# Rigidity theorems for best Sobolev inequalities

Francesco Maggi  Department of Mathematics, The University of Texas at Austin,
2515 Speedway, Stop C1200, Austin TX 78712-1202, United States of America
<EMAIL_ADDRESS>,  Robin Neumayer  Department of Mathematical Sciences,
Carnegie Mellon University, Pittsburgh, PA 15213, United States of America
<EMAIL_ADDRESS>and  Ignacio Tomasetti  Department of Mathematics, The
University of Texas at Austin, 2515 Speedway, Stop C1200, Austin TX
78712-1202, United States of America<EMAIL_ADDRESS>
###### Abstract.

For $n\geq 2$, $p\in(1,n)$, the “best $p$-Sobolev inequality” on an open set
$\Omega\subset\mathbb{R}^{n}$ is identified with a family $\Phi_{\Omega}$ of
variational problems with critical volume and trace constraints. When $\Omega$
is bounded we prove: (i) for every $n$ and $p$, the existence of generalized
minimizers that have at most one boundary concentration point, and: (ii) for
$n>2\,p$, the existence of (classical) minimizers. We then establish rigidity
results for the comparison theorem “balls have the worst best Sobolev
inequalities” by the first named author and Villani, thus giving the first
affirmative answers to a question raised in [MV05].

###### Contents

  1. 1 Introduction
  2. 2 Notation and preparations
  3. 3 Boundary concentrations
  4. 4 Existence of minimizers
  5. 5 Rigidity theorems for best Sobolev inequalities
  6. A Proof of Lemma 2.1
  7. B Proof of Lemmas 2.2 and 2.3

##  1\. Introduction

###  1.1. Overview

The goal of this paper is to answer some basic open questions concerning a
“doubly critical” family $\\{\Phi_{\Omega}(T)\\}_{T\geq 0}$ of minimization
problems on Sobolev functions, which, in a precise sense to be clarified
below, can be interpreted as collectively defining the best Sobolev inequality
on an open set $\Omega\subset\mathbb{R}^{n}$ with $C^{1}$-boundary. Given an
integer $n\geq 2$ and $p\in(1,n)$, these problems are defined as

$\Phi_{\Omega}(T)=\inf\Big{\\{}\Big{(}\int_{\Omega}|\nabla
u|^{p}\Big{)}^{1/p}:\int_{\Omega}|u|^{p^{\star}}=1\,,\,\int_{\partial\Omega}|u|^{p^{\\#}}=T^{p^{\\#}}\Big{\\}}\,,$
(1.1)

and their minimizers, whenever they exist, satisfy the Euler–Lagrange equation

$\left\\{\begin{split}&-\Delta_{p}u=\lambda\,u^{{p^{\star}}-1}\,,\quad\hskip
28.45274pt\mbox{on $\Omega$}\,,\\\ &|\nabla u|^{p-2}\,\frac{\partial
u}{\partial\nu_{\Omega}}=\sigma\,u^{{p^{\\#}}-1}\,,\quad\mbox{on
$\partial\Omega$}\,,\end{split}\right.$ (1.2)

for suitable Lagrange multipliers $\lambda,\sigma\in\mathbb{R}$. We call
$\int_{\Omega}|u|^{p^{\star}}=1$ and
$\int_{\partial\Omega}|u|^{p^{\\#}}=T^{p^{\\#}}$ the “volume” and “trace”
constraints of $\Phi_{\Omega}(T)$. The critical Sobolev exponents associated
to $n$ and $p$, ${p^{\star}}$ and ${p^{\\#}}$, are defined by

${p^{\star}}=\frac{np}{n-p}\,,\qquad{p^{\\#}}=\frac{(n-1)p}{n-p}=\frac{n-1}{n}\,{p^{\star}}\,,$

and their precise values guarantee the scale invariance of $\Phi_{\Omega}$,
i.e.

$\Phi_{x+r\,\Omega}(T)=\Phi_{\Omega}(T)\qquad\forall
x\in\mathbb{R}^{n}\,,r>0\,,T\geq 0\,.$

When $\Omega$ is bounded, the $C^{1}$-regularity of $\partial\Omega$
guarantees that every $u\in L^{1}_{\rm loc}(\Omega)$ with $\nabla u\in
L^{p}(\Omega;\mathbb{R}^{n})$ lies in the competition class of
$\Phi_{\Omega}(T)$ for some $T\geq 0$. In particular,

${\rm Epi}(\Phi_{\Omega})=\big{\\{}(T,G)\in\mathbb{R}^{2}:T\geq
0\,,G\geq\Phi_{\Omega}(T)\big{\\}}\,,$

(the epigraph of $\Phi_{\Omega}$) collects the best possible information on
the range of values achievable by $\|\nabla u\|_{L^{p}(\Omega)}$ when
$\|u\|_{L^{p^{\star}}(\Omega)}$ is fixed: from this peculiar viewpoint, which
is reminiscent of the one adopted in the study of Blaschke–Santaló diagrams,
${\rm Epi}(\Phi_{\Omega})$ is “the best Sobolev inequality on $\Omega$”. The
following list of results, summarized in

$\Phi_{H}(T)$$\Phi_{B}(T)$$G$$T_{0}$$S$$T$${\rm
ISO}(B)^{1/{p^{\\#}}}$$\Phi_{\Omega}(T)$$ET$${\rm
ISO}(\Omega)^{1/{p^{\\#}}}$$T^{p^{\\#}}/{p^{\\#}}$$\frac{S}{2^{1/n}}$$T_{E}$$T=0$

Figure 1.1. The state of the art about $\Phi_{\Omega}$. The “exclusion zone”
for the values of $\|\nabla u\|_{L^{p}(\Omega)}$ (under the constraint
$\|u\|_{L^{p^{\star}}(\Omega)}=1$) is depicted in gray. It always contains the
subgraph of $\Phi_{B}(T)$ on $T\in[0,{\rm ISO}(B)^{1/{p^{\\#}}}]$, see (1.12),
but is always smaller than the one of a half-space $H$, see (1.19). The
Sobolev inequality on $\mathbb{R}^{n}$ is equivalent to
$\Phi_{\Omega}(0)=S(n,p)$, the Euclidean isoperimetric inequality to the fact
that the only zero of $\Phi_{\Omega}$ (i.e. $T={\rm
ISO}(\Omega)^{1/{p^{\\#}}}$) is achieved to the right of the only zero of
$\Phi_{B}$, and the Escobar inequality (1.6) is equivalent to the linear bound
$\Phi_{H}(T)\geq E\,T$. Both $\\{(T,\Phi_{B}(T)):T\in[0,{\rm
ISO}(B)^{1/{p^{\\#}}}]\\}$ and $\\{(T,\Phi_{H}(T)):T\geq 0\\}$ can be
implicitly parametrized by looking at the explicit families of minimizers
given in (1.11), (1.15), (1.16) and (1.18).

Figure 1.1, aims to provide a hopefully complete state of the art on
$\Phi_{\Omega}$, and illustrates the wealth of information stored in this
family of variational problems. As a disclaimer: here we are definitely not
attempting to exhaustively frame the study of $\Phi_{\Omega}$ into the
incredibly vast and layered context of the theory of Sobolev-type inequalities
(see e.g. [Maz85]), as that would be a long and delicate exercise, lying well
beyond the scope of this introduction.

(1) Sobolev inequality on $\mathbb{R}^{n}$: A scaling and localization
argument shows that, for every open set $\Omega$, one has

$\Phi_{\Omega}(0)=S(n,p):=\inf\Big{\\{}\Big{(}\int_{\mathbb{R}^{n}}|\nabla
u|^{p}\Big{)}^{1/p}:\int_{\mathbb{R}^{n}}|u|^{p^{\star}}=1\Big{\\}}\,,$ (1.3)

that is, $\Phi_{\Omega}(0)$ is the best constant in the $L^{p}$-Sobolev
inequality on $\mathbb{R}^{n}$. Minimizers of (1.3) are exactly given by the
family
$\\{\tau_{x_{0}}[U_{S}^{(\alpha)}]\\}_{x_{0}\in\mathbb{R}^{n}\,,\alpha>0}$
generated by

$U_{S}(x)=\big{(}1+|x|^{p/(p-1)}\big{)}^{1-(n/p)}\,,\qquad
x\in\mathbb{R}^{n}\,,$ (1.4)

see [Aub76, Tal76, CENV04], we see that $\Phi_{\Omega}(0)$ is attained if and
only if $\Omega=\mathbb{R}^{n}$. Here and in the following, we set

$\tau_{x_{0}}[v](x)=v(x-x_{0})\,,\qquad
v^{(\alpha)}(x)=\alpha^{-n/{p^{\star}}}\,v(x/\alpha)$ (1.5)

whenever $x_{0}\in\mathbb{R}^{n}$ and $\alpha>0$.

(2) Escobar inequality: The Escobar inequality ([Esc88, for $p=2$], [Naz06,
for $p\in(1,n)$]) states that if $H$ is an (open) half-space in
$\mathbb{R}^{n}$ with outer unit normal $\nu_{H}$, then

$\Big{(}\int_{H}|\nabla u|^{p}\Big{)}^{1/p}\geq E(n,p)\,\Big{(}\int_{\partial
H}|u|^{p^{\\#}}\Big{)}^{1/{p^{\\#}}}$ (1.6)

with equality if and only if111The “only if” statement for $p\neq 2$ was left
open in [Naz06], but was proven in [MN17, Theorem 2.3].
$u=\tau_{x_{0}}[U_{E}^{(\alpha)}]$ for some
$x_{0}\in\mathbb{R}^{n}\setminus\overline{H}$, and where

$U_{E}(x)={|x|^{-(n-p)/(p-1)}}\,,\qquad x\in\mathbb{R}^{n}\setminus\\{0\\}\,,$
(1.7)

is a multiple of the fundamental solution of the $p\,$-Laplacian. The quantity

$T_{E}=\|\tau_{x_{0}}[U_{E}^{(\alpha)}]\|_{L^{{p^{\\#}}}(\partial
H)}\,\Big{/}\,\|\tau_{x_{0}}[U_{E}^{(\alpha)}]\|_{L^{{p^{\star}}}(H)}$ (1.8)

is independent of $x_{0}\in\mathbb{R}^{n}\setminus\overline{H}$, and is such
that

$\Phi_{H}(T_{E})=E(n,p)\,.$ (1.9)

The Escobar inequality (1.6) can be equivalently reformulated in the
“$\Phi$-setting” as a linear lower bound for $\Phi_{H}$, i.e.

$\Phi_{H}(T)\geq E(n,p)\,T\,,\qquad\forall T\geq 0\,.$

This bound is sharp only if $T=T_{E}$, and is nearly optimal only if $T$ is
close to $T_{E}$; but it largely suboptimal away from $T_{E}$, see Figure 1.1.

(3) Euclidean isoperimetry: It is easily seen that $\Phi_{\Omega}(T)=0$ for
some $T>0$ if and only if $|\Omega|<\infty$ and $T={\rm
ISO}(\Omega)^{1/{p^{\\#}}}$, where ${\rm
ISO}(\Omega)=\mathcal{H}^{n-1}(\partial\Omega)/|\Omega|^{(n-1)/n}$ stands for
the isoperimetric ratio of $\Omega$. Since the Euclidean isoperimetric
inequality states that

${\rm ISO}(\Omega)\geq{\rm ISO}(B)\,,$ (1.10)

(with equality if and only if $\Omega$ is a ball), in the $\Phi$-setting,
(1.10) is equivalent to saying that $\Phi_{B}$ has the left-most zero among
all $\Phi_{\Omega}$.

(4) Balls have the worst best Sobolev inequalities: In [CL90, CL94, $p=2$] (by
symmetrization methods and conformal invariance) and in [MV05, $p\in(1,n)$]
(via the mass transportation method pioneered in [Kno57, MS86, CENV04]) it is
shown that if $B$ is a ball, then for every $T\in(0,{\rm
ISO}(B)^{1/{p^{\\#}}})$ there is a unique $\alpha>0$ such that

$\Phi_{B}(T)=\|\nabla
U_{S}^{(\alpha)}\|_{L^{p}(B)}\,\Big{/}\,\|U_{S}^{(\alpha)}\|_{L^{p^{\star}}(B)}\,,$
(1.11)

with $U_{S}$ defined in (1.4). Further elaborating on the proof of this
partial characterization of $\Phi_{B}$, again in [MV05] the comparison theorem
that balls have the worst best Sobolev inequalities

$\Phi_{\Omega}(T)\geq\Phi_{B}(T)\,,\qquad\forall T\in[0,{\rm
ISO}(B)^{1/{p^{\\#}}}]$ (1.12)

is proved. This sharp lower bound, combined with (1.11), allows one to infer
some sharp and more traditional-looking Sobolev-type inequalities, like the
following sharp interpolation between (1.3) and (1.10)

$\frac{\|\nabla
u\|_{L^{p}(\Omega)}}{S(n,p)}+\frac{\|u\|_{L^{p^{\\#}}(\partial\Omega)}}{{\rm
ISO}(B)^{1/{p^{\\#}}}}\geq\|u\|_{L^{p^{\star}}(\Omega)}\,,$ (1.13)

and the following sharp Sobolev inequality, additive in the domain of the
$p$-Dirichlet energy,

$\frac{\|\nabla
u\|_{L^{p}(\Omega)}^{p}}{S(n,p)^{p}}+\frac{\|u\|_{L^{p^{\\#}}(\partial\Omega)}^{p}}{C(n,p)}\geq\|u\|_{L^{p^{\star}}(\Omega)}^{p}\,,$
(1.14)

which was first conjectured by Brezis and Lieb in [BL85].

(5) Full characterization of $\Phi_{H}$: In [CL90, CL94, for $p=2$] and, again
by optimal mass transport arguments, in [MN17, for $p\in(1,n)$], it is proven
that if $H$ is half-space in $\mathbb{R}^{n}$, then: for each $T\in(0,T_{E})$
(Sobolev range) there is a unique $t_{T}\in\mathbb{R}$ such that

$\Phi_{H}(T)=\|\nabla(\tau_{t_{T}\,\nu_{H}}U_{S})\|_{L^{p}(H)}\,\Big{/}\,\|\tau_{t_{T}\,\nu_{H}}U_{S}\|_{L^{p^{\star}}(H)}\,;$
(1.15)

if $T=T_{E}$ (Escobar point), then

$\Phi_{H}(T_{E})=E(n,p)=\|\nabla(\tau_{\nu_{H}}U_{E})\|_{L^{p}(H)}\,\Big{/}\,\|\tau_{\nu_{H}}U_{E}\|_{L^{p^{\star}}(H)}\,;$
(1.16)

for each $T>T_{E}$ (beyond Escobar range), there is a unique $s_{T}>1$ s.t.

$\displaystyle\Phi_{H}(T)=\|\nabla(\tau_{s_{T}\,\nu_{H}}U_{BE})\|_{L^{p}(H)}\,\Big{/}\,\|\tau_{s_{T}\,\nu_{H}}U_{BE}\|_{L^{p^{\star}}(H)}\,,$
(1.18)
$\displaystyle\mbox{where}\,\,U_{BE}(x)=(|x|^{p/(p-1)}-1)^{1-(n/p)}\,,\qquad|x|>1\,.$

Up to the natural dilation and translation invariances, these functions are
the unique minimizers of $\Phi_{H}(T)$. Moreover, again by [MN17]: (a):
$\inf_{T\geq 0}\Phi_{H}(T)$ is achieved at $T=T_{0}\in(0,T_{E})$, where
$t_{T_{0}}=0$ and $\Phi_{H}(T_{0})=S(n,p)/2^{1/n}$; (b): by the divergence
theorem, $\Phi_{H}(T)>T^{{p^{\\#}}}/{p^{\\#}}$ for every $T>0$, and this lower
bound is sharp as $T\to\infty$; (c): finally, $\Phi_{H}$ has the best best
Sobolev inequality, i.e.

$\Phi_{\Omega}(T)\leq\Phi_{H}(T)\,,\qquad\forall T\geq 0\,.$ (1.19)

###  1.2. Statements of the main results

With this summary on the state of the art for $\Phi_{\Omega}$ in mind, there
are two fundamental open questions that form the subject of our paper:

Question 1: When does $\Phi_{\Omega}(T)$ ($T>0$) admit minimizers?

Question 2: Does rigidity hold in the comparison theorem (1.12)?

The main idea of this paper is attacking these two closely related questions
by systematically exploiting the complete characterization of $\Phi_{H}$
obtained in [MN17].

Concerning Question 1, a classical concentration-compactness argument
characterizes the limit behavior of minimizing sequences of $\Phi_{\Omega}(T)$
as the superposition of a standard weak limit plus at most countably many
concentration points, located either in the interior of $\Omega$, or on its
boundary. By exploiting properties of $\Phi_{H}$ we are able to (i): exclude
all interior concentrations and all but at most one boundary concentration,
thus proving existence of minimizers for a suitable “relaxed problem”
$\Phi_{\Omega}^{*}(T)$; and (ii): completely exclude concentrations, and thus
establish the existence of minimizers of $\Phi_{\Omega}(T)$, as soon as
$\partial\Omega$ is of class $C^{2}$ and $n>2p\,$. To give precise statements,
it is convenient to let $\mathcal{X}_{\Omega}(T)$ denote the competition class
of $\Phi_{\Omega}(T)$, and let $\mathcal{Y}_{\Omega}(T)$ denote the set of
triples $(u,\mathsf{v},\mathsf{t})$ with either $u\in\mathcal{X}_{\Omega}(T)$
and $\mathsf{v}=\mathsf{t}=0$, or $u\in W^{1,p}(\Omega)$,
$\mathsf{v}\in(0,1]$, $\mathsf{t}\in(0,T]$, and

$\mathsf{v}^{p^{\star}}+\int_{\Omega}u^{p^{\star}}=1\,,\qquad\mathsf{t}^{p^{\\#}}+\int_{\partial\Omega}u^{p^{\\#}}=T^{p^{\\#}}\,.$
(1.20)

The relaxed problem associated to $\Phi_{\Omega}(T)$ is then given by

$\Phi_{\Omega}^{*}(T)=\inf_{\mathcal{Y}_{\Omega}(T)}\,\mathcal{E}\,,\,\,\,\mbox{where}\,\,\,\mathcal{E}(u,\mathsf{v},\mathsf{t})^{p}=\int_{\Omega}|\nabla
u|^{p}+\mathsf{v}^{p}\,\Phi_{H}\Big{(}\frac{\mathsf{t}}{\mathsf{v}}\Big{)}^{p}\,,$
(1.21)

with the convention that $\mathsf{v}^{p}\Phi_{H}(\mathsf{t}/\mathsf{v})^{p}=0$
if $(\mathsf{v},\mathsf{t})=(0,0)$.

######  Theorem 1.1 (Existence of minimizers of $\Phi_{\Omega}$).

If $n\geq 2$, $p\in(1,n)$, and $\Omega$ is a bounded open set with
$C^{1}$-boundary in $\mathbb{R}^{n}$, then:

(i): for every $T>0$, there is a minimizer $(u,\mathsf{v},\mathsf{t})$ of
$\Phi_{\Omega}^{*}(T)$, and

$\Phi_{\Omega}(T)=\Phi_{\Omega}^{*}(T)\,;$ (1.22)

moreover, if $\int_{\Omega}u^{p^{\star}}>0$, then
$u/\|u\|_{L^{p^{\star}}(\Omega)}$ is a minimizer of
$\Phi_{\Omega}(\|u\|_{L^{p^{\\#}}(\partial\Omega)}/\|u\|_{L^{p^{\star}}(\Omega)})$;

(ii): if $\Omega$ has boundary of class $C^{2}$, $n>2p$, $T>0$, and
$(u,\mathsf{v},\mathsf{t})$ is a minimizer of $\Phi_{\Omega}^{*}(T)$, then
$\mathsf{v}=\mathsf{t}=0$, and thus $u$ is a minimizer of $\Phi_{\Omega}(T)$.

######  Remark 1.2.

Minimizers of $\Phi_{\Omega}(T)$ solve the Euler–Lagrange equation (1.2). For
the Euler–Lagrange equation satisfied by minimizers
$(u,\mathsf{v},\mathsf{t})$ of the relaxed problem $\Phi_{\Omega}^{*}(T)$, see
Theorem 4.1 below.

Question 2 is motivated by the various rigidity statements associated to
comparison theorems in Riemannian geometry (see, e.g. [CE08]). In that
setting, a certain model space provides a universal bound on a certain global
geometric quantity (comparison theorem), which is then shown to be saturated
by the model space alone (rigidity statement). With this analogy in mind, we
can reformulate more precisely Question 2 as follows:

Question 2, weak form: Does $\Phi_{\Omega}=\Phi_{B}$ on $(0,{\rm
ISO}(B)^{1/{p^{\\#}}})$ imply that $\Omega$ is a ball?

Question 2, strong form: Does $\Phi_{\Omega}(T)=\Phi_{B}(T)$ at just one value
of $T\in(0,{\rm ISO}(B)^{1/{p^{\\#}}})$ imply that $\Omega$ is a ball?

Concerning the weak form of Question 2, through a careful use of the
properties of $\Phi_{H}$ we answer affirmatively whenever $\Omega$ is bounded
and connected. These conditions are optimal, as shown by unbounded or
disconnected non-rigidity examples presented in [MV05]. In fact, the argument
we propose gives rigidity under the mere assumption that
$\Phi_{\Omega}=\Phi_{B}$ holds on an open neighborhood of $T=0$. Concerning
the strong form of Question 2, which was originally formulated in [MV05,
Section 1.9], we can answer in the affirmative as a direct by-product of our
existence result for minimizers of $\Phi_{\Omega}(T)$ (thus, when $\Omega$ has
$C^{2}$-boundary and $n>2p$) thanks to the following “conditional rigidity”
statement, which is proved in [MV05] as a direct by-product of the proof of
(1.12):

$\displaystyle\mbox{{\it if $\Omega$ is connected (possibly unbounded)}}\,,$
$\displaystyle\mbox{{\it if $\Phi_{\Omega}(T)=\Phi_{B}(T)$ for a value of
$T\in(0,{\rm ISO}(B)^{1/{p^{\\#}}})$}}\,,$ (1.23) $\displaystyle\mbox{{\it and
if $\Phi_{\Omega}(T)$ is known to admit minimizers (possibly just for that
$T$)}}\,,$ $\displaystyle\mbox{{\it then $\Omega$ is a ball}}\,.$

(We notice for future use an important consequence of (1.23), namely, we have

$\Phi_{B}(T)<\Phi_{H}(T)\,,\qquad\forall T\in\big{(}0,{\rm
ISO}(B)^{1/{p^{\\#}}}\big{)}\,;$ (1.24)

indeed, by [MN17], $\Phi_{H}(T)$ admits minimizers for every $T>0$.) With
these premises, we now state our main results concerning Question 2.

######  Theorem 1.3 (Rigidity of “Balls have the worst best Sobolev
inequalities”).

Let $n\geq 2$, $p\in(1,n)$, $\Omega$ an open, bounded, connected set with
$C^{1}$-boundary in $\mathbb{R}^{n}$, and assume that one of the following two
conditions holds:

(i): there is $T_{*}>0$ such that $\Phi_{\Omega}(T)=\Phi_{B}(T)$ for every
$T\in(0,T_{*})$; or

(ii): $n>2p$, the boundary of $\Omega$ is of class $C^{2}$, and there is
$T\in(0,{\rm ISO}(B)^{1/{p^{\\#}}})$ such that $\Phi_{\Omega}(T)=\Phi_{B}(T)$.

Then, $\Omega$ is a ball.

###  1.3. Strategy of proof

Concentration-compactness arguments and the use of sharp Sobolev-type
inequalities (like the Sobolev and Escobar inequalities (1.3) and (1.6)) are
the standard tools of the trade in the analysis of variational problems with
critical growth. As seen, if interpreted as assertions about $\Phi_{H}$, (1.3)
and (1.6) contain only very partial information (respectively,
“$\Phi_{H}(0)=S(n,p)$” and “$\Phi_{H}(T)\geq E(n,p)\,T$ for every $T>0$”).
From this viewpoint, our arguments provide an interesting example of the
potentialities of using, in the familiar context of concentration-compactness,
the full characterization of $\Phi_{H}$ obtained in [CL94, MN17]. We now
explain how this characterization is used in this paper.

We have already mentioned how the mere knowledge of the existence of
minimizers in $\Phi_{H}(T)$ for every $T>0$ allows one to reduce the analysis
of concentrations to the simplest possible case of a single boundary
concentration (thus leading to Theorem 1.1-(i)). Finer properties of
$\Phi_{H}$ are exploited in the proof of Theorem 1.1-(ii), which goes as
follows. We consider the existence of a minimizer $(u,\mathsf{v},\mathsf{t})$
of $\Phi_{\Omega}^{*}(T)$ with $\mathsf{v}>0$, and, keeping in mind that
$\Phi_{\Omega}(T)=\Phi_{\Omega}^{*}(T)$, we aim to obtain a contradiction to
$\mathsf{v}>0$ by constructing a competitor $v$ of $\Phi_{\Omega}(T)$ with

$\int_{\Omega}|\nabla v|^{p}<\int_{\Omega}|\nabla
u|^{p}+\mathsf{v}^{p}\,\Phi_{H}\Big{(}\frac{\mathsf{t}}{\mathsf{v}}\Big{)}^{p}\,.$

We seek $v$ in the form $v=u_{\varepsilon}$, for the Ansatz given by

$u_{\varepsilon}(x)=(1-\varphi_{\varepsilon}(x))\,u(x)+\varphi_{\varepsilon}(x)\,\big{(}U^{(\varepsilon)}\circ
g\big{)}(x)\,,\qquad x\in\Omega\,.$ (1.25)

Here $x_{0}\in\partial\Omega$ is a boundary point of $\Omega$ with positive
mean curvature222Our convention is that the scalar mean curvature of
$\partial\Omega$ is computed with respect to the outer unit normal to
$\Omega$, so that every bounded open set with $C^{2}$-boundary has at least
one boundary point of positive mean curvature., i.e. ${\rm
H}_{\partial\Omega}(x_{0})>0$; $\varphi_{\varepsilon}$ is a cut-off function
between $B_{\varepsilon^{\beta}}(x_{0})$ and
$B_{2\,\varepsilon^{\beta}}(x_{0})$ for $\beta=\beta(n,p)\in(0,1)$ to be
suitably chosen (the condition $n>2\,p$ enters in this choice); $g$ is a
boundary flattening diffeomorphism near $x_{0}$; and, finally,
$U=U_{\tau}+b\,\varepsilon\,V_{\tau}$ for $\tau=\mathsf{t}/\mathsf{v}$,
$V_{\tau}$ a standard perturbation of $U_{\tau}$, and $b$ a constant suitably
chosen depending on $n$, $p$, $H_{\partial\Omega}(x_{0})$ and $\tau$. The
energy, volume and trace expansions for $u_{\varepsilon}$ as $\varepsilon\to
0^{+}$ are computed to be

$\displaystyle\int_{\Omega}|\nabla
u_{\varepsilon}|^{p}\leq\int_{\Omega}|\nabla
u|^{p}+\mathsf{v}^{p}\,\Phi_{H}\Big{(}\frac{\mathsf{t}}{\mathsf{v}}\Big{)}^{p}$
(1.26) $\displaystyle\hskip
85.35826pt-\Big{\\{}\mathcal{L}(U_{\tau})-\frac{(n-p)}{n}\,\lambda_{H}(\tau)\,\mathcal{M}(U_{\tau})\Big{\\}}\,{\rm
H}_{\partial\Omega}(x_{0})\,\mathsf{v}^{p}\varepsilon+{\rm o}(\varepsilon)\,,$
$\displaystyle\int_{\Omega}u_{\varepsilon}^{p^{\star}}=1+{\rm
o}(\varepsilon)\,,\qquad\int_{\partial\Omega}u_{\varepsilon}^{p^{\\#}}=T^{p^{\\#}}+{\rm
o}(\varepsilon)\,,$ (1.27)

where $\lambda_{H}(T)$ is the volume Lagrange multiplier of $U_{T}$ (see (3.7)
below), and where $\mathcal{L}$ and $\mathcal{M}$ are functionals defined on
$U:H\to\mathbb{R}$ by

$\displaystyle\mathcal{L}(U)=\int_{H}x_{n}\,|\nabla
U|^{p}-p\,x_{n}\,(\partial_{1}U)^{2}\,|\nabla U|^{p-2}\,,$ (1.28)
$\displaystyle\mathcal{M}(U)=\int_{H}x_{n}\,U^{p^{\star}}\,.$ (1.29)

Modulo ${\rm o}(\varepsilon)$-perturbations of $u_{\varepsilon}$ aimed at
correcting the volume and trace constraints to the exact values needed for
inclusion in $\mathcal{X}_{\Omega}(T)$, we have constructed the required
competitors, and proved Theorem 1.1-(ii), if we can show the existence of
$c(n,p,T)>0$ such that

$\mathcal{L}(U_{T})-\frac{(n-p)}{n}\,\lambda_{H}(T)\,\mathcal{M}(U_{T})\geq
c(n,p,T)\,,\qquad\forall T>0\,.$ (1.30)

Of course, the full characterization of $\Phi_{H}$ plays a crucial role in our
proof of (1.30), see Lemma 3.1 below.

While Theorem 1.3-(ii) is immediate from Theorem 1.1-(ii) thanks to the
rigidity criterion (1.23), the proof of Theorem 1.3-(i) requires an additional
argument, which once more exploits several fine properties of $\Phi_{B}$ and
$\Phi_{H}$: these include (1.23), (1.24), and information on the signs of the
Lagrange multipliers $\lambda_{H}(T)$ and $\sigma_{H}(T)$ for minimizers
$U_{T}$ of $\Phi_{H}(T)$ (see (3.12) and (3.11) below).

###  1.4. Organization of the paper

After collecting a few preliminary results in section 2, in section 3 we study
in detail various properties of $\Phi_{H}$ and of its minimizers: in
particular, we prove the key inequality (1.30) (see Lemma 3.1), and discuss in
detail the Ansatz (1.25) (see Lemma 3.4). Sections 4 and 5 contain,
respectively, the proofs of Theorem 1.1 and Theorem 1.3. Finally, we collect
some auxiliary, routine proofs in an appendix.

Acknowledgements: FM was supported by NSF-DMS RTG 1840314, NSF-DMS FRG
1854344, and NSF-DMS 2000034. RN was supported by NSF-DMS 2200886.

##  2\. Notation and preparations

Some basic notation is presented in section 2.1. We then discuss, in separate
subsections, four useful technical lemmas: a concentration-compactness lemma
with boundary terms (Lemma 2.1); a second order expansion for the boundary
flattening diffeomorphisms used in the Ansatz (1.25) (Lemma 2.3); some basic
regularity information on minimizers of $\Phi_{\Omega}(T)$ (Lemma 2.5); and
the basic technique of “volume/trace correcting variations” (Lemma 2.6). Some
proofs are postponed to the appendix.

###  2.1. Notation

Throughout the paper we always assume that $n\geq 2$ and $p\in(1,n)$. We
denote by $\mathcal{L}^{n}$ and $\mathcal{H}^{k}$ the Lebesgue measure and the
$k$-dimensional Hausdorff measure of $\mathbb{R}^{n}$, although we simply set
$|E|$ in place of $\mathcal{L}^{n}(E)$. We denote by $B_{r}(x)$ the open ball
of center $x\in\mathbb{R}^{n}$ and radius $r>0$, and set $B_{r}=B_{r}(0)$,
while $B$ denotes a ball of unspecified center and radius.

Following a standard shorthand notation, by “$f(x)=g(x)+{\rm O}_{a,b}(|x|)$
for $|x|>R$” we mean that $|f(x)-g(x)|\leq C(a,b)\,|x|$ if $|x|>R$; by
“$f(x)=g(x)+{\rm o}_{a,b}(|x|)$ as $|x|\to 0$” we mean that $\lim_{|x|\to
0}|f(x)-g(x)|/|x|=0$ at a rate that is uniform with respect to the parameters
$a$ and $b$.

In general, we will use capital letters (e.g. $U,V,\Psi$) to denote functions
defined on the half space $H$ and lowercase letters (e.g. $u,v,\varphi$) to
denote functions defined on an open bounded domain $\Omega$.

###  2.2. Concentration-compactness

The following lemma is a version of Lions’ celebrated concentration-
compactness lemma and provides a natural starting point to study minimizing
sequences of $\Phi_{\Omega}(T)$.

######  Lemma 2.1 (Concentration-compactness).

Let $n\geq 2$, $p\in(1,n)$, and let $\Omega\subset\mathbb{R}^{n}$ be open and
bounded with $C^{1}$-boundary. If $\\{u_{j}\\}_{j}$ is a sequence in
$L^{1}_{\rm loc}(\Omega)$, $\\{\nabla u_{j}\\}_{j}$ is bounded in
$L^{p}(\Omega;\mathbb{R}^{n})$ and $u_{j}\rightharpoonup u$ as distributions
in $\Omega$, then the Radon measures on $\mathbb{R}^{n}$ defined by

$\displaystyle\mu_{j}=|\nabla
u_{j}|^{p}\,\mathcal{L}^{n}\llcorner\Omega\,,\qquad\nu_{j}=|u_{j}|^{p^{\star}}\,\mathcal{L}^{n}\llcorner\Omega\,,\qquad\tau_{j}=|u_{j}|^{{p^{\\#}}}\,\mathcal{H}^{n-1}\llcorner\partial\Omega\,,$
(2.1)

have subsequential weak-star limits $\mu$, $\nu$ and $\tau$ which satisfy

$\displaystyle\nu\\!$ $\displaystyle=$
$\displaystyle\\!|u|^{{p^{\star}}}\mathcal{L}^{n}\llcorner\Omega+\sum_{i\in
I}\mathsf{v}_{i}^{p^{\star}}\,\delta_{x_{i}}\,,$ (2.2) $\displaystyle\tau\\!$
$\displaystyle=$
$\displaystyle\\!|u|^{{p^{\\#}}}\mathcal{H}^{n-1}\llcorner\partial\Omega+\sum_{i\in
I}\mathsf{t}_{i}^{p^{\\#}}\,\delta_{x_{i}}\,,$ (2.3) $\displaystyle\mu\\!$
$\displaystyle\geq$ $\displaystyle\\!|\nabla
u|^{p}\mathcal{L}^{n}\llcorner\Omega+\sum_{i\in
I}\mathsf{g}_{i}^{p}\,\delta_{x_{i}},$ (2.4)

where $\\{x_{i}\\}_{i\in I}\subset\overline{\Omega}$ is at most countable set,
$\mathsf{v}_{i}>0$ and $\mathsf{t}_{i}\geq 0$ for every $i\in I$,
$\mathsf{t}_{i}>0$ only if $x_{i}\in\partial\Omega$, and

$\mathsf{g}_{i}\geq\mathsf{v}_{i}\,\Phi_{H}\Big{(}\frac{\mathsf{t}_{i}}{\mathsf{v}_{i}}\Big{)}\,,\qquad\forall
i\in I\,.$ (2.5)

In particular, $\mathsf{g}_{i}\geq S\,\mathsf{v}_{i}$ whenever
$x_{i}\in\Omega$.

###### Proof.

See appendix A. ∎

###  2.3. Near-boundary coordinates

In this section, we introduce two types of coordinates for a neighborhood of a
boundary point of a domain $\Omega$: one that requires minimal regularity of
the boundary of $\Omega$ and will suffice in the proofs of Theorem 1.1(i) and
Theorem 1.3(i), and a second that requires $C^{2}$ regularity of the boundary
of $\Omega$ and will be used in the proof of Theorem 1.1(ii) and Theorem
1.3(ii).

Given an open set $\Omega$ with $C^{1}$-boundary, we denote by $\nu_{\Omega}$
its outer unit normal and by $T_{x}(\partial\Omega)$ the tangent space to
$x\in\partial\Omega$. When $\Omega$ has $C^{2}$-boundary, we denote by ${\rm
A}_{\Omega}$ and ${\rm H}_{\Omega}$ the second fundamental form and the scalar
mean curvature of $\partial\Omega$ defined by $\nu_{\Omega}$. To define
coordinates near boundary points of $\Omega$, for $x\in\mathbb{R}^{n}$ we set
$\mathbf{p}(x)=x-x_{n}\,e_{n}$,
$\textbf{{D}}_{r}=\\{x:x_{n}=0\,,|\mathbf{p}x|<r\\}$, and
$\textbf{{C}}_{r}=\big{\\{}x:|x_{n}|<r\,,|\mathbf{p}(x)|<r\big{\\}}$. In
particular, if $\Omega$ is an open set with $C^{1}$-boundary such that

$0\in\partial\Omega\,,\qquad
T_{0}\,(\partial\Omega)=\\{x_{n}=0\\}\,,\qquad\nu_{\Omega}(0)=-e_{n}\,,$ (2.6)

then we can find $r_{0}>0$ and $\ell:\textbf{{D}}_{r_{0}}\to(-r_{0},r_{0})$
such that $\ell(0)=0$, $\nabla\ell(0)=0$, and

$\displaystyle\Omega\cap\textbf{{C}}_{r_{0}}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\big{\\{}x+t\,e_{n}:x\in\textbf{{D}}_{r_{0}}\,,r_{0}>t>\ell(x)\big{\\}}\,,$
$\displaystyle(\partial\Omega)\cap\textbf{{C}}_{r_{0}}\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\big{\\{}x+\ell(x)\,e_{n}:x\in\textbf{{D}}_{r_{0}}\big{\\}}\,.$

We then define the maps $F:\textbf{{D}}_{r_{0}}\to\partial\Omega$,
$f:\textbf{{C}}_{r_{0}}\to\mathbb{R}^{n}$ and
$\hat{f}:\textbf{{C}}_{r_{0}}\to\mathbb{R}^{n}$ by setting

$\displaystyle F(x)\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!x+\ell(x)\,e_{n}\,,\qquad\qquad\qquad\quad\,x\in\textbf{{D}}_{r_{0}}\,,$
(2.7) $\displaystyle\hat{f}(x)\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!F(\mathbf{p}x)+x_{n}e_{n}\,,\qquad\qquad\qquad\\!x\in\textbf{{C}}_{r_{0}}\,.$
(2.8) $\displaystyle f(x)\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!F(\mathbf{p}x)-x_{n}\,\nu_{\Omega}(F(\mathbf{p}x))\,,\qquad
x\in\textbf{{C}}_{r_{0}}\,.$ (2.9)

In this way, for every $y\in(\partial\Omega)\cap\textbf{{C}}_{r_{0}}$, if we
set $y=F(\mathbf{p}x)$, then

$\nu_{\Omega}(y)=\frac{\nabla\ell(x)-e_{n}}{\sqrt{1+|\nabla\ell(x)|^{2}}}\,,\qquad{\rm
H}_{\Omega}(y)={\rm
div}\,\Big{(}\frac{\nabla\ell}{\sqrt{1+|\nabla\ell|^{2}}}\Big{)}(x)\,.$

Notice that the map $f$ need not be of class $C^{1}$ if the boundary of
$\Omega$ is only of class $C^{1}$, while the map $\hat{f}$ will be as regular
as the boundary of $\Omega$. The following lemma summarizes basic properties
about the map $\hat{f}$.

######  Lemma 2.2 (Near-boundary coordinates, one).

If $H=\\{x_{n}>0\\}$, $\Omega$ is an open set with $C^{1}$-boundary and (2.6)
holds, then there exist $r_{0}$ and $C_{0}$ positive such that the map
$\hat{f}$ in (2.8) defines a $C^{1}$-diffeomorphism from
$\textbf{{C}}_{r_{0}}$ to its image, taking $\textbf{{D}}_{r_{0}}$ into
$\partial\Omega$ and with

$\displaystyle\hat{f}(\textbf{{C}}_{r/C_{0}}\cap H)\subset\Omega\cap
B_{r}\subset\hat{f}(\textbf{{C}}_{C_{0}\,r}\cap H)\qquad\forall
r<r_{0}/C_{0}\,.$ (2.10)

Moreover, letting $\hat{g}=\hat{f}^{-1}$ denote the inverse of $\hat{f}$, we
have

$\displaystyle\nabla\hat{f}={\rm Id}\,_{\mathbb{R}^{n}}+{\rm
o}(1)\,,\qquad(\nabla\hat{g})^{*}\circ\hat{f}={\rm Id}\,_{\mathbb{R}^{n}}+{\rm
o}(1)\,,$ (2.11) $\displaystyle J\hat{f}=1+{\rm o}(1)\,,\qquad\qquad 1\leq
J^{\partial H}\hat{f}\leq 1+{\rm o}(1)\,,\quad\quad\mbox{for
$x\in\textbf{{C}}_{r_{0}}$.}$ (2.12)

The orders in (2.11) and (2.12) depend on $\partial\Omega$ and on
$0\in\partial\Omega$.

###### Proof.

See appendix B. ∎

The map $f$ defined in (2.9) has the advantage that, when the boundary of
$\Omega$ is at least of class $C^{2}$, curvature quantities appear in
expansions of the metric coefficients and the volume form in these
coordinates. These properties are the content of the following lemma.

######  Lemma 2.3 (Near-boundary coordinates, two).

If $H=\\{x_{n}>0\\}$, $\Omega$ is an open set with $C^{2}$-boundary and (2.6)
holds, then there exist $r_{0}$ and $C_{0}$ positive such that the map $f$ in
(2.9) defines a $C^{1}$-diffeomorphism from $\textbf{{C}}_{r_{0}}$ to its
image, taking $\textbf{{D}}_{r_{0}}$ into $\partial\Omega$ and with

$\displaystyle f(\textbf{{C}}_{r/C_{0}}\cap H)\subset\Omega\cap B_{r}\subset
f(\textbf{{C}}_{C_{0}\,r}\cap H)\qquad\forall r<r_{0}/C_{0}\,.$ (2.13)

Moreover, for $x\in\textbf{{C}}_{r_{0}}$ and $x\in\textbf{{D}}_{r_{0}}$
respectively, we have

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!Jf(x)=1-x_{n}\,{\rm
H}_{\Omega}(0)+{\rm O}(|x|^{2})\,,\qquad 1\leq J^{\partial H}f(x)\leq 1+{\rm
O}(|x|^{2})\,,$ (2.14)

and if $\\{e_{i}\\}_{i=1}^{n-1}$ is an orthonormal basis of
$\mathbb{R}^{n-1}\subset\mathbb{R}^{n}$ of eigenvectors of $\nabla^{2}\ell(0)$
and $\\{\kappa_{i}\\}_{i=1}^{n-1}$ denote the corresponding eigenvalues (so
that, by (2.6), they are the principal curvatures of $\partial\Omega$ with
respect to $\nu_{\Omega}$ computed at $0\in\partial\Omega$, and in particular
${\rm H}_{\Omega}(0)=\sum_{i=1}^{n-1}\kappa_{i}$), then, letting $g=f^{-1}$
denote the inverse of $f$, we have

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!(\nabla
g)^{*}\circ f={\rm Id}\,_{\mathbb{R}^{n}}\\!+\big{(}\nabla\ell\otimes
e_{n}\\!-\\!e_{n}\otimes\nabla\ell)+x_{n}\sum_{i=1}^{n-1}\kappa_{i}e_{i}\otimes
e_{i}+\\!{\rm O}(|x|^{2})\,.$ (2.15)

The orders in (2.14) and (2.15) depend on $\partial\Omega$ and on
$0\in\partial\Omega$.

###### Proof.

See appendix B. ∎

######  Remark 2.4.

Given $x_{0}\in\partial\Omega$, we denote by $\pi_{x_{0}}$ the rigid motion of
$\mathbb{R}^{n}$ that maps $x_{0}$ to $0$ such that (2.6) holds with
$\pi_{x_{0}}(\Omega)$ in place of $\Omega$. Then we set, for $\hat{f}$ and $f$
defined as in (2.8) and (2.9) respectively but with $\pi_{x_{0}}(\Omega)$ in
place of $\Omega$,

$\hat{f}_{x_{0}}=\pi_{x_{0}}^{-1}\circ\hat{f}\,,\qquad
f_{x_{0}}=\pi_{x_{0}}^{-1}\circ f\,.$

Clearly these maps are diffeomorphisms on $\textbf{{C}}_{r_{0}}$, mapping
$H\cap\textbf{{C}}_{r_{0}}$ into a neighborhood of $x_{0}$ in $\Omega$ and
$\textbf{{D}}_{r_{0}}=(\partial H)\cap\textbf{{C}}_{r_{0}}$ into a
neighborhood of $x_{0}$ in $\partial\Omega$, and satisfies proper
reformulations of the estimates in Lemmas 2.2 and 2.3. Here $r_{0}$ and
$C_{0}$ depend also on the choice of $x_{0}$, and can of course be assumed
uniform across $x_{0}\in\partial\Omega$ if $\partial\Omega$ is bounded.

###  2.4. Properties of minimizers

The following lemma gathers some fundamental properties of minimizers of
$\Phi_{\Omega}$ that will be needed in the sequel.

######  Lemma 2.5.

If $n\geq 2$, $p\in(1,n)$, $T>0$, $\Omega$ is a bounded open set with
$C^{2}$-boundary, and $u$ is a minimizer of $\Phi_{\Omega}(T)$, then $u$ is
bounded and Lipschitz continuous in $\Omega$, there are $\lambda$ and $\sigma$
such that the Euler–Lagrange equation (1.2) holds in the weak sense, and the
balance condition

$\lambda\,\int_{\Omega}u^{{p^{\star}}-1}+\sigma\,\int_{\partial\Omega}u^{{p^{\\#}}-1}=0\,,$
(2.16)

holds.

###### Proof.

By a standard argument, based on similar considerations to the one presented
in Lemma 2.6 below, one sees that a minimizer $u$ of $\Phi_{\Omega}(T)$ is a
$W^{1,p}(\Omega)$-distributional solution of the Euler-Lagrange equation (1.2)
for some $\lambda,\sigma\in\mathbb{R}$. As soon as $\Omega$ is bounded and has
Lipschitz boundary, one can exploit (1.2) in conjunction with a Moser
iteration argument to prove that $u\in L^{\infty}(\Omega)$ (see, e.g. [MW19,
Theorem 3.1]; their result applies to (1.2) by taking, in the notation of
their paper, $\mathcal{A}(x,u,{\nabla}u)=|{\nabla}u|^{p-2}{\nabla}u$,
$\mathcal{B}(x,u,{\nabla}u)=\lambda u^{{p^{\star}}-1}$, and
$\mathcal{C}(x,u)=\sigma u^{{p^{\\#}}-1}$). On further assuming that
$\partial\Omega$ is of class $C^{2}$, then the classical result [Lie92,
Theorem 1.7] can be applied to deduce that $u\in
C^{1,\beta}(\overline{\Omega})$ for a suitable $\beta=\beta(n,p)\in(0,1)$ (for
more details, see [MW19, Theorem 3.9]). In particular, $u$ is bounded and
Lipschitz continuous on $\Omega$, as claimed. ∎

###  2.5. Volume/trace correcting variations

At various stages in our arguments we will need to slightly modify certain
competitors so to restore the volume and trace constraints defining
$\mathcal{X}_{\Omega}(T)$. The following lemma describes the basic mechanism
used to this end.

######  Lemma 2.6 (Volume/trace correcting variations).

If $n\geq 2$, $p\in(1,n)$, $M>0$, $\Omega$ is an open set with
$C^{1}$-boundary, $v\in L^{1}_{\rm loc}(\Omega)$ with $\nabla v\in
L^{p}(\Omega;\mathbb{R}^{n})$, and if $x_{0}\in\mathbb{R}^{n}$ and $r>0$ are
such that

$\mbox{$\int_{\Omega\setminus B_{r}(x_{0})}v^{p^{\star}}$ and
$\int_{(\partial\Omega)\setminus B_{r}(x_{0})}v^{p^{\\#}}$ are positive and
finite}\,,$ (2.17)

then there exist positive constants $\eta$ and $C$, and functions $\varphi\in
C^{\infty}_{c}(\mathbb{R}^{n}\setminus\overline{B}_{r}(x_{0}))$ and $\psi\in
C^{\infty}_{c}(\Omega\setminus\overline{B}_{r}(x_{0}))$, all depending on $n$,
$p$, $v$ and $M$ only, and with the following property.

If $\\{v_{\varepsilon}\\}_{\varepsilon<\varepsilon_{0}}\subset L^{1}_{{\rm
loc}}(\Omega)$ is such that, for every $\varepsilon<\varepsilon_{0}$,

$\mbox{$v_{\varepsilon}=v$ on $\Omega\setminus
B_{r}(x_{0})$}\,,\qquad\int_{\Omega}|\nabla v_{\varepsilon}|^{p}\leq M\,,$
(2.18)

then for every $(a,b)$ with $|a|,|b|<\eta/C$, we can find $(s,t)$ with
$|s|,|t|<\eta$ such that

$w_{\varepsilon}=v_{\varepsilon}+s\,\varphi+t\,\psi$

satisfies

$\int_{\partial\Omega}|w_{\varepsilon}|^{p^{\\#}}=a+\int_{\partial\Omega}|v_{\varepsilon}|^{p^{\\#}}\,,\qquad\int_{\Omega}|w_{\varepsilon}|^{p^{\star}}=b+\int_{\Omega}|v_{\varepsilon}|^{p^{\star}}\,,$
(2.19) $\Big{|}\int_{\Omega}|\nabla w_{\varepsilon}|^{p}-\int_{\Omega}|\nabla
v_{\varepsilon}|^{p}\Big{|}\leq C\,\big{(}|a|+|b|\big{)}\,.$ (2.20)

###### Proof.

By (2.17) there are $\xi\in
C^{\infty}_{c}(\mathbb{R}^{n}\setminus\overline{B}_{r}(x_{0}))$ and $\psi\in
C^{\infty}_{c}(\Omega\setminus\overline{B}_{r}(x_{0}))$ such that

$\int_{\partial\Omega}v^{{p^{\\#}}-1}\,\xi=1\,,\qquad\int_{\Omega}v^{{p^{\star}}-1}\,\psi=1\,.$
(2.21)

Setting $\varphi=\xi-(\int_{\Omega}v^{{p^{\star}}-1}\xi)\,\psi$, we have
$\varphi\in C^{\infty}_{c}(\mathbb{R}^{n}\setminus\overline{B}_{r}(x_{0}))$
with

$\int_{\Omega}v^{{p^{\star}}-1}\,\varphi=0\,,\qquad\int_{\partial\Omega}v^{{p^{\\#}}-1}\,\varphi=1\,.$
(2.22)

We now define $h_{\varepsilon}:\mathbb{R}^{2}\to\mathbb{R}^{2}$ by

$h_{\varepsilon}(s,t)\\!=\\!\Big{(}\int_{\partial\Omega}|v_{\varepsilon}+s\,\varphi+t\,\psi|^{p^{\\#}}-\\!\int_{\partial\Omega}|v_{\varepsilon}|^{p^{\\#}},\
\int_{\Omega}|v_{\varepsilon}+s\,\varphi+t\,\psi|^{p^{\star}}-\\!\int_{\Omega}|v_{\varepsilon}|^{p^{\star}}\Big{)}\,.$

By (2.18) we have $h_{\varepsilon}\in
C^{1,\alpha}(\mathbb{R}^{2};\mathbb{R}^{2})$ for some
$\alpha=\alpha(n,p)\in(0,1)$, with

$\sup_{\varepsilon<\varepsilon_{0}}\|h_{\varepsilon}\|_{C^{1,\alpha}(\mathbb{R}^{2};\mathbb{R}^{2})}<\infty\,;$

moreover, $h_{\varepsilon}(0,0)=0$ and, by (2.21) and (2.22),

$\nabla h_{\varepsilon}(0,0)=\left(\begin{array}[]{c c
c}{p^{\\#}}\,\int_{\partial\Omega}|v_{\varepsilon}|^{{p^{\\#}}-1}\,\varphi&&{p^{\\#}}\,\int_{\partial\Omega}|v_{\varepsilon}|^{{p^{\\#}}-1}\,\psi\\\
&\\\
{p^{\star}}\,\int_{\Omega}|v_{\varepsilon}|^{{p^{\star}}-1}\,\varphi&&{p^{\star}}\,\int_{\Omega}|v_{\varepsilon}|^{{p^{\star}}-1}\,\psi\end{array}\right)=\Big{(}\begin{array}[]{c
c}{p^{\\#}}&0\\\ 0&{p^{\star}}\end{array}\Big{)}\,.$

We can thus apply the inverse function theorem uniformly in $\varepsilon$, to
find positive constants $\eta$ and $C_{1}$ depending on $n$, $p$, and $v$ so
that each $h_{\varepsilon}$ is invertible on $E=\\{(s,t):|s|,|t|<\eta\\}$,
with $\\{(a,b):|a|,|b|<\eta/C_{1}\\}\subset h_{\varepsilon}(E)$, and $\nabla
h_{\varepsilon}^{-1}(a,b)\geq{\rm Id}\,_{2\times 2}/C_{1}$ (in the sense of
positive definite matrices) for every $(a,b)\in h_{\varepsilon}(E)$. In
particular, if we let $(s,t)=h_{\varepsilon}^{-1}(a,b)$ for a pair $(a,b)$
with $|a|,|b|<\eta/C_{1}$, then the function
$w_{\varepsilon}=v_{\varepsilon}+s\,\varphi+t\,\psi$ satisfies (2.19) and
$|(a,b)|=|h_{\varepsilon}^{-1}(s,t)|\geq|(s,t)|/C_{1}$. Moreover, by the
elementary inequality

$\big{|}|X+Y|^{p}-|X|^{p}\big{|}\leq
p\,\max\big{\\{}|X|,|Y|\big{\\}}^{p-1}\,|Y|\,\qquad\forall
X,Y\in\mathbb{R}^{n}\,,$

we see that, setting $\gamma=\max\big{\\{}|\nabla
v_{\varepsilon}|,|\nabla\varphi|,|\nabla\psi|\big{\\}}^{p}$,

$\begin{split}\Big{|}\int_{\Omega}|\nabla
w_{\varepsilon}|^{p}&-\int_{\Omega}|\nabla v_{\varepsilon}|^{p}\Big{|}\leq
p\,\int_{\Omega}\gamma^{(p-1)/p}\,\big{(}|s|\,|\nabla\varphi|+|t|\,|\nabla\psi|\big{)}\\\
&\leq
C\,\Big{(}\int_{\Omega}\gamma\,\Big{)}^{(p-1)/p}\,\Big{(}\int_{\Omega}\big{(}|s|^{p}\,|\nabla\varphi|^{p}+|t|^{p}\,|\nabla\psi|^{p}\big{)}\Big{)}^{1/p}\\\
&\leq C\,|(s,t)|\leq C_{2}\,|(a,b)|\,,\end{split}$ (2.23)

for a constant $C_{2}$ depending on $n$, $p$, $v$, and $M$. Letting
$C=\max\\{C_{1},C_{2}\\}$ concludes the proof of the lemma. ∎

##  3\. Boundary concentrations

###  3.1. Properties of $\Phi_{H}$-minimizers

We recall some facts proved in [MN17] about $\Phi_{H}$ and its minimizers.
Recall that we denote by $T_{0}$ the minimum point of $\Phi_{H}$, so that

$T_{0}\in(0,T_{E})\,,\qquad\Phi_{H}(T_{0})=2^{-1/n}\,S(n,p)\,,$ (3.1)

where $T_{E}$ is the “Escobar trace” defined in (1.8). If we set
$H=\\{x_{n}>0\\}$, the minimizers of $U_{T}$ of $\Phi_{H}(T)$ for $T>0$ are
characterized (modulo the obvious scaling and translation invariance of
$\Phi_{H}$) as

$\displaystyle
U_{T}(x)=c_{T}\\!\\!\left\\{\begin{split}&\tau_{t_{T}\,e_{n}}U_{S}(x)=(1+|x-t_{T}\,e_{n}|^{p^{\prime}})^{1-(n/p)}\,,\qquad\\!\\!\\!\\!\hskip
8.5359ptT\in(0,T_{E})\,,\\\
&\tau_{e_{n}}U_{E}(x)=|x+e_{n}|^{(p-n)/(p-1)}\,,\qquad\\!\\!\\!\\!\hskip
48.36958ptT=T_{E}\,,\\\
&\tau_{s_{T}\,e_{n}}U_{BE}(x)=(|x-s_{T}\,e_{n}|^{p^{\prime}}-1)^{1-(n/p)}\,,\\!\\!\\!\\!\qquad
T>T_{E}\,,\end{split}\right.$ (3.2)

where the constants $c_{T}$, $t_{T}$ and $s_{T}$ are chosen in such a way that

$\int_{H}\,U_{T}^{p^{\star}}=1\,,\qquad\int_{\partial
H}\,U_{T}^{p^{\sharp}}=T^{p^{\sharp}}\,,\qquad\int_{H}|\nabla
U_{T}|^{p}=\Phi_{H}(T)^{p}\,.$ (3.3)

It is convenient to keep in mind that the various formulas for $U_{T}$ listed
in (3.2) all share the same decay behavior at infinity, that is (see (3.20)
below), we have

$U_{T}(x)\sim|x|^{(p-n)/(p-1)},\qquad|{\nabla}U_{T}|\sim|x|^{(1-n)/(p-1)}\qquad\text{
as }|x|\to\infty.$ (3.4)

(where the rate depends on the specific value of $T$ under consideration). The
constants $t_{T}$ and $s_{T}$ have the following properties:
$T\in(0,T_{E})\mapsto t_{T}$ is continuous and strictly decreasing, with
$t_{T}>0$ if and only if $T\in(0,T_{0})$, and

$\lim_{T\to
0^{+}}t_{T}=+\infty\,,\qquad\lim_{t\to(T_{E})^{-}}t_{T}=-\infty\,,\qquad
t_{T_{0}}=0\,,$ (3.5)

while $T\in(T_{E},\infty)\mapsto s_{T}$ is continuous, negative, strictly
increasing, with

$\lim_{T\to(T_{E})^{+}}s_{T}=-\infty\,,\qquad\lim_{T\to+\infty}s_{T}=-1\,.$
(3.6)

Denoting by $\Delta_{p}v={\rm div}\,(|\nabla v|^{p-2}\nabla v)$ the
$p\,$-Laplace operator, we have

$\left\\{\begin{split}&-\Delta_{p}U_{T}=\lambda_{H}(T)\,U_{T}^{p^{\star}-1}\,\quad\hskip
28.45274pt\mbox{on $H$}\,,\\\ &|\nabla U_{T}|^{p-2}\,\frac{\partial
U_{T}}{\partial\nu_{H}}=\sigma_{H}(T)\,U_{T}^{p^{\sharp}-1}\,\quad\mbox{on
$\partial H$}\,,\end{split}\right.\qquad\forall T>0\,,$ (3.7)

where $\lambda_{H},\sigma_{H}:(0,\infty)\to\mathbb{R}$ are continuous and
satisfy the relations

$\Phi_{H}(T)^{p}=\lambda_{H}(T)+\sigma_{H}(T)\,T^{p^{\\#}}\,,\qquad\sigma_{H}(T)=\frac{\Phi_{H}(T)^{p-1}\,\Phi_{H}^{\prime}(T)}{T^{{p^{\\#}}-1}}\,,$
(3.8)

(see [MN17, Lemma 3.3]333Notice that in [MN17, (3.16)] it is incorrectly
stated that
$\sigma_{H}(T)=\Phi_{H}(T)^{p-1}\,\Phi_{H}^{\prime}(T)/({p^{\\#}}\,T^{{p^{\\#}}-1})$,
where the extra $1/{p^{\\#}}$-factor is wrongly introduced in the penultimate
displayed equation in the proof of Lemma 3.3, where
$\tau^{\prime}(0)=T^{1-{p^{\\#}}}/{p^{\\#}}$ should be replaced by
$\tau^{\prime}(0)=T^{1-{p^{\\#}}}$. This error is inconsequential for the
arguments in [MN17], since this expression for $\sigma_{H}(T)$ is only used in
equation (3.22) and subsequent displayed equations, and since, in all these
subsequent identities, a generic multiplicative factor $c(n,p)$ is used (in
particular, two functions of $(n,p)$ differing by a $1/{p^{\\#}}$-factor are
both $c(n,p)$). It will instead be important in the proof of (4.3) to work
with correct expression for $\sigma_{H}(T)$.) as well as

$\displaystyle\lim_{T\to
0^{+}}\sigma_{H}(T)=-\infty\,,\qquad\lim_{T\to+\infty}\sigma_{H}(T)=+\infty$
(3.9) $\displaystyle\lim_{T\to
0^{+}}\lambda_{H}(T)>0\,,\qquad\lim_{T\to+\infty}\lambda_{H}(T)=-\infty\,.$
(3.10)

The signs of $\sigma_{H}$ and $\lambda_{H}$ can be easily deduced from (3.2),
and satisfy

$\displaystyle(0,T_{0})=\\{\sigma_{H}<0\\}\,,\qquad(T_{0},\infty)=\\{\sigma_{H}>0\\}\,,\qquad\sigma_{H}(T_{0})=0\,,$
(3.11)
$\displaystyle(0,T_{E})=\\{\lambda_{H}>0\\}\,,\qquad(T_{E},\infty)=\\{\lambda_{H}<0\\}\,,\qquad\lambda_{H}(T_{E})=0\,.$
(3.12)

###  3.2. A key inequality and further properties of $U_{T}$

In this section, we prove the key inequality (3.13) for the functions
$\mathcal{L}$ and $\mathcal{M}$ introduced in (1.28) and (1.29), namely

$\displaystyle\mathcal{L}(U)=\int_{H}x_{n}\,|\nabla
U|^{p}-p\,x_{n}\,(\partial_{1}U)^{2}\,|\nabla U|^{p-2}\,,$
$\displaystyle\mathcal{M}(U)=\int_{H}x_{n}\,U^{p^{\star}}\,.$

Whenever $U$ satisfies the decay properties (3.4) (e.g., when $U$ is a
compactly supported perturbation of some $U_{T}$), we have that
$\mathcal{M}(U)<\infty$; however, $\mathcal{L}(U)<\infty$ under (3.4) if and
only if $n>2\,p-1$; see (3.24) and (3.25) below.

######  Lemma 3.1 (Key inequality).

If $n\geq 2$, $p\in(1,n)$, $n>2p-1$, and $T>0$, then there is a positive
constant $c(n,p,T)$ such that

$\displaystyle\mathcal{L}(U_{T})-\frac{n-p}{n}\,\lambda_{H}(T)\,\mathcal{M}(U_{T})\geq
c(n,p,T)\,.$ (3.13)

The following lemma will be useful in proving Lemma 3.1.

######  Lemma 3.2.

If $H=\\{x_{n}>0\\}$, $U:H\to\mathbb{R}$ is radially symmetric with respect to
$t\,e_{n}$ for some $t\in\mathbb{R}$, and $\int_{H}x_{n}|{\nabla}U|^{p}$ is
finite, then

$\int_{H}x_{n}\,|\nabla
U|^{p-2}\big{\\{}(\partial_{n}U)^{2}-(\partial_{1}U)^{2}\big{\\}}>0\,.$

###### Proof of Lemma 3.2.

We have $U(x)=\eta(|x-t\,e_{n}|)$, $y=x-t\,e_{n}$, $r=|y|$, and
$\hat{y}=y/|y|$, so that

$x_{n}\,|\nabla
U|^{p-2}\big{\\{}(\partial_{n}U)^{2}-(\partial_{1}U)^{2}\big{\\}}=(y_{n}+t)\,|\eta^{\prime}(r)|^{p}\big{\\{}(\hat{y}_{n})^{2}-(\hat{y}_{1})^{2}\big{\\}}\,,$

and, setting $y=r\,z$,

$\displaystyle\int_{H}x_{n}\,|\nabla
U|^{p-2}\big{\\{}(\partial_{n}U)^{2}-(\partial_{1}U)^{2}\big{\\}}=\int_{\\{y_{n}>-t\\}}(y_{n}+t)\,|\eta^{\prime}(r)|^{p}\big{\\{}(\hat{y}_{n})^{2}-(\hat{y}_{1})^{2}\big{\\}}$
$\displaystyle=\int_{0}^{\infty}\,|\eta^{\prime}(r)|^{p}\,dr\int_{\\{y_{n}>-t\\}\cap\partial
B_{r}}(y_{n}+t)\,\big{\\{}(\hat{y}_{n})^{2}-(\hat{y}_{1})^{2}\big{\\}}\,d\mathcal{H}^{n-1}_{y}$
$\displaystyle=\int_{0}^{\infty}\,r^{n}\,|\eta^{\prime}(r)|^{p}\,dr\int_{\\{z_{n}>-t/r\\}\cap\partial
B_{1}}\Big{(}z_{n}+\frac{t}{r}\Big{)}\,\big{\\{}z_{n}^{2}-z_{1}^{2}\big{\\}}\,d\mathcal{H}^{n-1}_{z}\,.$

We conclude the proof by showing that

$\int_{\\{z_{n}>-s\\}\cap\partial
B_{1}}\big{(}z_{n}+s\big{)}\,\big{\\{}z_{n}^{2}-z_{1}^{2}\big{\\}}\,d\mathcal{H}^{n-1}_{z}>0\quad\text{
for all }s\in(-1,1),$ (3.14)

noting that for each $s\in[1,\infty)$, this integral vanishes by symmetry
while for $s\in(-\infty,-1]$ the domain of integration is empty. To see
(3.14), let $\mathbf{p}:\mathbb{R}^{n}\to\mathbb{R}^{n-2}$ denote the
projection map $\mathbf{p}(x)=(x_{2},...,x_{n-1})$ (if $n=2$ there is no need
to introduce $\mathbf{p}$). The tangential coarea factor of $\mathbf{p}$ along
$\partial B_{1}$ defines a positive function $K:\partial B_{1}\to(0,\infty]$
which is $\mathcal{H}^{n-1}$-a.e. finite on $\partial B_{1}$, and which is
independent of the variables $(x_{1},x_{n})$, i.e. $K(x_{1},w,x_{n})=K(w)$ for
every $(x_{1},w,x_{n})\in\partial B_{1}$. Therefore, setting for brevity
$M_{s}=\\{z_{n}>-s\\}\cap\partial B_{1}$,

$\int_{M_{s}}\big{(}z_{n}+s\big{)}\,\big{\\{}z_{n}^{2}-z_{1}^{2}\big{\\}}\,d\mathcal{H}^{n-1}_{z}=\int_{\mathbf{p}(M_{s})}\,\frac{d\mathcal{L}^{n-2}_{w}}{K(w)}\int_{M_{s}\cap\mathbf{p}^{-1}(w)}\big{(}z_{n}+s\big{)}\,\big{\\{}z_{n}^{2}-z_{1}^{2}\big{\\}}\,d\mathcal{H}^{1}_{(z_{1},z_{n})}\,,$

where

$M_{s}\cap\mathbf{p}^{-1}(w)=\Big{\\{}(z_{1},z_{n}):z_{1}^{2}+z_{n}^{2}=1-|w|^{2}\,,z_{n}>-s\Big{\\}}\,,\qquad
s\in(-1,1)\,.$

As before, the inner integral above vanishes when $1-|w|^{2}\leq s$ by
symmetry and the domain of integration is empty when $-s\geq 1-|w|^{2}$. We
are thus left to prove that

$\int_{-\alpha}^{\pi+\alpha}\big{(}\sin\theta+\sin\alpha\big{)}\,\big{\\{}\sin^{2}\theta-\cos^{2}\theta\big{\\}}\,d\theta>0$

for $\alpha\in(0,\pi/2)$ (corresponding to the case when $s\geq 0$) and

$\int_{\alpha}^{\pi-\alpha}\big{(}\sin\theta-\sin\alpha\big{)}\,\big{\\{}\sin^{2}\theta-\cos^{2}\theta\big{\\}}\,d\theta>0$

for $\alpha\in(0,\pi/2)$ (corresponding to the case when $s<0$). Direct
computation shows that both of these integrals are equal to the positive
quantity $(2/3)\,(\cos\alpha)^{3}$. ∎

We now prove Lemma 3.1.

###### Proof of Lemma 3.1.

Testing (3.7) with $x_{n}\,U_{T}$ we find

$\displaystyle\lambda_{H}(T)\,\int_{H}x_{n}\,U_{T}^{p^{\star}}=-\int_{H}x_{n}\,U_{T}\,\Delta_{p}U_{T}=\int_{H}|\nabla
U_{T}|^{p-2}\,\nabla U_{T}\cdot\nabla(x_{n}\,U_{T})\,.$

Here the integration by parts is justified since $x_{n}=0$ on $\partial H$ and
since by (3.4),

$\Big{|}\int_{H\cap\partial B_{R}}x_{n}\,U_{T}\,|\nabla
U_{T}|^{p-2}(\nu_{B_{R}}\cdot\nabla U_{T})\Big{|}\leq
C\,\frac{R^{n-1}\,R}{R^{(n-p)/(p-1)}}\,\Big{(}\frac{1}{R^{(n-1)/(p-1)}}\Big{)}^{p-1}\to
0$

like $R^{-(n+1)/(p-1)}$ as $R\to\infty$. We thus find that

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\mathcal{L}(U_{T})-\frac{n-p}{n}\,\lambda_{H}(T)\,\mathcal{M}(U_{T})=\int_{H}x_{n}\,|\nabla
U_{T}|^{p}-p\,x_{n}\,(\partial_{1}U_{T})^{2}\,|\nabla U_{T}|^{p-2}$
$\displaystyle\hskip
113.81102pt-\frac{n-p}{n}\,\Big{\\{}\int_{H}x_{n}\,|\nabla
U_{T}|^{p}+\int_{H}U_{T}\,|\nabla U_{T}|^{p-2}\,\partial_{n}U_{T}\Big{\\}}\,.$

Now, since $|\nabla U_{T}|$ is symmetric by reflection with respect to the
hyperplanes $\\{x_{i}=0\\}$, $i=1,...,n-1$, we see that

$\displaystyle\int_{H}x_{n}\,|\nabla U_{T}|^{p}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n-1}\int_{H}\,x_{n}\,(\partial_{i}U_{T})^{2}|\nabla
U_{T}|^{p-2}+\int_{H}\,x_{n}\,(\partial_{n}U_{T})^{2}|\nabla U_{T}|^{p-2}$
$\displaystyle=$
$\displaystyle(n-1)\int_{H}\,x_{n}\,(\partial_{1}U_{T})^{2}|\nabla
U_{T}|^{p-2}+\int_{H}\,x_{n}\,(\partial_{n}U_{T})^{2}|\nabla U_{T}|^{p-2}$

so that continuing from above we have

$\displaystyle\mathcal{L}(U_{T})-\frac{n-p}{n}\,\lambda_{H}(T)\,\mathcal{M}(U_{T})$
$\displaystyle=\frac{p}{n}\,\int_{H}x_{n}\,|\nabla
U_{T}|^{p}-p\,\int_{H}\,x_{n}\,(\partial_{1}U_{T})^{2}\,|\nabla
U_{T}|^{p-2}-\frac{n-p}{n}\,\int_{H}U_{T}\,|\nabla
U_{T}|^{p-2}\,\partial_{n}U_{T}$
$\displaystyle=\frac{p}{n}\,\int_{H}x_{n}\,|\nabla
U_{T}|^{p-2}\Big{\\{}(\partial_{n}U_{T})^{2}-(\partial_{1}U_{T})^{2}\Big{\\}}+\frac{n-p}{n}\,\int_{H}U_{T}\,|\nabla
U_{T}|^{p-2}\,\big{(}-\partial_{n}U_{T}\big{)}\,.$

In particular, the lemma is proved by showing that

$\displaystyle\int_{H}x_{n}\,|\nabla
U_{T}|^{p-2}\big{\\{}(\partial_{n}U_{T})^{2}-(\partial_{1}U_{T})^{2}\big{\\}}>0\,,$
(3.15) $\displaystyle\int_{H}U_{T}\,|\nabla
U_{T}|^{p-2}\,\big{(}-\partial_{n}U_{T}\big{)}>0\,,$ (3.16)

where the first inequality, (3.15), is immediate from Lemma 3.2 (recall that
$n>2p-1$ and $U_{T}$ is radially symmetric with respect to $te_{n}$ for some
$t\in\mathbb{R}$).

We are thus left to prove (3.16). This is immediate in the case when $T\geq
T_{0}$, because in that case, by (3.5) and (3.6), $U_{T}$ has center of
symmetry at $t\,e_{n}$ for some $t\leq 0$, and thus $\partial_{n}U_{T}<0$ on
$H$. By (3.5), if $T\in(0,T_{0})$, then $U_{T}$ has center of symmetry at
$t\,e_{n}$ for some $t>0$. Correspondingly, $U_{T}\,\partial_{n}U_{T}$ is odd
with respect to $\\{x_{n}=t\\}$, with $U_{T}\,\partial_{n}U_{T}<0$ on
$\\{x_{n}>t\\}$ and $U_{T}\,\partial_{n}U_{T}>0$ on $\\{0<x_{n}<t\\}$: in
particular, if $p_{t}$ denotes the reflection with respect to $\\{x_{n}=t\\}$,
then

$\displaystyle\int_{2t>x_{n}>t}\,x_{n}\,U_{T}\,(-\partial_{n}U_{T})=\int_{t>x_{n}>0}\,(p_{t}(x)\cdot
e_{n})\,\,[U_{T}\,(-\partial_{n}U_{T})](p_{t}(x))\,dx$
$\displaystyle=\int_{t>x_{n}>0}\,(p_{t}(x)\cdot
e_{n})\,\,[U_{T}\,\partial_{n}U_{T})](x)\,dx\geq\int_{t>x_{n}>0}\,x_{n}\,\,U_{T}\,\partial_{n}U_{T}\,,$

so that

$\int_{H}U_{T}\,|\nabla
U_{T}|^{p-2}\,\big{(}-\partial_{n}U_{T}\big{)}\geq\int_{\\{x_{n}>2t\\}}U_{T}\,|\nabla
U_{T}|^{p-2}\,\big{(}-\partial_{n}U_{T}\big{)}\,,$

and the latter integral is positive because $\partial_{n}U_{T}<0$ on
$\\{x_{n}>t\\}$. ∎

###  3.3. Standard variations of $\Phi_{H}$-minimizers

We now introduce a “class of standard variations” of minimizers of $\Phi_{H}$.
With $H=\\{x_{n}>0\\}$, we define
$\zeta=\zeta(r,T):[0,\infty)\times(0,\infty)\to[0,\infty)$, so that, setting
$V_{T}(x)=\zeta(|x-e_{n}|,T)$ for $x\in\mathbb{R}^{n}$, we have

$V_{T}\in
C^{\infty}_{c}(H;[0,\infty))\,,\qquad\int_{H}\,U_{T}^{{p^{\star}}-1}\,V_{T}=1\,,\qquad\int_{\partial
H}\,U_{T}^{{p^{\\#}}-1}\,V_{T}=0\,.$ (3.17)

Given $T>0$ we denote by

$\mathcal{U}_{T}$ (3.18)

the family of functions $U:H\to\mathbb{R}$ of the form

$U=U_{T}+t\,V_{T}\,,\qquad|t|\leq 1\,.$

The following lemma contains some basic properties of functions in
$\mathcal{U}_{T}$. We notice that

$\begin{split}&\mbox{every $U\in\mathcal{U}_{T}$ is symmetric by
reflection}\\\ &\mbox{with respect to the coordinates $x_{1}$, ...,
$x_{n-1}$}\,.\end{split}$ (3.19)

######  Lemma 3.3 (Standard variations of $U_{T}$).

If $n\geq 2$, $p\in(1,n)$, and $T>0$, then there are positive constants
$R_{0}$ and $C_{0}$ depending on $n$, $p$, $T$, and $V_{T}$ such that the
following properties hold:

(i): if $U\in\mathcal{U}_{T}$, then for every $|x|>R_{0}$ we have

$\displaystyle\begin{split}\hskip 31.2982pt\frac{1}{C_{0}|x|^{(n-p)/(p-1)}}\
&\leq U(x)\,\leq\frac{C_{0}}{|x|^{(n-p)/(p-1)}}\,,\\\
\frac{1}{C_{0}|x|^{(n-1)/(p-1)}}&\leq|\nabla
U(x)|\leq\frac{C_{0}}{|x|^{(n-1)/(p-1)}}\,,\end{split}$ (3.20)

and for every $R>R_{0}$,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\int_{H\setminus
B_{R}}U^{{p^{\star}}}\leq\frac{C_{0}}{R^{n/(p-1)}}\,,\qquad\int_{H\setminus
B_{R}}U^{{p^{\star}}-1}\leq\frac{C_{0}}{R^{p/(p-1)}}\,,$ (3.21)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\int_{H\cap(B_{2R}\setminus
B_{R})}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!U^{p}\leq\frac{C_{0}}{R^{(n-p^{2})/(p-1)}}\,,\qquad\\!\\!\\!\\!\\!\\!\int_{H\setminus
B_{R}}\\!\\!\\!|\nabla U|^{p}\leq\frac{C_{0}}{R^{(n-p)/(p-1)}}\,,$ (3.22)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\int_{(\partial H)\setminus
B_{R}}\\!\\!\\!\\!\\!\\!U^{p^{\\#}}\leq\frac{C_{0}}{R^{(n-1)/(p-1)}}\,,\qquad\int_{(\partial
H)\setminus B_{R}}\\!\\!\\!\\!\\!\\!U^{{p^{\\#}}-1}\leq\frac{C_{0}}{R}\,,$
(3.23) $\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\int_{H\setminus
B_{R}}|x|\,U^{{p^{\star}}}\leq\frac{C_{0}}{R^{[1+n-p]/(p-1)}}\,,$ (3.24)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\int_{H\setminus
B_{R}}|x|\,|\nabla U|^{p}\leq\frac{C_{0}}{R^{(n+1-2p)/(p-1)}}\,,\qquad\mbox{if
$n>2\,p-1$}\,.$ (3.25)

(ii): for every $U\in\mathcal{U}_{T}$ we have

$\displaystyle\int_{H}|\nabla U|^{p}=\Phi_{H}(T)^{p}+p\,\lambda_{H}(T)\,t+{\rm
o}(t)\,,$ (3.26) $\displaystyle\int_{H}U^{p^{\star}}=1+{p^{\star}}\,t+{\rm
o}(t)\,,$ (3.27) $\displaystyle\int_{\partial H}U^{p^{\\#}}=T^{p^{\\#}}\,.$
(3.28)

###### Proof of Lemma 3.3.

Since $V_{T}$ is assumed to be compactly supported, statement (i) follows
immediately from the corresponding properties for $U_{T}$. More specifically,
we deduce (3.20) from (3.2), and (3.21)–(3.25) from (3.20). Statement (ii)
follows from (3.7). ∎

###  3.4. The Ansatz for boundary concentrations

We next use the standard variations of minimizers of $\Phi_{H}$ described in
Lemma 3.3 to define certain competitors for $\Phi_{\Omega}$ that provide us
with a notion of “standard boundary concentration.” Recall the notation
$U^{(\varepsilon)}$ for dilations introduced in (1.5).

######  Lemma 3.4.

Fix $n\geq 2$, $p\in(1,n)$, $T>0$, $U\in\mathcal{U}_{T}$. Let $\Omega$ be an
open set with $C^{1}$-boundary, $x_{0}\in\partial\Omega$, and let
$\hat{f}=\hat{f}_{x_{0}}$, $\hat{g}=\hat{f}^{-1}$, $f=f_{x_{0}}$ and
$g=f^{-1}$ be determined as in Remark 2.4 starting from $\Omega$ and $x_{0}$.
Then the following statements hold:

(i): If $v\in W^{1,p}(\Omega)$, $\beta\in(0,1)$, $r_{1}=\varepsilon^{\beta}$,
$r_{2}=2\,\varepsilon^{\beta}$, and $\varphi_{\varepsilon}$ is a cut-off
function between $B_{r_{1}}(x_{0})$ and $B_{r_{2}}(x_{0})$ with
$|{\nabla}\varphi_{\varepsilon}|\leq C\varepsilon^{-\beta}$, then

$\displaystyle
v_{\varepsilon}(x)=(1-\varphi_{\varepsilon}(x))\,v(x)+\varphi_{\varepsilon}(x)\,\big{(}U^{(\varepsilon)}\circ\hat{g}\big{)}(x)\,,\qquad
x\in\Omega\,,$ (3.29)

satisfies

$\displaystyle\lim_{\varepsilon\to 0^{+}}\int_{\Omega}\\!\\!\\!|\nabla
v_{\varepsilon}|^{p}=\int_{H}\,|\nabla U|^{p}+\int_{\Omega}|\nabla v|^{p}\,,$
(3.31) $\displaystyle\lim_{\varepsilon\to
0^{+}}\int_{\Omega}\\!\\!\\!v_{\varepsilon}^{p^{\star}}=\int_{H}U^{p^{\star}}+\int_{\Omega}v^{p^{\star}}\,,$
$\displaystyle\lim_{\varepsilon\to
0^{+}}\int_{\partial\Omega}\\!\\!\\!v_{\varepsilon}^{p^{\\#}}=\int_{\partial
H}U^{p^{\\#}}+\int_{\partial\Omega}v^{p^{\\#}}\,.$ (3.32)

(ii): If $n>2p$, $v\in{\rm Lip}(\Omega)$ and $\Omega$ has $C^{2}$-boundary,
then there exists a choice of $\beta=\beta(n,p)\in(0,1)$ (used in the
definition of $r_{1}=\varepsilon^{\beta}$ and $r_{2}=2\,\varepsilon^{\beta}$),
such that the function

$\displaystyle
v_{\varepsilon}(x)=(1-\varphi_{\varepsilon}(x))\,v(x)+\varphi_{\varepsilon}(x)\,\big{(}U^{(\varepsilon)}\circ
g\big{)}(x)\,,\qquad x\in\Omega\,,$ (3.33)

satisfies (3.31), (3.31), and (3.32) in the more precise form

$\displaystyle\int_{\Omega}|\nabla v_{\varepsilon}|^{p}$
$\displaystyle=\int_{H}|\nabla U|^{p}+\int_{\Omega}|\nabla v|^{p}-{\rm
H}_{\partial\Omega}(x_{0})\,\mathcal{L}(U)\,\varepsilon+{\rm
o}(\varepsilon)\,,$ (3.34)
$\displaystyle\int_{\Omega}v_{\varepsilon}^{p^{\star}}$
$\displaystyle=\int_{H}U^{p^{\star}}+\int_{\Omega}v^{p^{\star}}-{\rm
H}_{\partial\Omega}(x_{0})\,\mathcal{M}(U)\,\varepsilon+{\rm
o}(\varepsilon)\,,$ (3.35)
$\displaystyle\int_{\partial\Omega}v_{\varepsilon}^{p^{\\#}}$
$\displaystyle=\int_{\partial
H}U^{{p^{\\#}}}+\int_{\partial\Omega}v^{p^{\\#}}+{\rm o}(\varepsilon)\,,$
(3.36)

as $\varepsilon\to 0^{+}$. Here $\mathcal{L}(U)$ and $\mathcal{M}(U)$ are
defined in (1.28) and (1.29) and the orders in (3.34), (3.35), and (3.36)
depend on $n$, $p$, $T$, and $v$.

###### Proof.

Without loss of generality we assume that $x_{0}=0\in\partial\Omega$,
$T_{0}\,(\partial\Omega)=\\{x_{n}=0\\}$ and $\nu_{\Omega}(0)=-e_{n}$. We carry
out the proof in several steps.

Step one: We start by noticing the following estimates for the energy, volume
and trace of $v_{\varepsilon}$ in transition region for the cut-off function
$\varphi_{\varepsilon}$. The estimates in this step hold in identical form
with the same proofs for $v_{\varepsilon}$ defined from $\hat{f}$ as in (3.29)
and for $v_{\varepsilon}$ defined from $f$ as in (3.33); we write the proof
for (3.33). First, with $v\in W^{1,p}(\Omega)$,

$\lim_{\varepsilon\to 0}\max\Big{\\{}\int_{\Omega\cap(B_{r_{2}}\setminus
B_{r_{1}})}|\nabla v_{\varepsilon}|^{p},\int_{\Omega\cap(B_{r_{2}}\setminus
B_{r_{1}})}v_{\varepsilon}^{p^{\star}},\int_{(\partial\Omega)\cap(B_{r_{2}}\setminus
B_{r_{1}})}v_{\varepsilon}^{p^{\\#}}\Big{\\}}=0\,,$ (3.37)

and, second, under the additional assumption that $v\in{\rm Lip}(\Omega)$,

$\displaystyle\int_{\Omega\cap(B_{r_{2}}\setminus B_{r_{1}})}|\nabla
v_{\varepsilon}|^{p}\leq
C\,\max\Big{\\{}\varepsilon^{(1-\beta)\,(n-p)/(p-1)},\varepsilon^{\beta(n-p)}\Big{\\}}\,,$
(3.38) $\displaystyle\int_{\Omega\cap(B_{r_{2}}\setminus
B_{r_{1}})}v_{\varepsilon}^{p^{\star}}\leq
C\,\max\Big{\\{}\varepsilon^{(1-\beta)\,n/(p-1)},\varepsilon^{\beta\,n}\Big{\\}}$
(3.39) $\displaystyle\int_{(\partial\Omega)\cap(B_{r_{2}}\setminus
B_{r_{1}})}v_{\varepsilon}^{p^{\\#}}\leq
C\,\max\Big{\\{}\varepsilon^{(1-\beta)\,(n-1)/(p-1)}\,,\varepsilon^{\beta(n-1)}\Big{\\}}\,.$
(3.40)

Indeed, we have $\nabla v_{\varepsilon}=a_{\varepsilon}+b_{\varepsilon}$ for

$a_{\varepsilon}=\varphi_{\varepsilon}\,(\nabla g)^{*}[(\nabla
U^{(\varepsilon)})\circ g]+(U^{(\varepsilon)}\circ
g)\,\nabla\varphi_{\varepsilon}\,,\qquad
b_{\varepsilon}=(1-\varphi_{\varepsilon})\,\nabla
v-v\,\nabla\varphi_{\varepsilon}\,.$

By (2.13), and thanks to $|\nabla g|,Jf\leq 2$ on $\textbf{{C}}_{r_{0}}$, we
find

$\displaystyle\int_{\Omega\cap(B_{r_{2}}\setminus
B_{r_{1}})}|a_{\varepsilon}|^{p}\leq C\,\int_{\Omega\cap(B_{r_{2}}\setminus
B_{r_{1}})}\\!\\!\\!|(\nabla g)^{*}[(\nabla U^{(\varepsilon)})\circ
g]|^{p}+\frac{U^{(\varepsilon)}(g)^{p}}{\varepsilon^{\beta\,p}}$
$\displaystyle\leq$ $\displaystyle C\,\int_{H\cap(B_{Cr_{2}}\setminus
B_{r_{1}/C})}\\!\\!\\!|\nabla
U^{(\varepsilon)}|^{p}+\frac{(U^{(\varepsilon)})^{p}}{\varepsilon^{\beta\,p}}$
$\displaystyle=$ $\displaystyle C\,\int_{H\cap(B_{Cr_{2}/\varepsilon}\setminus
B_{r_{1}/C\varepsilon})}\\!\\!\\!|\nabla
U|^{p}+\varepsilon^{np/{p^{\star}}}\,\frac{U^{p}}{\varepsilon^{\beta\,p}}\,\varepsilon^{n}$
$\displaystyle\leq$
$\displaystyle\\!\\!\\!C\,\Big{\\{}\varepsilon^{(1-\beta)(n-p)/(p-1)}+\frac{\varepsilon^{p}\,\varepsilon^{(\beta-1)(p^{2}-n)/(p-1)}}{\varepsilon^{\beta\,p}}\Big{\\}}=C\,\varepsilon^{(1-\beta)(n-p)/(p-1)}\,,$

where in the last inequality we have used (3.22). Concerning
$b_{\varepsilon}$, we notice that if we only know that $v\in W^{1,p}(\Omega)$
then by $v\in L^{p^{\star}}(\Omega)$ and $\nabla v\in L^{p}(\Omega)$ we find
that

$\int_{\Omega\cap(B_{r_{2}}\setminus
B_{r_{1}})}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!|b_{\varepsilon}|^{p}\leq\int_{\Omega\cap
B_{r_{2}}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!|\nabla v|^{p}+\int_{\Omega\cap
B_{r_{2}}}\frac{|v|^{p}}{\varepsilon^{\beta\,p}}\leq\int_{\Omega\cap
B_{r_{2}}}|\nabla v|^{p}+\Big{(}\int_{\Omega\cap
B_{r_{2}}}\\!\\!\\!\\!|v|^{p^{\star}}\Big{)}^{p/{p^{\star}}}$

where the latter quantity converges to $0$ at an non-quantified rate as
$\varepsilon\to 0^{+}$ (as stated in (3.37)); while, if $v\in{\rm
Lip}(\Omega)$, then

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\int_{\Omega\cap(B_{r_{2}}\setminus
B_{r_{1}})}\\!\\!\\!\\!\\!\\!\\!\\!\\!|b_{\varepsilon}|^{p}\leq
C\int_{\Omega\cap(B_{r_{2}}\setminus
B_{r_{1}})}\\!\\!\\!\\!\\!\\!\\!\\!|v|^{p}\,|\nabla\varphi_{\varepsilon}|^{p}+|\nabla
v|^{p}\leq C\,r_{2}^{n}\,{\rm Lip}(\varphi_{\varepsilon})^{p}\leq
C\varepsilon^{\beta\,(n-p)}\,,$

and (3.38) is proved. The other two limits in (3.37) follow similarly (with
non-quantified rates), while if $v\in{\rm Lip}(\Omega)$, then (3.39) and
(3.40) follow from (3.21), (3.23), and

$\displaystyle\int_{\Omega\cap(B_{r_{2}}\setminus
B_{r_{1}})}u_{\varepsilon}^{p^{\star}}$ $\displaystyle\leq C\varepsilon^{\beta
n}+C\,\int_{H\cap(B_{Cr_{2}/\varepsilon}\setminus
B_{r_{1}/C\varepsilon})}U^{p^{\star}}\leq C\varepsilon^{\beta
n}+C\,\varepsilon^{(1-\beta)n/(p-1)}$
$\displaystyle\int_{(\partial\Omega)\cap(B_{r_{2}}\setminus
B_{r_{1}})}u_{\varepsilon}^{p^{\\#}}$ $\displaystyle\leq
C\varepsilon^{\beta(n-1)}+C\,\int_{(\partial
H)\cap(B_{Cr_{2}/\varepsilon}\setminus B_{r_{1}/C\varepsilon})}U^{p^{\\#}}\leq
C\varepsilon^{\beta(n-1)}+C\,\varepsilon^{(1-\beta)\frac{(n-1)}{(p-1)}}\,.$

Step two: We prove statement (i). By (3.37),

$\int_{\Omega}|\nabla v_{\varepsilon}|^{p}=\int_{\Omega\cap
B_{r_{1}}}\\!\\!\\!|(\nabla\hat{g})^{*}[(\nabla
U^{(\varepsilon)})\circ\hat{g}]|^{p}+\int_{\Omega\setminus B_{r_{2}}}|\nabla
v|^{p}+{\rm o}(1)\,,$

and, similarly,

$\int_{\Omega}|\nabla v|^{p}\geq\int_{\Omega\setminus B_{r_{2}}}|\nabla
v|^{p}\geq\int_{\Omega}|\nabla v|^{p}+{\rm o}(1)\,.$ (3.41)

Moreover, if we set $E_{\varepsilon}=g(B_{r_{1}})\subset H$ and
$\tilde{E}_{\varepsilon}=E_{\varepsilon}/\varepsilon\subset H$, then keeping
in mind (2.13), (2.11), (2.12), and (3.22), we have

$\displaystyle\int_{\Omega\cap B_{r_{1}}}\\!\\!\\!|(\nabla\hat{g})^{*}[(\nabla
U^{(\varepsilon)})\circ\hat{g}]|^{p}=$
$\displaystyle\int_{E_{\varepsilon}}\\!\\!|((\nabla\hat{g})\circ\hat{f})^{*}[\nabla
U^{(\varepsilon)}]|^{p}\,J\hat{f}=(1+{\rm
o}(1))\int_{E_{\varepsilon}}\\!\\!|\nabla U^{(\varepsilon)}|^{p}$
$\displaystyle=(1+{\rm o}(1))$ $\displaystyle\Big{\\{}\int_{H}\\!\\!|\nabla
U|^{p}-\int_{H\setminus\tilde{E_{\varepsilon}}}|\nabla U|^{p}\Big{\\}}=(1+{\rm
o}(1))\int_{H}\\!\\!|\nabla U|^{p}\,.$

This proves (3.31). Entirely analogous arguments prove (3.31) and (3.32).

Step three: We now start the proof of statement (ii); in particular, from now
on, $\Omega$ has $C^{2}$-boundary, $n>2p$, and $v_{\varepsilon}$ is defined as
in (3.33); moreover, for the sake of brevity, we set $h={\rm
H}_{\partial\Omega}(0)$. In this step, we discuss the choice of
$\beta=\beta(n,p)\in(0,1)$, which is determined by the rates in (3.38), (3.39)
and (3.40), and by the fact that in (3.34), (3.35) and (3.36) we want errors
of size ${\rm o}(\varepsilon)$: therefore, by

$\displaystyle\beta\,\min\Big{\\{}n,n-1,n-p\Big{\\}}>1\qquad$ iff
$\displaystyle\qquad\beta>\frac{1}{n-p}\,,$
$\displaystyle(1-\beta)\,\min\Big{\\{}\frac{n-p}{p-1},\frac{n-1}{p-1},\frac{n}{p-1}\Big{\\}}>1\,\qquad$
iff $\displaystyle\qquad\beta<\frac{n+1-2\,p}{n-p}\,,$

we are led to choose

$\beta\in\Big{(}\frac{1}{n-p},\min\Big{\\{}1,\frac{n+1-2\,p}{n-p}\Big{\\}}\Big{)}\,,$
(3.42)

(where the interval appearing in (3.42) is non-empty thanks to $n>2p$). With
this choice of $\beta$, we have
$\min\\{\beta\,(n-p),(1-\beta)\,\frac{n-p}{p-1}\\}>1\,,$ and thus deduce from
(3.38), (3.39) and (3.40) that

$\max\Big{\\{}\int_{\Omega\cap(B_{r_{2}}\setminus B_{r_{1}})}|\nabla
v_{\varepsilon}|^{p},\ \int_{\Omega\cap(B_{r_{2}}\setminus
B_{r_{1}})}v_{\varepsilon}^{p^{\star}},\
\int_{(\partial\Omega)\cap(B_{r_{2}}\setminus
B_{r_{1}})}v_{\varepsilon}^{p^{\\#}}\Big{\\}}={\rm o}(\varepsilon)\,.$ (3.43)

Step four: We prove (3.34). We first notice that by (3.43) and (3.41) we have

$\int_{\Omega}|\nabla v_{\varepsilon}|^{p}=\int_{\Omega\cap
B_{r_{1}}}\\!\\!\\!|(\nabla g)^{*}[(\nabla U^{(\varepsilon)})\circ
g]|^{p}+\int_{\Omega}|\nabla v|^{p}+{\rm o}(\varepsilon)\,.$ (3.44)

Now, by (2.15) we have

$\displaystyle\nabla(U^{(\varepsilon)}\circ g)\circ f=[(\nabla g)\circ
f]^{*}\,\nabla U^{(\varepsilon)}$ $\displaystyle=$ $\displaystyle\nabla
U^{(\varepsilon)}+\partial_{n}U^{(\varepsilon)}\,\nabla\ell-(\nabla\ell\cdot\nabla
U^{(\varepsilon)})\,e_{n}+x_{n}\,\sum_{i=1}^{n-1}\kappa_{i}\,\partial_{i}U^{(\varepsilon)}\,e_{i}+{\rm
O}(|x|^{2})\,|\nabla U^{(\varepsilon)}|\,,$

so that, recalling that $|\nabla\ell|={\rm O}(|x|)$,

$\displaystyle\Big{|}\nabla(U^{(\varepsilon)}\circ g)\circ f\Big{|}^{2}$
$\displaystyle=$ $\displaystyle|\nabla
U^{(\varepsilon)}|^{2}+2\,\big{(}(\nabla\ell\cdot\nabla
U^{(\varepsilon)})\,e_{n}-\partial_{n}U^{(\varepsilon)}\,\nabla\ell\big{)}\cdot\nabla
U^{(\varepsilon)}$
$\displaystyle+2\,x_{n}\,\sum_{i=1}^{n-1}\kappa_{i}\,(\partial_{i}U^{(\varepsilon)})^{2}+{\rm
O}(|x|^{2})\,|\nabla U^{(\varepsilon)}|^{2}$ $\displaystyle=$
$\displaystyle|\nabla
U^{(\varepsilon)}|^{2}+2\,x_{n}\,\sum_{i=1}^{n-1}\kappa_{i}\,(\partial_{i}U^{(\varepsilon)})^{2}+{\rm
O}(|x|^{2})\,|\nabla U^{(\varepsilon)}|^{2}\,.$

Now set $a=|\nabla U^{(\varepsilon)}|$ and
$b=[2\,\sum_{i=1}^{n-1}\kappa_{i}\,(\partial_{i}U^{(\varepsilon)})^{2}]^{1/2}$,
so that $0\leq b\leq C\,a$ for a constant depending on
$|A_{\partial\Omega}(x_{0})|$. Since $z\mapsto(1+z)^{p/2}$ is smooth in a
neighborhood of $z=0$, we see that if $|x|<1/C$ for a constant $C$ depending
on $|A_{\partial\Omega}(x_{0})|$, then

$\displaystyle\big{(}a^{2}+x_{n}\,b^{2}+{\rm
O}(|x|^{2})\,a^{2}\big{)}^{p/2}=a^{p}\,\big{(}1+(b/a)^{2}\,x_{n}+{\rm
O}(|x|^{2})\big{)}^{p/2}$
$\displaystyle=a^{p}\,\Big{(}1+\frac{p}{2}(b/a)^{2}\,x_{n}+{\rm
O}(|x|^{2})\Big{)}=a^{p}+\frac{p}{2}\,a^{p-2}\,b^{2}\,x_{n}+{\rm O}(|x|^{2})$

and thus

$\displaystyle\Big{|}\nabla(U^{(\varepsilon)}\circ g)\circ f\Big{|}^{p}\,Jf$
$\displaystyle=|\nabla
U^{(\varepsilon)}|^{p}\Big{(}1+p\,x_{n}\,\sum_{i=1}^{n-1}\kappa_{i}\frac{(\partial_{i}U^{(\varepsilon)})^{2}}{|\nabla
U^{(\varepsilon)}|^{2}}+{\rm O}(|x|^{2})\Big{)}\Big{(}1-x_{n}\,h+{\rm
O}(|x|^{2})\Big{)}$ $\displaystyle=|\nabla
U^{(\varepsilon)}|^{p}-x_{n}\,\Big{(}h-p\,\sum_{i=1}^{n-1}\kappa_{i}\,\frac{(\partial_{i}U^{(\varepsilon)})^{2}}{|\nabla
U^{(\varepsilon)}|^{2}}\Big{)}\,|\nabla U^{(\varepsilon)}|^{p}+{\rm
O}(|x|^{2})\,|\nabla U^{(\varepsilon)}|^{p}\,.$

Then, by (2.13),

$\displaystyle\int_{\Omega\cap B_{r_{1}}}\\!\\!\\!|(\nabla g)^{*}[(\nabla
U^{(\varepsilon)})\circ g]|^{p}\leq\int_{H\cap B_{C\,r_{1}}}\big{|}[(\nabla
g)\circ f]^{*}\,(\nabla U^{(\varepsilon)})\big{|}^{p}\,Jf$
$\displaystyle\leq\int_{H\cap B_{C\,r_{1}}}|\nabla
U^{(\varepsilon)}|^{p}-h\,\int_{H\cap B_{C\,r_{1}}}\\!\\!\\!\\!x_{n}\,|\nabla
U^{(\varepsilon)}|^{p}$
$\displaystyle+p\,\sum_{i=1}^{n-1}\kappa_{i}\,\int_{H\cap
B_{C\,r_{1}}}\\!\\!\\!\\!x_{n}\,(\partial_{i}U^{(\varepsilon)})^{2}\,|\nabla
U^{(\varepsilon)}|^{p-2}+C\,\int_{H\cap
B_{C\,r_{1}}}\\!\\!\\!\\!|x|^{2}\,|\nabla U^{(\varepsilon)}|^{p}$
$\displaystyle=\int_{H\cap B_{Cr_{1}/\varepsilon}}\\!\\!\\!\\!|\nabla
U|^{p}-h\,\varepsilon\,\int_{H\cap
B_{Cr_{1}/\varepsilon}}\\!\\!\\!\\!x_{n}\,|\nabla U|^{p}$ (3.45)
$\displaystyle+p\,\varepsilon\,\sum_{i=1}^{n-1}\kappa_{i}\,\int_{H\cap
B_{Cr_{1}/\varepsilon}}\\!\\!\\!\\!x_{n}\,(\partial_{i}U)^{2}\,|\nabla
U|^{p-2}+C\,\varepsilon^{2}\,\int_{H\cap
B_{Cr_{1}/\varepsilon}}\\!\\!\\!\\!|x|^{2}\,|\nabla U|^{p}\,.$

Now, by the reflection symmetries of $U$ with respect to $\\{x_{i}=0\\}$,
$i=1,...,n-1$ (recall (3.19)), we have

$\int_{H\cap B_{R}}x_{n}\,(\partial_{i}U)^{2}\,|\nabla U|^{p-2}=\int_{H\cap
B_{R}}x_{n}\,(\partial_{1}U)^{2}\,|\nabla U|^{p-2}\,,\qquad\forall
i=1,...,n-1\,,\forall R>0\,,$

and therefore

$\displaystyle\sum_{i=1}^{n-1}\kappa_{i}\,\int_{H\cap
B_{Cr_{1}/\varepsilon}}x_{n}\,(\partial_{i}U)^{2}\,|\nabla
U|^{p-2}=h\,\int_{H\cap
B_{Cr_{1}/\varepsilon}}x_{n}\,\big{(}\partial_{1}U)^{2}\,|\nabla U|^{p-2}\,.$

Setting

$\displaystyle\mathcal{L}(U,B_{R})=\int_{H\cap B_{R}}x_{n}\,|\nabla
U|^{p}-p\,x_{n}\,(\partial_{1}U)^{2}\,|\nabla U|^{p-2}\,,$ (3.46)

we can thus rewrite (3.45) as

$\int_{\Omega\cap B_{r_{1}}}\\!\\!\\!\\!\\!|(\nabla g)^{*}[(\nabla
U^{(\varepsilon)})\circ g]|^{p}\leq\int_{H}|\nabla
U|^{p}-h\,\varepsilon\,\mathcal{L}(U,B_{C\,r_{1}/\varepsilon})+C\,\varepsilon^{2}\,\int_{H\cap
B_{C\,r_{1}/\varepsilon}}\\!\\!\\!\\!\\!\\!|x|^{2}\,|\nabla U|^{p}\,.$

At the same time, $\varepsilon^{2}\,|x|^{2}\leq C\,\varepsilon\,r_{1}\,|x|$
for any $x\in B_{C\,r_{1}/\varepsilon}$, so

$\displaystyle\varepsilon^{2}\,\int_{H\cap
B_{Cr_{1}/\varepsilon}}\,|x|^{2}\,|\nabla U|^{p}\leq
C\,\varepsilon\,r_{1}\,\int_{H\cap B_{Cr_{1}/\varepsilon}}\,|x|\,|\nabla
U|^{p}\,$ $\displaystyle\leq C\,\varepsilon^{1+\beta}\,\int_{H}\,|x|\,|\nabla
U_{T}|^{p}+C\,\varepsilon^{1+\beta}\,\int_{H}\,|x|\,|\nabla V_{T}|^{p}\,\leq
C(n,p,T)\,\varepsilon^{1+\beta},$

where we have used the facts that $V_{T}$ is compactly supported and that
$n>2p-1$ to guarantee the convergence of the integrals in the final line.
Hence,

$\displaystyle\int_{\Omega\cap B_{r_{1}}}\\!\\!\\!\\!\\!|(\nabla
g)^{*}[(\nabla U^{(\varepsilon)})\circ g]|^{p}$ $\displaystyle=\int_{H}|\nabla
U|^{p}-h\,\mathcal{L}(U,B_{C\,r_{1}/\varepsilon})\,\varepsilon+{\rm
o}(\varepsilon)\,$ $\displaystyle=\int_{H}|\nabla
U|^{p}-h\,\mathcal{L}(U)\,\varepsilon+{\rm o}(\varepsilon)\,,$

where the ${\rm o}(\varepsilon)$ term depends on $n$, $p$, and $T$, and in the
second line we have applied (3.25). By (3.44) we deduce (3.34).

Step five: We prove (3.35). We first notice that by (3.43), $v\in{\rm
Lip}(\Omega)$, $r_{2}^{n}=\varepsilon^{\beta\,n}={\rm o}(\varepsilon)$ and our
choice of $\beta$ we have

$\int_{\Omega}v_{\varepsilon}^{p^{\star}}=\int_{\Omega\cap
B_{r_{1}}}\\!\\!\\!(U^{(\varepsilon)}\circ
g)^{p^{\star}}+\int_{\Omega}v^{p^{\star}}+{\rm o}(\varepsilon)\,.$ (3.47)

Let $E_{\varepsilon}=g(B_{r_{1}}\cap\Omega)\subset H$ and
$\tilde{E}_{\varepsilon}=E_{\varepsilon}/\varepsilon\subset E$ as in step two.
Then keeping in mind (2.13) and (2.14),

$\displaystyle\int_{\Omega\cap B_{r_{1}}}$
$\displaystyle\\!\\!\\!(U^{(\varepsilon)}\circ
g)^{p^{\star}}=\int_{E_{\varepsilon}}(U^{(\varepsilon)})^{p^{\star}}-h\,\int_{E_{\varepsilon}}x_{n}\,(U^{(\varepsilon)})^{p^{\star}}+\,\int_{E_{\varepsilon}}{\rm
O}(|x|^{2})\,(U^{(\varepsilon)})^{p^{\star}}$
$\displaystyle=\int_{\tilde{E}_{\varepsilon}}U^{{p^{\star}}}-h\,\varepsilon\,\int_{\tilde{E}_{\varepsilon}}x_{n}\,U^{{p^{\star}}}+\,\int_{E_{\varepsilon}}{\rm
O}(|x|^{2})\,(U^{(\varepsilon)})^{p^{\star}}$
$\displaystyle=\int_{H}U^{p^{\star}}-h\,\varepsilon\mathcal{M}(U)+\Big{\\{}-\int_{H\setminus\tilde{E}_{\varepsilon}}\\!\\!\\!\\!U^{{p^{\star}}}+h\varepsilon\int_{H\setminus\tilde{E}_{\varepsilon}}x_{n}U^{{p^{\star}}}+\,\int_{E_{\varepsilon}}{\rm
O}(|x|^{2})\,(U^{(\varepsilon)})^{p^{\star}}\Big{\\}}\,.$

By (3.21) and (3.24), along with our choice of $\beta$, we see that

$-\int_{H\setminus\tilde{E}_{\varepsilon}}U^{{p^{\star}}}=o(\varepsilon),\qquad
h\,\varepsilon\,\int_{H\setminus\tilde{E}_{\varepsilon}}x_{n}U^{{p^{\star}}}=o(\varepsilon)\,.$

Moreover, since $U=U_{T}+tV_{T}$ with $V_{T}$ compactly supported in $H$ and
$|t|\leq 1$, we have

$\displaystyle\int_{E_{\varepsilon}}|x|^{2}\,(U^{(\varepsilon)})^{p^{\star}}$
$\displaystyle\leq
C\,r_{1}\,\varepsilon\,\int_{\tilde{E}_{\varepsilon}}|x|\,U^{{p^{\star}}}\leq\varepsilon^{1+\beta}\int_{H}|x|\,U^{{p^{\star}}}$
$\displaystyle\leq
C\varepsilon^{1+\beta}\int_{H}|x|\,U_{T}^{{p^{\star}}}+C\varepsilon^{1+\beta}\int_{H}|x|V^{{p^{\star}}}={\rm
o}(\varepsilon),$

with ${\rm o}(\varepsilon)$ depending on $n$, $p$, and $T$. So, the entire
term in brackets above can be written as ${\rm o}(\varepsilon)$. Combining
this estimate with (3.47), we deduce (3.35).

Step six: We finally prove (3.36). Notice that, by (3.43), $v\in{\rm
Lip}(\Omega)$, $r_{2}^{n-1}=\varepsilon^{\beta\,(n-1)}={\rm o}(\varepsilon)$
(by the choice of $\beta$), we have

$\int_{\partial\Omega}v_{\varepsilon}^{p^{\\#}}=\int_{(\partial\Omega)\cap
B_{r_{1}}}\\!\\!\\!(U^{(\varepsilon)}\circ
g)^{p^{\\#}}+\int_{\partial\Omega}v^{p^{\\#}}+{\rm o}(\varepsilon)\,.$ (3.48)

Now, by $J^{\partial H}f\geq 1$, (3.23) and our choice of $\beta$ we have

$\displaystyle\int_{(\partial\Omega)\cap
B_{r_{1}}}\\!\\!\\!\\!(U^{(\varepsilon)}\circ g)^{p^{\\#}}\geq\int_{(\partial
H)\cap B_{r_{1}/C}}\\!\\!\\!\\!(U^{(\varepsilon)})^{{p^{\\#}}}=\int_{\partial
H}U^{{p^{\\#}}}-\int_{(\partial H)\setminus
B_{r_{1}/C\varepsilon}}\\!\\!\\!\\!\\!\\!U^{{p^{\\#}}}\geq\int_{\partial
H}U^{{p^{\\#}}}+{\rm o}(\varepsilon)\,.$

At the same time, by $J^{\partial H}f\leq 1+C\,|x|^{2}$, we have

$\displaystyle\int_{(\partial\Omega)\cap
B_{r_{1}}}\\!\\!\\!(U^{(\varepsilon)}\circ g)^{p^{\\#}}\leq\int_{(\partial
H)\cap B_{C\,r_{1}}}(1+C\,|x|^{2})\,(U^{(\varepsilon)})^{{p^{\\#}}}$
$\displaystyle\leq\int_{\partial
H}U^{{p^{\\#}}}+C\,\varepsilon^{2}\,\int_{(\partial H)\cap
B_{C\,r_{1}/\varepsilon}}|x|^{2}\,U^{{p^{\\#}}}$

where

$\displaystyle\varepsilon^{2}\,\int_{(\partial H)\cap
B_{C\,r_{1}/\varepsilon}}|x|^{2}\,U^{{p^{\\#}}}\leq
C\,\varepsilon^{2}\,\int_{0}^{C\,r_{1}/\varepsilon}\,\frac{r^{2}\,r^{n-2}\,dr}{(r^{(n-p)/(p-1)})^{p^{\\#}}}$
$\displaystyle\leq
C\,\varepsilon^{2}\,\big{(}r_{1}/\varepsilon\big{)}^{1+n-(n-1)\,[p/(p-1)]}$
$\displaystyle\leq
C\,\varepsilon^{2}\,\big{(}r_{1}/\varepsilon\big{)}^{(2p-n-1)/(p-1)}\leq
C\,\varepsilon^{2}\,\varepsilon^{(1-\beta)\,(n+1-2p)/(p-1)}\leq
C\,\varepsilon^{2}={\rm o}(\varepsilon)\,,$

thanks to $n>2p-1$. This completes the proof. ∎

##  4\. Existence of minimizers

We first establish the existence of generalized minimizers. Recall that
$\Phi_{\Omega}^{*}(T)$ was defined in (1.21).

######  Theorem 4.1.

Let $n\geq 2$, $p\in(1,n)$, and let $\Omega$ be a bounded open set with
$C^{1}$-boundary in $\mathbb{R}^{n}$. Then:

(i): for every $T>0$, $\Phi_{\Omega}(T)=\Phi_{\Omega}^{*}(T)$;

(ii): there is a minimizer $(u,\mathsf{v},\mathsf{t})$ of
$\Phi_{\Omega}^{*}(T)$;

(iii): if $(u,\mathsf{v},\mathsf{t})$ is a minimizer of $\Phi_{\Omega}^{*}(T)$
with $\int_{\Omega}u^{p^{\star}}>0$, then
$\int_{\partial\Omega}u^{p^{\\#}}>0$,

$u\Big{/}\|u\|_{L^{p^{\star}}(\Omega)}\,\,\mbox{is a minimizer
of}\,\,\,\Phi_{\Omega}\Big{(}\|u\|_{L^{p^{\\#}}(\partial\Omega)}\Big{/}\|u\|_{L^{p^{\star}}(\Omega)}\Big{)}\,,$
(4.1)

and there exists $\lambda,\sigma\in\mathbb{R}$ such that

$\left\\{\begin{split}&-\Delta_{p}u=\lambda\,u^{{p^{\star}}-1}\,,\quad\hskip
28.45274pt\mbox{on $\Omega$}\,,\\\ &|\nabla u|^{p-2}\,\frac{\partial
u}{\partial\nu_{\Omega}}=\sigma\,u^{{p^{\\#}}-1}\,,\quad\mbox{on
$\partial\Omega$}\,,\end{split}\right.$ (4.2)

In particular, $u\in{\rm Lip}(\Omega)$. If, in addition, $\mathsf{v}>0$, then
$\lambda$ and $\sigma$ are given by

$\lambda=\mathsf{v}^{p-{p^{\star}}}\,\lambda_{H}(\mathsf{t}/\mathsf{v})\,,\qquad\sigma=\mathsf{v}^{p-{p^{\\#}}}\,\sigma_{H}(\mathsf{t}/\mathsf{v})\,,$
(4.3)

and, in particular,

$\frac{\mathsf{t}}{\mathsf{v}}\in(0,T_{0})\cup(T_{E},\infty)\,.$ (4.4)

###### Proof.

Step one: Since $(u,0,0)\in\mathcal{Y}_{\Omega}(T)$ if
$u\in\mathcal{X}_{\Omega}(T)$, we have
$\Phi_{\Omega}^{*}(T)\leq\Phi_{\Omega}(T)$. To prove the converse inequality
it is enough to show that for every
$(u,\mathsf{v},\mathsf{t})\in\mathcal{Y}_{\Omega}(T)$,

$\mbox{$\exists\,u_{j}\in\mathcal{X}_{\Omega}(T)$
s.t.}\,\,\,\lim_{j\to\infty}\int_{\Omega}|\nabla
u_{j}|^{p}=\int_{\Omega}|\nabla
u|^{p}+\mathsf{v}^{p}\,\Phi_{H}\Big{(}\frac{\mathsf{t}}{\mathsf{v}}\Big{)}^{p}\,.$
(4.5)

Looking back at the definition of $\mathcal{Y}_{\Omega}(T)$ in the paragraph
preceding the statement of Theorem 1.1, we can assume without loss of
generality that $\mathsf{v}>0$ and $\mathsf{t}>0$. Moreover, given
$(u,\mathsf{v},\mathsf{t})\in\mathcal{Y}_{\Omega}(T)$ with $\mathsf{v}$ and
$\mathsf{t}$ positive we can easily find
$(u_{j},\mathsf{v}_{j},\mathsf{t}_{j})\in\mathcal{Y}_{\Omega}(T)$ with
$\mathsf{v}_{j}$, $\mathsf{t}_{j}$, $\int_{\Omega}u_{j}^{p^{\star}}$, and
$\int_{\partial\Omega}u_{j}^{p^{\\#}}$ positive and such that
$\mathcal{E}(u_{j},\mathsf{v}_{j},\mathsf{t}_{j})\to\mathcal{E}(u,\mathsf{v},\mathsf{t})$.
By a diagonal argument, it is thus sufficient proving (4.5) under the
assumption that $\int_{\Omega}u^{p^{\star}}$ and
$\int_{\partial\Omega}u^{p^{\\#}}$ are positive. This said, we apply Lemma
3.4(i) with

$v=\frac{u}{\mathsf{v}}\,,\qquad U=U_{\mathsf{t}/\mathsf{v}}\,,$

to find functions $v_{j}$ with $v_{j}=v$ on $\Omega\setminus
B_{2\,\varepsilon_{j}}(x_{0})$ for some $x_{0}\in\partial\Omega$ and
$\varepsilon_{j}\to 0^{+}$, and with

$\displaystyle\int_{\Omega}|\nabla
v_{j}|^{p}=\frac{1}{\mathsf{v}^{p}}\int_{\Omega}|\nabla
u|^{p}+\Phi_{H}(\mathsf{t}/\mathsf{v})^{p}+G_{j}\,,$
$\displaystyle\int_{\Omega}v_{j}^{p^{\star}}=\frac{1}{\mathsf{v}^{p^{\star}}}\int_{\Omega}u^{p^{\star}}+1+V_{j}$
$\displaystyle\int_{\partial\Omega}v_{j}^{p^{\\#}}=\frac{1}{\mathsf{v}^{p^{\\#}}}\int_{\partial\Omega}u^{p^{\\#}}+(\mathsf{t}/\mathsf{v})^{p^{\\#}}+T_{j}\,,$

where $G_{j}$, $V_{j}$, $T_{j}\to 0$ as $j\to\infty$ at a rate depending on
$n$, $p$, $\Omega$, $\mathsf{t}/\mathsf{v}$ and $u$ only. By Lemma 2.6, there
exist $\eta$ and $C$ depending on $n$, $p$, $\Omega$, $\mathsf{t}/\mathsf{v}$
and $u$, but independent from $j$, such that for any $(a_{j},b_{j})$ with
$|a_{j}|+|b_{j}|<\eta,$ we have functions $w_{j}$ such that

$\int_{\partial\Omega}|w_{j}|^{p^{\\#}}=a_{j}+\int_{\partial\Omega}|v_{j}|^{p^{\\#}}\,,\qquad\int_{\Omega}|w_{j}|^{p^{\star}}=b_{j}+\int_{\Omega}|v_{j}|^{p^{\star}}\,,$
(4.6) $\Big{|}\int_{\Omega}|\nabla w_{j}|^{p}-\int_{\Omega}|\nabla
v_{j}|^{p}\Big{|}\leq C\,\big{(}|a_{j}|+|b_{j}|\big{)}\,.$ (4.7)

For $j$ large enough we can apply this statement with $a_{j}=-T_{j}$ and
$b_{j}=-V_{j}$ to find a sequence $\\{w_{j}\\}_{j}$ with

$\int_{\partial\Omega}|w_{j}|^{p^{\\#}}=\frac{1}{\mathsf{v}^{p^{\\#}}}\int_{\partial\Omega}u^{p^{\\#}}+(\mathsf{t}/\mathsf{v})^{p^{\\#}}=\frac{T^{p^{\\#}}}{\mathsf{v}^{p^{\\#}}}\,,\qquad\int_{\Omega}|w_{j}|^{p^{\star}}=\frac{1}{\mathsf{v}^{p^{\star}}}\int_{\Omega}u^{p^{\star}}+1=\frac{1}{\mathsf{v}^{p^{\star}}}\,,$
(4.8) $\Big{|}\int_{\Omega}|\nabla
w_{j}|^{p}-\frac{1}{\mathsf{v}^{p}}\int_{\Omega}|\nabla
u|^{p}-\Phi_{H}(\mathsf{t}/\mathsf{v})^{p}-G_{j}\Big{|}\leq
C\,\big{(}|T_{j}|+|V_{j}|\big{)}\,.$ (4.9)

Setting $u_{j}=\mathsf{v}\,w_{j}$, we obtain a sequence in
$\mathcal{X}_{\Omega}(T)$ that satisfies (4.5).

Step two: We prove that there is a minimizer for the generalized problem
$\Phi_{\Omega}^{*}(T)$. By the argument in step one we can find a sequence
$\\{u_{j}\\}_{j}$ in $\mathcal{X}_{\Omega}(T)$ such that $\int_{\Omega}|\nabla
u_{j}|^{p}\to\Phi_{\Omega}^{*}(T)^{p}$. By Lemma 2.1, the measures $\mu_{j}$,
$\nu_{j}$ and $\tau_{j}$ defined in (2.1) have subsequential weak-star limits
$\mu$, $\nu$ and $\tau$ satisfying (2.2), (2.3) and (2.4) and (2.5). In
particular, there is an at most countable set $\\{x_{i}\\}_{i\in
I}\subset\overline{\Omega}$ and corresponding $\mathsf{v}_{i}>0$ and
$\mathsf{t}_{i}\geq 0$ for every $i\in I$, such that

$\Phi_{\Omega}^{*}(T)^{p}=\lim_{j\to\infty}\int_{\Omega}|\nabla
u_{j}|^{p}\geq\int_{\Omega}|\nabla u|^{p}+S^{p}\,\sum_{i\in I\setminus I_{{\rm
bd}}}\mathsf{v}_{i}^{p}+\sum_{i\in I_{\rm
bd}}\mathsf{v}_{i}^{p}\,\Phi_{H}(\mathsf{t}_{i}/\mathsf{v}_{i})^{p}\,,$ (4.10)

where $u$ is the subsequential weak limit of $u_{j}$, and

$1=\int_{\Omega}u^{p^{\star}}+\sum_{i\in
I}\,\mathsf{v}_{i}^{p^{\star}}\,,\qquad
T^{p^{\\#}}=\int_{\partial\Omega}u^{p^{\\#}}+\sum_{i\in I_{\rm
bd}}\,\mathsf{t}_{i}^{p^{\\#}}\,.$ (4.11)

Now set

$\mathsf{v}_{c}^{p^{\star}}=\sum_{i\in
I}\,\mathsf{v}_{i}^{p^{\star}}\,,\qquad\mathsf{t}_{c}^{p^{\\#}}=\sum_{i\in
I_{\rm bd}}\,\mathsf{t}_{i}^{p^{\\#}}\,.$

By an immediate adaptation of the proof of Lemma 3.4 we can easily construct a
sequence $\\{W_{j}\\}_{j}$ in $\mathcal{X}_{H}(\mathsf{t}_{c}/\mathsf{v}_{c})$
with the property that

$\int_{H}|\nabla W_{j}|^{p}\to\sum_{i\in
I}\Big{(}\frac{\mathsf{v}_{i}}{\mathsf{v}_{c}}\Big{)}^{p}\,\Phi_{H}(\mathsf{t}_{i}/\mathsf{v}_{i})^{p}\,.$

Since $W_{j}\in\mathcal{X}_{H}(\mathsf{t}_{c}/\mathsf{v}_{c})$ implies
$\int_{H}|\nabla W_{j}|^{p}\geq\Phi_{H}(\mathsf{t}_{c}/\mathsf{v}_{c})^{p}$,
we deduce from (4.10) that

$\Phi_{\Omega}^{*}(T)^{p}\geq\int_{\Omega}|\nabla
u|^{p}+\mathsf{v}_{c}^{p}\,\Phi_{H}(\mathsf{t}_{c}/\mathsf{v}_{c})^{p}\,,$

while (4.11) gives
$(u,\mathsf{v}_{c},\mathsf{t}_{c})\in\mathcal{Y}_{\Omega}(T)$. This proves
that $(u,\mathsf{v}_{c},\mathsf{t}_{c})$ is a minimizer of
$\Phi_{\Omega}^{*}(T)$.

Step three: We finally prove statement (iii). If $(u,\mathsf{v},\mathsf{t})$
is a minimizer of $\Phi_{\Omega}^{*}(T)$ with $\int_{\Omega}u^{p^{\star}}>0$,
it is immediate to deduce (4.1), and since $\Phi_{\Omega}(0)$ does not admit
minimizers, it must also be $\int_{\partial\Omega}u^{p^{\\#}}>0$. By Lemma
2.5, the Euler-Lagrange equation (4.2) for $u$ holds for some
$\lambda,\sigma\in\mathbb{R}$ and $u\in{\rm Lip}(\Omega)$. Assuming now that
$\mathsf{v}>0$, we can prove (4.3) by noticing that, given $\varphi\in
C^{\infty}_{c}(\mathbb{R}^{n})$, if we define

$\alpha(\delta)=\Big{(}1-\int_{\Omega}(u+\delta\,\varphi)^{p^{\star}}\Big{)}^{1/{p^{\star}}}-\mathsf{v}\,,\qquad\beta(\delta)=\Big{(}T^{p^{\\#}}-\int_{\partial\Omega}(u+\delta\,\varphi)^{p^{\\#}}\Big{)}^{1/{p^{\\#}}}-\mathsf{t}\,,$

then there is $\delta_{0}>0$ such that
$(u+\delta\,\varphi,\mathsf{v}+\alpha(\delta),\mathsf{t}+\beta(\delta))\in\mathcal{Y}_{\Omega}(T)$
for every $|\delta|<\delta_{0}$. In particular,

$\displaystyle
0=\frac{d}{d\delta}\Big{|}_{\delta=0}\,\int_{\Omega}|\nabla(u+\delta\varphi)|^{p}+(\mathsf{v}+\alpha(\delta))^{p}\,\Phi_{H}\Big{(}\frac{\mathsf{t}+\beta(\delta)}{\mathsf{v}+\alpha(\delta)}\Big{)}^{p}\,,$

and exploiting (4.2) as well as

$\alpha^{\prime}(0)=-\mathsf{v}^{1-{p^{\star}}}\,\int_{\Omega}u^{{p^{\star}}-1}\varphi\,,\qquad\beta^{\prime}(0)=-\mathsf{t}^{1-{p^{\\#}}}\,\int_{\partial\Omega}u^{{p^{\\#}}-1}\varphi\,,$

and (3.8) (i.e. $\Phi_{H}(T)=\lambda_{H}(T)+\sigma_{H}(T)\,T^{p^{\\#}}$ and
$\sigma_{H}(T)\,T^{{p^{\\#}}-1}=\Phi_{H}(T)^{p-1}\Phi_{H}^{\prime}(T)$ for
every $T>0$), we see that

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\int_{\Omega}|\nabla
u|^{p-2}\nabla
u\cdot\nabla\varphi+\mathsf{v}^{p-1}\,\Phi_{H}(\mathsf{t}/\mathsf{v})^{p-1}\,\Phi_{H}^{\prime}(\mathsf{t}/\mathsf{v})\,\beta^{\prime}(0)$
$\displaystyle\hskip
96.73918pt+\Big{\\{}\mathsf{v}^{p-1}\,\Phi_{H}(\mathsf{t}/\mathsf{v})^{p}-\mathsf{t}\,\mathsf{v}^{p-2}\,\Phi_{H}(\mathsf{t}/\mathsf{v})^{p-1}\,\Phi_{H}^{\prime}(\mathsf{t}/\mathsf{v})\Big{\\}}\alpha^{\prime}(0)$
$\displaystyle=$
$\displaystyle\lambda\,\int_{\Omega}u^{{p^{\star}}-1}\,\varphi+\sigma\,\int_{\partial\Omega}u^{{p^{\\#}}-1}\varphi-\mathsf{v}^{p-1}\,\mathsf{t}^{1-{p^{\\#}}}\,\Phi_{H}(\mathsf{t}/\mathsf{v})^{p-1}\,\Phi_{H}^{\prime}(\mathsf{t}/\mathsf{v})\,\int_{\partial\Omega}u^{{p^{\\#}}-1}\varphi$
$\displaystyle-\mathsf{v}^{p-{p^{\star}}}\,\Big{\\{}\Phi_{H}(\mathsf{t}/\mathsf{v})^{p}-(\mathsf{t}/\mathsf{v})\,\Phi_{H}(\mathsf{t}/\mathsf{v})^{p-1}\,\Phi_{H}^{\prime}(\mathsf{t}/\mathsf{v})\Big{\\}}\int_{\Omega}u^{{p^{\star}}-1}\varphi$
$\displaystyle=$
$\displaystyle\big{(}\sigma-\mathsf{v}^{p-{p^{\\#}}}\sigma_{H}(\mathsf{t}/\mathsf{v})\big{)}\,\int_{\partial\Omega}u^{{p^{\\#}}-1}\varphi+\big{(}\lambda-\mathsf{v}^{p-{p^{\star}}}\lambda_{H}(\mathsf{t}/\mathsf{v})\big{)}\,\int_{\Omega}u^{{p^{\star}}-1}\varphi\,.$

Testing with $\varphi\in C^{\infty}_{c}(\Omega)$ we find
$\lambda=\mathsf{v}^{p-{p^{\star}}}\lambda_{H}(\mathsf{t}/\mathsf{v})$, and
testing with $\varphi=1$ on $\overline{\Omega}$ then gives
$\sigma=\mathsf{v}^{p-{p^{\\#}}}\sigma_{H}(\mathsf{t}/\mathsf{v})$. We finally
prove (4.4), i.e. $\mathsf{t}/\mathsf{v}\not\in[T_{0},T_{E}]$. Indeed,
combining the balance condition (2.16) with (4.3) we find

$\mathsf{v}^{p-{p^{\star}}}\,\lambda_{H}(\mathsf{t}/\mathsf{v})\,\int_{\Omega}u^{{p^{\star}}-1}+\mathsf{v}^{p-{p^{\\#}}}\,\sigma_{H}(\mathsf{t}/\mathsf{v})\,\int_{\partial\Omega}u^{{p^{\\#}}-1}=0\,,$
(4.12)

where by (3.11) and (3.12) we have $\lambda_{H}>0$ in $[T_{0},T_{E})$ and
$\sigma_{H}>0$ on $(T_{0},T_{E}]$ (with
$\lambda_{H}(T_{E})=\sigma_{H}(T_{0})=0$ by continuity). If
$\mathsf{t}/\mathsf{v}\in[T_{0},T_{E})$, then $\eqref{exclusion proof}$
implies $\int_{\Omega}u^{p^{\star}}=0$, a contradiction; if
$\mathsf{t}/\mathsf{v}=T_{E}$, then (4.12) gives $u=0$ on $\partial\Omega$, so
that, by (4.1), $u$ is a minimizer of $\Phi_{\Omega}(0)$, and thus $u$ is
optimal in the Sobolev inequality on $\mathbb{R}^{n}$, so that
$\Omega=\mathbb{R}^{n}$, contradicting the fact that $\Omega$ is bounded. ∎

We are now ready to prove Theorem 1.1.

###### Proof of Theorem 1.1.

Statement (i) is an immediate consequence of Theorem 4.1. We thus focus on
statement (ii), and assume that $\Omega$ is of class $C^{2}$ and that $n>2p$.
We want to prove that if $(u,\mathsf{v},\mathsf{t})$ is a minimizer of
$\Phi_{\Omega}^{*}(T)$, then $\mathsf{v}=\mathsf{t}=0$. We assume by way of
contradiction that either $\mathsf{v}>0$ or $\mathsf{t}>0$; recalling the
definition of $\mathcal{Y}_{\Omega}(T)$, this implies that $\mathsf{v}>0$ and
$\mathsf{t}>0$. We apply Lemma 3.4 (ii) with the choice
$(v,T)=(u/\mathsf{v},\tau)$ at a point $x_{0}\in\partial\Omega$ of positive
mean curvature, noting that if $\mathsf{v}=1$ then $v\equiv 0$ and that $v$ is
Lipschitz continuous if $\mathsf{v}\in(0,1)$ thanks to (4.1) and Lemma 2.5.
Then, for every $U\in\mathcal{U}_{\tau}$, we have

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\int_{\Omega}|\nabla
v_{\varepsilon}|^{p}\leq\int_{H}|\nabla U|^{p}+\int_{\Omega}|\nabla
v|^{p}-{\rm H}_{\partial\Omega}(0)\,\mathcal{L}(U)\,\varepsilon+{\rm
o}(\varepsilon)\,,$ (4.13)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\int_{\Omega}v_{\varepsilon}^{p^{\star}}=\int_{H}U^{p^{\star}}+\int_{\Omega}v^{p^{\star}}\\!-{\rm
H}_{\partial\Omega}(0)\,\mathcal{M}(U)\,\varepsilon+{\rm o}(\varepsilon)\,,$
(4.14)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\int_{\partial\Omega}v_{\varepsilon}^{p^{\\#}}=\int_{\partial
H}U^{{p^{\\#}}}+\int_{\partial\Omega}v^{p^{\\#}}+{\rm o}(\varepsilon)\,,$
(4.15)

where $\mathcal{L}(U)$ and $\mathcal{M}(U)$ are defined in (3.46) and (1.29).
We apply this with $U\in\mathcal{U}_{\tau}$ given by

$U=U_{\tau}+b\,\varepsilon\,V_{\tau}\,,\qquad|\varepsilon|<\frac{1}{|b|}\,,\qquad
b=\frac{{\rm H}_{\partial\Omega}(0)\,\mathcal{M}(U_{\tau})}{{p^{\star}}}\,,$

The reason for the choice of $b$ will become apparent in a moment. Indeed,
thanks to (3.26), (3.27) and (3.28), we have

$\displaystyle\int_{H}|\nabla
U|^{p}=\Phi_{H}(\tau)^{p}+p\,b\,\lambda_{H}(\tau)\varepsilon+{\rm
o}\big{(}\varepsilon\big{)}\,,$
$\displaystyle\int_{H}U^{p^{\star}}=1+{p^{\star}}\,b\,\varepsilon+{\rm
o}\big{(}\varepsilon\big{)}\,,\qquad\int_{\partial
H}U^{p^{\\#}}=\tau^{p^{\\#}}\,,$

which, combined with (4.13), (4.14), (4.15) and

$\frac{\Phi_{\Omega}(T)^{p}}{\mathsf{v}^{p}}=\Phi_{H}(\tau)^{p}+\int_{\Omega}|\nabla
v|^{p},\qquad\frac{1}{\mathsf{v}^{p^{\star}}}=1+\int_{\Omega}\,v^{p^{\star}}\,,\qquad(T/\mathsf{v})^{p^{\\#}}=\tau^{p^{\\#}}+\int_{\partial\Omega}v^{p^{\\#}}\,,$

implies that $w_{\varepsilon}=\mathsf{v}\,v_{\varepsilon}$ satisfies

$\displaystyle\int_{\Omega}|\nabla
w_{\varepsilon}|^{p}\leq\Phi_{\Omega}(T)^{p}+\Big{\\{}p\,b\,\lambda_{H}(\tau)-{\rm
H}_{\partial\Omega}(0)\,\mathcal{L}(U)\Big{\\}}\,\mathsf{v}^{p}\varepsilon+{\rm
o}(\varepsilon)\,,$ (4.16)
$\displaystyle\int_{\Omega}w_{\varepsilon}^{p^{\star}}=1\\!+\big{\\{}{p^{\star}}\,b-{\rm
H}_{\partial\Omega}(0)\,\mathcal{M}(U)\big{\\}}\,\mathsf{v}^{{p^{\star}}}\varepsilon+{\rm
o}(\varepsilon)=1+{\rm o}(\varepsilon)$ (4.17)
$\displaystyle\int_{\partial\Omega}w_{\varepsilon}^{p^{\\#}}=T^{p^{\\#}}+{\rm
o}(\varepsilon)\,,$ (4.18)

where in (4.16) we have used the choice of $b$ to deduce

$\displaystyle\mathcal{M}(U)=\mathcal{M}(U_{\tau})+{p^{\star}}\int_{H}x_{n}\,U_{\tau}^{{p^{\star}}-1}\,V_{\tau}\,b\,\varepsilon+{\rm
o}(\varepsilon)\,,\quad\big{\\{}{p^{\star}}\,b-{\rm
H}_{\partial\Omega}(0)\,\mathcal{M}(U)\big{\\}}\,\varepsilon={\rm
o}(\varepsilon)\,.$

In the same spirit, by

$\lim_{\varepsilon\to
0^{+}}|\mathcal{L}(U_{\tau}+b\,\varepsilon\,V_{\tau})-\mathcal{L}(U_{\tau})|=0\,,$

we deduce from (4.16) that

$\begin{split}\int_{\Omega}|\nabla
w_{\varepsilon}|^{p}&\leq\Phi_{\Omega}(T)^{p}-\Big{\\{}\mathcal{L}(U_{\tau})-\frac{(n-p)}{n}\,\mathcal{M}(U_{\tau})\,\lambda_{H}(\tau)\Big{\\}}\,{\rm
H}_{\partial\Omega}(0)\,\mathsf{v}^{p}\varepsilon+{\rm o}(\varepsilon)\\\
&\leq\Phi_{\Omega}(T)^{p}-C(n,p,\tau)\,{\rm
H}_{\partial\Omega}(0)\,\mathsf{v}^{p}\varepsilon+{\rm
o}(\varepsilon)\,,\end{split}$ (4.19)

where in the second line we apply Lemma 3.1. We thus conclude that

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\int_{\Omega}|\nabla
w_{\varepsilon}|^{p}\leq\int_{\Omega}|\nabla
u|^{p}+\mathsf{v}^{p}\,\Phi_{H}(\mathsf{t}/\mathsf{v})^{p}-M\,\mathsf{v}^{p}\,\varepsilon+{\rm
o}(\varepsilon)\,.$ (4.20)

It remains to modify the functions $w_{\varepsilon}$ to obtain
$w_{\varepsilon}^{*}\in\mathcal{X}_{\Omega}(T)$ also satisfying (4.20),
allowing us to conclude the proof of the theorem by choosing $\varepsilon$
sufficiently small. We will distinguish between two cases, applying Lemma 2.6
in different ways in the two cases.

Case one: Suppose first that $\mathsf{v}<1$ and thus
$\int_{\Omega}u^{{p^{\star}}}>0$. This also implies that
$\int_{\partial\Omega}u^{p^{\\#}}>0$ by Theorem 4.1-(iii). Taking into account
(4.17) and (4.18), we can thus apply Lemma 2.6 in an analogous way to step one
of the proof of Theorem 4.1 in order to slightly modify $w_{\varepsilon}$ into
$w_{\varepsilon}^{*}\in\mathcal{X}_{\Omega}(T)$ with

$\displaystyle\Phi_{\Omega}(T)^{p}\leq\int_{\Omega}|\nabla
w_{\varepsilon}^{*}|^{p}=\int_{\Omega}|\nabla
w_{\varepsilon}|^{p}+\,(\mathsf{v}^{p^{\\#}}+\mathsf{v}^{p^{\star}})\,{\rm
o}(\varepsilon)$ $\displaystyle\leq\int_{\Omega}|\nabla
u|^{p}+\mathsf{v}^{p}\,\Phi_{H}(\mathsf{t}/\mathsf{v})^{p}-\frac{M}{2}\,\mathsf{v}^{p}\,\varepsilon=\Phi_{\Omega}(T)^{p}-\frac{M}{2}\,\mathsf{v}^{p}\,\varepsilon<\Phi_{\Omega}(T)^{p}\,,$

thus reaching a contradiction.

Case two: Next, suppose that $\mathsf{v}=1$. So,
$\Phi_{\Omega}(T)=\Phi_{H}(T)$, $u=v\equiv 0$ and
$v_{\varepsilon}=\varphi_{\varepsilon}\,(U^{(\varepsilon)}\circ g).$ In this
case, we will pull the relevant quantities back to the half space $H$ and
apply Lemma 2.6 there to correct the volume and trace constraints. More
specifically, for $\varepsilon<\varepsilon_{0}$, the support of
$\varphi_{\varepsilon}$ is entirely contained in the domain of the
diffeomorphism $f_{x_{0}}$, and so we can define
$\Psi_{\varepsilon}:H\to\mathbb{R}$ by
$\Psi_{\varepsilon}(y)=\varphi_{\varepsilon}\circ f(\varepsilon\,y)$. In this
way, we can rewrite

$w_{\varepsilon}=v_{\varepsilon}=(\Psi_{\varepsilon}\,U)^{(\varepsilon)}\circ
g.$

Thanks to (2.13), $\Psi_{\varepsilon}$ is identically equal to one in
$B_{\varepsilon^{\beta-1}/C}\cap H$ and vanishes outside of
$B_{C\,\varepsilon^{\beta-1}}\cap H$. Using the area formula, we rewrite
(4.17), (4.18), and (4.20) as

$\displaystyle\int_{H}(\Psi_{\varepsilon}\,U)^{{p^{\star}}}\,m_{\varepsilon}$
$\displaystyle=\int_{\Omega}w_{\varepsilon}^{{p^{\star}}}dx=1+o(\varepsilon)\,,$
$\displaystyle\int_{\partial
H}(\Psi_{\varepsilon}\,U)^{p^{\sharp}}\,\hat{m}_{\varepsilon}$
$\displaystyle=\int_{\partial\Omega}w_{\varepsilon}^{p^{\sharp}}=T^{p^{\sharp}}+o(\varepsilon).$
$\displaystyle\int_{H}|A_{\varepsilon}\,[{\nabla}(\Psi_{\varepsilon}\,U)]|^{p}\,m_{\varepsilon}=\int_{\Omega}|{\nabla}w_{\varepsilon}|^{p}$
$\displaystyle\leq\Phi_{H}(T)-M\varepsilon+o(\varepsilon),$

where we have set

$m_{\varepsilon}(x)=Jf(\varepsilon\,x)\,,\qquad\hat{m}_{\varepsilon}(x)=J^{\partial
H}f(\varepsilon\,x)\,,\qquad A_{\varepsilon}(x)=({\nabla}g\circ
f(\varepsilon\,x))^{*}\,.$

We now repeat the argument used in the proof of Lemma 2.6: exploiting the fact
that

$\Psi_{\varepsilon}\,U=U_{T}\qquad\mbox{on $(H\cap B_{R})\setminus({\rm
spt}V_{T})$}\,,\qquad R=\frac{\varepsilon_{0}^{\beta-1}}{C}\,,$ (4.21)

as well as that both
$\int_{H}(\Psi_{\varepsilon}\,U)^{{p^{\star}}}\,m_{\varepsilon}$ and
$\int_{\partial H}(\Psi_{\varepsilon}\,U)^{p^{\sharp}}\,\hat{m}_{\varepsilon}$
are positive and finite, we can easily find $\psi\in
C^{\infty}_{c}((H\cap{B}_{R})\setminus({\rm spt}V_{T}))$ and $\varphi\in
C^{\infty}_{c}((\mathbb{R}^{n}\cap{B}_{R})\setminus({\rm spt}V_{T}))$ such
that

$\displaystyle\int_{H}(\Psi_{\varepsilon}\,U_{T})^{{p^{\star}}-1}\,m_{\varepsilon}\,\psi=\int_{\partial
H}(\Psi_{\varepsilon}\,U_{T})^{{p^{\\#}}-1}\,\hat{m}_{\varepsilon}\varphi=1\,,$
(4.22)
$\displaystyle\int_{H}(\Psi_{\varepsilon}\,U_{T})^{{p^{\star}}-1}\,m_{\varepsilon}\,\varphi=\int_{\partial
H}(\Psi_{\varepsilon}\,U_{T})^{{p^{\\#}}-1}\,\hat{m}_{\varepsilon}\,\psi=0\,.$

Correspondingly, we consider the maps
$h_{\varepsilon}:\mathbb{R}^{2}\to\mathbb{R}^{2}$ by

$h_{\varepsilon}(s,t)\\!=\\!\Big{(}\int_{\partial
H}\big{(}|v_{\varepsilon}+s\,\varphi+t\,\psi|^{p^{\\#}}-|v_{\varepsilon}|^{p^{\\#}}\big{)}\,\hat{m}_{\varepsilon},\
\int_{H}\big{(}|v_{\varepsilon}+s\,\varphi+t\,\psi|^{p^{\star}}-|v_{\varepsilon}|^{p^{\star}}\big{)}\,m_{\varepsilon}\Big{)}\,.$

By (4.21), $h_{\varepsilon}\in C^{1,\alpha}(\mathbb{R}^{2};\mathbb{R}^{2})$
for some $\alpha=\alpha(n,p)\in(0,1)$, with

$\sup_{\varepsilon<\varepsilon_{0}}\|h_{\varepsilon}\|_{C^{1,\alpha}(\mathbb{R}^{2};\mathbb{R}^{2})}<\infty\,;$

moreover, $h_{\varepsilon}(0,0)=0$ and, by (4.22),

$\nabla h_{\varepsilon}(0,0)=\Big{(}\begin{array}[]{c c}{p^{\\#}}&0\\\
0&{p^{\star}}\end{array}\Big{)}\,.$

We can thus apply the inverse function theorem uniformly in $\varepsilon$, to
obtain functions $W_{\varepsilon}^{*}:H\to\mathbb{R}$ with support in
$B_{C\varepsilon^{\beta-1}}$ such that

$\begin{split}\int_{H}(W_{\varepsilon}^{*})^{{p^{\star}}}m_{\varepsilon}=1\,,&\qquad\int_{\partial
H}(W_{\varepsilon}^{*})^{p^{\sharp}}\hat{m}_{\varepsilon}=T^{p^{\sharp}}\,,\\\
\bigg{|}\int_{H}|A_{\varepsilon}[{\nabla}(\Psi_{\varepsilon}\,U)]|^{p}m_{\varepsilon}\
&-\
\int_{H}|A_{\varepsilon}[{\nabla}W_{\varepsilon}^{*}]|^{p}m_{\varepsilon}\bigg{|}=o(\varepsilon).\end{split}$
(4.23)

Finally, for $\varepsilon<\varepsilon_{0}$, define
$w_{\varepsilon}^{*}:\Omega\to\mathbb{R}$ by
$w_{\varepsilon}^{*}=(W_{\varepsilon}^{*})^{(\varepsilon)}\circ g$. Changing
variables once again, (4.23) tells us that
$w_{\varepsilon}\in\mathcal{X}_{\Omega}(T)$ and that

$\displaystyle\Phi_{\Omega}(T)^{p}$
$\displaystyle\leq\int_{\Omega}|{\nabla}w_{\varepsilon}^{*}|^{p}=\int_{\Omega}|{\nabla}w_{\varepsilon}|^{p}+o(\varepsilon)$
$\displaystyle\leq\Phi_{H}(T)^{p}-M\varepsilon+{\rm
o}(\varepsilon)\leq\Phi_{H}(T)^{p}-\frac{M\varepsilon}{2}<\Phi_{H}(T)=\Phi_{\Omega}(T),$

giving us a contradiction in this case as well. This completes the proof of
the theorem. ∎

##  5\. Rigidity theorems for best Sobolev inequalities

In this section, we prove Theorem 1.3.

###### Proof of Theorem 1.3.

Rigidity under assumption (ii) is immediate by combining Theorem 1.1-(ii) with
(1.23) and (1.24). Let us now consider assumption (i), namely, there is
$T_{*}>0$ such that

$\Phi_{\Omega}(T)=\Phi_{B}(T)\qquad\forall T\in(0,T_{*})\,.$ (5.1)

Without loss of generality we can assume that $T_{*}<{\rm
ISO}(B)^{1/{p^{\\#}}}$. We argue by contradiction and assume that $\Omega$ is
not a ball.

By Theorem 4.1, for every $T>0$ there is
$(u_{T},\mathsf{v}_{T},\mathsf{t}_{T})$ a minimizer of $\Phi_{\Omega}^{*}(T)$.
The basic idea of the proof will be to show that the trace-to-volume ratio of
$u_{T}$ must be, on one hand, uniformly positive and, on the other hand,
tending to zero as $T\to 0$, giving us a contradiction. Since $\Omega$ is
connected and is not a ball by assumption, the rigidity criterion (1.23)
together with (1.24) and (5.1) tell us that a classical minimizer for
$\Phi_{\Omega}(T)$ cannot exist for $T\in(0,T_{*})$, and so we immediately
deduce that $\mathsf{v}_{T}<1$ for all such $T$. In other words, if we set

$\nu_{T}=\big{(}1-\mathsf{v}_{T}^{p^{\star}})^{1/{p^{\star}}}=\|u_{T}\|_{L^{p^{\star}}(\Omega)}\,,\qquad\tau_{T}=(T^{p^{\\#}}-\mathsf{t}^{p^{\\#}})^{1/{p^{\\#}}}=\|u\|_{L^{p^{\\#}}(\partial\Omega)}\,,$

then $\nu_{T}>0$ for all $T\in(0,T_{*})$. So, Theorem 4.1-(iii) implies that

$\mbox{$u_{T}/\nu_{T}$ is a minimizer of
$\Phi_{\Omega}(\tau_{T}/\nu_{T})$}\,.$ (5.2)

In particular, this means that

$\frac{\tau_{T}}{\nu_{T}}\geq T_{*}\qquad\text{for all }T\in(0,T_{*})\,,$
(5.3)

since as we noted above, no minimizer of $\Phi_{\Omega}(\tilde{T})$ can exist
for $\tilde{T}=\tau_{T}/\nu_{T}<T_{*}$. Since $T\geq\tau_{T}$, (5.3) tells us
that $T/\nu_{T}\geq T_{*}$; rearranging this inequality gives us the following
lower bound on $\mathsf{v}_{T}$:

$\displaystyle\mathsf{v}_{T}^{p^{\star}}\geq
1-(T/T_{*})^{p^{\star}}\,\qquad\forall T\in(0,T_{*})\,.$ (5.4)

From this, an upper bound on the ratio $\mathsf{t}_{T}/\mathsf{v}_{T}$ follows
immediately:

$\displaystyle\frac{\mathsf{t}_{T}}{\mathsf{v}_{T}}\leq\big{(}1-(T/T_{*})^{p^{\star}}\big{)}^{-1/{p^{\star}}}\,T\,\qquad\forall
T\in(0,T_{*})\,.$ (5.5)

In particular $\mathsf{t}_{T}/\mathsf{v}_{T}\to 0$ as $T\to 0^{+}$.

On the other hand, we will now use the Euler-Lagrange equation for $u_{T}$ to
show that

$\lim_{T\to 0^{+}}\frac{\tau_{T}}{\nu_{T}}=0\,,$ (5.6)

which is a clear contradiction to (5.3). Indeed, by Theorem 4.1-(iii) and
(5.2), $u_{T}$ satisfies the Euler-Lagrange equation

$\left\\{\begin{split}&-\Delta_{p}u_{T}=\mathsf{v}_{T}^{p-{p^{\star}}}\,\lambda_{H}(\mathsf{t}_{T}/\mathsf{v}_{T})\,u_{T}^{{p^{\star}}-1}\,\quad\hskip
28.45274pt\mbox{on $\Omega$}\,,\\\ &|\nabla u_{T}|^{p-2}\,\frac{\partial
u}{\partial\nu_{\Omega}}=\mathsf{v}_{T}^{p-{p^{\\#}}}\,\sigma_{H}(\mathsf{t}_{T}/\mathsf{v}_{T})\,u_{T}^{{p^{\\#}}-1}\,\quad\mbox{on
$\partial\Omega$}\,;\end{split}\right.$ (5.7)

see (4.2), (4.3) and (3.7) (for the definition of$\lambda_{H}(T)$ and
$\sigma_{H}(T)$). Testing (5.7) with $u_{T}$, we find that

$\displaystyle\int_{\Omega}|\nabla u_{T}|^{p}$ $\displaystyle=$
$\displaystyle\mathsf{v}_{T}^{p-{p^{\star}}}\,\lambda_{H}\Big{(}\frac{\mathsf{t}_{T}}{\mathsf{v}_{T}}\Big{)}\,\int_{\Omega}u_{T}^{p^{\star}}+\mathsf{v}_{T}^{p-{p^{\\#}}}\,\sigma_{H}\Big{(}\frac{\mathsf{t}_{T}}{\mathsf{v}_{T}}\Big{)}\,\int_{\partial\Omega}u_{T}^{p^{\\#}}$
$\displaystyle=$
$\displaystyle\mathsf{v}_{T}^{p-{p^{\star}}}\,\lambda_{H}\Big{(}\frac{\mathsf{t}_{T}}{\mathsf{v}_{T}}\Big{)}\,\nu_{T}^{p^{\star}}+\mathsf{v}_{T}^{p-{p^{\\#}}}\,\sigma_{H}\Big{(}\frac{\mathsf{t}_{T}}{\mathsf{v}_{T}}\Big{)}\,\tau_{T}^{p^{\\#}}\,.$

After rearranging and multiplying through by
$\mathsf{v}_{T}^{p^{\sharp}-p}\nu_{T}^{-p^{\sharp}}>0$, we arrive at the
inequality

$-\sigma_{H}\Big{(}\frac{\mathsf{t}_{T}}{\mathsf{v}_{T}}\Big{)}\,\Big{(}\frac{\tau_{T}}{\nu_{T}}\Big{)}^{p^{\\#}}\leq\lambda_{H}\Big{(}\frac{\mathsf{t}_{T}}{\mathsf{v}_{T}}\Big{)}\,\Big{(}\frac{\nu_{T}}{\mathsf{v}_{T}}\Big{)}^{{p^{\star}}-{p^{\\#}}}\,.$
(5.8)

By (3.12) and the continuity of $T\mapsto\lambda_{H}(T)$, there are $C>0$ and
$T_{**}>0$ such that $\lambda_{H}(T)\in(1/C,C)$ for every $T\in(0,T_{**})$.
Moreover, thanks to (5.4), we can ask that $\mathsf{v}_{T}\geq 1/C$ for
$T\in(0,T_{**})$. In particular, by (5.5) and up to further increasing $C$, if
$T<1/C$, then $\mathsf{t}_{T}/\mathsf{v}_{T}<T_{**}$ and thus (5.8),
$\nu_{T}\leq 1$ and $\mathsf{v}_{T}\geq 1/C$ give

$-\sigma_{H}\Big{(}\frac{\mathsf{t}_{T}}{\mathsf{v}_{T}}\Big{)}\,\Big{(}\frac{\tau_{T}}{\nu_{T}}\Big{)}^{p^{\\#}}\leq
C\,.$ (5.9)

By (3.11) and the fact that $\mathsf{t}_{T}/\mathsf{v}_{T}\to 0$ as $T\to 0$,
we see that $\sigma_{H}(T)\to-\infty$ as $T\to 0^{+}$, so that (5.9) implies
(5.6). We reach a contradiction to (5.3), completing the proof. ∎

##  Appendix A Proof of Lemma 2.1

We will use the Brezis–Lieb lemma (if $(X,\mu)$ is a measure space, $q\geq 1$,
and $\\{f_{j}\\}_{j}$ is bounded in $L^{q}(X)$, then

$\int_{X}|f|^{q}\,d\mu=\lim_{j\to\infty}\int_{X}|f_{j}|^{q}\,d\mu-\int_{X}|f_{j}-f|^{q}\,d\mu\,,$
(A.1)

provided $f$ is a $\mu$-a.e. limit of $\\{f_{j}\\}_{j}$ on $X$), and the two
Sobolev-type inequalities

$\displaystyle\|u\|_{L^{p^{\star}}(\Omega)}\leq C_{1}\,\Big{(}\|\nabla
u\|_{L^{p}(\Omega)}+\|u\|_{L^{p}(\Omega)}\Big{)}\,,$ (A.2)
$\displaystyle\|u\|_{L^{p^{\\#}}(\partial\Omega)}\leq C_{2}\,\Big{(}\|\nabla
u\|_{L^{p}(\Omega)}+\|u\|_{L^{p}(\Omega)}\Big{)}\,,$ (A.3)

which are valid, with constants $C_{1}$ and $C_{2}$ depending on $n$, $p$ and
$\Omega$ only, as soon as $\Omega$ is bounded and has Lipschitz boundary.

###### Proof of Lemma 2.1.

Step one: Since $\Omega$ is a bounded open set with Lipschitz boundary,
$u_{j}\rightharpoonup u$ as distributions in $\Omega$, and $\\{\nabla
u_{j}\\}_{j}$ is bounded in $L^{p}(\Omega)$, standard considerations show that
$u\in W^{1,p}(\Omega)$, $\\{u_{j}\\}_{j}$ is bounded in
$L^{p^{\star}}(\Omega)$ and in $L^{p^{\\#}}(\partial\Omega)$, and, up to
extracting subsequences, $u_{j}\to u$ in $L^{q}(\Omega)$ for every
$q\in[1,{p^{\star}})$, $u_{j}\to u$ in $L^{r}(\partial\Omega)$ for every
$r<{p^{\\#}}$, $u_{j}\to u$ pointwise $\mathcal{L}^{n}$-a.e. on $\Omega$ and
$\mathcal{H}^{n-1}$-a.e. on $\partial\Omega$, and that the sequences of Radon
measures $\\{\nu_{j}\\}_{j}$, $\\{\tau_{j}\\}_{j}$ and $\\{\mu_{j}\\}_{j}$
defined in (2.1) admits weak-$*$ limits $\nu$, $\tau$ and $\mu$, with ${\rm
spt}\,\nu$ and ${\rm spt}\,\mu$ contained in $\overline{\Omega}$, and ${\rm
spt}\tau$ contained in $\partial\Omega$.

Step two: We let $\tilde{\mu}$ denote the weak-$*$ limit of
$|\nabla(u_{j}-u)|^{p}\,\mathcal{L}^{n}\llcorner\Omega$ (which exists up to
possibly extracting a further subsequence), and claim that
$\tilde{\mu}(\\{x\\})=\mu(\\{x\\})$ for every $x\in\mathbb{R}^{n}$. Indeed, by
the elementary inequality

$\big{|}|v+w|^{p}-|w|^{p}\big{|}\leq\varepsilon\,|v|^{p}+C(p,\varepsilon)\,|w|^{p}\,,\qquad
v,w\in\mathbb{R}^{n}\,,\varepsilon>0\,,$

given $x\in\mathbb{R}^{n}$ and $r>0$ we see that

$\displaystyle\Big{|}\int_{B_{r}(x)}|\nabla
u_{j}|^{p}-\int_{B_{r}(x)}|\nabla(u_{j}-u)|^{p}\Big{|}\leq\varepsilon\,\int_{B_{r}(x)}|\nabla(u_{j}-u)|^{p}+C(p,\varepsilon)\,\int_{B_{r}(x)}|\nabla
u|^{p}\,,$

so that letting first $j\to\infty$ (for $r>0$ such that $\mu(\partial
B_{r}(x))=\tilde{\mu}(\partial B_{r}(x))=0$) and then $r\to 0^{+}$ (along a
generic sequence of radii), we find indeed
$|\mu(\\{x\\})-\tilde{\mu}(\\{x\\})|\leq\varepsilon\,\tilde{\mu}(\\{x\\})$ for
every $\varepsilon>0$.

Step three: If now pick $\eta\in C^{\infty}_{c}(\mathbb{R}^{n})$ and set
$\tilde{\nu}=\nu-|u|^{p^{\star}}\,\mathcal{L}^{n}\llcorner\Omega$ (notice
that, by lower semicontinuity, $\tilde{\nu}$ is a Radon measure), then, by
exploiting, in order,
$\nu_{j}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\nu$, (A.1), (A.2), and
$u_{j}\to u$ in $L^{p}(\Omega)$, we find

$\displaystyle\int_{\mathbb{R}^{n}}\,|\eta|^{p^{\star}}\,d\tilde{\nu}\leq\liminf_{j\to\infty}\int_{\Omega}\,|\eta|^{p^{\star}}\,(|u_{j}|^{p^{\star}}-|u|^{p^{\star}})\,=\lim_{j\to\infty}\int_{\Omega}\,|\eta\,(u_{j}-u)|^{p^{\star}}$
$\displaystyle\leq
C_{1}^{p^{\star}}\,\lim_{j\to\infty}\Big{(}\|\nabla(\eta(u_{j}-u))\|_{L^{p}(\Omega)}+\|\eta(u_{j}-u)\|_{L^{p}(\Omega)}\Big{)}^{{p^{\star}}}$
$\displaystyle=C_{1}^{p^{\star}}\,\lim_{j\to\infty}\,\|\nabla(\eta(u_{j}-u))\|_{L^{p}(\Omega)}^{p^{\star}}=C_{1}^{p^{\star}}\,\lim_{j\to\infty}\,\|\eta\,\nabla(u_{j}-u))\|_{L^{p}(\Omega)}^{p^{\star}}$
$\displaystyle\leq
C_{1}^{p^{\star}}\,\Big{(}\int_{\mathbb{R}^{n}}|\eta|^{p}\,d\tilde{\mu}\Big{)}^{{p^{\star}}/p}\,.$

In particular, for every $x\in\mathbb{R}^{n}$ one has

$\tilde{\nu}(B_{r}(x))\leq
C_{1}^{p^{\star}}\,\tilde{\mu}(B_{r}(x))^{{p^{\star}}/p}\qquad\mbox{for a.e.
$r>0$}\,.$ (A.4)

By (A.4), $\tilde{\nu}$ is absolutely continuous with respect to
$\tilde{\mu}$, so that $\tilde{\nu}=f\,\tilde{\mu}$ where $f$ is such that,
for $\tilde{\mu}$-a.e. $x\in\mathbb{R}^{n}$,

$f(x)=\lim_{r\to 0^{+}}\frac{\tilde{\nu}(B_{r}(x))}{\tilde{\mu}(B_{r}(x))}\,;$

in particular, again by (A.4), for $\tilde{\mu}$-a.e. $x\in\mathbb{R}^{n}$ we
have

$f(x)\leq C_{1}^{p^{\star}}\,\lim_{r\to
0^{+}}\tilde{\mu}(B_{r}(x))^{({p^{\star}}/p)-1}=C_{1}^{p^{\star}}\,\mu(\\{x\\})^{({p^{\star}}/p)-1}\,.$

In particular, as ${p^{\star}}>p$, if
$X=\\{x\in\mathbb{R}^{n}:\mu(\\{x\\})>0\\}=\\{x_{i}\\}_{i\in
I}\subset\overline{\Omega}$ ($I$ at most countable) denotes the set of atoms
of $\mu$, then $f(x)=0$ $\mu$-a.e. on $\mathbb{R}^{n}\setminus X$, and we have
proved that

${\rm spt}\tilde{\nu}\subset\\{x_{i}\\}_{i\in I}\,.$ (A.5)

An entirely analogous argument, this time based on (A.3) rather than on (A.2),
shows that, if
$\tilde{\tau}=\tau-|u|^{p^{\\#}}\,\mathcal{H}^{n-1}\llcorner\partial\Omega$,
then

${\rm spt}\tilde{\tau}\subset\\{x_{i}\\}_{i\in I}\cap\partial\Omega\,.$ (A.6)

We have thus proved the validity of (2.2), (2.3) and (2.4) for suitable
$\mathsf{v}_{i},\mathsf{t}_{i}\geq 0$ and $\mathsf{g}_{i}>0$: and of course we
can discard possible points $x_{i}$ with $\mathsf{v}_{i}=0$ from these
decompositions, and directly assume that $\mathsf{v}_{i}>0$ for every $i$. The
fact that $\mathsf{g}_{i}\geq S\,\mathsf{v}_{i}$ if $x_{i}\in\Omega$ is
immediate by repeating the above argument with arbitrary $\eta\in
C^{\infty}_{c}(\Omega)$ (in which case $C_{1}$ can be replaced by $1/S$). An
analogous argument, this time using Lemma 2.2, shows that
$\mathsf{g}_{i}\geq\mathsf{v}_{i}\,\Phi_{H}(\mathsf{t}_{i}/\mathsf{v}_{i})$ if
$x_{i}\in\partial\Omega$. ∎

##  Appendix B Proof of Lemmas 2.2 and 2.3

This section is dedicated to the proof of Lemmas 2.2 and 2.3. We recall the
standard Taylor expansions for the inverse and determinant of a matrix that is
a perturbation of the identity:

$\displaystyle(\text{Id}_{\mathbb{R}^{n}}+tA)^{-1}$
$\displaystyle=\text{Id}_{\mathbb{R}^{n}}-tA+O(t^{2})\,,$ (B.1)
$\displaystyle\det(\text{Id}_{\mathbb{R}^{n}}+tA)$
$\displaystyle=1+t\,\text{trace}A+O(t^{2});$ (B.2)

see for instance [Mag12, Lemma 17.4].

###### Proof of Lemma 2.2.

Note that $\hat{f}$ can equivalently be written as $\hat{f}(x)=x+\ell({\bf
p}(x))e_{n}$. We directly see that $\hat{f}$ maps $C^{1}$-diffeomorphically
onto its image with inverse $\hat{g}(y)=y-\ell({\bf p}(y))x_{n}$ and (2.10)
holds because $\ell(0)=|{\nabla}\ell(0)|=0$ and $\ell$ is $C^{1}$. We compute,
in the standard basis for $\mathbb{R}^{n}$, that

${\nabla}\hat{f}({\bf p}(x),x_{n})=\left(\begin{array}[]{ccc}Id&0\\\
{\nabla}\ell({\bf
p}(x))&1\end{array}\right)=Id+\left(\begin{array}[]{ccc}0&0\\\
{\nabla}\ell({\bf p}(x))&0\end{array}\right)\,.$

Since $\ell$ is $C^{1}$ with ${\nabla}\ell(0)=0$, this gives the first
estimate in (2.11). The second estimate in (2.11) follows in the same way
using the explicit form of $g$ above. The first estimate in (2.12) follows
from (B.2) and the expression for ${\nabla}\hat{f}$ above. The second estimate
in (2.12) follows because $\hat{f}=F$ on ${\bf D}_{r_{0}}$ (compare with
(2.7)), and so

$J^{\partial H}\hat{f}=J^{\partial H}F=\sqrt{1+|{\nabla}\ell|^{2}}\,.$

This completes the proof of the lemma. ∎

###### Proof of Lemma 2.3.

Step one: We compute some geometric quantities for $\partial\Omega$ using the
graphical coordinates defined by the map $F$ given in (2.7). We start with
first order quantities. For $x\in{\bf D}_{r_{0}}$ and $i=1,\dots,n-1$, we set
$\tau_{i}:=dF_{x}(e_{i})$, i.e.,

$\tau_{i}=\partial_{i}F(x)=e_{i}+\partial_{i}\ell(x)e_{n},$ (B.3)

so that $\\{\tau_{1},\dots,\tau_{n-1}\\}$ forms a basis for
$T_{F(x)}\partial\Omega$. Since $\ell$ is $C^{2}$ and ${\nabla}\ell(x)=0$, we
have

$g_{ij}=\langle\tau_{i},\tau_{j}\rangle_{\mathbb{R}^{n}}=\delta_{ij}+\partial_{i}\ell\,\partial_{j}\ell=\delta_{ij}+{\rm
O}(|x|^{2})\,.$ (B.4)

In other words, the metric coefficients in graphical coordinates are Euclidean
up to second order in $|x|$. The volume measure of $\partial\Omega$ is given
by

$J^{\partial H}F=\sqrt{\det g_{ij}}=\sqrt{1+|{\nabla}\ell|^{2}}\,;$

this immediately gives the second estimate in (2.14) since $f=F$ for $x\in{\bf
D}_{r_{0}}$. The inverse metric coefficients are

$g^{ij}=\delta^{ij}+\frac{\delta^{ia}\,\delta^{ib}\,\partial_{a}\ell\,\partial_{b}\ell}{1+|{\nabla}\ell|^{2}}\delta^{ij}+O(|x|^{2})\,.$
(B.5)

Recall that, without loss of generality, we have chosen an orthonormal basis
for $\mathbb{R}^{n-1}\subset\mathbb{R}^{n}$ that diagonalizes the Hessian of
$\ell$ at $x=0$. Let $\\{\kappa_{1},\dots,\kappa_{n-1}\\}$ denote the
eigenvalues. The second fundamental form of $\partial\Omega$ at $x$ is defined
by $A_{x}(v,w)=-\langle d\nu_{\Omega}(v),w\rangle$ for tangent vectors $v,w\in
T_{x}(\partial\Omega)$. Using the shorthand $\nu:=\nu_{\Omega}\circ F:{\bf
D}_{r_{0}}\to S^{n-1}$, the coefficients of the second fundamental form in the
coordinates defined by $F$ are given by
$A_{ij}=\langle\partial_{ij}F,\nu\rangle_{\mathbb{R}^{n}}.$ Differentiating
(B.3) above, we have
$\partial_{ij}F=\partial_{j}\tau_{i}=\partial_{ij}\ell\,e_{n}$, and thus for
$i,j\in\\{1,\dots,n-1\\}$ we have

$A_{ij}=\langle\partial_{ij}F,\nu\rangle=\partial_{ij}\ell\,\langle
e_{n},\nu\rangle=\frac{-\partial_{ij}\ell}{\sqrt{1+|{\nabla}\ell|^{2}}}=-\partial_{ij}\ell+{\rm
O}(|x|^{2})\,.$ (B.6)

We have

$\partial_{i}\nu=-A_{i}^{j}\tau_{j}\,,$ (B.7)

and from (B.5) and (B.6) we directly compute that $A_{i}^{j}=g^{ik}A_{kj}$ is
given by

$\displaystyle
A_{i}^{j}=\frac{-\partial_{ij}\ell}{\sqrt{1+|{\nabla}\ell|^{2}}}+O(|x|^{2})$
$\displaystyle=-\partial_{ij}\ell+O(|x|^{2})$
$\displaystyle=-\partial_{ij}\ell(0)+O(|x|^{2})=-\kappa_{j}\delta_{i}^{j}+O(|x|^{2}).$

Step 2: Next, we use the previous step to compute geometric quantities
associated to the coordinates defined by $f$. For $x\in{\bf C}_{r_{0}}$ and
$i=1,\dots,n-1$, we note that $\partial_{i}{\bf p}(x)=e_{i}$ and thus from
(B.7), In what follows we will suppress the composition with ${\bf p}$ in our
notation, writing for instance $\tau_{i}$ in place of $\tau_{i}\circ{\bf
p}(x)$. For $i=1,\dots,n-1$, we have

$\displaystyle\partial_{i}f(x)$
$\displaystyle=\tau_{i}-x_{n}\partial_{i}\nu({\bf p}(x))$ (B.8)
$\displaystyle=e_{i}+\partial_{i}\ell\,e_{n}+x_{n}A_{i}^{j}\tau_{j}=e_{i}+\partial_{i}\ell\,e_{n}-x_{n}\kappa_{i}e_{i}+O(|x|^{2}),$
$\displaystyle\text{ and }\qquad\partial_{n}f(x)$
$\displaystyle=-\nu=\frac{e_{n}-{\nabla}\ell}{\sqrt{1+|{\nabla}\ell|^{2}}}=e_{n}-{\nabla}\ell+O(|x|^{2}).$
(B.9)

Together (B.8) and (B.9) can be expressed in consolidated form as

$\displaystyle{\nabla}f=\sum_{i=1}^{n}\partial_{i}f\otimes e_{i}$
$\displaystyle=\sum_{i=1}^{n}e_{i}\otimes
e_{i}+e_{n}\otimes{\nabla}\ell-{\nabla}\ell\otimes
e_{n}-x_{n}\sum_{i,j=1}^{n-1}\kappa_{i}(0)e_{i}\otimes e_{i}+O(|x|^{2})$
$\displaystyle=\text{Id}_{\mathbb{R}^{n}}+e_{n}\otimes{\nabla}\ell-{\nabla}\ell\otimes
e_{n}-x_{n}\sum_{i,j=1}^{n-1}\kappa_{i}(0)e_{i}\otimes e_{i}+O(|x|^{2}).$

In particular, from (B.2) we see that the volume form is given by

$Jf=\sqrt{\det g_{ij}}=1-x_{n}H_{\Omega}(0)+O(|x|^{2})\,,$

giving us the first estimate in (2.14). We see directly from the definition
that $f$ is a $C^{1}$ map, and since we see from the expression for
${\nabla}f$ above that ${\nabla}f(0)=\text{Id}_{\mathbb{R}^{n}}$, we may apply
the inverse function theorem to see that, up to decreasing $r_{0}$, $f$
defines a $C^{1}$ diffeomorphism onto its image. Letting $g=f^{-1}$ and using
the expansion of the inverse (B.1), we find

$\displaystyle({\nabla}g)\circ f$
$\displaystyle=({\nabla}f)^{-1}=\text{Id}_{\mathbb{R}^{n}}-e_{n}\otimes{\nabla}\ell+{\nabla}\ell\otimes
e_{n}+x_{n}\sum_{i,j=1}^{n-1}\kappa_{i}(0)e_{i}\otimes e_{i}+O(|x|^{2}).$

Finally, (2.13) follows from these expressions for ${\nabla}f$ and
${\nabla}g$, along with the assumptions that ${\nabla}\ell(0)=0$. This
completes the proof of the lemma. ∎

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