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\controldates{30-JAN-2003,30-JAN-2003,30-JAN-2003,30-JAN-2003}
 
\RequirePackage[warning,log]{snapshot}
\documentclass{era-l}
\issueinfo{9}{03}{}{2003}
\dateposted{February 3, 2003}
\pagespan{19}{25}
\PII{S 1079-6762(03)00108-2}

\copyrightinfo{2003}{American Mathematical Society}

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{xca}[theorem]{Exercise}

\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}


\begin{document}

\title[Blow-up theory for elliptic equations]
{A $C^0$-theory for the blow-up of second order elliptic equations of 
critical Sobolev growth}

\author{Olivier Druet}
\address{D\'epartement de Math\'ematiques, Ecole Normale Sup\'erieure de 
Lyon, 46 all\'ee d'Italie, 69364 Lyon cedex 07, France}
\email{Olivier.Druet@umpa.ens-lyon.fr}

\author{Emmanuel Hebey}
\address{D\'epartement de Math\'ematiques, Universit\'e de Cergy-Pontoise, 2 
avenue Adolphe Chauvin, 
95302 Cergy-Pontoise cedex, France}
\email{Emmanuel.Hebey@math.u-cergy.fr}

\author{Fr\'ed\'eric Robert}
\address{Department of Mathematics, ETH Z\"urich, CH-8092 Z\"urich, 
Switzerland}
\email{Frederic.Robert@math.ethz.ch}

\subjclass[2000]{Primary 35J60; Secondary 58J05}

\date{November 4, 2002 and, in revised form, December 16, 2002}

\commby{Tobias Colding}

\keywords{Critical elliptic equations, blow-up behaviour, bubbles}

\begin{abstract}
Let $(M,g)$ be a smooth compact Riemannian manifold 
of dimension $n \ge 3$, and $\Delta_g = -div_g\nabla$ the Laplace-Beltrami 
operator. Also let $2^\star$ be the critical Sobolev exponent 
for the embedding of the Sobolev space $H_1^2(M)$ into Lebesgue spaces, 
and $h$ a smooth function on $M$. 
Elliptic equations of critical Sobolev growth like
\[\Delta_gu + hu = u^{2^\star-1}\]
have been the target of investigation 
for decades. A very nice $H_1^2$-theory for the 
asymptotic behaviour of solutions of such an equation
is available since the 1980's. 
In this announcement we present the $C^0$-theory we have recently developed. 
Such a theory provides 
sharp pointwise estimates for the asymptotic behaviour of solutions of the 
above equation.
\end{abstract}

\maketitle

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n \ge 3$. 
We denote by $H_1^2(M)$ 
the standard Sobolev space of functions in $L^2$ with one derivative in 
$L^2$. We consider equations like
\begin{equation}\label{Eq1}
\Delta_gu + hu = u^{2^\star-1},
\end{equation}
where $\Delta_g = -div_g\nabla$ is the Laplace-Beltrami operator, $2^\star = 
2n/(n-2)$ is the critical
Sobolev exponent for the embedding of the Sobolev space $H_1^2(M)$ into 
Lebesgue spaces, $h$ is a 
$C^{0,\theta}$ function on $M$, $0 < \theta < 1$, and 
$u$ is required to be positive. We regard (\ref{Eq1}) as a possible model 
for second order elliptic equations 
of critical Sobolev growth. 
We let $(h_\alpha)$ be a sequence of $C^{0,\theta}$ functions on $M$, $0 < 
\theta < 1$, and let 
$(u_\alpha)$ be a bounded sequence in $H_1^2(M)$ of solutions of (\ref{Eq1}) 
in the sense that  
for any $\alpha$,
\begin{equation}\label{Eqref}
\Delta_gu_\alpha + h_\alpha u_\alpha = u_\alpha^{2^\star-1}
\end{equation}
and $\Vert u_\alpha\Vert_{H_1^2} \le \Lambda$, where $\Lambda > 0$ is 
independent of $\alpha$. We also assume 
that the $h_\alpha$'s are uniformly bounded and that they converge in $L^2$ 
to some limiting 
function $h_\infty$. Then, thanks to 
Struwe \cite{Str}, we know how to describe the asymptotic behaviour 
of the $u_\alpha$'s as $\alpha \to +\infty$. More precisely, it follows from 
Struwe \cite{Str} that, up to a subsequence,
\begin{equation}\label{Eq2}
u_\alpha\hskip.1cm = u^0 + \sum B_\alpha^i + R_\alpha,
\end{equation}
where $u^0$ is a solution of the limit equation
\[\Delta_gu + h_\infty u = u^{2^\star-1},\]
the sum on the right hand side of (\ref{Eq2}) is a finite sum over $i$, 
$B_\alpha^i$ is 
a bubble 
obtained by rescaling fundamental positive solutions of the Euclidean 
equation $\Delta u = u^{2^\star-1}$, 
and the $R_\alpha$'s are lower order terms which 
converge strongly to $0$ in $H_1^2(M)$. This asymptotic description provides 
a very satisfactory $H_1^2$-theory for the asymptotic behaviour of solutions 
of equations like (\ref{Eq1}). 
Let us assume now that the $h_\alpha$'s converge $C^{0,\theta}$ to 
$h_\infty$ for some $0 < \theta < 1$.  
An important issue in the study of equations like (\ref{Eq1}) is to get a 
theory in which 
the above asymptotic 
description holds also in the $C^0$-space, where pointwise estimates are 
involved. 
Such a $C^0$-theory was developed in Druet, Hebey and Robert 
\cite{DruHebRob}. We present the 
theory in this announcement. 

 We know from the Euclidean Sobolev inequality that there exists $K 
> 0$ such that for 
any smooth function $u$ with compact support in ${\mathbb R}^n$,
\[\left(\int_{{\mathbb R}^n}\vert u\vert^{2^\star}dx\right)^{1/2^\star} \le 
K\left(\int_{{\mathbb R}^n}\vert\nabla u\vert^2dx\right)^{1/2}\hskip.1cm .\]
The sharp constant $K$ in this inequality is
\[K_n = \sqrt{\frac{4}{n(n-2)\omega_n^{2/n}}},\]
where $\omega_n$ is the volume of the unit $n$-sphere. We let $u$ be the 
function 
on ${\mathbb R}^n$ given by
\[u(x) = \left(1 + \frac{\vert 
x\vert^2}{n(n-2)}\right)^{1-\frac{n}{2}}\hskip.1cm .\]
It is easily seen that $u$ is an extremal function for the sharp Euclidean 
Sobolev inequality. Thanks to 
Caffarelli, Gidas and Spruck \cite{CafGidSpr}, $u$ is also the unique 
positive solution 
of the critical Euclidean equation
\[\Delta u = u^{2^\star-1}\]
which is such that $u(0) = \max_{{\mathbb R}^n}u = 1$. Its energy $E(u) = 
\Vert u\Vert_{2^\star}$ is given by 
$E(u) = K_n^{-(n-2)/2}$. From now on, we 
let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n \ge 3$. 
We also let 
$(x_\alpha)$ be a converging sequence of points in $M$, and $(\mu_\alpha)$ 
a sequence of positive real numbers 
converging to $0$ as $\alpha \to +\infty$. We define the standard bubble 
with respect to the $x_\alpha$'s and $\mu_\alpha$'s as the 
sequence $(B_\alpha)$ of functions on $M$ given by
\[B_\alpha(x) = \Biggl(\frac{\mu_\alpha}{\mu_\alpha^2 + 
\frac{d_g(x_\alpha,x)^2}{n(n-2)}}\Biggl)^{\frac{n-2}{2}},\]
where $d_g$ is the distance with respect to $g$. 
In other words, standard bubbles are rescalings of fundamental solutions of 
the critical Euclidean equation
$\Delta u = u^{2^\star-1}$. Now we return to equation (\ref{Eqref}), 
and we assume that there exist $0 < \theta < 1$ and 
a $C^{0,\theta}$ function $h_\infty$ on $M$ such that
\begin{equation}\label{Eq3}
\begin{split}
&\hbox{the operator}\hskip.1cm \Delta_g + h_\infty
\hskip.1cm\hbox{is coercive}\hskip.1cm ,\hskip.1cm\hbox{and}\\
&h_\alpha \to h_\infty\hskip.1cm\hbox{in}\hskip.1cm C^{0,\theta}(M)\hskip.1cm
\hbox{as}\hskip.1cm \alpha \to +\infty\hskip.1cm .
\end{split}
\end{equation}
We let $(u_\alpha)$ be a sequence of solutions of (\ref{Eqref}), $u_\alpha 
\in C^{2,\theta}(M)$, $u_\alpha$ positive. The existence 
of $u_\alpha$ implies the coercivity of the operator $\Delta_g + h_\alpha$. 
An estimate like the one in Theorem \ref{Th} below 
implies the coercivity of the operator $\Delta_g + h_\infty$ when $N \ge 1$. 
We also assume that there exists $\Lambda > 0$ such that
$E(u_\alpha) \le \Lambda$ for all $\alpha$, where $E(u) = \Vert 
u\Vert_{2^\star}$. Multiplying $(\ref{Eqref})$ by 
$u_\alpha$ and integrating over $M$, it is easily seen that the $u_\alpha$'s 
are bounded in $H_1^2(M)$. Up to a 
subsequence, we may therefore assume that for some $u^0 \in H_1^2(M)$, 
\begin{equation}\label{Eq4}
u_\alpha \rightharpoonup 
u^0\hskip.2cm\hbox{weakly}\hskip.1cm\hbox{in}\hskip.1cm H_1^2(M)
\end{equation}
as $\alpha \to +\infty$. Then $u^0$ is a solution of the limit equation
\begin{equation}\label{Eq5}
\Delta_gu + h_\infty u = u^{2^\star-1}\hskip.1cm .
\end{equation}
By the maximum principle and regularity theory, $u^0 \in C^{2,\theta}(M)$ 
and either $u^0 \equiv 0$ or 
$u^0 > 0$ everywhere in $M$. 
As already mentioned, we know from Struwe \cite{Str} that the $u_\alpha$'s 
express as $u^0$, plus a sum of standard 
bubbles, plus lower order terms in $H_1^2(M)$. We want to prove that the 
$u_\alpha$'s are $C^0$-controlled, from below and from 
above, by $u^0$ and standard bubbles. Our theorem is as follows:

\begin{theorem}\label{Th} Let $(M,g)$ be a smooth compact Riemannian 
manifold of dimension $n \ge 3$, $(h_\alpha)$ a 
sequence of $C^{0,\theta}$ functions on $M$ such that \eqref{Eq3} is 
satisfied, and 
$(u_\alpha)$ a sequence of positive solutions to \eqref{Eqref} such that 
$E(u_\alpha) \le \Lambda$ 
for some $\Lambda > 0$ and all $\alpha$. Then there exist 
$N \in {\mathbb N}$, converging sequences $(x_{i,\alpha})$ in $M$, and 
sequences $(\mu_{i,\alpha})$ 
of positive real numbers converging to $0$, $i = 1,\dots,N$, such that, up 
to a
subsequence,
\begin{eqnarray*}
&&\left(1 - \varepsilon_\alpha\right)u^0(x) + 
\frac{1}{C} \sum_{i=1}^N
\Biggl(\frac{\mu_{i,\alpha}}{\mu_{i,\alpha}^2 + 
\frac{d_g(x_{i,\alpha},x)^2}{n(n-2)}}\Biggr)^{\frac{n-2}{2}}\\
&&\qquad \le u_\alpha(x) \le \left(1 + \varepsilon_\alpha\right)u^0(x) + 
C \sum_{i=1}^N
\Biggl(\frac{\mu_{i,\alpha}}{\mu_{i,\alpha}^2 + 
\frac{d_g(x_{i,\alpha},x)^2}{n(n-2)}}\Biggr)^{\frac{n-2}{2}}
\end{eqnarray*}
for all $x \in M$ and all $\alpha$, where $u^0$ is the solution of the limit 
equation \eqref{Eq5} given 
by \eqref{Eq4}, $C > 1$ is independent of $\alpha$ and $x$, and 
$(\varepsilon_\alpha)$, independent of $x$, 
is a sequence of positive real numbers converging to $0$ as $\alpha \to +
\infty$. 
In particular, 
the $u_\alpha$'s are $C^0$-controlled, on both sides, by $u^0$ and standard 
bubbles.
\end{theorem}
 
A complement to Theorem \ref{Th} is that $C$ can be chosen as close as we 
want to 
1 if we restrict the equation in Theorem \ref{Th} to small neighbourhoods of 
the geometrical blow-up points, defined as the limits of the 
$x_{i,\alpha}$'s. For instance, if $u^0 \not\equiv 0$, 
or if the $u_\alpha$'s just have one geometrical blow-up point, then 
for any $\varepsilon > 0$, there exists $\delta_\varepsilon > 0$ such that, 
up to a subsequence,
\begin{eqnarray*}
&&\left(1 - \varepsilon_\alpha\right)u^0(x) + 
\frac{1}{1+\varepsilon} \sum_{i=1}^N 
\Biggl(\frac{\mu_{i,\alpha}}{\mu_{i,\alpha}^2 + 
\frac{d_g(x_{i,\alpha},x)^2}{n(n-2)}}\Biggr)^{\frac{n-2}{2}}\\
&&\qquad \le u_\alpha(x) \le \left(1 + \varepsilon_\alpha\right)u^0(x) + 
\left(1 + \varepsilon\right) \sum_{i=1}^N
\Biggl(\frac{\mu_{i,\alpha}}{\mu_{i,\alpha}^2 + 
\frac{d_g(x_{i,\alpha},x)^2}{n(n-2)}}\Biggr)^{\frac{n-2}{2}}
\end{eqnarray*}
for all $\alpha$, all $x_0 \in {\mathcal S}$, and all $x \in 
B_{x_0}\left(\delta_\varepsilon\right)$, where 
${\mathcal S}$ is the set consisting of the limits of the $x_{i,\alpha}$'s 
as $\alpha \to +\infty$, 
and $B_{x_0}\left(\delta_\varepsilon\right)$ is the geodesic ball of center 
$x_0$ and radius $\delta_\varepsilon$. 
Outside the $B_x(\delta_\varepsilon)$'s, $x \in {\mathcal S}$, the 
$u_\alpha$'s converge $C^{2,\theta}$ to $u^0$. The 
estimate then extends to $M$ in the particular case where $u^0 \not\equiv 
0$. A refined estimate on the $u_\alpha$'s is given below in 
$(\ref{Asympt})$.
Another complement to Theorem \ref{Th} is that the bubbles in this theorem 
satisfy the Struwe $H_1^2$-decomposition 
\cite{Str}. More precisely, 
we also have that for any $x \in M$ and any $\alpha$,
\[u_\alpha(x) = u^0(x) + \sum_{i=1}^N
\Biggl(\frac{\mu_{i,\alpha}}{\mu_{i,\alpha}^2 + 
\frac{d_g(x_{i,\alpha},x)^2}{n(n-2)}}\Biggl)^{\frac{n-2}{2}} 
+ R_\alpha(x),\]
where the $R_\alpha$'s, 
$R_\alpha \in H_1^2(M)$ for all $\alpha$, are such that $R_\alpha \to 0$ 
strongly in $H_1^2(M)$ as $\alpha \to +\infty$. 
Moreover, 
\[E(u_\alpha)^{2^\star} = E(u^0)^{2^\star} + NK_n^{-n} + r_\alpha\]
for all $\alpha$, where the $r_\alpha$'s, $r_\alpha \in {\mathbb R}$, 
are such that $r_\alpha \to 0$ as $\alpha \to +\infty$. In other words, the 
energies split also.

 Such a theorem has important applications when dealing with sharp 
Sobolev inequalities. 
In this case $N = 1$, and the estimate in Theorem \ref{Th} appeared to be a 
key point when discussing 
the validity and the existence of extremal functions for sharp inequalities 
like
\[\Vert u\Vert_{2^\star}^2 \le K_n^2\Vert\nabla u\Vert_2^2 + B \Vert 
u\Vert_2^2,\]
where $B$ is a constant independent of $u$. 
Monographs on sharp constant problems are Druet-Hebey \cite{DruHeb}, and 
Hebey \cite{Heb1}. 
Other directions of research are dealing with the energy function
(see for instance Hebey \cite{Heb2}) or with 
compactness results as in Schoen \cite{Sch1,Sch2}. Applications of the 
theorem in such directions are 
in Druet \cite{Dru3}, where compactness results and low dimension phenomena 
are discussed, and
the lower semicontinuity of the energy function for equations like 
$(\ref{Eqref})$ is proved. The energy function 
for equations like $(\ref{Eqref})$ is defined by
\[E(\alpha) = \inf_{u \in {\mathcal S}_\alpha}E(u),\]
where $E(u) = \Vert u\Vert_{2^\star}$ and ${\mathcal S}_\alpha$ is the set 
consisting of the solutions 
of $(\ref{Eqref})$. From the historical viewpoint, without any pretention to 
exhaustivity, 
Atkinson-Peletier \cite{AtkPel} and Brezis-Peletier \cite{BrePel} have been 
concerned 
with the description of the pointwise
behaviour of sequences of solutions of equations like (\ref{Eq1}), dealing 
with radially 
symmetrical solutions $u_\varepsilon$ of the semicritical equations $\Delta 
u = u^{2^\star-1-\varepsilon}$ 
on the unit ball of the Euclidean space. More recent developments 
in this specific direction are in Robert \cite{Rob1,Rob2}. An estimate 
like in Theorem \ref{Th} in the case $N = 1$, stating that solutions of 
minimal energy of equations like (\ref{Eq1}) are controlled from above by a 
standard bubble, 
appeared then in Han \cite{Han} when dealing 
with solutions $u_\varepsilon$ of the equations $\Delta u = 
u^{2^\star-1-\varepsilon}$ on 
bounded open subsets of the Euclidean space, in Hebey-Vaugon \cite{HebVau} 
when dealing with (\ref{Eq1}) and arbitrary Riemannian manifolds, and in Li 
\cite{Li1,Li2} when dealing with 
equations like (\ref{Eq1}) on the unit sphere. Improvements, still in the 
case $N = 1$, are 
in Druet \cite{Dru1,Dru2} and Druet-Robert \cite{DruRob}. We refer also to 
Schoen-Zhang \cite{SchZha}. 
Examples of blowing-up sequences of solutions of equations like 
$(\ref{Eq1})$ are in 
Druet-Hebey \cite{DruHebEx} and Druet-Hebey-Robert 
\cite{DruHebRob}.

\section{Proof of the theorem}

We only present a very brief sketch of the proof. 
Details on the special case where $N = 1$, which is actually due to Druet 
and Robert, can be found in 
the monograph \cite{DruHeb} by Druet and Hebey. 
The intricate general case where $N$ is arbitrary is in Druet, Hebey, and 
Robert \cite{DruHebRob}. 
We assume in what follows 
that $\max u_\alpha \to +\infty$ as $\alpha \to +\infty$. If not the case, 
up to a subsequence, the $u_\alpha$'s 
converge $C^2$ to $u^0$, and the theorem is true. As a remark, 
the proof of the theorem goes through the proof of a slightly stronger 
result. 
All that follows is up to a subsequence. We let $G$ be the Green's function of 
the operator $\Delta_g + h_\infty$, and let 
$\Phi$ be the continuous function on $M\times M$ given by
\[\Phi(x,y) = (n-2)\omega_{n-1}d_g(x,y)^{n-2}G(x,y)\]
if $x \not= y$, and $\Phi(x,y) = 1$ if $x = y$, where $\omega_{n-1}$ is the 
volume of the unit $(n-1)$-sphere. 
Then we claim that for any converging sequence $(x_\alpha)$ of points in $M$, 
and any $\alpha$,
\begin{equation}\label{Asympt}
u_\alpha(x_\alpha) = \Bigl(1+o(1)\Bigr)u^0(x_\alpha) + 
\sum_{i=1}^N\Bigl(\Phi(x_i,x) + o(1)\Bigr)B_\alpha^i(x_\alpha),
\end{equation}
where $x$ is the limit of the $x_\alpha$'s, $x_i$ is the limit of the 
$x_{i,\alpha}$'s, and $B_\alpha^i$ is the standard 
bubble with respect to the $x_{i,\alpha}$'s and $\mu_{i,\alpha}$'s given by 
Theorem \ref{Th}. It is easily seen that 
the theorem and the remarks after this theorem follow from such an 
asymptotic description. 
The proof of these asymptotics splits into four main steps. The first 
preliminary step consists in getting rescaling 
invariant estimates. Such estimates basically state that there exist 
$N \in {\mathbb N}^\star$, converging sequences $(x_{i,\alpha})$ in $M$, and 
sequences $(\mu_{i,\alpha})$ 
of positive real numbers converging to $0$, $i = 1,\dots,N$, such that
\[R_\alpha^N(x)^{\frac{n}{2}-1}u_\alpha(x) \le C\]
for all $x$ and all $\alpha$, where
$R_\alpha^N(x)$ is the minimum over $i$ of the distances from the 
$x_{i,\alpha}$'s to $x$. The key idea here is that 
if such an estimate is false, then we can construct another blow-up point. 
This weak estimate comes with an important refinement 
and complementary informations on the limit of the $u_\alpha$'s 
outside blow-up points. Then we 
need to prove that the upper estimate in Theorem \ref{Th} holds. For that 
purpose, we rearrange the $x_{i,\alpha}$'s in families. 
Inside a family, blow-up points are close to each other. Two families are 
far from each other. If the $y_{i,\alpha}$'s 
are the representatives of such families having the largest 
$\mu_{i,\alpha}$, $i = 1,\dots, k$, the second step in the proof consists in 
proving that an upper estimate like in Theorem \ref{Th} holds 
with respect to the $y_{i,\alpha}$'s. We prove that there exist $C > 0$ and 
$R > 0$ such that
\[u_\alpha(x) \le \bigl(1+o(1)\bigr)u^0(x) + C\sum_{i=1}^kB_\alpha^i(x)\]
for all $\alpha$ and all $x \in M\backslash 
\bigcup_{i=1}^kB_{y_{i,\alpha}}(R\mu_{i,\alpha})$, where $B_\alpha^i$ is the 
standard 
bubble with respect to $y_{i,\alpha}$ and its corresponding 
$\mu_{i,\alpha}$. The proof of such an estimate goes through 
the establishment of a scale of intermediate estimates, referred to as
$\varepsilon$-sharp estimates with $0 < \varepsilon < \frac{n-2}{2}$, the 
weakest of these when 
$\varepsilon = \frac{n-2}{2}$ being like the weak estimate we discussed 
above. The proof of the upper estimate as in the theorem 
then reduces to the proof that this estimate holds inside the 
$B_{y_{i,\alpha}}(R\mu_{i,\alpha})$'s. This is the third step in the proof. 
We proceed here by induction. We 
consider subfamilies of blow-up points, and prove that the estimate holds in 
$B_{y_{i,\alpha}}(R\mu_{i,\alpha})$, 
outside smaller balls of sub-representatives, and so on up to the 
point where we have exhausted all the blow-up points. In each step of this 
induction 
process, we pass through 
$\varepsilon$-sharp estimates, and, in some sense, 
let then $\varepsilon \to 0$ to get the sharp estimate. Once we have proved 
that the upper estimate 
of Theorem \ref{Th} holds in $M$, the argument becomes simpler and more 
conventional. The fourth step in the proof consists in 
proving that the above asymptotics follow from  
the Green representation formula
\begin{eqnarray*}
u_\alpha(x_\alpha) - u^0(x_\alpha)
& = & \int_MG_\alpha(x_\alpha,x)\bigl(u_\alpha(x)^{2^\star-1}-u^0(x)^{2^%
\star-1}\bigr)dv_g\\
&&\hskip.2cm + \int_MG_\alpha(x_\alpha,x)\bigl(h_\infty(x) - 
h_\alpha(x)\bigr)u^0(x)dv_g
\end{eqnarray*}
where $G_\alpha$ is the Green's function for the operator $\Delta_g + 
h_\alpha$. 
The different terms that are involved in this formula are controlled 
thanks to the upper estimate we have just discussed. The asymptotics follow 
from rather standard developments. As already mentioned, 
since $\Phi$ is continuous, the theorem and the remarks after the theorem 
are then easy consequences of the asymptotics. 
We refer to Druet, Hebey and Robert \cite{DruHebRob} for more details, and 
also to Druet and Hebey 
\cite{DruHeb} for the special case where $N = 1$.


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