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\begin{document}
\title[Bernoulli jets]{Bernoulli jets and \\
the zero mean curvature
equation}
\author{Enrico Valdinoci}%
\address{
Dipartimento di Matematica\\
Universit\`a di Roma Tor Vergata\\
Via della Ricerca Scientifica, 1\\
I-00133 Roma (Italy)
}
\email{valdinoci@mat.uniroma2.it}%
\thanks{
I would like to thank Arshak Petrosyan for having
pointed out to me some results from the literature, which have been
exploited here.
This paper has been presented in the occasion
of a graduate course in Torvergata: I would
like to thank the participants for their valuable feedback.
This research has been
partially supported by MIUR {\sl Variational
Methods and
Nonlinear Differential Equations}.
}
\begin{abstract}
We consider an elliptic PDE problem related with fluid
mechanics.
We show that
level sets of rescaled solutions
satisfy
the zero mean curvature equation in a suitable weak
viscosity
sense.
In particular, such level sets
cannot be touched by
below (above)
by a
convex (concave) paraboloid in a suitably
small neighborhood.
\end{abstract}
\maketitle
{\footnotesize
\bigskip
\noindent {\bf
Keywords:} Variational and PDE models for fluid dynamics,
$p$-Laplacian operator,
sliding methods, geometric and qualitative
properties of
solutions.
\medskip
\noindent {\bf 2000 MS
Classification:} 35J60, 35J70, 35R35, 76M30, 76B10.}
\bigskip
\bigskip
\tableofcontents
\bigskip
\bigskip
\newpage
\section{Introduction}
Given $p\in (1,+\infty)$, we consider here a
problem driven by
the $p$-Laplacian operator\footnote{As usual, we denote here
the $p$-Laplacian by $\Delta_p u:= \,{\rm div\,}(
|\nabla u|^{p-2} \nabla u
)$, in the distributional sense.}
and related with fluid dynamics.
Some mean curvature estimates
on the level sets of the solutions will be obtained,
following a technique recently developed in \cite{S}
for smooth phase transition models.
The setting in which these mean curvature estimates
arise is a weak, quantitative, viscosity sense.
Roughly speaking, we will consider a homogeneous $\e$ rescaling
of the solution and prove that the level sets of
such rescaled solutions cannot be touched by a curved paraboloid
in a neighborhood of order $\sqrt\e$.
More formally,
the main result dealt with in this paper is the following:
\begin{thm}\label{main2}
Let $\varpi>0$ and $u\in W^{1,p}_{\rm loc}(\R^N)\cap C(\R^N)$
be so that:\footnote{We will often denote
a point $x\in \R^N$ by
$x=(x',x_N)
\in\R^{N-1}\times\R$. Quantities depending
only on $N$, $p$ and $\varpi$ will be referred to as
(universal) constants.}
\begin{eqnarray}
\label{udizero}
&& u(0)=0\,,\\
&& \label{1.2} |u|\leq 1\,,\\
%% && \label{1.3} {\mbox{$u$ is continuous}}\\
&&\label{1.4}
{\mbox{$\Delta_p u=0$ for any $x\in \{|u|<1\}$,}}
\end{eqnarray}
Let us suppose that the following two free boundary growth
conditions hold:
\begin{itemize}
\item if there is an open ball $B_+\subseteq
\{|u|<1\}$ touching $\partial\{u<1\}$ at $x_+$, then
\begin{equation}\label{FB-piu}
u(x_+ +t_j\nu_0)\geq 1-\varpi t_j -\fgot_+(t_j)
\,,
\end{equation}
for an infinitesimal positive sequence $t_j\searrow 0^+$,
where $\nu_0$ is the interior normal of $B_+$ at $x_+$
and $\fgot_+:(0,1)\rightarrow \R$ is so that
\begin{equation}\label{FB-piu:fgot}
\lim_{t\rightarrow 0^+}\frac{\fgot_+ (t)}{t}=0\,;
\end{equation}
\item
if there is an open ball $B_-\subseteq
\{u=-1\}$ touching $\partial\{u<-1\}$ at $x_-$, then
\begin{equation}\label{FB-meno}
u(x_- -t_j\nu_0)\geq -1+\varpi t_j -\fgot_-(t_j)
\,,
\end{equation}
for an infinitesimal positive sequence $t_j\searrow 0^+$,
where $\nu_0$ is the interior normal of $B_-$ at $x_-$
and $\fgot_-:(0,1)\rightarrow \R$ is so that
\begin{equation}\label{FB-meno:fgot}
\lim_{t\rightarrow 0^+}\frac{\fgot_- (t)}{t}=0\,.
\end{equation}
\end{itemize}
Assume also that $u$ satisfies the following decay property:
there exists a universal $L>0$ such that:
\begin{equation}\label{the deacy}\begin{split}
&{\mbox{if $\{u=0\}\cap\{|x'|\leq l\}\subseteq \{x_N\geq -l/100 \}$ }} \\
&{\mbox{then $u(x)=-1$ for any $x$ so that $|x'|\leq l/2$ and
$x_N\leq -l/10$,}}\end{split}
\end{equation}
provided $l\geq L$.
Let $\dgot\in(0,1)$ and $M\in {\rm Mat}((N-1) \times(N-1))$
with
$${\rm tr}\,M>\dgot\|M\|\qquad{\mbox{and}}\qquad
\|M\|\leq \dgot^{-1}\,.
$$
Let
$$\Gamma:=\left\{
x=(x',x_N)\in
\R^{N-1}\times\R\;\;{\mbox{s.t.}}\;\;
x_N
=\frac{1}{2}x'\cdot Mx'\right\}\,.$$
Let also
$u_\e(x):= u(x/\e)$.
Then, there exist a universal $\dgot^\star>0$ and a function
$\sigma_0:(0,1)\longrightarrow(0,1)$
such that if $\e\in (0,\sigma_0(\dgot))$ and $\dgot\in
(0,\dgot^\star)$,
then $\Gamma$ cannot touch $\{u_\e=0\}$ by below
in $B_{\dgot\sqrt\e/
\sqrt{{
{\rm tr}\,M}}
}$: more explicitly,
$$ \{ u_\e=0\}\,\cap\,\left\{ x_N<
\frac{1}{2}x'\cdot Mx'
\right\}\,\cap\, \left\{ |x|< \frac{\dgot\,\sqrt\e}{\sqrt{
{\rm tr}\,M}}\right\}\;\not=\;\emptyset\,.
$$\end{thm}
Using an informal language, one may describe the content
of
Theorem~\ref{main2} here above by stating that
solutions of the problem considered enjoy a {\em weak viscosity
zero mean curvature property}, in the sense that the level set
$\{u_\e=0\}$ cannot be touched from below by a
convex\footnote{Of course, an analogous statement
holds for concave paraboloids touching from
above.} paraboloid in a
neighborhood
of the origin (which gets small with $\e$).
It is known (see \cite{Bo}) that level sets of
rescaled minimizers approach a minimal surface
in the $\Gamma$-convergence setting.
Thus,
in a way, we may think that Theorem~\ref{main2}
says that the level set
$\{u_\e=0\}$ attains a zero mean curvature property
(though in a weak, quantitative, viscosity sense) even ``before"
converging to a limit surface.
The fact
that level sets inherit further properties from
the minimal surface
limit case may be related with the flat
regularity of low dimensional level sets
first conjectured by De~Giorgi (see \cite{DG}).
Also, Theorem~\ref{main2} here holds for more general
solutions than minimizers, differently from the
$\Gamma$-convergence results in \cite{Bo}, and it
provides a geometric and quantitative
connection between the problem discussed here
and the minimal surfaces.
\bigskip
Let us now briefly discuss some physical motivation behind
the model considered here.
The problem dealt with in Theorem~\ref{main2}
is inspired by ideal fluid jets.
For instance, if $N=p=2$,
then $u$ may be seen as the stream function\footnote{In such setting,
we remark that the level sets of $u$, which we study here,
have some physical relevance, since
the particles of the fluid move along them.}
of a fluid, that is,
the particles of the fluid move along the level sets
$\{u=\theta\}$, for $\theta\in(-1,1)$. In this sense,~(\ref{1.4})
is just the continuity equation.
On the free boundary $\partial\{|u|<1\}$, Bernoulli's law
states that the speed of
the fluid (which agrees with $|\nabla u|$), must be balanced
by the exterior pressure. This is the physical meaning of~(\ref{FB-piu})
and~(\ref{FB-meno}),
in the sense that these assumptions are just weak versions
of the Bernoulli condition ``$|\nabla u|=\varpi$ on the free boundary".
\medskip
More precisely,
the solutions dealt with in Theorem~\ref{main2}
are related with the minimizers of a functional
widely studied in the literature both from the
pure and applied mathematics point of view. Namely,
for $\lambda>0$,
let us define\footnote{As a matter of fact, the quantities
$\lambda$ and $\varpi$ will be related by~(\ref{DP:rel}).}
the following
functional
on $W^{1,p}(\Omega)$:
$$ {\mathcal{F}}_\Omega (u)=\int_\Omega
|\nabla u(x)|^p+\lambda \chi_{(-1,1)}\big(u(x)\big)
\,dx\,.$$
The functional ${\mathcal{F}}$ is\footnote{Here
above and in the sequel, we use the standard
notation for the characteristic function of a set $E$, namely
$$ \chi_E (\xi)\,=\,\left\{
\begin{matrix}
1 & \;& {\mbox{if $\xi\in E$,}}\cr
0 & \;& {\mbox{if $\xi\not\in E$.}}
\end{matrix}
\right.$$
}
a model for ideal fluid
jets and cavitation problems
(see, e.g., \cite{AlC}, \cite{ACF},
\cite{ACF3} and \cite{PV1}); roughly speaking,
the
``kinetic'' part
$|\nabla u|^p$ leads to the PDE equation satisfied by the stream function
of the ideal fluid, while the free boundary imposed by the discontinuity
of the characteristic function yields the balance with the exterior
pressure, according to Bernoulli's law. We refer to
the above cited papers for further discussions upon these facts.
Also, similar functionals provide models
for flame propagation, combustion
and electro-chemical processes
(see, e.g., \cite{DaP}, \cite{DPS}
and references therein).
In this setting, we derive from Theorem~\ref{main2} the following
result:
\begin{thm}\label{main2:minima}
Let $u\in W^{1,p}_{\rm loc}(\R^N)$, $|u|\leq 1$, be a Class A
minimizer
of ${\mathcal{F}}$, i.e., assume that
$$
{\mathcal{F}}_{\Omega}(u)\,\leq\,
{\mathcal{F}}_{\Omega}(u+\phi)\,,$$
for any $\phi\in C^\infty_0(\Omega)$, for any
bounded domain $\Omega$. Suppose that
$u(0)=0$.
Let $u_\e(x):= u(x/\e)$ and
$$\Gamma:=\left\{
x=(x',x_N)\in
\R^{N-1}\times\R\;\;{\mbox{s.t.}}\;\;
x_N
=\frac{1}{2}x'\cdot Mx'\right\}\,.$$
Assume also that $\{|u|<1 \}$ possesses
the inner ball property, i.e., suppose that
\begin{equation}\label{st.inner}
\begin{split}
&{\mbox{for any $\bar x\in \partial\{|u|<1 \}$, there exists
}}\\ &{\mbox{
an
open ball
$B\subseteq\{|u|<1 \}$}}\\ &{\mbox{
which touches $\partial\{|u|<1 \}$ at
$\bar x$.
}}
\end{split}\end{equation}
Then, there exist a universal $\dgot^\star>0$ and a function
$\sigma_0:(0,1)\longrightarrow(0,1)$
such that if $\e\in (0,\sigma_0(\dgot))$ and $\dgot\in
(0,\dgot^\star)$,
then $\Gamma$ cannot touch $\{u_\e=0\}$ by below
in $B_{\dgot\sqrt\e/
\sqrt{{
{\rm tr}\,M}}
}$.
\end{thm}
Theorems~\ref{main2} and~\ref{main2:minima}
have also the following consequence:
{if $\{u_\e=0\}$ converges uniformly to
a hypersurface, then this surface satisfies
the zero mean curvature equation in
the (standard)
viscosity sense}. We omit here
the details on this, referring the interested
reader to Theorem~2.1
of~\cite{SV} (see also~\cite{PV1}
for conditions under which the uniform convergence
of level sets holds).\medskip
The proof of the results of this paper
is deeply inspired by the geometric
construction developed in the masterpiece~\cite{S} (see also~\cite{SV}
and~\cite{VSS} for related results). Also, Theorems~\ref{main2}
and~\ref{main2:minima} will be bridged together via some results
in~\cite{DaP} and~\cite{PV1}.\medskip
We organize this paper in the following way.
In \S~\ref{barriers}, we will construct
suitable barriers, in order to estimate the curvatures
of the level sets.
Barriers are like ships and solutions are like
the land where the ships are going to dock.
Inspired by the case $N=1$, one suspects that
such lands look like
flat hills of slope $\varpi$.
Thus, a first ship will be constructed with a protruding
zero level set (as a little rostrum), in such a way
the dock will occur on it.
The second ship is a modification of the flat
distance function: on the one hand, the distance function
is expected to play a r\^ole, since it encodes curvature
information; on the other hand, we need to bend
the distance function a bit
to get apart from the free boundary.
In \S~\ref{mth}, the ships will sail the sea to touch
the land: barriers will be slided towards the solution
to check the curvatures of its zero
level set: a Comparison Principle of~\cite{DAM} will
be employed for this.
The proof of Theorem~\ref{main2}
will then follow at once, by a scaling argument presented
in \S~\ref{ss2}.
The proof of Theorem~\ref{main2:minima}
is contained in \S~\ref{ss3}.
\section{Useful barriers}\label{barriers}
Following the ideas in \cite{S} and
\cite{SV}, we now construct
some barriers in order to trap our solution.
Roughly, the crucial idea, which goes back to De~Giorgi,
is that one-dimensional solutions are the ones which
encode much information of the system. We will therefore modify
the one-dimensional broken line to get suitable (one-dimensional
or rotation) supersolutions. For other heuristic justification
of a similar construction, see also \S~5 in \cite{SV}.
The first barrier we introduce is smooth but on the levels~$0$
and~$\pm 1$. This will confine touching
points on these levels. Actually, a free boundary analysis will
avoid touching points to occur at $\pm 1$-levels, and this will
localize the touching points on the zero level set
of the solution.
\begin{lemma}\label{cop.func.}
Let $\kappa,\,\kappa^\star>0$ be suitably small and
define
$$ \varpi_\pm \,:=\,\sqrt{(\varpi\pm\kappa^\star)^2
\pm 4\kappa}$$
and
$$ \agot_\pm\,:=\,\frac{\varpi_\pm-\sqrt{\varpi_\pm^2\mp
4\kappa}}{2\kappa}\,.$$
Let
$$ g(s)\,:=\,\left\{
\begin{matrix}
\varpi_+ s-\kappa s^2&\qquad&{\mbox{ if $s\in[
0,\,\agot_+
]$,}}\cr
\varpi_- s-\kappa s^2&\qquad&{\mbox{ if $s\in [
\agot_-,\,0)$,}}\cr
-1&\qquad&{\mbox{ if $s\in (-\infty,\,
\agot_-)$.}}
\end{matrix}
\right.
$$
Let\footnote{
Note that $g$ is defined and
continuous in
$\left(-\infty,\,\agot_+
\right]$,
that
$$ g( a_\pm)=\pm 1$$
and that $g$ is smooth (with $g'>0$) except on the level sets $0$ and
$-1$. Furthermore,
\begin{eqnarray*}
g'(\agot_-)&=&
\varpi-\kappa^\star<\\ &<& \varpi <\\ &<& \varpi+\kappa^\star=\\&=&
g'(\agot_+)\,.
\end{eqnarray*}
Also,
$$ \agot_\pm
\,\sim\, \pm \frac{1}{\varpi}
$$
for small positive
$\kappa$ and $\kappa^\star$.
%% In fact, a better estimate is obtained by using that
%% $$ \sqrt{1+x}\,\leq\,1+\frac x 2\,\qquad{\forall|x|<1}\,,$$
%% which shows that
%% $$ a_+\leq \frac{1}{\varpi+\kappa^\star}<\frac 1 \varpi\,.$$
We will freely use these
elementary
observations in the sequel.\label{K.kk:k:ll:pp}
Notice also that the r\^ole of $\kappa$ and $\kappa^\star$
with respect to $l$ is quite different: while $\kappa^\star$
is $l$-independent, $\kappa$ will be taken of the order
of $1/l$.}
also $y\in\R^N$, $l> 1$ and
$$ \Psi^{y,l}(x)\,:=\, g(|x-y|-l)\,,$$
for any $x\in
B_{l+\agot_+}(y)$.
Then,
there exists a uniform constant $\bar c\in(0,1)$,
so that, if $l\geq1/(\bar c \kappa)$ and $\kappa,\,\kappa^\star
\in (0,\bar
c)$,
$$ \Delta_p \Psi^{y,l} (x) \,\leq \, -\bar c \kappa\,<\,0\,,$$
for any $x$ so that $ \Psi^{y,l} (x)\neq 0,\pm 1$.
\begin{proof}
We take $x$ in the interior of
the domain of $\Psi^{y,l}$ and such that
$ \Psi^{y,l} (x)\neq 0,\pm 1$ and
we
use the short hand notation $t:=|x-y|-l$.
Since $0<|g(t)|<1$, we have that
$$ \frac \varpi 2\,\leq\, g'(t)\,\leq 2 \varpi\,,$$
if $\kappa$ and $\kappa^\star$ are small enough.
Then, a direct computation of the $p$-Laplacian
%% (see, e.g., Lemma~4.4 in \cite{SV})
shows that
$$ \Delta_p \Psi^{y,l}(x)=(g'(t))^{p-2}\Big( (p-1)\,g''(t)+(N-1)\,
g'(t)/|x-y|\Big)\,.$$
Also, if $\Psi^{y,l}(x)>-1$, then $|x-y|\geq l/2$ by
construction.
Thus,
$$ \Delta_p \Psi^{y,l}(x)\,\leq\, (g'(t))^{p-2}\,\Big(
-2\kappa\,(p-1)+\frac{4\varpi\,(N-1)}{l}
\Big)\,,$$
from which the desired result follows.
\end{proof} \end{lemma}
\begin{lemma}\label{lemma:barriere:1}
Let $\Omega$ be an open domain,
let $u\in W^{1,p}(\Omega)\cap C(\R^N)$ satisfy~(\ref{1.2}),
(\ref{1.4}), (\ref{FB-piu}) and~(\ref{FB-meno})
and let $\Psi^{y,l}$
as in Lemma~\ref{cop.func.}. Let us assume that
$\Psi^{y,l}\geq u$ in their common domain of definition.
If $x^\star\in \Omega$ is so that
$\Psi^{y,l}(x^\star)=u(x^\star)$, then either
$x^\star$ is in the interior of $\{ u=-1\}$
or $u(x^\star)=0$.
\begin{proof}
Assume that $u(x^\star)\neq 0$.
If also $|u(x^\star)|<1$, then $\Delta_p\Psi^{y,l}<0=
\Delta_p u$ and $\nabla \Psi^{y,l}\neq 0$
in a neighborhood of $x^\star$ (by
Lemma~\ref{cop.func.}),
and a contradiction then
follows from the Strong Comparison Principle
(see, e.g., Theorem~1.4 in~\cite{DAM} or Theorem~3.2 in~\cite{SV}). Thus,
$|u(x^\star)|=1$.
If $x^\star$ is in the interior of $\{u=1\}$, then
$u\equiv 1$ in a neighborhood $U$ of $x^\star$; thus, since
$\{\Psi^{y,l}=1\}$ is an $(N-1)$-dimensional sphere,
it would exist $\hat x\in U$ so that $1=u(\hat x)>
\Psi^{y,l}(\hat x)$, contradicting our hypothesis.
Therefore, $x^\star$ is either
in the interior of $\{ u=-1\}$, as claimed,
or $x^\star$ lies on the free boundary
$\partial\{ u=\pm 1\}$.
We
exclude this possibility by arguing as follows.
First, we exclude that $x^\star\in\partial\{ u=1\}$.
For this, we argue
by contradiction
and we assume that
$x^\star\in \partial\{ u=1\}$.
Then,
from the free boundary
growth~(\ref{FB-piu}),
if $\nu_0\in {\rm S}^{N-1}$ is the interior normal of
$\{\Psi^{y,l}=1\}$ at $x^\star$
(note that $\nu_0$ points towards
$\{ u<1\}$, since $u\leq \Psi^{y,l}$),
we have that
$$
u(x^\star +t_j\nu_0)\,\geq\, 1-\varpi t_j -\fgot_+(t_j)
\,,$$
for an infinitesimal positive sequence $t_j\searrow 0^+$.
On the other hand, recalling the footnote on page~\pageref{K.kk:k:ll:pp},
the fact that $\Psi^{y,l}(x^\star)=1$ gives that
$$ \frac{\partial \Psi^{y,l}}{\partial \nu_0} (x^\star)=
-(\varpi+\kappa^\star)\,,$$
thence
$$ { \Psi^{y,l}}(x^\star +t_j \nu_0)\leq
1-\left( \varpi +\frac{\kappa^\star}{2}\right)\,t_j\,.$$
The fact that $\Psi^{y,l}\geq u$ and the above estimates
thus imply that
$$ 1-\varpi t_j-\fgot_+(t_j)\leq 1-\left( \varpi
+\frac{\kappa^\star}{2}\right)\,t_j\,,$$
which easily yields a contradiction
with~(\ref{FB-piu:fgot}).
This shows that
$x^\star$ does not lie on the free boundary
$\partial\{ u= 1\}$.
\medskip
Thus, to complete the proof of the desired claim,
we need to exclude that
$x^\star\in
\partial\{ u= -1\}$. For this, assume, by contradiction,
that $x^\star\in\partial\{ u= -1\}$. Then, the fact
that $\Psi^{y,l}\geq u$ implies that also
$x^\star\in\partial\{ \Psi^{y,l}= -1\}$. Thus, let $\nu_0$
be the inner normal of $\partial\{ \Psi^{y,l}= -1\}$
at $x^\star$. Since $\Psi^{y,l}\geq u$,
it follows that $\nu_0$ points towards
$\{ u= -1\}$. Thus,
the free boundary
growth~(\ref{FB-meno}) yields that
$$ u(x^\star -t_j\nu_0)\geq -1+\varpi t_j -\fgot_-(t_j)$$
for $t_j\searrow 0^+$.
Also, recalling
the footnote on page~\pageref{K.kk:k:ll:pp} agin, we infer from
the fact that $x^\star\in\partial\{ \Psi^{y,l}= -1\}$ that
%% $$ -\frac{\partial \Psi^{y,l}}{\partial \nu_0} (x^\star)=
%% \varpi-\kappa^\star$$ and so
$$ \Psi^{y,l} (x^\star -t_j\nu_0)\leq -1+\Big(\varpi-
\frac{\kappa^\star}{2}
\Big)\,t_j\,.$$
Finally, the fact that $\Psi^{y,l}\geq u$, together
with the above estimates,
gives that
$$ -1+\varpi t_j -\fgot_-(t_j)\leq
-1+\Big(\varpi-
\frac{\kappa^\star}{2}
\Big)\,t_j\,,$$
contradicting~(\ref{FB-meno:fgot}) and
thus ending the proof of the desired
result.
\end{proof}\end{lemma}
We now introduce another barrier, in order to deal with the
distance function and control the curvature of the level sets
of the solution.
\begin{lemma}\label{dgammasolvisc} Let $0<\e\leq \sigma\leq \delta
<1$,
$\xi\in \R^{N-1}$, $M\in {\rm Mat}((N-1)\times (N-1))$.
Let $\Gamma $ be the hypersurface defined as
$$ \Gamma :=
\{\, x_N =\frac{\e}{2}x'\cdot M
x' +\sigma \xi\cdot x'\, \}\cap \{\,|x'|<\frac{\sigma}{\e} \,\}$$
and assume that $$ {\rm tr}\, M\geq \delta\,, \qquad
\|M\|\leq\frac{3}{\delta}\,,\qquad |\xi|\leq\frac{3}{\delta}\,.$$
We define $d_{\Gamma}(x)$ as
the signed distance
from $x$ to $\Gamma$, with the assumption that
$d_\Gamma$ is positive\footnote{Of course, here ``above" is intended
with respect to the $e_N$-direction.
For some properties of the distance function, see \cite{GT}.
As usual, $\{e_1,\dots,e_N\}$ denotes here
the standard base of $\R^N$.}
above $\Gamma$.
Let also $c_1>0$ be a suitably small constant.
Let us
define
$$ \bgot_\pm\,:=\,\frac{-\varpi+\sqrt{\varpi^2\pm 4c_1\e\delta}}{
2c_1\e\delta}$$
and
$$ {\mbox{$\breve g(s)\,:=\, \varpi s+c_1\e\delta s^2\,,\quad$
for any $s\in [\bgot_-,\,\bgot_+]$.}}$$
Let us also define $\breve g(s)=-1$ for any $s<\bgot_-$.
Then\footnote{Note that $\breve g$ is continuous
in $(-\infty,\bgot_+]$ and smooth (with $\breve g'>0$)
in $(\bgot_-,\,\bgot_+)$. Also,
$\bgot_\pm\sim \pm 1/\varpi$ for small positive
$c_1$,\label{K.kk:k:ll:pp:BISA}
$\e$ and
$\delta$.
What is more,
$\breve g(\bgot_\pm)=\pm 1$ and $\breve g'(\bgot_\pm)=\sqrt{\varpi^2
\pm 4c_1\e\delta}$. In particular,
$$ \breve g'(\bgot_+)\,>\, \varpi\,>\,
\breve g'(\bgot_-)\,.$$
A second order Taylor expansion
shows that
$$ \bgot_+\leq\frac 1\varpi-\frac{c_1\e\delta}{\varpi^3}+
{\,\rm const\,}\frac{(c_1\e\delta)^2}{\varpi^5}\,,$$
thus
$$ \bgot_+\,<\,\frac 1 \varpi$$
for small positive $c_1$, $\delta$ and $\e$.
These elementary properties of $\breve g$ will
be freely used in what follows.},
there exists a functions $\sigma _0 :(0,+\infty) \longrightarrow
(0,1)$
such that, if
$\e\leq\sigma\leq\sigma _0 (\delta)$,
then $$\Delta_p \Big(\breve g(d_\Gamma(x)\Big)<- \tilde c c_1
\e\delta<0 $$
at any point $x$ for which $d_\Gamma(x)\in
(\bgot_-,\,\bgot_+)$, for a
suitable small positive constant $\tilde c$.
\begin{proof} If $d_\Gamma(x)\in (\bgot_-,\,\bgot_+)$,
\begin{equation}\label{oPPOll09899}
|d_\Gamma (x)|\,\leq\, \frac2\varpi\,.
\end{equation}
Also, if $s:=d_\Gamma (x)$,
\begin{equation}\label{oPPOll09899:bis}
2\varpi\geq \breve g'(s)\geq \frac\varpi 2\,,
\end{equation}
for small $c_1$, $\e$ and $\delta$.\medskip
In an appropriate system of coordinates we
have that
\begin{equation}\label{equation:012333}
D^2 d_\Gamma ={\rm diag}\left(\frac{-k_1}{1-d_\Gamma k_1},\ldots
,\frac{-k_{N-1}}{1-d_\Gamma k_{N-1}}, 0\right)\in{\rm Mat}\,(N\times
N)\,,\end{equation}
where the $k_i$'s are the principal curvatures of $\Gamma$ at the point
where the distance is realized (see \S~14.6 in
\cite{GT} for further details). By construction,
$$|k_i|\leq C_1 (\delta)\e\,,$$
for a suitable $C_1(\delta)$.
We denote by $P$ the paraboloid describing $\Gamma$, i.e.,
$$P(x'):=\frac{\e}{2}x'\cdot M
x' +\sigma \xi\cdot x'\,.$$
Note that
\begin{equation}\label{GP:nab}
|\nabla P|\,\leq\,{\rm const}\,\frac{\sigma}{\delta}
\end{equation}
is a small quantity.
Therefore, by the mean curvature
equation (see, for instance, equation~(14.103)
of \cite{GT}), it follows
that
\begin{eqnarray*}
\sum_{i=1}^{N-1} k_i &=&\sum_{i=1}^{N-1}
\partial_i \left( \frac{\partial_i P}{\sqrt{1+|\nabla
P|^2}}\right)\\
&=&
\frac{\Delta P}{\sqrt{
1+|\nabla P|^2
}} -\frac{(D^2 P \,\nabla
P)\cdot\nabla P}{
(1+|\nabla P|^2)^{3/2}}
\\
&\geq&\frac 1 2
\Delta P- {\rm const}\,|\nabla P|^2 \|D^2
P\|
\,.\end{eqnarray*}
Thus,
by using also~(\ref{oPPOll09899}),
(\ref{equation:012333}) and~(\ref{GP:nab}),
we infer that
\begin{eqnarray*}
\Delta d_\Gamma &=&
\sum_{i=1}^{N-1}\frac{-k_i}{1-d_\Gamma k_i}=\\&=&
-\sum_{i=1}^{N-1} k_i -\sum_{i=1}^{N-1}\frac{d_\Gamma k_i^2}{
1-d_\Gamma k_i}\leq \\ &\leq& -\frac 1 2\Delta P+
C_2(\delta) \,\Big(
|\nabla P|^2 \|D^2
P\|+\e^2
\Big)\leq\\ &\leq&
-\frac{\e\delta}2+C_3(\delta)\,(\e\sigma^2+\e^2)\,,
\end{eqnarray*}
for suitable $C_i(\delta)$. In particular,
$$ \Delta d_\Gamma \,\leq\,
-\frac{\e\delta}4\,.$$
Then, a direct computation on the $p$-Laplacian
%% (see, e.g., Lemma~4.5 in \cite{SV})
and~(\ref{oPPOll09899:bis}) give that
\begin{eqnarray*}&&
\Delta_p \Big(\breve g(d_\Gamma(x)\Big)=\\ &=&\Big(
\breve g'(d_\Gamma(x))\Big)^{p-2}\,\Big[
(p-1)\,\breve g''(d_\Gamma(x))+\breve g'(d_\Gamma(x)) \Delta
d_\Gamma(x)\Big]\leq\\ &\leq&
\Big( \frac \varpi 2\Big)^{p-2}\,\Big[
2c_1 \e \delta \,(p-1)
-\frac{\varpi \e \delta}{8}
\Big]\,,
\end{eqnarray*}
from which the desired result follows by taking $c_1$
conveniently small.\footnote{The last passage in the proof
also gives a good hint on how such barrier has been constructed:
namely, the curvature of $\Gamma$, which is of order $-\e\delta$,
is going to compensate the one of $\breve g$, which is of
order $c_1 \e\delta$.}
\end{proof}
\end{lemma}
In analogy with Lemma~\ref{lemma:barriere:1}
we point out the following result for
the barrier $\breve g\circ d_\Gamma$ constructed here above.
Though the proof is similar in spirit to the one of
Lemma~\ref{lemma:barriere:1}, we provide full details
of it for the reader's convenience.
\begin{lemma}\label{lemma:omi}
Let $\Omega$ be an open domain,
let $u\in W^{1,p}(\Omega)\cap C(\R^N)$ satisfy~(\ref{1.2}),
(\ref{1.4}), (\ref{FB-piu}) and~(\ref{FB-meno})
and let $\Gamma$ and $\breve g$
be as in Lemma~\ref{dgammasolvisc}. Let us assume that
$\breve g\circ d_\Gamma\geq u$
in their common domain of definition (which is supposed to be
nonempty).
Then,
if $x^\star\in\Omega$
is so that $\breve g(d_\Gamma(x^\star))=u(x^\star)$, then
$x^\star$ lies in the interior of
$\{ u=-1\}$.
\begin{proof}
Let us first observe that $x^\star$ cannot lie on the
free boundary $\partial\{u=\pm 1 \}$. First, we show
that $x^\star\not\in
\partial\{u= -1 \}$.
We argue by contradiction,
assuming that $x^\star\in\partial\{ u= -1\}$. Then, since
$\breve g\circ d_\Gamma\geq u$, we have that
also
$x^\star\in\partial\{ \breve g\circ
d_\Gamma= -1\}$. Thus, let $\nu_0$
be the normal of $\partial\{
\breve g\circ d_\Gamma= -1\}$
at $x^\star$ pointing towards
$\{ \breve g\circ
d_\Gamma= -1\}$. The fact that $
\breve g\circ d_\Gamma\geq u$ also implies
that $\nu_0$ points towards
$\{ u= -1\}$. Thus,
the fact that $x^\star\in\partial\{ u= -1\}$ and
the free boundary
growth~(\ref{FB-meno}) yield that
$$u(x_- -t_j\nu_0)\geq -1+\varpi t_j -\fgot_-(t_j)\,,$$
with $t_j\searrow 0^+$.
Also, recalling
the footnote on page~\pageref{K.kk:k:ll:pp:BISA},
the fact that $x^\star\in\partial\{
\breve g\circ d_\Gamma= -1\}$ gives that
%% $$ -\frac{\partial \breve g\circ d_\Gamma
%% }{\partial \nu_0} (x^\star)<
%% \sqrt{\varpi^2-4c_1\e\delta}$$
%% and so
$$ \breve g\big(d_\Gamma(x^\star-t_j\nu_0)\big)\leq
-1+
\sqrt{\varpi^2
-2c_1\e\delta}
\;t_j\,. $$
Using again that $\breve g\circ d_\Gamma\geq u$, one thus gets
that
$$
-1+\varpi t_j -\fgot_-(t_j)
\leq
-1+
\sqrt{\varpi^2
-2c_1\e\delta}
\;t_j\,.$$
The latter estimate
and~(\ref{FB-meno:fgot}) lead to
a contradiction,
thus $x^\star\not\in\partial\{ u= -1\}$. \medskip
We now show that $x^\star\not\in\partial\{ u= 1\}$.
To see this, let
us argue by contradiction and let us
assume that $x^\star\in \partial\{u= 1 \}$.
By construction, $x^\star$ also belongs to the
$(N-1)$-dimensional surface
$$ \Sigma\,=\,\Big\{ x\in\R^N\;\,\Big|\;\, d_\Gamma(x)=\bgot_+
\Big\}\,.$$
Let
$\nu_0\in {\rm S}^{N-1}$ be the interior normal of
$\Sigma$ at $x^\star$.
Note that $\nu_0$ points towards
$\{ |u|<1\}$, since $u\leq {\breve g\circ d_\Gamma}$.
Then, by the free boundary growth~(\ref{FB-piu}),
$$ u(x^\star +t_j\nu_0)\geq 1-\varpi t_j -\fgot_+(t_j)
\,,
$$ for an infinitesimal positive sequence $t_j$.
Also, by construction, $u\leq\breve g\circ
d_\Gamma$
and thus
$$ \breve g\big( d_\Gamma(x^\star+t_j\nu_0)\big)
\geq 1-\varpi t_j
-\fgot_+(t_j)
\,.$$
By the elementary properties described in the footnote on
page~\pageref{K.kk:k:ll:pp:BISA}, we also have that
$$ \frac{\partial (\breve g\circ d_\Gamma)}{\partial\nu_0}(x^\star)\,=\,
-\sqrt{\varpi^2+ 4c_1 \e\delta}$$
and therefore
$$ \breve g\big( d_\Gamma(x^\star+t_j\nu_0)\big)\leq 1-
\sqrt{\varpi^2+ 2c_1 \e\delta} \;t_j$$
if $j$ is large enough.
These estimates give that
$$1-\varpi t_j
-\fgot_+(t_j)
\,\leq\,
1-
\sqrt{\varpi^2+ 2c_1 \e\delta}\; t_j\,$$
which easily provides a contradiction with~(\ref{FB-piu:fgot}) for $j$
large.\medskip
We thus have that $x^\star$ does not lie on the
free boundary $\partial\{u=\pm 1 \}$.
Furthermore, $x^\star$ cannot lie in the interior
of $\{u= 1 \}$. To see this, let us assume, by contradiction,
that $x^\star\in U\subset\{u= 1 \}$, for a suitable open set $U$.
Then, since $\Sigma$, as defined here above, is an
$(N-1)$-dimensional surface, there would exist
$x_0\in U$ such that $\breve g(d_\Gamma(x_0))<1$.
Therefore,
$$ \breve g(d_\Gamma(x_0))<1=u(x_0)\,,$$
in contradiction with our assumptions.\medskip
We have thus proved that either $x^\star$ is in the
interior of $\{u= -1 \}$ or $u(x^\star)\in (-1,1)$.
The latter possibility, however, cannot hold, due
to the Strong Comparison Principle
(see, e.g., Theorem~1.4 in~\cite{DAM} or Theorem~3.2 in~\cite{SV}).
\end{proof}
\end{lemma}
\section{Sliding methods}\label{mth}
We now use the barriers introduced here above
in \S~\ref{barriers}
to
deduce an estimate on the curvature of
the paraboloids which may touch our solution.
In particular, we will show that a zero mean
curvature property
is attained by the level sets of our solution,
though in a weak viscosity
sense. Next result, which, in our opinion, is interesting
in itself, will play a crucial r\^ole
in the proof of Theorem~\ref{main2}, which indeed will follow
via a natural rescaling.
\begin{thm}\label{lemma 322b}
Let
$$\Omega:=
\Big\{ (x',x_N)\in\R^{N-1}\times\R\;\,\Big|\,\;
|x'|0$ and $M_1\in{\rm
Mat}((N-1)\times( N-1))$. Let $u$ be as in Theorem~\ref{main2}.
Assume that $u(x) < 0$ for any
$x=(x',x_N)\in \Omega$ so that $$ x_N <
\frac{\theta}{2l^2}x'\cdot M_1 x'+\frac{\theta}{l} \xi\cdot x' \,.$$
Then, there exist a
universal constant
$\delta_0>0$ and a function
$\sigma:(0,1)\longrightarrow(0,1)$, so that, if $$ \delta\in
(0,\delta_0]\,, \quad \delta\leq \theta\,,\quad
\frac{\theta}{l}\in\Big(0, \sigma(\delta)\Big]\,, \quad\|M_1\|\leq
\frac{1}{\delta}\quad{\mbox{and}}\quad |\xi|\leq \frac{1}{\delta}\,,$$ then
$$ {\rm tr} M_1\leq \delta\,.$$
\begin{proof} The proof is
similar to the one of Lemma~3.2.2 in~\cite{S}
and
Lemma~6.6 in~\cite{SV}.
However, due to the technicalities\footnote{
In particular, several quantitative estimates here
differ from similar ones in~\cite{S}
and~\cite{SV}.
The main reason for such difference is that
the smooth transitions considered in~\cite{S}
and~\cite{SV} lead to exponentially deacying barriers,
while the barriers constructed here
have ``something like" a linear decay.}
involved, we provide
full details for the reader's convenience.
We will apply Lemmas~\ref{cop.func.} and~\ref{dgammasolvisc}
by choosing
\begin{eqnarray}\kappa&:=&\frac{1}{\bar c\,l}\nonumber\\
\label{eqna876HHy} {{\mbox {and }}}\,\, \e &:=&\frac{\theta}{2l^2}
\,.\end{eqnarray}
Note that, by our assumptions, $l\geq
\delta/\sigma(\delta)$
may and will be assumed to be a large quantity.
In particular, we may and do assume $\kappa$ and $\kappa^\star$
to be suitably small with respect to $\delta$.
Let also
\begin{equation}\label{gatt} {l}_+\,:=\,
{l/4+\agot_+}
\,,\end{equation}
so that,
by Lemma~\ref{cop.func.}, the barrier
$\Psi^{y,l/4}$ is defined in $B_{{l}_+}
(y)$.
Define also
\begin{eqnarray*}
\Gamma_1&:=&\,\Big\{ x=(x',x_N)\in\R^N
\;\;{\mbox{s.t.}}\;\;|x'|< l\\ &&{\mbox{and}}\;\;
x_N= \frac{\theta}{2l^2}x'\cdot M_1
x'+\frac{\theta}{l} \xi\cdot x' \Big\}\,.\end{eqnarray*}
Let us make some elementary observations upon the
above paraboloid.
First of all, by construction,
$u$ is negative below $\Gamma_1$ in $[-l,l]^N$.
What is more, the principal curvatures of $\Gamma_1$
are bounded by $\,{\rm const}\,{\sigma(\delta)}/({l\delta})$:
thus,
if ${\sigma(\delta)}{/\delta}$ is
sufficiently small, then, given any $\breve x\in\Gamma_1$,
there exists a ball
of radius $l/4$ which touches $\Gamma_1$
from below at $\breve x$.
Given $\breve x\in\Gamma_1$,
let $\nu_{\breve x}$ be the normal direction of $\Gamma_1$ at ${\breve x}$ pointing
downwards.
Let
\begin{equation}\label{u990012}
\Kgot\,:=\,
\{|x'|\leq l/4\}\,\cap\,
\left\{ d_{\Gamma_1}(x)\in \Big[-\frac l 8,\,
\agot_+
\Big]\right\}\,.
\end{equation}
\medskip
We now claim that
\begin{equation}\label{aim...} u(x)\leq g
(d_{\Gamma_1}(x)) \end{equation} for any $x\in \Kgot$.
To prove (\ref{aim...}),
first notice that, by construction,
the zero level set of $u$ is above $\Gamma_1$,
hence above the hyperplane $\{x_N=-l/100\}$; thus,
from~(\ref{the deacy}),
$$u(x)\,< \, \Psi^{-(l/2)e_N,l/4}
(x)$$
for any $x$ in their common domain of definition, provided
$$\Psi^{-(l/2)e_N,l/4}(x)\neq -1\,.$$
Then, for a given ${\breve x}\in\Gamma_1$
we
define \begin{equation}\label{can}
x_0\,=\,x_0({\breve x})\,:=\,{\breve x}+ (l/4) \nu_{\breve x}\,,\end{equation}
where
we have
denoted by $\nu_{\breve x}$
the normal direction of $\Gamma_1$ at ${\breve x}$ pointing
downwards.
In particular, from the above observation,
$B_{l/4}(x_0)$ touches $\Gamma_1$
from below at ${\breve x}$.
\medskip
We now slide the
surface $\Psi^{-(l/2)e_N,l/4}$ in the direction of the
vector $$ v\,=\,v({\breve x})\,:=\,x_0+(l/2)e_N\,,$$
that is, we will consider the
surface $\Psi^t:=\Psi^{-(l/2)e_N+tv, l/4}$ for $t>0$. We
will show that \begin{equation}\label{slide} \Psi^t(
\xgot)>u(\xgot)
\end{equation} for any $t\in[0,1)$ and any $\xgot$
in their common domain of definition,
provided
$\Psi^t(\xgot)\neq -1$.
%% Indeed, if $t\in [0,1)$,
%% the domain of $\Psi^t$ lies below $\Gamma_1$
%% and therefore the domain of $\Psi^t$
%% lies in $\{u<0\}$
Indeed, as a consequence of Lemma~\ref{cop.func.},
we have that
$\Psi^t$ and $u$ cannot
touch each other on the free boundary $\partial\{u=-1\}$.
In the
light of this observation, we may
take $t\in[0,1)$ as the first time (if
any)
on which $\Psi^t$ touches $u$ at a point in $\{\Psi^t\neq-1\}$.
Note that, since $t<1$, we have
that
the domain of $\Psi^t$ lies below $\Gamma_1$
and thence the domain of $\Psi^t$
lies in $\{u<0\}$.
Therefore, $u$
cannot be equal to $\Psi^t$ and touching points between
$u$ and $\Psi^t$ cannot occur on
$\{
\Psi^t=0\}$.
On the other hand, Lemma~\ref{lemma:barriere:1}
says that
touching points cannot occur anywhere else.
This proves (\ref{slide}). \medskip
We are now in the position
to complete the proof of (\ref{aim...}), by arguing as follows.
We deduce from (\ref{slide}) that $ \Psi^1(\xgot)\geq u(\xgot)$ for any
$\xgot$ in the common domain of definition of $\Psi^1$ and $u$, that is,
for any $\xgot\in B_{{l}_+}(x_0)$.
Take now any $x\in\Kgot$ and let ${\breve x}$ realize
$d_{\Gamma_1}(x)$. Let also
$x_0$ be as in~(\ref{can}),~so that $x\in B_{{l}_+}
(x_0)$: then,
\begin{eqnarray*}
g \Big( d_{\Gamma_1}(x)\Big) &=& g (|x-x_0|- l/4)\nonumber \\
&=& \Psi^{x_0, l/4}(x)\nonumber \\ &=& \Psi^1(x)\nonumber \\&\geq& u(x)\,.
\end{eqnarray*} This proves (\ref{aim...}). \bigskip
We now complete the proof of the desired result
arguing by contradiction and supposing that $ {\rm tr}
M_1>\delta$. We define
\begin{eqnarray*}
\Gamma_2 &:=& \Big\{
x=(x',x_N)\in [-l,l]^N\;\;{\mbox{s.t.}}\;\;\\
&& x_N = \frac{\theta}{2l^2}x'\cdot M_1 x'+\frac{\theta}{l}
\xi\cdot x'-\frac{\e\delta}{2(N-1)}|x'|^2 \Big\}
\,,\end{eqnarray*} where $\e$ has been
introduced in (\ref{eqna876HHy}). By
Lemma~\ref{dgammasolvisc}, we get that $\breve g\circ
d_{\Gamma_2}$ is strictly $p$-superharmonic.\footnote{The
quantities $M$, $\sigma$ and $\xi$ in Lemma~\ref{dgammasolvisc}
correspond here to $2M_1-\delta/(N-1)$, $\theta/(2l)$ and $2\xi$,
respectively.}
\medskip
Furthermore, by the definitions of $\Gamma_1$ and $\Gamma_2$,
if $|x'|= l/4$, then
\begin{equation}\label{equa:09:01:01:oo}
d_{\Gamma_2}(x) \geq d_{\Gamma_1} (x)
+c(\delta)
\end{equation}
for a suitable $c(\delta)\in(0,1)$.
We now take $\kappa$ and $\kappa^\star$ in Lemma~\ref{cop.func.}
to be
positive and suitably\footnote{
Note that we required $l\sim 1/\kappa$,
thus the fact that
$\kappa$ is small in dependence of $\delta$
is warranted by the fact that $1/l$ is small
in dependence of $\delta$: recall (\ref{eqna876HHy}).}
small (possibly in dependence of
$\delta$) in such a way that
\begin{equation}\label{eqna765tt}
|\varpi_\pm -\varpi|\,\leq\,\frac{c(\delta)\,\varpi^2}{
4}\,.\end{equation}
For further use, recalling the footnote on page~\pageref{K.kk:k:ll:pp},
we will also assume that
$$ \agot_+\,\geq\,\frac 1 \varpi-\frac{c(\delta)}{2}\,.$$
We now
recall
the footnote on page~\pageref{K.kk:k:ll:pp:BISA} and note that
the latter assumption and~(\ref{equa:09:01:01:oo}) thus
imply the following estimate: if $\hat x$ is so that
$d_{\Gamma_1} (\hat x)\geq \agot_+$ and $|\hat x'|=l/4$, then
\begin{equation}\label{equ:999}
d_{\Gamma_2} (\hat x)\geq \agot_+
+c(\delta)\geq\frac{1}{\varpi}+
\frac{c(\delta)}{2}>\bgot_+
\,,\end{equation}
which will be of later use.
The choice in (\ref{eqna765tt})
implies that, if $s$ is in the domain of $g$,
\begin{eqnarray*}
g(s)&\leq& \varpi s +\frac{c(\delta)\,\varpi^2\,|s|}{4}-\kappa
s^2\leq\\
&\leq& \varpi s +\frac{c(\delta)\,\varpi}{2}\,.
\end{eqnarray*}
Thus, in the
light of~(\ref{equa:09:01:01:oo}), we have that
\begin{equation}\label{dom:con:kk}
\begin{split}
g\big(d_{\Gamma_1}(x)\big) \leq&\varpi d_{\Gamma_1}(x)
+\frac{c(\delta)\,\varpi}{2}\leq\\ \leq&\varpi d_{\Gamma_2}(x)
-\frac{c(\delta)\,\varpi}{2}\leq\\ \leq&
\breve g\big(d_{\Gamma_2}(x)\big)
-\frac{c(\delta)\,\varpi}{2}<\\<&
\breve g\big(d_{\Gamma_2}(x)\big)\,.
\end{split}
\end{equation}
for any $x$ for which the above functions are defined, so that
$|x'|=l/4$.
\medskip
We point out that if $x\in\Kgot$, then, by~(\ref{u990012}), we have that
$$d_{\Gamma_1}(x)\leq\agot_+$$
and so
$d_{\Gamma_1}(x)$ is in the domain of $g$.
>From this observation,~(\ref{aim...}) and~(\ref{dom:con:kk}),
we gather
that
\begin{equation}\label{side}
u(x)\,<\, \breve g(d_{\Gamma_2}(x))\,,\end{equation}
for any $x\in\Kgot$
so that $|x'|=l/4$ and $d_{\Gamma_2}(x)$ is
in the domain of~$\breve g$.\bigskip
With these estimates, we are now ready to deduce the contradiction that
will finish the proof of the desired result. To this end, we
define\footnote{Note that, by (\ref{u990012}), $\Kgot^\star\supset\Kgot$.}
$$ \Kgot^\star\,:=\,
\{|x'|\leq l/4\}\,\cap\,
\left\{ d_{\Gamma_1}(x) \geq-\frac l 8\right\}$$
and (recalling Lemma~\ref{lemma:omi})
we slide
$\breve g\circ d_{\Gamma_2}$
from $-\infty$ in the $e_N$-direction until we
touch\footnote{This touching must indeed occur sooner or
later, since $u(0)=0$, due to~(\ref{udizero}).} $u$
in the closed domain $\Kgot^\star$
at some point, say $x^\star$, with $u(x^\star)>-1$.
For this, we consider, for $t\in \R$, $$ g^t(x):=
\breve g\Big(d_{\Gamma_2} (x-t e_N) \Big)$$
and we increase $t$ from $-\infty$.
By Lemma~\ref{lemma:omi}, we have that
the first touching points between $g^t$ and $u$
at a level greater than $-1$
must occur on $\partial \Kgot^\star$.
Note that, by definition,
$\partial \Kgot^\star$ is composed by two
parts:
the ``side'', given by the cylinder $|x'|=l/4$
and the ``bottom", given by
$$\big\{ d_{\Gamma_1}(x)=-l/8\big\}\,.$$
We now show that no first touching points between $g^t$ and $u$
at a level greater than $-1$ may occur on $\partial\Kgot^\star$
and this
will give the desired contradiction.\medskip
By the definition of $\Gamma_1$, one sees that
the bottom of $\Kgot^\star$ lies in $\{x_N\leq -l/9\}$,
thus,
by~(\ref{the deacy}), $u=-1$ on the bottom of $\Kgot^\star$.
This excludes that
first touching points between $g^t$ and $u$
at a level greater than $-1$
may occur on the bottom of $\partial\Kgot^\star$.
But these touching points cannot occur on the side
of $\partial\Kgot^\star$ either, thanks to the following argument.
We assume, by contradiction that
there exists
a first touching point $x^\star$ between $g^t$ and $u$
lying on the side of
$\partial\Kgot^\star$ (that is, $|(x^\star)'|=l/4$).
There are two cases: either $d_{\Gamma_1}(x^\star)\leq\agot_+$
or the converse.
Let us first assume that $d_{\Gamma_1}(x^\star)\leq\agot_+$.
Then, by (\ref{u990012}), $x^\star\in\Kgot$.
Observe also that the fact that $u(0)=0$ implies
$t\leq0$; thus, an elementary observation
%% (see, e.g., Lemma~4.8 in~\cite{SV})
gives that
$$ d_{\Gamma_2}(x^\star-te_N
)\geq d_{\Gamma_2}(x^\star)\,.$$
But then, since
$\breve g$ is non-decreasing, we deduce from the fact that
$x^\star\in\Kgot$ and~(\ref{side}) that
$$ \breve g (d_{\Gamma_2}(x^\star-te_N)) =g^t(x^\star)
=u(x^\star)< \breve g(d_{\Gamma_2}(x^\star))
\leq \breve g(d_{\Gamma_2}(x^\star-te_N))
\,.
$$
This contradiction shows that only the second
case may occur, that is, $d_{\Gamma_1}(x^\star)>\agot_+$.
But even this last case cannot hold, since, from (\ref{equ:999}),
we would get that $x^\star$ is strictly
above the domain of $\breve g\circ
d_{\Gamma_2}$ and thus, since $t\leq 0$,
strictly above the domain of $g^t$.\medskip
This contradiction
concludes the proof of Theorem~\ref{lemma 322b}.
\end{proof}
\end{thm}
\section{Proof of Theorem~\ref{main2}}\label{ss2}
The proof of Theorem~\ref{main2} can be now
completed by arguing as follows.
We will apply Theorem~\ref{lemma 322b}
by making use of the following choice
of parameters:
$$ l:=\frac{\dgot}{\sqrt{\e\,{\rm tr}\,M}}\,,\quad
\delta:=\theta:={\dgot^2}\,,\quad
M_1:=\frac{1}{{\rm tr}\,M}\,M\,,\quad \xi:=0\,.$$
If, by contradiction, the claim of
Theorem~\ref{main2}
were false, by scaling back and using the above parameters,
we would have
that $\Gamma_1$ touches the zero level set of
$u$ by below ,
where
$$ \Gamma_1=\left\{
x=(x',x_N)\in
\R^{N-1}\times\R\;\;{\mbox{s.t.}}\;\; x_N = \frac{\theta}{2l^2}x'\cdot
M_1
x'+\frac{\theta}{l} \xi\cdot x'
\right\}\,.$$
By Theorem~\ref{lemma 322b}, we gather that
$1>\delta\geq {\rm tr}\,M_1=1$,
which is the contradiction
that proves Theorem~\ref{main2}.
\section{Proof of Theorem~\ref{main2:minima}}\label{ss3}
The proof of Theorem~\ref{main2:minima}
will be accomplished once we show that
either the function $u$ in Theorem~\ref{main2:minima}
or the function $\tilde u:=-u$
satisfies the assumptions\footnote{Note that $\tilde u$
also is a Class A
minimizer
of ${\mathcal{F}}$.}
of Theorem~\ref{main2}.
Namely, we have to prove the continuity of $u$,
(\ref{1.4}), (\ref{FB-piu}),
(\ref{FB-meno}) and~(\ref{the deacy})
for either $u$ or $\tilde u$.
\medskip
First of all, $u$ is uniformly continuous by the celebrated
result in~\cite{GiG}. More precisely, it is uniformly
Lipschitz continuous:
see Theorem~2.1 in \cite{PV1}.
\medskip
The proof of~(\ref{1.4}) is standard and we omit it.
\medskip
In order to proof~(\ref{FB-piu}) we argue as follows.
First observe that the continuity of $u$
implies that the free
boundary
$\partial\{u=-1\}$
is uniformly separated from the free
boundary $\partial\{u=1\}$.
Therefore, by elementary observations (see, e.g., Lemma~3.3
in \cite{PV1}), there exists a universal $\rho>0$ so that,
if $$ x_\pm\in\partial\{u=\pm 1\}=\partial\{\tilde u=\mp 1\}\,,$$
then $1\mp u=1\pm \tilde u$ is an absolute minimizer for
the functional
$$ \widetilde{\mathcal{F}}(w)\,=\,
\int_{B_\rho(x_\pm)}
|\nabla w(x)|^p+\lambda \chi_{(0,+\infty)}\big(w(x)\big)
\,dx\,.$$
Then, by Theorem~7.1 in~\cite{DaP}, if $x\in\R^N$ and
$r>0$ are so that
$B_r(x)\subset B_\rho(x_\pm)$ and $B_r(x)\cap
\partial\{\pm u<1\}\neq \emptyset$, we have that
\begin{equation}\label{DP:fu}
\sup_{B_r(x)}|\nabla u|\,\leq\,\varpi +Cr^\alpha\,,
\end{equation}
for suitable universal positive $C$ and $\alpha$
and
\begin{equation}\label{DP:rel}
\varpi\,:=\,\frac{\lambda}{(p-1)^{1/p}}\,.
\end{equation}
In particular, if there is a ball $B\subseteq\{|u|<1\}$
touching $\partial\{ \pm u=1\}$ at $x_\pm$,
we define $\nu_0$ as the interior normal of
$B$ at $x^\mp$, we fix small
positive $s\,0$$
and
$$ \Lgot^N \Big( B_C(x)\cap\{u=-1\}\Big)\,>\,0\,,$$
where $\Lgot^N$ is the Lebesgue measure and $C$ is a
suitable positive constant.
In particular, $B_C(x)$ contains points $x_\pm$ so that
$u(x_\pm)=\pm 1$. Thus, by the continuity of $u$,
there must be a point $x_*$ on the segment joining $x_+$ and $x_-$
so that $u(x_*)=0$. By construction, $x_*\in B_C(x)$,
thus
$$ |x_*'|\leq |x'|+C\leq \frac l 2+C