\documentclass[reqno]{amsart}
\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 11, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2005/11\hfil An Aleksandrov-type estimate]
{Aleksandrov-type estimates for a parabolic Monge-Amp\`ere equation}
\author[D. Hartenstine\hfil EJDE-2005/11\hfilneg]
{David Hartenstine}
\address{Department of Mathematics \\
Western Washington University\\
516 High Street, Bond Hall 202\\
Bellingham, WA 98225-9063, USA}
\email{david.hartenstine@wwu.edu}
\date{}
\thanks{Submitted January 12, 2005. Published January 27, 2005.}
\thanks{Partially supported by grant DMS-0091675 from the University
of Utah's NSF VIGRE}
\subjclass[2000]{35K55, 35B45, 35D99}
\keywords{Parabolic Monge-Amp\`ere measure; pointwise estimates}
\begin{abstract}
A classical result of Aleksandrov allows us to estimate the size of
a convex function $u$ at a point $x$ in a bounded domain $\Omega$
in terms of the distance from $x$ to the boundary of $\Omega$ if
$\int_{\Omega} \det D^{2}u \, dx < \infty$. This estimate plays a
prominent role in the existence and regularity theory of the
Monge-Amp\`ere equation. Jerison proved an extension of Aleksandrov's
result that provides a similar estimate, in some cases for which this
integral is infinite. Guti\'{e}rrez and Huang proved a variant of
the Aleksandrov estimate, relevant to solutions of a parabolic
Monge-Amp\`{e}re equation. In this paper, we prove Jerison-like
extensions to this parabolic estimate.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\section{Introduction}\label{intro}
In studying the regularity and existence of weak solutions
(in the sense of Aleksandrov) to the Dirichlet problem for
the Monge-Amp\`{e}re equation:
\begin{equation} \label{dpMA}
\begin{gathered}
\det D^2 u = \mu \quad \text{in } \Omega, \\
u|_{\partial \Omega} = g,
\end{gathered}
\end{equation}
where $\mu$ is a Borel measure on the convex domain $\Omega$ and
$g \in C(\partial \Omega)$, the following estimate of Aleksandrov plays a
critical role. For its applications to this problem, see, for example,
\cite{rauch:dpm77}, \cite{caffarelli:srp91}, and \cite{gutierrez:mae01}.
A variant of this estimate appears in \cite{caffarelli:lpv90}.
\begin{theorem}[Aleksandrov's estimate] \label{alekest}
Let $\Omega$ be a bounded convex domain in $\mathbb{R}^n$, and let
$u \in C(\bar \Omega)$ be convex, with $u = 0$ on $\partial \Omega$.
Then for all $x \in \Omega$,
\begin{equation}\label{eq:alekest}
|u(x)|^n \leq C_n (\mathop{\rm diam} \Omega)^{n-1}
\mathop{\rm dist}(x, \partial \Omega) \, Mu(\Omega),
\end{equation}
where $C_n$ is a dimensional constant and $Mu$ is the Monge-Amp\`{e}re measure associated to $u$.
\end{theorem}
This estimate allows one to estimate the size of $u$ at a point $x$ in
terms of the distance from $x$ to the boundary of the domain.
However, if $u$ is such that $Mu(\Omega) = \infty$ (which can occur
if $|Du| \to \infty$ at $\partial \Omega$), (\ref{eq:alekest})
does not give any information about the size of $u(x)$.
Jerison, in \cite{jerison:mpe96}, extended this inequality, using an
affine-invariant normalized distance to the boundary, to an estimate
(Theorem~\ref{jer2}) that is useful even if $Mu (\Omega) = \infty$,
provided $Mu$ does not blow up too quickly at the boundary.
This result allows for a Caffarelli-style regularity theory for such problems,
provided $Mu$ satisfies a technical requirement, weaker than the doubling
condition, on the cross-sections of $u$; see \cite{jerison:mpe96}
and \cite{gutierrez:rws03}.
The parabolic Monge-Amp\`{e}re operator $u_t \det D^{2}_{x}u$ was
introduced in \cite{krylov:scf76}. It is related to the problem
of deformation of surfaces by Gauss curvature (see
\cite{tso:dhg85}). This operator is also considered in the
following works:
\cite{wang:gmt93, gutierrez:gtc98,chen:cfg02,chen:gsf01,wang:fbv87,
gutierrez:wep01,wang:eur92}.
When studying entire solutions of the parabolic Monge-Amp\`{e}re equation \break
$-u_t \det D^{2}_{x}u = 1$, Guti\'{e}rrez and Huang (\cite{gutierrez:gtc98})
extended the Aleksandrov estimate (Theorem \ref{alekest}) to parabolically
convex functions on bounded bowl-shaped domains. This estimate again
degenerates when the parabolic Monge-Amp\`{e}re measure associated to $u$
of the entire domain is infinite. The purpose of this note is to extend
the estimates of Jerison to the parabolic setting. These estimates are
given below in Lemma \ref{parabol1} and Theorem \ref{parabol2}.
Because Jerison's estimates allow for a regularity theory for problem
(\ref{dpMA}) when $\mu(\Omega) = \infty$, it is our hope that the
estimates presented here will allow one to deduce regularity properties of
parabolically convex solutions of the Dirichlet problem:
\begin{gather*}
-u_t \, \det D^2 u = f \quad\mbox{in } E \\
u\big|_{\partial_{p} E} = g,
\end{gather*}
where $f \geq 0$ may fail to be in $L^{1}(E)$, $E \subset
\mathbb{R}^{n+1}$ is bowl-shaped, and $\partial_{p}E$ is the
parabolic boundary of $E$. This would extend the regularity
theory found in \cite{chen:cfg02, chen:gsf01,wang:fbv87, wang:eur92},
all of which assume that $f$ is bounded.
\section{Preliminaries}\label{prelim}
We begin this section by reviewing the basic theory of weak or generalized
solutions, in the Aleksandrov sense, to the (elliptic) Monge-Amp\`{e}re equation.
Proofs of these results and historical notes indicating their original sources
can be found in the books \cite{bakelman:can94} and \cite{gutierrez:mae01}.
Given $u:\Omega \to \mathbb{R}$ we recall that the
normal mapping (or subgradient) of $u$
is defined by
\[
\partial u(x_{0}) = \{p \in \mathbb{R}^{n}: u(x) \geq u(x_{0}) + p \cdot (x -
x_{0}), \; \forall \,x \in \Omega\};
\]
and if $E \subset \Omega$, then we set $\partial u(E) = \bigcup_{x
\in E} \partial u(x)$. Note that the normal map of $u$ at a point $x_0$
is the set of points $p$ which are normal vectors for supporting hyperplanes
to the graph of $u$ at $x_0$.
If $\Omega$ is open and $u \in C(\Omega)$ then the family of sets
\[
S = \{E \subset \Omega: \partial u(E) \text{ is Lebesgue
measurable} \}
\]
is a Borel $\sigma$--algebra. The map $M\!u:S \to
\bar{\mathbb{R}}$ defined by $M\!u(E) = |\partial u(E)|$ (where $|S|$
indicates the Lebesgue measure of the set $S$) is a measure,
finite on compact subsets, called the Monge--Amp\`ere measure
associated with the function $u$. The convex function $u$ is a
weak (Aleksandrov) solution of $\det
D^{2}u = \nu$ if the Monge--Amp\`ere measure $M\!u$ associated with
$u$ equals the Borel measure $\nu$.
We use the notation $B_{r}(y)$ for the open Euclidean ball of radius $r$
with center $y$. The dimension of $B_r (y)$ should be clear from context.
\begin{definition}\label{defnormalized} \rm
A convex domain $\Omega \subset \mathbb{R}^{n}$ with center of mass at the
origin is said to be normalized if $B_{\alpha_n}(0) \subset \Omega \subset B_1(0)$,
where $\alpha_n = n^{-3/2}$.
\end{definition}
The following lemma allows us to carry out our analysis in a normalized setting.
It is a consequence of a result of John on ellipsoids of minimum volume.
See Section 1.8 of \cite{gutierrez:mae01} and its references for more detail.
\begin{lemma}\label{fritzjohn}
If $\Omega$ is a bounded convex domain, there exists an affine transformation
$T$ such that $T(\Omega)$ is normalized.
\end{lemma}
We now introduce the normalized distance to the boundary used
by Jerison in \cite{jerison:mpe96}.
\begin{definition}\label{dist} \rm
Let $\Omega \subset \mathbb{R}^{n}$ be bounded, open and convex.
The normalized distance from $x \in \Omega$ to the boundary of $\Omega$ is
\[
\delta(x,\Omega) = \min \big\{\frac{|x-x_1|}{|x-x_2|}: x_1, \, x_2 \in
\partial \Omega \mbox{ and $x, x_1, x_2$ are collinear} \big\}.
\]
\end{definition}
The most important properties of this distance for our purposes are summarized
in the following lemma.
\begin{lemma}\label{deltaprop}
Let $\Omega$ be a bounded convex domain.
\begin{itemize}
\item[(a)] If $T$ is an invertible affine transformation on $\mathbb{R}^{n}$, then
\[
\delta(x,\Omega) = \delta(Tx, T(\Omega)).
\]
\item[(b)] If $\Omega$ is normalized, $\delta(x,\Omega)$ is equivalent to
$\mathop{\rm dist}(x,\partial \Omega)$, i.e. there exist constants $C_1$ and
$C_2$ (depending only on the dimension) such that
\[
C_1 \delta(x,\Omega) \leq \mathop{\rm dist}(x,\partial \Omega)
\leq C_2 \delta(x,\Omega)
\]
for all $x \in \Omega$, where $\mathop{\rm dist}$ is the Euclidean distance.
\item[(c)] For all $x \in \Omega$, $\mathop{\rm dist}(x,\partial \Omega)
\leq\mathop{\rm diam}(\Omega) \delta(x,\Omega)$.
\end{itemize}
\end{lemma}
We now state Jerison's estimates. The first (Lemma~\ref{jer1})
is \cite[Lemma 7.2]{jerison:mpe96}.
Estimate (\ref{eq:jer1}) is similar to Aleksandrov's estimate (\ref{eq:alekest}),
with the normalized notion of distance replacing the standard one,
and the Lebesgue measure of $\Omega$ replacing the diameter term.
\begin{lemma}\label{jer1}
Let $\Omega$ be an open convex set and suppose $u \in C(\overline{\Omega})$ is
convex and zero on $\partial \Omega$. Then, for all $x \in \Omega$,
\begin{equation}\label{eq:jer1}
|u(x)|^n \leq C \delta(x,\Omega)|\Omega| Mu(\Omega)
\end{equation}
where $C$ is a constant depending only on the dimension.
\end{lemma}
Note that the estimate (\ref{eq:jer1}) gives no information when
$Mu(\Omega) = \infty$. If this is the case, $Mu$ must blow up near
$\partial \Omega$, but this is precisely where $\delta(\cdot,\Omega)$ is small.
As a consequence, the estimate in the next result
(\cite[Lemma 7.3]{jerison:mpe96}) may be meaningful.
\begin{theorem}\label{jer2}
Let $\Omega$ be bounded, open, convex and normalized, and suppose
$u \in C(\overline{\Omega})$ is convex and zero on $\partial \Omega$.
For each $\epsilon \in (0,1]$, there exists a constant $C(n,\epsilon)$
such that
\begin{equation}\label{eq:jer2}
|u(x_0)|^n \leq C(n,\epsilon) \delta(x_0,\Omega)^{\epsilon}
\, \int_{\Omega} \delta(x,\Omega)^{1-\epsilon} \, dMu(x)
\end{equation}
for all $x_0 \in \Omega$.
\end{theorem}
We now introduce some terminology and notation for the parabolic problem.
Let $D\subset \mathbb{R}^{n+1}$ and let $t \in \mathbb{R}$. Then define
$$
D(t) = \{x \in \mathbb{R}^n: (x,t) \in D \}.
$$
\begin{definition} \rm
The domain $D$ is said to be bowl-shaped if $D(t)$ is convex for every $t$
and $D(t_1) \subset D(t_2)$ whenever $t_1 \leq t_2$. If $D$ is
bounded, let $t_0 = \inf \{t:D(t) \not = \emptyset \}$. Then the parabolic
boundary of $D$ is defined to be
$$
\partial_p D = (\bar D(t_0) \times \{t_0\}) \cup \Big(\bigcup_{t \in
\mathbb{R}} (\partial D(t) \times \{t\}) \Big).
$$
\end{definition}
For a bowl-shaped domain $D$ we define the set $D_{t_0}$ to be $D_{t_0} = D
\cap \{(x,t):t \leq t_0\}$.
\begin{definition} \rm
A function $u: \mathbb{R}^n \times \mathbb{R} \to \mathbb{R}$,
$u =u(x,t)$, is called parabolically convex (or convex-monotone) if it is
continuous, convex in $x$ and non-increasing in $t$.
\end{definition}
We now define the parabolic normal map and parabolic Monge-Amp\`ere measure.
As in the elliptic case, this will lead to the notion of weak solution for
this operator. Let $D \subset \mathbb{R}^{n+1}$ be an open, bounded
bowl-shaped domain, and $u$ be a continuous real-valued function on $D$.
The parabolic normal mapping
of $u$ at a point $(x_0,t_0)$ is the set-valued function $P_u(x_0,t_0)$ given by
\begin{gather*}
\{(p,h):u(x,t) \geq u(x_0,t_0) + p \cdot(x-x_0) \; \mathrm{for} \;
\mathrm{all} \; t\leq t_0 \; \mathrm{and} \; x \in D(t), \\
h=p \cdot x_0 - u(x_0,t_0)\}.
\end{gather*}
As before, the parabolic normal mapping of a set $E \subset D$ is defined to
be the union of the parabolic normal maps of each point in the set. The
family of subsets $E$ of $D$ for which $P_u(E)$ is Lebesgue measurable is a
Borel $\sigma$-algebra and the map $M_{p}(E) = |P_u (E)|$ is a measure,
called the parabolic Monge-Amp\`ere measure associated to
the function $u$. These results are proved in \cite{wang:gmt93}.
We remark that, because of the translation invariance of the Lebesgue measure,
the parabolic Monge-Amp\`ere measure of a function $u$ is identical to the
parabolic Monge-Amp\`ere measure of $u - \lambda$ for any constant $\lambda$.
We conclude this section with a parabolic analog of Aleksandrov's estimate
(Theorem \ref{alekest}) due to Guti\'errez and
Huang (\cite{gutierrez:gtc98}).
\begin{theorem}\label{parAleksandrov}
Let $D \subset \mathbb{R}^{n+1}$ be an open bounded bowl-shaped
domain, and let $u \in C(\bar D)$ be a parabolically convex function with $u = 0$
on $\partial_{p} D$. If $(x_0,t_0) \in D$, then
$$
|u(x_0,t_0)|^{n+1} \leq C_n \mathop{\rm dist}(x_0,\partial D(t_0))
\mathop{\rm diam}(D(t_0))^{n-1}
M_{p}(D_{t_{0}})
$$
where $C_n$ is a dimensional
constant, and $M_p$ is the parabolic Monge-Amp\`ere measure associated to $u$.
\end{theorem}
\section{Parabolic estimates}\label{parest}
In this section, we prove parabolic versions of Jerison's estimates.
We adapt the arguments given in \cite{jerison:mpe96} to our situation.
The first is the analog of Lemma~\ref{jer1}.
\begin{lemma}\label{parabol1}
Let $D$ be a bounded, open bowl-shaped domain in
$\mathbb{R}^{n+1}$. Suppose $u \in C(\bar D)$ is parabolically convex and
$u|_{\partial_p D} = 0$. Then there exists a dimensional constant $C_n$
such that
$$
|u(x_0,t_0)|^{n+1} \leq C_n \delta(x_0, D(t_0)) |D(t_0)||P_{u}(D_{t_0})|
$$
for all $(x_0,t_0) \in D$, where $\delta(x_0,D(t_0))$ is the normalized
distance from $x_0$ to the boundary of the n-dimensional convex set
$D(t_0)$, and $|P_{u}(D_{t_0})| = M_{p}(D_{t_0})$ is the Lebesgue measure of
the set $P_{u}(D_{t_0}) \subset \mathbb{R}^{n+1}$.
\end{lemma}
\begin{proof} $D(t_0)$ is a bounded convex subset of $\mathbb{R}^n$.
By Lemma \ref{fritzjohn}, we may choose an affine transformation $T$ of
$\mathbb{R}^n$ that normalizes $D(t_0)$. Define $\tilde T:\mathbb{R}^{n+1}
\to \mathbb{R}^{n+1}$ by $\tilde T(x,t) = (Tx,t)$. Then $\tilde
T(D_{t_0}) \subset B_1(0) \times (-\infty,t_0]$. Let $v(z) =
u(\tilde T^{-1}z)$ for $z \in \tilde T(D).$ Then $\tilde T(D)$ is a bowl-shaped domain, $v$ is continuous on the closure of $\tilde T(D)$, is
parabolically convex, and is zero on $\partial_{p}\tilde T(D)$.
Now apply the parabolic Aleksandrov estimate (Theorem \ref{parAleksandrov}) to
$v$ in $\tilde T(D)$ to obtain
\begin{equation}\label{parAlekest}
\begin{aligned}
|u(x_0,t_0)|^{n+1} &= |v(\tilde T(x_0,t_0))|^{n+1} \\
&\leq C_n \mathop{\rm dist} (Tx_0,
\partial \tilde T(D(t_0)))[\mathop{\rm diam} (\tilde T(D(t_0)))]^{n-1} |P_{v}(\tilde
T(D_{t_0}))|.
\end{aligned}
\end{equation}
Next, we establish the change-of-variable formula
\begin{equation}\label{changevar}
|P_{v}(\tilde T(D_{t_0}))| = |\det T^{-1}| \;|P_{u}(D_{t_0})|.
\end{equation}
For simplicity, we make the following abuse of notation:
when we write $u$ or $v$ as functions of $x$ only, we mean the restrictions
of $u$ and $v$ to $D(t_0)$. Let $p \in \partial u(x_0)$. Then
\[
u(x,t_0) \geq u(x_0,t_0) + p \cdot (x - x_0)
\]
for all $x \in D(t_0)$. Since $u$ is non-increasing in $t$,
\[
u(x,t) \geq u(x,t_0) \geq u(x_0,t_0) + p \cdot (x - x_0)
\]
for all $t\leq t_0$ and $x \in D(t)$, so $(p,h) \in P_{u}(x_0,t_0)$ where $h =
p \cdot x_0 - u(x_0,t_0)$. If $p \not \in \partial u(x_0)$, then $(p,h) \not \in
P_{u}(x_0,t_0)$; therefore, $p \in \partial u(x_0)$ if and only if $(p,h) \in P_{u}(x_0,t_0)$. It is not hard to see that $p \in \partial u(x_0)$ if and only if $(T^{-1})^{t}p \in
\partial v(Tx_0)$. Then as above, for $t \leq t_0$ and $y \in \tilde T(D)(t)$,
\[
v(y,t) \geq v(y,t_0) + (T^{-1})^{t}p \cdot(y - Tx_0).
\]
Hence, $(T^{-1})^{t}p \in \partial v(Tx_0)$ if and only if
$((T^{-1})^{t}p,\tilde h)\in P_{v}(Tx_0,t_0)$, where $\tilde h = (T^{-1})^{t}p
\cdot Tx_0 - v(Tx_0,t) = p \cdot x_0 - u(x_0)= h$. In other words, $(p,h) \in
P_{u}(x_0,t_0)$ if and only if $((T^{-1})^{t}p,h) \in P_{v}(Tx_0,t_0)$. We also have $((T^{-1})^{t}p,h) = (\tilde T^{-1})^{t}(p,h)$ which implies that
\[
(\tilde
T^{-1})^{t}P_{u}(E) = P_{v}(\tilde T(E))
\]
for any Borel set $E \subset D$. In
particular, $(\tilde T^{-1})^{t}P_{u}(D_{t_0}) = P_{v}(\tilde T(D_{t_0}))$.
This implies that
$$
|\det \tilde T^{-1}| \; |P_{u}(D_{t_0})| = |P_{v}(\tilde T(D_{t_0}))|,
$$
but $\det \tilde T^{-1} = \det T^{-1}$, showing (\ref{changevar}).
Then using equation (\ref{changevar}), Lemma \ref{deltaprop},
inequality (\ref{parAlekest}), and the fact that
$|\det T^{-1}| \leq C(n) |D(t_0)|_n$, we prove
the claimed estimate:
\begin{align*}
|u(x_0,t_0)|^{n+1}
&\leq C_n \delta(Tx_0,T(D(t_0))) |P_{v}(\tilde T(D_{t_0}))|\\
&= C_n \delta(x_0,D(t_0)) |P_{v}(\tilde T(D_{t_0}))|\\
&= C_n \delta(x_0,D(t_0)) |\det T^{-1}| |P_{u}(D_{t_0})|\\
&\leq C_n \delta(x_0,D(t_0)) |D(t_0)| |P_{u}(D_{t_0})|.
\end{align*}
\end{proof}
The next result extends Theorem \ref{jer2} to the parabolic setting.
\begin{theorem}\label{parabol2}
Let $0 < \epsilon \leq 1$. Let $E$ be a bounded open bowl-shaped
domain in $\mathbb{R}^{n+1}$, such that
$E \subset B_1(0) \times (-\infty, \infty)$. Suppose $u \in C(\bar E)$ is
parabolically convex and zero on $\partial_{p} E$.
Let $M_{p}$ be the parabolic Monge-Amp\`ere measure associated to $u$.
Then there exists $C = C(\epsilon,n)$ such that
$$
|u(x_0,t_0)|^{n+1} \leq C \delta(x_0,E(t_0))^{\epsilon} \int_{E_{t_0}}
\delta(x,E(t_0))^{1- \epsilon} \; dM_{p}(x,t).
$$
for all $(x_0,t_0) \in E$.
\end{theorem}
\begin{proof} Without loss of generality, we may assume that
$u(x_0,t_0) = -1$ (if this is not the case, multiply
$u$ by a suitably chosen positive constant). Let $s_k = s 2^{-k
\beta}$ where $s$ and $\beta$ are positive and chosen to satisfy $\beta(n+1)
\leq \epsilon$ and $\sum_{k=1}^{\infty} s_k \leq 1/2$.
\[
A := \delta(x_0,E(t_0))^{\epsilon} \int_{E_{t_0}}
\delta(x,E(t_0))^{1- \epsilon} \; dM_{p}(x,t).
\]
It suffices to show that $A \geq C(s)$, a constant depending on $s$ and $\epsilon$.
For $k = 1, 2, \dots $, let $E_k = \{(x,t) \in E: u(x,t) \leq \lambda_k = -1
+s_1 + \dots + s_k\}$. Define $E_0 = \{(x,t) \in E: u(x,t) \leq
-1\}$. Note that $E_k \subset E_{k+1}$ for $k=1,2, \dots$, and that
$E_0 \neq \emptyset$. Each of the sets $E_k$ is bowl-shaped and
$u|_{\partial_{p} E_k} = \lambda_k$ (taking $\lambda_0 = -1$).
Fix $t$ and let $\delta_k(t) =
\mathop{\rm dist}(\partial E_k(t),\partial E(t))$.
Since $\delta_k(t) \not \to 0$ as $k \to \infty$ (if
$\delta_k(t) \to 0$, then $u$ would be smaller than $-\frac{1}{2}$
somewhere on $\partial_{p}E$), we may choose $k$ to be the smallest
nonnegative integer for which $\delta_{k+1}(t) > \frac{1}{2}\delta_{k}(t)$.
Let $x_k \in \partial E_k(t)$ be a point closest to $\partial E(t)$.
Then we have that
\begin{equation}\label{dist1}
\mathop{\rm dist}(x_k,\partial E_{k+1}(t)) < \frac{1}{2}\delta_{k}(t)
< \delta_{k+1}(t).
\end{equation}
The second of these inequalities holds because of the choice of $k$.
The first inequality requires the following geometric argument.
Let $L$ be a line segment of length $\delta_k$ from $x_k$ to $\partial E(t)$.
The segment $L$ meets $\partial E_{k+1}(t)$ at a point, $x_{k+1}$.
Let $\ell$ represent the length of the part of $L$ that connects
$\partial E_{k+1}(t)$ to $\partial E(t)$. Then
\begin{align*}
\delta_k &= |x_k - x_{k+1}| + \ell \\
&\geq |x_k - x_{k+1}| + \mathop{\rm dist}(\partial E_{k+1}(t), \partial E(t)) \\
&= |x_k - x_{k+1}| + \delta_{k+1}\\
&> |x_k - x_{k+1}| + \frac{1}{2}\delta_{k}.
\end{align*}
Therefore, $\frac{1}{2} \delta_k > |x_k - x_{k+1}|
\geq \mathop{\rm dist}(x_k, \partial E_{k+1}(t))$.
Now we apply Lemma \ref{parabol1} to the function $u(x,t) - \lambda_{k+1}$ on the
set $E_{k+1}$ to get
$$
|u(x_k,t) - \lambda_{k+1}|^{n+1} \leq C_n \delta (x_k,E_{k+1}(t))|E_{k+1}(t)|\,
M_{p}((E_{k+1})_{t}).
$$
The point $x_k \in \partial E_{k}(t)$, so $u(x_k,t) = \lambda_k$ and
$|u(x_k,t) - \lambda_{k+1}| = |\lambda_k - \lambda_{k+1}| = s_{k+1}$. Thus,
\begin{equation}\label{s(k+1)ineq}
s_{k+1}^{n+1} \leq C_n \delta(x_k,E_{k+1}(t)) |E_{k+1}(t)|\,
M_{p}((E_{k+1})_{t}).
\end{equation}
Let $L_t$ be a shortest segment from $x_k$ to $\partial E_{k+1}(t)$ and
let $z \in \partial E_{k+1}(t)$ be the other endpoint of $L_t$.
Let $\rho$ denote
\begin{equation}\label{eqrho}
\rho = |L_t|= |x_k - z| = \mathop{\rm dist}(x_k,\partial E_{k+1}(t)).
\end{equation}
Since the set $E_{k+1}(t)$ is convex, the hyperplane $\Pi$ (of dimension $n-1$) normal to $L_t$ through $z$ is a
support plane for $E_{k+1}(t)$. Let $\Pi'$ be the support plane
parallel to $\Pi$ on the opposite side of $E_{k+1}(t)$, so that $E_{k+1}(t)$
is contained between the two planes, and let $r = \mathop{\rm dist} (\Pi,
\Pi')$. Then since $E_{k+1}(t) \subset B_1(0)$, there
exists a constant $C=C(n)$ such that
\begin{equation}\label{ek+1est}
|E_{k+1}(t)| \leq Cr.
\end{equation}
We remark that the $C$ in (\ref{ek+1est}) can be chosen to be the volume
of the unit ball in $\mathbb{R}^{n-1}$.
Let $T:\mathbb{R}^{n} \to \mathbb{R}^n$ be an affine transformation
normalizing $E_{k+1}(t)$. Then $\mathop{\rm dist}(T(\Pi),T(\Pi'))$ is bounded between two dimensional constants $C_1$ and $C_2$, with $C_1 -1$ on $E(t)$. Then for any $t \geq t_0$, by the choice
of $k$, we have $\delta_{k+1}(t) < \delta_k (t) \leq 2^{-k} \delta_0 (t)$.
Since $\delta_0(t_0) \leq \mathop{\rm dist}
(x_0,\partial E(t_0))$ and $\mathop{\rm diam}(E(t_0)) \leq 2$, we may conclude by Lemma~\ref{deltaprop}\textit{(c)} that $2^{-k} \delta_0(t_0) \leq C 2^{-k} \delta(x_0,E(t_0))$ for a
dimensional constant $C$.
Therefore,
\begin{equation} \label{eq:lastest}
\begin{aligned}
\delta_{k+1}(t_0)\, M_{p}((E_{k+1})_{t_0})
&= \delta_{k+1}(t_0)^{\epsilon}
\int_{(E_{k+1})_{t_0}}\delta_{k+1}(t_0)^{1-\epsilon}\; dM_{p}(y,s) \\
&\leq C2^{-k \epsilon} \delta(x_0,E(t_0))^{\epsilon}
\int_{(E_{k+1})_{t_0}}\delta_{k+1}(t_0)^{1-\epsilon}\; dM_{p}(y,s)\\
&\leq C2^{-k \epsilon}
\delta(x_0,E(t_0))^{\epsilon}\int_{(E_{k+1})_{t_0}}
\delta(y,E(t_0))^{1-\epsilon} dM_{p}(y,s).
\end{aligned}
\end{equation}
The last inequality holds since
\[
\delta_{k+1}(t_0) = \mathop{\rm dist} (\partial
E_{k+1}(t_0),\partial E(t_0)) \leq \mathop{\rm dist}(y,\partial E(t_0))
\leq C \delta(y,E(t_0))
\]
for all $y \in E_{k+1}(t_0)$. Then from
(\ref{s(k+1)ineq2}) and (\ref{eq:lastest}) we obtain that
\[
s_{k+1}^{n+1} \leq C2^{-k \epsilon}
\delta(x_0,E(t_0))^{\epsilon}\int_{(E_{k+1})_{t_0}}
\delta(y,E(t_0))^{1-\epsilon} \; dM_{p}(y,s)
\leq C2^{-k \epsilon} A.
\]
Recall that
\[
s_{k+1}^{n+1} = s^{n+1} 2^{-(n+1)(k+1)\beta} \geq s^{n+1} 2^{-\epsilon(k+1)}
\]
since $\beta(n+1) \leq \epsilon$. Hence
\[
s^{n+1} 2^{-\epsilon(k+1)} \leq C2^{-k \epsilon} A \Rightarrow s^{n+1} \leq C A,
\]
where $C$ depends on $\epsilon$, so $A \geq C(s)$ as desired.
\end{proof}
\subsection*{Acknowledgement} Much of this work appeared in the
author's Ph.D. thesis, completed under the direction of Professor Cristian
Guti\'{e}rrez, to whom the author is grateful.
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\end{document}