x}+O\left(\eta\right)+O\left(\frac{1}{R^{1+s}}\right).\end{split}$

Actually, we can now take $R$ as large as we wish and invade all the space
outside the singular set by the “good” domains ${\mathcal{G}}$,
${\mathcal{G}}^{\prime}$ and ${\mathcal{G}}^{\prime\prime}$. In this way, we
find that, on the regular part $\operatorname{Reg}{\mathcal{C}}$ of
$\partial{\mathcal{C}}$,

$\begin{split}&0\leqslant\int_{\operatorname{Reg}{\mathcal{C}}}\big{(}v_{\infty}(x_{0})-v_{\infty}(x)\big{)}\,K_{1,\infty}(x,x_{0})\,d{\mathcal{H}}^{n-1}_{x}\\\
&\qquad\qquad-
v_{\infty}(x_{0})\int_{\operatorname{Reg}{\mathcal{C}}}\big{(}1-\nu(x_{0})\cdot\nu(x)\big{)}\,K_{2,\infty}(x,x_{0})\,d{\mathcal{H}}^{n-1}_{x}.\end{split}$

Now, we wish to apply Lemma 4.1. To this end, we observe that (4.1) is
satisfied in our case by ${\mathcal{R}}:=\operatorname{Reg}{\mathcal{C}}$,
thanks to the perimeter estimates for nonlocal minimal surfaces, see
[MR3981295, equation (1.16)].

Hence, from Lemma 4.1, we deduce that

$\iint_{(\operatorname{Reg}{\mathcal{C}}\cap
B_{2})\times\operatorname{Reg}{\mathcal{C}}}\frac{(v_{Q}(x)-v_{Q}(y))^{2}}{|x-y|^{n+s}}\,d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}<+\infty$

and, for every $\phi\in C^{\infty}_{0}(\mathbb{R}^{n},[0,+\infty))$,

$\iint_{\operatorname{Reg}{\mathcal{C}}\times\operatorname{Reg}{\mathcal{C}}}\frac{(v_{Q}(x)-v_{Q}(y))(\phi(x)-\phi(y))}{|x-y|^{n+s}}\,d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}\geqslant
0,$

where $v_{Q}:=\min\\{v_{\infty},Q\\}$.

Moreover,

(8.12) $v_{\infty}(x_{\star,\infty})=\lim_{k\to+\infty}v_{k}(x_{\star,k})=1.$

It follows from the Harnack Inequality in [MR3934589, Theorem 1.7] that, for
every compact subset ${\mathcal{K}}$ of $\operatorname{Reg}{\mathcal{C}}$,

$\inf_{{\mathcal{K}}\cap B_{1}}v_{Q}\geqslant c\left(\int_{{\mathcal{K}}\cap
B_{1}}v_{Q}(x)\,d{\mathcal{H}}^{n-1}_{x}+\int_{{\mathcal{K}}\setminus
B_{1}}\frac{v_{Q}(x)}{|x|^{n+s}}\,d{\mathcal{H}}^{n-1}_{x}\right),$

for some $c>0$, depending only on $n$ and $s$.

Sending $Q\to+\infty$ and using (8.12), we thus obtain

(8.13) $m:=\inf_{{\mathcal{K}}\cap B_{1}}v_{\infty}\geqslant
c\int_{{\mathcal{K}}\cap B_{1}}v_{\infty}(x)\,d{\mathcal{H}}^{n-1}_{x}>0.$

As a result, for every compact subset ${\mathcal{K}}$ of
$\operatorname{Reg}{\mathcal{C}}\cap B_{3/4}$, we have that

$\inf_{x\in{\mathcal{K}}}\frac{w_{k}(x)}{w_{k}(x_{\star,k})}=\inf_{x\in{\mathcal{K}}}v_{k}(x)\geqslant\frac{m}{2},$

as long as $k$ is large enough (possibly in dependence of ${\mathcal{K}}$).

In particular,

$\frac{w_{k}(y_{\star,k})}{w_{k}(x_{\star,k})}\geqslant\frac{m}{2},$

which gives a contradiction with (8.9) when $\delta$ is sufficiently small.∎

##  Appendix A Proof of Lemma 4.1

Here we give a self-contained energy argument to prove Lemma 4.1.

###### Proof of Lemma 4.1.

For the convenience of the reader, we split this proof into independent steps.

Step 1. Proof of (4.3). First of all, we have that, for every
$x\in{\mathcal{R}}$,

(A.1)
$\int_{\mathcal{R}}\frac{u_{Q}(x)-u_{Q}(y)}{|x-y|^{n+s}}\,d{\mathcal{H}}^{n-1}_{y}\geqslant
0.$

Indeed, if $u(x)\geqslant Q$, the claim is obvious, since in this case
$u_{Q}(x)=Q\geqslant u_{Q}(y)$.

Therefore, we may suppose that $u(x)<Q$. In this case,

$\displaystyle\int_{\mathcal{R}}\frac{u_{Q}(x)-u_{Q}(y)}{|x-y|^{n+s}}\,d{\mathcal{H}}^{n-1}_{y}=\int_{\mathcal{R}}\frac{u(x)-u_{Q}(y)}{|x-y|^{n+s}}\,d{\mathcal{H}}^{n-1}_{y}$
$\displaystyle\qquad\qquad=\int_{{\mathcal{R}}\cap\\{u\geqslant
Q\\}}\frac{u(x)-Q}{|x-y|^{n+s}}\,d{\mathcal{H}}^{n-1}_{y}+\int_{{\mathcal{R}}\cap\\{u<Q\\}}\frac{u(x)-u(y)}{|x-y|^{n+s}}\,d{\mathcal{H}}^{n-1}_{y}$
$\displaystyle\qquad\qquad=\int_{{\mathcal{R}}\cap\\{u\geqslant
Q\\}}\frac{u(x)-Q}{|x-y|^{n+s}}\,d{\mathcal{H}}^{n-1}_{y}-\int_{{\mathcal{R}}\cap\\{u\geqslant
Q\\}}\frac{u(x)-u(y)}{|x-y|^{n+s}}\,d{\mathcal{H}}^{n-1}_{y}+\int_{{\mathcal{R}}}\frac{u(x)-u(y)}{|x-y|^{n+s}}\,d{\mathcal{H}}^{n-1}_{y}$
$\displaystyle\qquad\qquad=\int_{{\mathcal{R}}\cap\\{u\geqslant
Q\\}}\frac{u(y)-Q}{|x-y|^{n+s}}\,d{\mathcal{H}}^{n-1}_{y}+\int_{{\mathcal{R}}}\frac{u(x)-u(y)}{|x-y|^{n+s}}\,d{\mathcal{H}}^{n-1}_{y}$
$\displaystyle\qquad\qquad\geqslant\int_{{\mathcal{R}}}\frac{u(x)-u(y)}{|x-y|^{n+s}}\,d{\mathcal{H}}^{n-1}_{y}$

and accordingly (A.1) follows from (4.2).

Now we prove (4.3). To this end, as a byproduct of (A.1), we find that

$\displaystyle\iint_{{\mathcal{R}}\times{\mathcal{R}}}\frac{(u_{Q}(x)-u_{Q}(y))(\zeta(x)-\zeta(y))}{|x-y|^{n+s}}\,d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}$
$\displaystyle\qquad=2\iint_{{\mathcal{R}}\times{\mathcal{R}}}\frac{(u_{Q}(x)-u_{Q}(y))\,\zeta(x)}{|x-y|^{n+s}}\,d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}\geqslant
0,$

which establishes (4.3).

Step 2. Covering arguments. Let now $\varepsilon\in(0,1)$, to be taken as
small as we wish here below. From our assumption on $S$, if $D\in(d,n-2-s]$,
we know that $S$ is contained in the union of some balls
$\\{B_{r_{j}}(p_{j})\\}_{j\in\mathbb{N}}$, with

(A.2) $\sum_{j\in\mathbb{N}}r_{j}^{D}\leqslant\varepsilon.$

We claim that we can reduce to the case in which

(A.3) the family of dilated balls $\\{B_{32r_{j}}(p_{j})\\}_{j\in\mathbb{N}}$
has a finite intersection property.

For this, we start by noticing that, without loss of generality, there exists
$q_{j}\in B_{r_{j}}(p_{j})\cap S$ (if not, such a ball can be freely removed
from the covering of $S$). Moreover, by the Besicovitch Covering Theorem, we
may suppose that

(A.4) the original family of balls $\\{B_{r_{j}}(p_{j})\\}_{j\in\mathbb{N}}$
has a finite intersection property.

Thus, to prove (A.3), we argue by contradiction and suppose, say, that there
are infinitely many $j$’s such that $B_{32r_{j}}(p_{j})\cap
B_{32r_{1}}(p_{1})$. In particular, all these $p_{j}$’s remain at a bounded
distance from $p_{1}$, and so do the corresponding $q_{j}$’s.

Therefore, up to a subsequence, we can assume that there exists
$q\in\mathbb{R}^{n}$ such that $q_{j}\to q$ as $j\to+\infty$. In fact, since
$S$ is closed, we know that $q\in S$. As a consequence, there exists
$j_{\star}\in\mathbb{N}$ such that $q\in B_{r_{j_{\star}}}(p_{j_{\star}})$.
This and the convergence of the $q_{j}$’s yield that $q_{j}\in
B_{r_{j_{\star}}}(p_{j_{\star}})$ for infinitely many $j$’s.

In particular, $q_{j}\in B_{r_{j_{\star}}}(p_{j_{\star}})\cap
B_{r_{j}}(p_{j})$ for infinitely many $j$’s, which is in contradiction with
(A.4). The proof of (A.3) is thereby complete.

Step 3. Bump functions. Let now $\tau_{j}\in
C^{\infty}_{0}(B_{4r_{j}}(p_{j}),[0,1])$ be such that $\tau_{j}=1$ in
$B_{3r_{j}}(p_{j})$ and

(A.5)
$|\nabla\tau_{j}|\leqslant\frac{C}{r_{j}}\,\chi_{B_{4r_{j}}(p_{j})\setminus
B_{3r_{j}}(p_{j})}.$

Let $\phi\in C^{\infty}_{0}(\mathbb{R}^{n},[0,+\infty))$ and assume that the
support of $\phi$ is contained in some ball $B_{R_{0}}$. Since
$S\cap\overline{B_{R_{0}}}$ is compact, we may suppose that

(A.6) $S\cap\overline{B_{R_{0}}}\subseteq\bigcup_{j=0}^{N}B_{r_{j}}(p_{j}).$

Thus, we define

(A.7)
$\varphi:=\phi\,\tau_{\star},\qquad{\mbox{where}}\quad\tau_{\star}:=\min_{j=0,\dots,N}\\{1-\tau_{j}\\}.$

We observe that $\varphi$ actually depends on $\varepsilon$, since so does the
family of balls for which (A.2) holds true, but, for short, we omit this
dependence in the notation (yet, we will consider the limit in $\varepsilon$
here below). Also, we note that $\varphi\in
W^{1,\infty}_{0}(\mathbb{R}^{n},[0,+\infty))$ and we also claim that

(A.8) the support of $\varphi\big{|}_{\mathcal{R}}$ is contained in
${\mathcal{R}}$.

For this, let $q_{k}\in{\mathcal{R}}$ be such that $\varphi(q_{k})>0$ and
$q_{k}\to q$ as $k\to+\infty$. Then, $q_{k}\in B_{R_{0}}$, whence
$q\in\overline{B_{R_{0}}}$. We point out that

(A.9) $|q-p_{j}|\geqslant 2r_{j}$ for all $j\in\\{0,\dots,N\\}$.

Indeed, if not, say $|q-p_{0}|<2r_{0}$, we would have that also
$|q_{k}-p_{0}|<2r_{0}$ for infinitely many $k$’s and therefore
$\tau_{0}(q_{k})=1$. But this would give that $\varphi(q_{k})>0$, which is a
contradiction and (A.9) is proved.

From this, we obtain that $q$ belongs to the complement of
$B_{2r_{0}}(p_{0})\cup\dots\cup B_{2r_{N}}(p_{N})$, and so to the complement
of $S$, from which (A.8) follows.

Step 4. Integral estimates. Thanks to (A.8), we can exploit (4.3) with
$\zeta:=\frac{\varphi^{2}}{u_{Q,\ell}+1}$, being $u_{Q,\ell}$ a mollified
sequence of $u_{Q}$, with $\ell\in\mathbb{N}$. In this way, symmetrizing the
integrands when necessary,

$\displaystyle 0$ $\displaystyle\leqslant$
$\displaystyle\iint_{{\mathcal{R}}\times{\mathcal{R}}}(u_{Q}(x)-u_{Q}(y))\left(\frac{\varphi^{2}(x)}{u_{Q,\ell}(x)+1}-\frac{\varphi^{2}(y)}{u_{Q,\ell}(y)+1}\right)\,\frac{\,d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}$
$\displaystyle=$
$\displaystyle\iint_{{\mathcal{R}}\times{\mathcal{R}}}\varphi^{2}(x)(u_{Q}(x)-u_{Q}(y))\left(\frac{1}{u_{Q,\ell}(x)+1}-\frac{1}{u_{Q,\ell}(y)+1}\right)\,\frac{\,d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}$
$\displaystyle\qquad+\iint_{{\mathcal{R}}\times{\mathcal{R}}}\frac{(u_{Q}(x)-u_{Q}(y))(\varphi^{2}(x)-\varphi^{2}(y))}{u_{Q,\ell}(y)+1}\,\frac{\,d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}$
$\displaystyle=$
$\displaystyle-\iint_{{\mathcal{R}}\times{\mathcal{R}}}\frac{\varphi^{2}(x)(u_{Q}(x)-u_{Q}(y))(u_{Q,\ell}(x)-u_{Q,\ell}(y))}{(u_{Q,\ell}(x)+1)(u_{Q,\ell}(y)+1)}\,\frac{\,d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}$
$\displaystyle\qquad+\iint_{{\mathcal{R}}\times{\mathcal{R}}}\frac{(u_{Q}(x)-u_{Q}(y))(\varphi^{2}(x)-\varphi^{2}(y))}{u_{Q,\ell}(y)+1}\,\frac{\,d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\iint_{{\mathcal{R}}\times{\mathcal{R}}}\frac{(\varphi^{2}(x)+\varphi^{2}(y))(u_{Q}(x)-u_{Q}(y))(u_{Q,\ell}(x)-u_{Q,\ell}(y))}{(u_{Q,\ell}(x)+1)(u_{Q,\ell}(y)+1)}\,\frac{\,d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}$
$\displaystyle\qquad+\frac{1}{2}\iint_{{\mathcal{R}}\times{\mathcal{R}}}\frac{(u_{Q}(x)-u_{Q}(y))(\varphi^{2}(x)-\varphi^{2}(y))}{u_{Q,\ell}(y)+1}\,\frac{\,d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}$
$\displaystyle\qquad+\frac{1}{2}\iint_{{\mathcal{R}}\times{\mathcal{R}}}\frac{(u_{Q}(x)-u_{Q}(y))(\varphi^{2}(x)-\varphi^{2}(y))}{u_{Q,\ell}(x)+1}\,\frac{\,d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\iint_{{\mathcal{R}}\times{\mathcal{R}}}\frac{(\varphi^{2}(x)+\varphi^{2}(y))(u_{Q}(x)-u_{Q}(y))(u_{Q,\ell}(x)-u_{Q,\ell}(y))}{(u_{Q,\ell}(x)+1)(u_{Q,\ell}(y)+1)}\,\frac{\,d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}$
$\displaystyle\qquad+\frac{1}{2}\iint_{{\mathcal{R}}\times{\mathcal{R}}}\frac{(u_{Q}(x)-u_{Q}(y))(u_{Q,\ell}(x)+u_{Q,\ell}(y)+2)(\varphi(x)-\varphi(y))(\varphi(x)+\varphi(y))}{(u_{Q,\ell}(x)+1)(u_{Q,\ell}(y)+1)}\,\frac{d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}.$

Thus, given $\alpha\in(0,1)$, to be chosen suitably small in what follows, by
a weighted Cauchy-Schwarz Inequality we deduce that

$\displaystyle\iint_{{\mathcal{R}}\times{\mathcal{R}}}\frac{(\varphi^{2}(x)+\varphi^{2}(y))(u_{Q}(x)-u_{Q}(y))(u_{Q,\ell}(x)-u_{Q,\ell}(y))}{(u_{Q,\ell}(x)+1)(u_{Q,\ell}(y)+1)}\,\frac{\,d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}$
$\displaystyle\leqslant$
$\displaystyle\alpha\iint_{{\mathcal{R}}\times{\mathcal{R}}}\frac{(u_{Q}(x)-u_{Q}(y))^{2}(\varphi(x)+\varphi(y))^{2}}{(u_{Q,\ell}(x)+1)(u_{Q,\ell}(y)+1)}\,\frac{d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}$
$\displaystyle\qquad+C_{\alpha}\iint_{{\mathcal{R}}\times{\mathcal{R}}}\frac{(u_{Q,\ell}(x)+u_{Q,\ell}(y)+2)^{2}(\varphi(x)-\varphi(y))^{2}}{(u_{Q,\ell}(x)+1)(u_{Q,\ell}(y)+1)}\,\frac{d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}.$

We also use the bound

$\displaystyle\iint_{{\mathcal{R}}\times{\mathcal{R}}}\frac{(u_{Q}(x)-u_{Q}(y))^{2}(\varphi(x)+\varphi(y))^{2}}{(u_{Q,\ell}(x)+1)(u_{Q,\ell}(y)+1)}\,\frac{d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}$
$\displaystyle\qquad\leqslant
4\iint_{{\mathcal{R}}\times{\mathcal{R}}}\frac{(u_{Q}(x)-u_{Q}(y))^{2}(\varphi^{2}(x)+\varphi^{2}(y))}{(u_{Q,\ell}(x)+1)(u_{Q,\ell}(y)+1)}\,\frac{d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}.$

Hence, sending $\ell\to+\infty$, and reabsorbing one term to the left-hand
side via a suitable choice of $\alpha$, we conclude that

$\displaystyle\iint_{{\mathcal{R}}\times{\mathcal{R}}}\frac{(\varphi^{2}(x)+\varphi^{2}(y))(u_{Q}(x)-u_{Q}(y))^{2}}{(u_{Q}(x)+1)(u_{Q}(y)+1)}\,\frac{\,d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}$
$\displaystyle\qquad\leqslant
C\iint_{{\mathcal{R}}\times{\mathcal{R}}}\frac{(u_{Q}(x)+u_{Q}(y)+2)^{2}(\varphi(x)-\varphi(y))^{2}}{(u_{Q}(x)+1)(u_{Q}(y)+1)}\,\frac{d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}.$

Since $u_{Q}\in[0,Q]$, we have that $u_{Q}+1\in[1,Q+1]$ and therefore

(A.10)
$\begin{split}&\iint_{{\mathcal{R}}\times{\mathcal{R}}}(\varphi^{2}(x)+\varphi^{2}(y))(u_{Q}(x)-u_{Q}(y))^{2}\,\frac{\,d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}\\\
&\qquad\leqslant
C_{Q}\iint_{{\mathcal{R}}\times{\mathcal{R}}}(\varphi(x)-\varphi(y))^{2}\,\frac{d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}.\end{split}$

Step 5. Further integral estimates towards the proof of (4.4). We observe that
$\varphi=\phi$ outside the set

$U:=\bigcup_{j=0}^{N}B_{4r_{j}}(p_{j})$

and, by (4.1) and (A.2),

(A.11) ${\mathcal{H}}^{n-1}({\mathcal{R}}\cap U)\leqslant\varepsilon.$

This and (A.7) say that, as $\varepsilon\searrow 0$, $\tau_{\star}$ approaches
$1$, and so $\varphi$ approaches $\phi$, up to negligible sets in
${\mathcal{R}}$ and therefore, as $\varepsilon\searrow 0$, the left-hand side
of (A.10) can be replaced (or minorized) by

(A.12)
$\iint_{{\mathcal{R}}\times{\mathcal{R}}}(\phi^{2}(x)+\phi^{2}(y))(u_{Q}(x)-u_{Q}(y))^{2}\,\frac{\,d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}.$

We claim that, as $\varepsilon\searrow 0$, the integral in the right-hand side
of (A.10) can be majorized by

(A.13)
$\iint_{{\mathcal{R}}\times{\mathcal{R}}}(\phi(x)-\phi(y))^{2}\,\frac{d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}.$

To check this, we remark that, given $\beta\in(0,1)$, to be taken as small as
we wish in what follows,

$\displaystyle(\varphi(x)-\varphi(y))^{2}$ $\displaystyle=$
$\displaystyle(\phi(x)\tau_{\star}(x)-\phi(y)\tau_{\star}(y))^{2}$
$\displaystyle=$
$\displaystyle\Big{(}(\phi(x)-\phi(y))\tau_{\star}(x)+\phi(y)(\tau_{\star}(x)-\tau_{\star}(y))\Big{)}^{2}$
$\displaystyle\leqslant$
$\displaystyle(1+\beta)(\phi(x)-\phi(y))^{2}\tau_{\star}^{2}(x)+C_{\beta}\phi^{2}(y)(\tau_{\star}(x)-\tau_{\star}(y))^{2}.$

The first term in the last line will produce the desired result in (A.13),
after sending $\beta\searrow 0$, therefore, to prove (A.13), we need to check
that, for a given $\beta>0$, the second term produces an integral that
vanishes as $\varepsilon\searrow 0$. To this end, without loss of generality
we can assume that the balls selected in (A.6) satisfy
$B_{r_{j}}(p_{j})\subseteq B_{2R_{0}}$, since the others, for small
$\varepsilon$ would not intersect $\overline{B_{R_{0}}}$ anyway. As a result,
if $x\in\mathbb{R}^{n}\setminus B_{2R_{0}}$, we have that $\tau_{\star}(x)=1$
and so, if $y\in B_{R_{0}}$,

$\tau_{\star}(x)-\tau_{\star}(y)=1-\tau_{\star}(y)=(1-\tau_{\star}(y))\chi_{U}(y).$

From this observation, (4.1) and (A.11), we infer that

$\displaystyle\iint_{({\mathcal{R}}\setminus
B_{2R_{0}})\times{\mathcal{R}}}\phi^{2}(y)(\tau_{\star}(x)-\tau_{\star}(y))^{2}\,\frac{d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}$
$\displaystyle\qquad\leqslant C\iint_{({\mathcal{R}}\setminus
B_{2R_{0}})\times({\mathcal{R}}\cap
B_{R_{0}})}(\tau_{\star}(x)-\tau_{\star}(y))^{2}\,\frac{d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}$
$\displaystyle\qquad=C\iint_{({\mathcal{R}}\setminus
B_{2R_{0}})\times({\mathcal{R}}\cap B_{R_{0}}\cap
U)}(1-\tau_{\star}(y))^{2}\,\frac{d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}$
$\displaystyle\qquad\leqslant
C\sum_{k=1}^{+\infty}\iint_{({\mathcal{R}}\cap(B_{2^{k+1}R_{0}}\setminus
B_{2^{k}R_{0}}))\times({\mathcal{R}}\cap B_{R_{0}}\cap
U)}\,\frac{d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{\big{(}(2^{k}-1)R_{0}\big{)}^{n+s}}$
$\displaystyle\qquad\leqslant
C\varepsilon\sum_{k=1}^{+\infty}\frac{(2^{k+1}R_{0})^{n-1}}{\big{(}(2^{k}-1)R_{0}\big{)}^{n+s}}$
$\displaystyle\qquad\leqslant C\varepsilon.$

Therefore, to estimate, as $\varepsilon\searrow 0$, the integral on the right-
hand side of (A.10), we can focus on the computation below:

$\displaystyle\iint_{({\mathcal{R}}\cap
B_{2R_{0}})\times{\mathcal{R}}}\phi^{2}(y)(\tau_{\star}(x)-\tau_{\star}(y))^{2}\,\frac{d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}$
$\displaystyle\qquad\leqslant C\iint_{({\mathcal{R}}\cap
B_{2R_{0}})\times({\mathcal{R}}\cap
B_{2R_{0}})}(\tau_{\star}(x)-\tau_{\star}(y))^{2}\,\frac{d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}.$

If both $x$ and $y$ lie outside $U$, we have that
$\tau_{\star}(x)-\tau_{\star}(y)=1-1=0$, hence, up to renaming $C$, we can
actually reduce to

(A.14) $\begin{split}&C\iint_{({\mathcal{R}}\cap B_{2R_{0}}\cap
U)\times({\mathcal{R}}\cap
B_{2R_{0}})}(\tau_{\star}(x)-\tau_{\star}(y))^{2}\,\frac{d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}\\\
&\qquad\leqslant C\sum_{j=0}^{N}\iint_{({\mathcal{R}}\cap B_{2R_{0}}\cap
B_{4r_{j}}(p_{j}))\times({\mathcal{R}}\cap
B_{2R_{0}})}(\tau_{\star}(x)-\tau_{\star}(y))^{2}\,\frac{d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}.\end{split}$

Furthermore, using again (4.1) and (A.11),

$\displaystyle\sum_{j=0}^{N}\iint_{({\mathcal{R}}\cap B_{2R_{0}}\cap
B_{4r_{j}}(p_{j}))\times(({\mathcal{R}}\cap B_{2R_{0}})\setminus
B_{16r_{j}}(p_{j}))}\,\frac{d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}$
$\displaystyle\qquad\leqslant\sum_{j=0}^{N}\sum_{k=2}^{+\infty}\iint_{({\mathcal{R}}\cap
B_{2R_{0}}\cap B_{4r_{j}}(p_{j}))\times(({\mathcal{R}}\cap
B_{2R_{0}})\cap(B_{4^{k+1}r_{j}}(p_{j})\setminus
B_{4^{k}r_{j}}(p_{j})))}\,\frac{d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}$
$\displaystyle\qquad\leqslant\sum_{j=0}^{N}\sum_{k=2}^{+\infty}\iint_{({\mathcal{R}}\cap
B_{4r_{j}}(p_{j}))\times({\mathcal{R}}\cap
B_{4^{k+1}r_{j}}(p_{j}))}\,\frac{d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{\big{(}(4^{k}-4)r_{j}\big{)}^{n+s}}$
$\displaystyle\qquad\leqslant
C\sum_{j=0}^{N}\sum_{k=2}^{+\infty}\,\frac{(4r_{j})^{n-1}(4^{k+1}r_{j})^{n-1}}{\big{(}(4^{k}-4)r_{j}\big{)}^{n+s}}$
$\displaystyle\qquad\leqslant C\sum_{j=0}^{N}r_{j}^{n-2-s}$
$\displaystyle\qquad\leqslant C\varepsilon.$

For this reason, as $\varepsilon\searrow 0$, we can reduce the calculation in
(A.14) to

(A.15) $\begin{split}&C\sum_{j=0}^{N}\iint_{({\mathcal{R}}\cap B_{2R_{0}}\cap
B_{16r_{j}}(p_{j}))\times({\mathcal{R}}\cap B_{2R_{0}}\cap
B_{16r_{j}}(p_{j}))}(\tau_{\star}(x)-\tau_{\star}(y))^{2}\,\frac{d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}.\end{split}$

Now we observe that, given real numbers $\\{a_{j}\\}_{j\in\\{0,\dots,N\\}}$
and $\\{b_{j}\\}_{j\in\\{0,\dots,N\\}}$, we have that

(A.16)
$\left|\min_{j\in\\{0,\dots,N\\}}a_{j}-\min_{j\in\\{0,\dots,N\\}}b_{j}\right|\leqslant\max_{j\in\\{0,\dots,N\\}}|a_{j}-b_{j}|.$

To check this, one can assume, up to swapping the two classes of numbers, that

$a_{j_{a}}=\min_{j\in\\{0,\dots,N\\}}a_{j}\geqslant\min_{j\in\\{0,\dots,N\\}}b_{j}=b_{j_{b}},$

for suitable $j_{a}$, $j_{b}\in\\{0,\dots,N\\}$.

Therefore,

$\displaystyle\left|\min_{j\in\\{0,\dots,N\\}}a_{j}-\min_{j\in\\{0,\dots,N\\}}b_{j}\right|=a_{j_{a}}-b_{j_{b}}\leqslant
a_{j_{b}}-b_{j_{b}}\leqslant|a_{j_{b}}-b_{j_{b}}|\leqslant\max_{j\in\\{0,\dots,N\\}}|a_{j}-b_{j}|,$

which proves (A.16).

As a byproduct of (A.16), we see that

$\displaystyle|\tau_{\star}(x)-\tau_{\star}(y)|=\left|\min_{j=0,\dots,N}\\{1-\tau_{j}(x)\\}-\min_{j=0,\dots,N}\\{1-\tau_{j}(y)\\}\right|$
$\displaystyle\qquad\leqslant\max_{j=0,\dots,N}\big{|}(1-\tau_{j}(x))-(1-\tau_{j}(y))\big{|}=\max_{j=0,\dots,N}|\tau_{j}(x)-\tau_{j}(y)|.$

Utilizing this information, we majorize (A.15) by

(A.17) $C\sum_{j=0}^{N}\iint_{({\mathcal{R}}\cap B_{2R_{0}}\cap
B_{16r_{j}}(p_{j}))\times({\mathcal{R}}\cap B_{2R_{0}}\cap
B_{16r_{j}}(p_{j}))}\max_{m=0,\dots,N}(\tau_{m}(x)-\tau_{m}(y))^{2}\,\frac{d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}.$

Now we observe that, for each $x$, $y\in B_{16r_{j}}(p_{j})$, we have that
$\tau_{m}(x)-\tau_{m}(y)=0$ if both $x$ and $y$ lie in the complement of
$B_{4r_{m}}(p_{m})$. Accordingly, (A.17) reduces to

$C\sum_{j=0}^{N}\iint_{({\mathcal{R}}\cap B_{2R_{0}}\cap
B_{16r_{j}}(p_{j}))\times({\mathcal{R}}\cap B_{2R_{0}}\cap
B_{16r_{j}}(p_{j}))}\;\max_{{m=0,\dots,N}\atop{B_{4r_{m}}(p_{m})\cap
B_{16r_{j}}(p_{j})\neq\varnothing}}(\tau_{m}(x)-\tau_{m}(y))^{2}\,\frac{d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}.$

Thus, recalling the gradient bound in (A.5), we majorize this quantity by

$\begin{split}&C\sum_{j,m=0}^{N}\iint_{{({\mathcal{R}}\cap B_{2R_{0}}\cap
B_{16r_{j}}(p_{j}))\times({\mathcal{R}}\cap B_{2R_{0}}\cap
B_{16r_{j}}(p_{j}))}\atop{B_{16r_{j}}(p_{j})\cap
B_{16r_{m}}(p_{m})\neq\varnothing}}\,\frac{d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{r_{m}^{2}\,|x-y|^{n+s-2}}\\\
&\qquad\leqslant C\sum_{j,m=0}^{N}\iint_{{({\mathcal{R}}\cap B_{2R_{0}}\cap
B_{32r_{m}}(p_{m}))\times({\mathcal{R}}\cap B_{2R_{0}}\cap
B_{32r_{m}}(p_{m}))}\atop{B_{16r_{j}}(p_{j})\cap
B_{16r_{m}}(p_{m})\neq\varnothing}}\,\frac{d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{r_{m}^{2}\,|x-y|^{n+s-2}}\end{split}$

and consequently, in light of the finite intersection property in (A.3), up to
renaming constants, by

(A.18) $C\sum_{m=0}^{N}\iint_{{({\mathcal{R}}\cap B_{2R_{0}}\cap
B_{32r_{m}}(p_{m}))\times({\mathcal{R}}\cap B_{2R_{0}}\cap
B_{32r_{m}}(p_{m}))}}\,\frac{d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{r_{m}^{2}\,|x-y|^{n+s-2}}.$

Now, given $x\in{\mathcal{R}}\cap B_{2R_{0}}\cap B_{32r_{m}}(p_{m})$, we
utilize (4.1) to see that

$\displaystyle\int_{{{\mathcal{R}}\cap B_{2R_{0}}\cap
B_{32r_{m}}(p_{m})}}\,\frac{d{\mathcal{H}}^{n-1}_{y}}{r_{m}^{2}\,|x-y|^{n+s-2}}\leqslant\sum_{k=0}^{+\infty}\int_{{{\mathcal{R}}\cap(B_{64r_{m}/2^{k}}(x)\setminus
B_{64r_{m}/2^{k+1}}(x))}}\,\frac{d{\mathcal{H}}^{n-1}_{y}}{r_{m}^{2}\,|x-y|^{n+s-2}}$
$\displaystyle\qquad\leqslant\sum_{k=0}^{+\infty}\frac{(r_{m}/2^{k})^{n-1}}{r_{m}^{2}\,(r_{m}/2^{k})^{n+s-2}}\leqslant\frac{C}{r_{m}^{1+s}}\,\sum_{k=0}^{+\infty}\frac{1}{2^{(1-s)k}}\leqslant\frac{C}{r_{m}^{1+s}}.$

Therefore, using again (4.1) and (A.2), the quantity in (A.18) can be bounded
from above by

$C\sum_{m=0}^{N}\int_{{({\mathcal{R}}\cap B_{2R_{0}}\cap
B_{32r_{m}}(p_{m})}}\,\frac{d{\mathcal{H}}^{n-1}_{x}}{r_{m}^{1+s}}\leqslant
C\sum_{m=0}^{N}r_{m}^{n-2-s}\leqslant C\varepsilon,$

as desired: this shows that, as $\varepsilon\searrow 0$, the integral in the
right-hand side of (A.10) can be majorized by the quantity in (A.13).

Step 6. Completion of the proof of (4.4). In view of the bounds in (A.10),
(A.12) and (A.13), we know that

$\iint_{{\mathcal{R}}\times{\mathcal{R}}}(\phi^{2}(x)+\phi^{2}(y))(u_{Q}(x)-u_{Q}(y))^{2}\,\frac{\,d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}\leqslant
C\iint_{{\mathcal{R}}\times{\mathcal{R}}}(\phi(x)-\phi(y))^{2}\,\frac{d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}}{|x-y|^{n+s}}.$

Assuming $\phi=1$ in $B_{R_{0}/2}$ (and possibly renaming $R_{0}$) we thereby
conclude that (4.4) holds true, as desired.

Step 7. Proof of (4.5). We observe that the left-hand side of (4.5) is finite,
thanks to (4.4). We employ (4.3) by taking $\zeta$ as the function defined in
(A.7). In this way, we have that

(A.19)
$\begin{split}0&\leqslant\iint_{{\mathcal{R}}\times{\mathcal{R}}}\frac{(u_{Q}(x)-u_{Q}(y))(\phi(x)\tau_{\star}(x)-\phi(y)\tau_{\star}(y))}{|x-y|^{n+s}}\,d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}\\\
&=\iint_{{\mathcal{R}}\times{\mathcal{R}}}\frac{\tau_{\star}(x)(u_{Q}(x)-u_{Q}(y))(\phi(x)-\phi(y))}{|x-y|^{n+s}}\,d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}\\\
&\qquad+\iint_{{\mathcal{R}}\times{\mathcal{R}}}\frac{\phi(y)(u_{Q}(x)-u_{Q}(y))(\tau_{\star}(x)-\tau_{\star}(y))}{|x-y|^{n+s}}\,d{\mathcal{H}}^{n-1}_{x}\,d{\mathcal{H}}^{n-1}_{y}.\end{split}$

The latter term vanishes as $\varepsilon\searrow 0$, owing to (4.4), and the
claim in (4.5) follows. ∎

##  Appendix B The set of “good” regular points

Here we describe the set of regular points of a nonlocal minimal set around
which the boundary is a hypersurface of class $C^{2}$ with bounded curvatures
in a conveniently scale invariant setting. In the classical case, this set was
introduced in [MR906394, Lemma 1] and we provide here a nonlocal version of
it.

######  Lemma B.1.

Let $E$ be $s$-minimal in $B_{1}$ and assume that the origin belongs to its
boundary.

Given $\theta>0$, let ${\mathcal{M}}_{\theta,E}$ be the set of points $x$
belonging to the regular part of $\partial E$ such that

(B.1) $\begin{split}&{\mbox{$(\partial E)\cap B_{\theta|x|}(x)$ belongs to the
regular part of\leavevmode\nobreak\ $\partial E$}}\\\ &{\mbox{and possesses
curvatures bounded in absolute value by }}\frac{1}{\theta|x|}.\end{split}$

Then, there exist $\rho_{0}$, $\theta_{0}>0$, depending only on $E$, $n$ and
$s$, such that, for every $\rho\in(0,\rho_{0}]$ and $\theta\in(0,\theta_{0}]$,
we have that

(B.2) ${\mathcal{M}}_{\theta,E}\cap(\partial B_{\rho})\neq\varnothing.$

###### Proof.

Suppose not. Then, there exist sequences $\theta_{j}\searrow 0$ and
$\rho_{j}\searrow 0$ for which ${\mathcal{M}}_{\theta_{j},E}\cap(\partial
B_{\rho_{j}})=\varnothing$. That is, if $x\in\partial B_{\rho_{j}}$ belongs to
the regular part of $\partial E$ and also $(\partial E)\cap
B_{\theta_{j}|x|}(x)$ belongs to the regular part of $\partial E$, then
necessarily its curvatures are not bounded in absolute value by
$\frac{1}{\theta_{j}|x|}=\frac{1}{\theta_{j}\rho_{j}}$.

Let $F_{j}:=E/\rho_{j}$. Up to a subsequence, we know that $F_{j}$ converges
locally uniformly to a cone ${\mathcal{C}}$ which is $s$-minimal (see
[MR2675483]). In fact, thanks to the improvement of flatness of nonlocal
minimal surfaces (see [MR2675483, Corollary 4.4 and Theorem 6.1]), this
convergence occurs in the $C^{1,\alpha}$ sense at the regular points of
${\mathcal{C}}$, and actually in the $C^{k}$ sense for any $k\geqslant 2$ (see
[MR3331523]).

Hence, we pick a regular point $y_{0}\in{\mathcal{C}}\cap(\partial B_{1})$
(whose existence is guaranteed by the regularity theory of nonlocal minimal
surfaces, see [MR3090533]). Then, we find a sequence $y_{j}\in\partial B_{1}$
of regular points of $F_{j}$ approaching $y_{0}$ as $j\to+\infty$ and
$(\partial F_{j})\cap B_{r_{0}}(y_{j})$ consists of regular points with
curvatures bounded in absolute value by $M_{0}$, for suitable $r_{0}$,
$M_{0}>0$.

Scaling back and setting $x_{j}:=\rho_{j}y_{j}\in\partial B_{\rho_{j}}$, we
find that $(\partial E)\cap B_{r_{0}\rho_{j}}(x_{j})$ lies in the regular set
of $\partial E$, with curvatures bounded in absolute value by
$\frac{M_{0}}{\rho_{j}}$, which, for large $j$, is strictly less than
$\frac{1}{\theta_{j}\rho_{j}}$, contradiction. ∎

With reference to Lemma B.1, we observe that, for all $r>0$,

(B.3) ${\mathcal{M}}_{\theta,\frac{E}{r}}=\frac{{\mathcal{M}}_{\theta,E}}{r}.$

Then, we have:

######  Corollary B.2.

Let $E_{1}$, $E_{2}$ be $s$-minimal sets in $B_{1}$ and assume that the origin
belongs to their boundary. Assume also that $E_{1}$ and $E_{2}$ have the same
tangent cone at the origin.

Let $\theta_{0}$, $\rho_{0}>0$ be as in Lemma B.1, used here with $E:=E_{1}$.
Let also ${\mathcal{M}}$ be the set ${\mathcal{M}}_{\theta_{0},E}$ in Lemma
B.1 with $E=E_{1}$.

Suppose that

$E_{2}\setminus
E_{1}=\Big{\\{}x+t\nu(x),\;x\in{\mathcal{M}},\;t\in\big{[}0,w(x)\big{)}\Big{\\}},$

for some function $w$ of class $C^{2}$, where $\nu$ denotes the external unit
normal of $E_{1}$ at its regular points.

Then,

(B.4) $\lim_{r\searrow 0}\frac{\displaystyle\sup_{x\in(\partial
B_{r})\cap{\mathcal{M}}}|w(x)|}{r}=0.$

In addition, if $w(x)\neq 0$ for all $x\neq 0$, then there exists an
infinitesimal sequence of points $z_{\star,k}\in{\mathcal{M}}$ such that

(B.5) $\frac{|w(x)|}{|x|}\leqslant\frac{2|w(z_{\star,k})|}{|z_{\star,k}|}$

for all $x\in{\mathcal{M}}$ with $|x|\leqslant|z_{\star,k}|/2$.

Finally, the distance of $z_{\star,k}$ from the singular set of $E_{1}$ is at
least $\theta_{0}|z_{\star,k}|$.

###### Proof.

Before proving (B.4), we stress that the supremum in this equation makes
sense, thanks to (B.2).

We now prove (B.4). Suppose not. Then, there exist $c>0$, an infinitesimal
sequence $r_{j}>0$ and $x_{j}\in(\partial B_{r_{j}})\cap{\mathcal{M}}$ such
that

(B.6) $|w(x_{j})|\geqslant cr_{j}.$

Thus, if $w_{j}:=\frac{w}{r_{j}}$ and $E_{i,j}:=\frac{E_{i}}{r_{j}}$ for
$i\in\\{1,2\\}$, we see that

$E_{2,j}\setminus
E_{1,j}=\left\\{x+t\nu_{j}(x),\;x\in\frac{{\mathcal{M}}}{r_{j}},\;t\in\big{[}0,w_{j}(x)\big{)}\right\\},$

where $\nu_{j}$ denotes the external unit normal of $E_{1,j}$.

Combining this and (B.3), we obtain

$E_{2,j}\setminus
E_{1,j}=\Big{\\{}x+t\nu_{j}(x),\;x\in{\mathcal{M}}_{\theta_{0},E_{1,j}},\;t\in\big{[}0,w_{j}(x)\big{)}\Big{\\}}.$

So, setting $y_{j}:=\frac{x_{j}}{r_{j}}\in(\partial
B_{1})\cap{\mathcal{M}}_{\theta_{0},E_{1,j}}$, we have that

(B.7) $y_{j}+w_{j}(y_{j})\nu_{j}(y_{j})\in\partial E_{2,j}.$

Notice in addition that, since the distance of $x_{j}$ from the singular set
of $E_{1}$ is at least $\theta_{0}|x_{j}|=\theta_{0}r_{j}$, we have that the
distance of $y_{j}$ from the singular set of $E_{1,j}$ is at least
$\theta_{0}$. The curvature bound thus allows us to pass to the limit as
$j\to+\infty$, up to a subsequence, and find that $y_{j}\to y_{\infty}$, with
$y_{\infty}$ belonging to the regular part of the tangent cone of $E_{1}$, and
also $\nu_{j}(y_{j})\to\nu_{\infty}(y_{\infty})$, where $\nu_{\infty}$ denotes
the normal of this tangent cone.

Hence, since $E_{2}$ shares the same tangent cone of $E_{1}$, it follows from
(B.7) that $w_{j}(y_{j})\to 0$ as $j\to+\infty$. But this is in contradiction
with (B.6) and the proof of (B.4) is thereby complete.

Now we prove (B.5). For this, let

$\sigma(r):=\frac{\displaystyle\sup_{x\in(\partial
B_{r})\cap{\mathcal{M}}}|w(x)|}{r}.$

Thanks to (B.4), we have that $\sigma(r)\to 0$ as $r\searrow 0$.

Also, we know that $\sigma(r)>0$ for all $r\neq 0$. Therefore, given
$\mu_{0}>1$ we pick an infinitesimal sequence $r_{k}>0$ and choose
$r_{\star,k}\in(0,r_{k}]$ such that

$\sigma(r_{\star,k})\geqslant\frac{\displaystyle\sup_{r\in(0,r_{k}]}\sigma(r)}{\mu_{0}}.$

As a result, for all $\mu\in(0,1)$,

$\displaystyle\frac{\displaystyle\sup_{x\in(\partial
B_{\mu{r_{\star,k}}})\cap{\mathcal{M}}}|w(x)|}{\mu{r_{\star,k}}}=\sigma({\mu{r_{\star,k}}})\leqslant\sup_{r\in(0,r_{k}]}\sigma(r)\leqslant\mu_{0}\,\sigma(r_{\star,k}).$

Besides, by (B.2), we can pick $z_{\star,k}\in(\partial
B_{r_{\star,k}})\cap{\mathcal{M}}$ with

$\frac{|w(z_{\star,k})|}{r_{\star,k}}\geqslant\frac{\displaystyle\sup_{x\in(\partial
B_{r_{\star,k}})\cap{\mathcal{M}}}|w(x)|}{\mu_{0}\,r_{\star,k}}=\frac{\sigma(r_{\star,k})}{\mu_{0}}.$

This gives that, for all $x\in{\mathcal{M}}$ with
$|x|\leqslant|z_{\star,k}|/2$, we have that

$\displaystyle\frac{|w(x)|}{|x|}\leqslant\sup_{\mu\in(0,1/2]}\frac{\displaystyle\sup_{x\in(\partial
B_{\mu{r_{\star,k}}})\cap{\mathcal{M}}}|w(x)|}{\mu
r_{\star,k}}\leqslant\mu_{0}\,\sigma(r_{\star,k})\leqslant\frac{\mu_{0}^{2}\,|w(z_{\star,k})|}{|z_{\star,k}|},$

which, choosing $\mu_{0}:=\sqrt{2}$, establishes (B.5), as desired.

Finally, the distance of $z_{\star,k}$ from the singular set is estimated from
below by the first line in (B.1). ∎

## References