\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: $$\label{dpMA} \begin{gathered} \det D^2 u = \mu \quad \text{in } \Omega, \\ u|_{\partial \Omega} = g, \end{gathered}$$ 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 $x \in \Omega$, $$\label{eq:alekest} |u(x)|^n \leq C_n (\mathop{\rm diam} \Omega)^{n-1} \mathop{\rm dist}(x, \partial \Omega) \, Mu(\Omega),$$ 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 $x \in \Omega$, $$\label{eq:jer1} |u(x)|^n \leq C \delta(x,\Omega)|\Omega| Mu(\Omega)$$ 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 $$\label{eq:jer2} |u(x_0)|^n \leq C(n,\epsilon) \delta(x_0,\Omega)^{\epsilon} \, \int_{\Omega} \delta(x,\Omega)^{1-\epsilon} \, dMu(x)$$ 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 \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} Next, we establish the change-of-variable formula $$\label{changevar} |P_{v}(\tilde T(D_{t_0}))| = |\det T^{-1}| \;|P_{u}(D_{t_0})|.$$ 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 $$\label{dist1} \mathop{\rm dist}(x_k,\partial E_{k+1}(t)) < \frac{1}{2}\delta_{k}(t) < \delta_{k+1}(t).$$ 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, $$\label{s(k+1)ineq} s_{k+1}^{n+1} \leq C_n \delta(x_k,E_{k+1}(t)) |E_{k+1}(t)|\, M_{p}((E_{k+1})_{t}).$$ 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 $$\label{eqrho} \rho = |L_t|= |x_k - z| = \mathop{\rm dist}(x_k,\partial E_{k+1}(t)).$$ 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 $$\label{ek+1est} |E_{k+1}(t)| \leq Cr.$$ 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, \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} 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. \begin{thebibliography}{00} \bibitem{bakelman:can94} {\sc I. J. Bakelman}; {\em Convex analysis and nonlinear geometric elliptic equations}, Springer-Verlag, Berlin, 1994. \newblock With an obituary for the author by William Rundell, Edited by Steven D. 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