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%% Translation via Omnimark script a2l, August 22, 2003 (all in one day!)


\newtheorem*{theorem1}{Theorem 2.1}
\newtheorem*{theorem2}{Proposition 2.2}
\newtheorem*{theorem3}{Lemma 2.3}
\newtheorem*{theorem4}{Lemma 2.4}
\newtheorem*{theorem5}{Theorem 3.1}
\newtheorem*{theorem6}{Theorem 3.2}
\newtheorem*{theorem7}{Theorem 4.1}
\newtheorem*{theorem8}{Proposition 4.2}
\newtheorem*{theorem9}{Corollary 4.3}
\newtheorem*{theorem10}{Proposition 4.4}
\newtheorem*{theorem11}{Theorem 4.5}


\newcommand{\RR}{\mathbb R}
\newcommand{\pa}{\partial }
\newcommand{\CC}{\mathbb C}
\newcommand{\bL}{\mathbf {L}}
\newcommand{\F}{\mathcal {F}}
\newcommand{\p}{\partial }
\newcommand{\ep}{\varepsilon }
\newcommand{\Ombar}{\overline {\Omega }}
\newcommand{\Ubar}{\overline {U}}
\newcommand{\Ric}{\operatorname {Ric}}
\newcommand{\bmo}{\operatorname {bmo}}
\newcommand{\Lip}{\operatorname {Lip}}
\newcommand{\End}{\operatorname {End}}
\newcommand{\Mbar}{\overline {M}}
\newcommand{\Exp}{\operatorname {Exp}}
\newcommand{\dist}{\operatorname {dist}}


\title[Geometric Regularity and Convergence,
and Inverse Problems]{Metric tensor estimates, geometric convergence,
and inverse boundary problems}
\author[M. Anderson]{Michael Anderson}
\address{Mathematics Department, State University of New York, Stony Brook, NY

\author[A. Katsuda]{Atsushi Katsuda}
\address{Mathematics Department, Okayama University, Tsushima-naka, Okayama,
700-8530, Japan}

\author[Ya. Kurylev]{Yaroslav Kurylev}
\address{Department of Mathematical Sciences, Loughborough University, 
Loughborough, LE11 3TU, UK}

\author[M. Lassas]{Matti Lassas}
\address{Rolf Nevanlinna Institute, University of Helsinki, FIN-00014, Finland}

\author[M. Taylor]{Michael Taylor}
\address{Mathematics Deptartment, University of North Carolina, 
Chapel Hill, NC 27599}
\email{met@math.unc.edu }

\dateposted{September 2, 2003}
\PII{S 1079-6762(03)00113-6}
\subjclass[2000]{Primary 35J25, 47A52, 53C21}
\keywords{Ricci tensor, harmonic coordinates, geometric convergence,
inverse problems, conditional stability}
\copyrightinfo{2003}{American Mathematical Society}
\date{December 17, 2002}
\commby{Tobias Colding}
Three themes are treated in the results announced here.  The first is the
regularity of a metric tensor, on a manifold with boundary,
on which there are given Ricci curvature bounds, on the manifold and
its boundary, and a Lipschitz bound on the mean curvature of the boundary.
The second is the geometric convergence of a (sub)sequence of
manifolds with boundary with such geometrical bounds and also an upper
bound on the diameter and a lower bound on injectivity and boundary
injectivity radius, making use of the first part.
The third theme involves the uniqueness and conditional stability
of an inverse problem proposed by Gel'fand, making essential
use of the results of the first two parts.

\section*{1. Introduction}

Here we announce results on regularity, up to the boundary,
of the metric tensor of a Riemannian manifold with boundary, under Ricci
curvature bounds and control of the boundary's mean curvature; an application
to results on Gromov compactness and geometric convergence in the category
of manifolds with boundary; and then an application of these results to the
study of an inverse boundary spectral problem introduced by I.~Gel'fand.
Details are given in \cite{AK2LT}.

Regularity of the metric tensor away from the boundary has been studied and
used in a number of papers, starting with \cite{DTK}.  One constructs
local harmonic coordinates and uses the fact that, in such harmonic
coordinates, the Ricci tensor has the form
\begin{equation*}\Delta g_{\ell m}-B_{\ell m}(g,\nabla g)=-2 \Ric _{\ell m}.
Here $\Delta $ is the Laplace-Beltrami operator, applied {\em componentwise}
to the components of the metric tensor, and $B_{\ell m}$ is a quadratic
form in $\nabla g_{ij}$, with coefficients that are smooth functions of
$g_{ij}$ as long as the metric tensor satisfies a bound $C_{1}|\eta |^{2}\le g_{jk}(x)\eta ^{j}\eta ^{k}\le C_{2}|\eta |^{2}$, with $00\}.$
Let $\Sigma =\mathcal{B}\cap \{x:x^{n}=0\}$ and set $\Ombar =\Omega \cup \Sigma $.
Let $g$ be a metric tensor on $\Ombar $, and denote by $h$ its restriction to
$\Sigma $.  We make the following hypotheses:
g_{jk}&\in H^{1,p}(\Omega ),\ \text{ for some }\ p>n,
\tag{{2.1}} \\
h_{jk}&\in H^{1,2}(\Sigma ),\quad 1\le j,k\le n-1,
\tag{{2.2}} \\
\Ric ^{\Omega }&\in L^{\infty }(\Omega ), \tag{{2.3}} \\
\Ric ^{\Sigma }&\in L^{\infty }(\Sigma ), \tag{{2.4}} \\
H&\in \Lip (\Sigma ). \tag{{2.5}}
Here $H$ denotes the mean curvature of $\Sigma \subset \Ombar $, i.e.,
$H=\text{Tr}\ A/(n-1)$, where $A$ is the Weingarten map, a section of
$\End (T\Sigma )$.  Our goal is to establish the following result.

\begin{theorem1} Under the hypotheses (2.1)--(2.5),
given $z\in \Ombar $,
there exist local harmonic coordinates
on a neighborhood $\Ubar $ of $z$ in $\Ombar $ with respect to which
\begin{equation*}g_{jk}\in C^{2}_{*}(\Ubar ).

Here $C^{2}_{*}(\Ubar )$ is a Zygmund space, as mentioned in \S {1}.
The harmonic coordinates for which (2.6) holds are arbitrary
coordinates $(u^{1},\dots ,u^{n})$ satisfying $\Delta u^{j}=0$ on a chart
not intersecting $\Sigma $.  On a neighborhood of a point in $\Sigma $,
these coordinates are ``boundary harmonic coordinates,'' which are
defined as follows.  We require $(u^{1},\dots ,u^{n})$ to be defined and
regular of class at least $C^{1}$ on a neighborhood of $z$ in $\Ombar $,
and $\Delta u^{j}=0$.  We require that $v^{j}=u^{j}|_{\Sigma }$ be harmonic
on $\Sigma $, i.e., annihilated by the Laplace-Beltrami operator of
$\Sigma $ with its induced metric tensor.  We require that $u^{n}$ vanish
on $\Sigma $, and that $(u^{1},\dots ,u^{n})$ map a neighborhood
of $z$ in $\Ombar $ diffeomorphically onto $\Ombar $.

Regarding the fact that the hypotheses (2.1)--(2.2) imply that various
curvature tensors are well defined, it can be shown that
g_{jk}\in C(\Ombar )\cap H^{1,2}(\Omega )&\Longrightarrow \Gamma \in L^{2}(\Omega ),\ R^{a}{}_{bjk}\in H^{-1,2}(\Omega )+L^{1}(\Omega ) \\
&\Longrightarrow \Ric _{bk}\in H^{-1,2}(\Omega )+L^{1}(\Omega ).
The hypothesis (2.1) is stronger than the hypothesis in (2.7).
It implies $g_{jk}\in C^{r}(\Ombar )$ for some $r>0$, so (2.7) is applicable
both to $g_{jk}$ on $\Omega $ and,
in view of (2.2), to $h_{jk}$ on $\Sigma $.
One also shows that the Weingarten map has the property
$A\in B^{-1/p}_{p,p}(\Sigma ),
as a consequence of (2.1).  Thus we have a priori that $H\in B^{-1/p}_{p,p}
(\Sigma )$, and the hypothesis (2.5) strengthens this condition on $H$,
in a fashion that is natural for the desired conclusion of Theorem 2.1.

Our approach to the proof of Theorem 2.1 is to obtain the result as a
regularity result for an elliptic boundary problem.  We use the PDE
(1.1) (the ``Ricci equation'') for the components of the metric tensor,
in boundary harmonic coordinates, and use Dirichlet boundary conditions on
some components of $g_{jk}$ and Neumann boundary conditions on complementary
components; see (2.9) and (2.12)--(2.13) for a more precise

The proof of Theorem 2.1 in \cite{AK2LT} is done in stages.  First it is shown
that the conclusion of Theorem 2.1 holds when the hypotheses
(2.1)--(2.2) are strengthened a \pagebreak bit.

\begin{theorem2} In the setting of Theorem 2.1, replace
hypotheses (2.1)--(2.2) by
\begin{equation*}g_{jk}\in C^{1+s}(\Ombar ),\ \text{ for some }\ s\in (0,1),
and retain hypotheses (2.3)--(2.5).  Then the conclusion (2.6) holds.

In addition to providing a first step toward establishing Theorem 2.1,
Proposition 2.2 is itself sufficiently strong for the application
to geometric convergence described in \S {3}.

The demonstration of Proposition 2.2 begins with a construction
of boundary harmonic coordinates, mentioned above.
In these new coordinates, (2.8) and (2.3)--(2.5) are preserved.
Now in harmonic coordinates the metric tensor satisfies the elliptic PDE
(1.1), and from (2.8) and (2.3) we have
$F_{\ell m}=B_{\ell m}-2 \Ric _{\ell m}\in L^{\infty }(\Omega ),
and the coefficients of $\Delta $ have the same degree of regularity as
$g_{jk}$ in (2.8).

Now, if $j,k\le n-1$, then well known local regularity results on $\Sigma $
following from (2.4) give
$g_{jk}\bigr |_{\Sigma }=h_{jk}\in H^{2,p}(\Sigma ),\ \forall \ p<\infty ,
but in fact there is the following refinement, established in
Proposition III.10.2 of \cite{T2}:
\begin{equation*}g_{jk}\bigr |_{\Sigma }=h_{jk}\in \mathfrak{h}^{2,\infty },\quad 1\le j,k\le n-1.
Here $\mathfrak{h}^{2,\infty }$ denotes the bmo-Sobolev space of functions whose
derivatives of order $\le 2$ belong to $\bmo $, the localized space of
functions of bounded mean oscillation.
We have the following (after perhaps shrinking $\Ombar $ to a smaller
neighborhood of $z$).  See \cite{AK2LT} for details.

\begin{theorem3} Under the hypotheses of Proposition 2.2, we have,
in boundary harmonic coordinates,
\begin{equation*}g_{jk}\in C^{2}_{*}(\Ombar ),\quad 1\le j,k\le n-1.

To continue, following \cite{An2}, we switch over to PDE for $g^{\ell m}$.
Parallel to (1.1), we have
\begin{equation*}\Delta g^{\ell m}=B^{\ell m}(g,\nabla g)+2(\Ric ^{\Omega })^{\ell m}=F^{\ell m},
and (2.8) and (2.3) give $F^{\ell m}\in L^{\infty }(\Omega )$.
We take $m=n$ and proceed to derive Neumann-type boundary
conditions for the components $g^{\ell n},\ 1\le \ell \le n$.
In fact, as shown in \cite{AK2LT},
\begin{equation*}N g^{nn}=-2(n-1)Hg^{nn},\quad \text{on }\ \Sigma ,
and, for $1\le \ell \le n-1$,
\begin{equation*}N g^{\ell n}=-(n-1)Hg^{\ell n}+\frac{1}{2}
\frac{1}{\sqrt {g^{nn}}}\, g^{\ell k}\pa _{k} g^{nn},\quad \text{on }\ \Sigma .
Here $H$ is the mean curvature of $\Sigma $, which we assume satisfies (2.5),
and $N$ is the unit normal field to $\Sigma $, pointing inside $\Omega $.

Having (2.11)--(2.13), we can establish further regularity of the
functions $g^{\ell n}$.  The following is proven in \cite{AK2LT}.

\begin{theorem4} In boundary harmonic coordinates, we have
\begin{equation*}g^{\ell n}\in C^{2}_{*}(\Ombar ),\quad 1\le \ell \le n.

One then verifies that Lemmas 2.3 and 2.4 yield
$g_{n\ell }=g_{\ell n}\in C^{2}_{*}(\Ombar ).
These results yield Proposition 2.2.
To establish Theorem 2.1 in full strength,
we need to work harder, especially
on the Neumann problem.  We continue to have (2.11);
however, this time it is not so straightforward to produce the Neumann-type
boundary conditions (2.12)--(2.13).
Consider (2.12).  The right side is well defined;
we have $Hg^{nn}|_{\Sigma }\in C^{s}(\Sigma )$, for some $s>0$.
As for $N$, the unit normal field to
$\Sigma $ is also H{\"{o}}lder continuous of class $C^{r}$.
But applying $N$ to $g^{nn}\in H^{1,p}(\Omega )$
does not yield an object that can be evaluated on $\Sigma $.  One has the
same problem with the left side of (2.13), and the right side of (2.13)
is also problematic.

However, we are able to show that a weak formulation of the Neumann boundary
condition is applicable, and we establish regularity results for weak
solutions to the Neumann problem strong enough 
to complete the proof of Theorem 2.1.
See \cite{AK2LT} for details.

\section*{3. Geometric convergence for manifolds with boundary}

A sequence $(\Mbar _{k},g_{k})$ of compact Riemannian manifolds with boundary
$\pa M_{k}$ is said to converge in the $C^{r}$-topology (given $00$ such that
\begin{equation*}r_{h}(\Mbar ,g,Q)\ge r_{\mathcal{M}},\quad \forall \ (\Mbar ,g)\in \mathcal{M}(R_{0},i_{0},S_{0},d_{0}).

The proof of Theorem 3.2 given in \cite{AK2LT} involves the following
ingredients: a blow-up argument, use of the regularity results of
\S {2} and of the fundamental equations of hypersurface theory,
and use of the Cheeger-Gromoll splitting theorem.

\begin{remark} Invoking the definition of the Gromov-Hausdorff topology
(cf.~\cite{Gr}) we can show that $\overline{\mathcal{M}(R_{0},i_{0},S_{0},d_{0})}$ is compact
in the Gromov-Hausdorff topology and $C^{r}$-convergence is equivalent to
Gromov-Hausdorff convergence on this compact set, for any $r\in [1,2)$.
\section*{4. Gel'fand inverse boundary problem}

In this section we discuss uniqueness and stability
for the
inverse boundary spectral problem. To fix notations, assume that
$(\Mbar ,g,\p M)$ is a compact, connected manifold, with nonempty
boundary, provided with a metric tensor $g$ with some limited
smoothness (specified more precisely below).
Let $\Delta ^{N}$ be the Neumann Laplacian.
Denote by $(\lambda _{k})_{k=0}^{\infty }$ its  eigenvalues
(counting multiplicity) and
$(\phi _{k})_{k=0}^{\infty }$
the corresponding  eigenfunctions.

The {\em Gel'fand inverse boundary problem} (in its spectral formulation)
is the problem of the
reconstruction of $(\Mbar ,g)$ from its boundary spectral data,
i.e., the collection
$(\p M,\{\lambda _{k}, \phi _{k}|_{\p M}\}_{k=1}^{\infty })$.
(We have mentioned Gel'fand's original formulation in the Introduction.)

The following uniqueness result is proven in \cite{AK2LT}.

\begin{theorem7}   Let $\Mbar $ be a compact, connected manifold with
nonempty boundary and $C^{2}_{*}$ metric tensor.  Then the boundary
spectral data $(\p M, \, \{\lambda _{k}, \phi _{k}|_{\p M}\}_{k=1}^{\infty })$
determine the manifold $\Mbar $ and its metric $g$ uniquely.

Such a result was established in the $C^{\infty }$ case in \cite{BK1}.
Some different techniques are required to treat the $C^{2}_{*}$ case.
We briefly describe how to determine $\Mbar $ as a topological space,
referring to \cite{AK2LT} for further details.

We start with the introduction of some useful geometric objects.
Let $\Gamma \subset \p M $ be open and take $t\geq 0$. Then we set
\begin{equation*}M(\Gamma ,t) = \{x \in M: d(x,\Gamma ) \leq t \},
the domain of influence of  $\Gamma $ at ``time'' $t$,
and define
\begin{equation*}{\bL }(\Gamma ,t)=
{\F }L^{2}(M(\Gamma ,t))
\subset \ell ^{2}.
Here, ${\F }$ stands for the Fourier transform of functions from
$L^{2}(M)$, i.e.,
${\F }(u) = \{u_{k}\}_{k=1}^{\infty }\in \ell ^{2},
\ u_{k}=(u,\phi _{k})_{L^{2}(M)},
and the subspace
$L^{2}(M(\Gamma ,t))$
consists of all functions in $L^{2}(M)$ with support in the set
$M(\Gamma ,t)$.

One ingredient in the proof of Theorem 4.1 is the following.
Consider the wave equation
&(\p _{t}^{2}-\Delta )u^{f}(x,t)=0\quad \hbox {in }M\times \RR _{+}, \\
&u^{f}|_{t=0}=0,\quad u_{t}^{f}|_{t=0}=0,\quad Nu^{f}|_{\p M\times \RR _{+} }=f
\in C^{1}_{0}(\Gamma \times (0,T)),
where $N$ is the exterior unit normal field to $\p M$.
Using Tataru's unique continuation theorem \cite{Ta}, it was shown in \cite{Be1}
(also Theorem 3.10 of \cite{KKL}) that the following holds.

\begin{theorem8} For each $T>0$, the set
$\{u^{f}(T):f\in L^{2}(\Gamma \times (0,T))\}$ is a
dense subspace of $L^{2}(M(\Gamma ,T))$.

Meanwhile, the following formula (due to Blagoveshchenskii) gives the Fourier
coefficients $u_{k}^{f}(t)$ of a wave $u^{f}(\cdotp ,t)$ in terms of the
boundary spectral data,
\begin{equation*}u_{k}^{f}(t) =
\int _{0}^{t} \int_{\p M} f(x,t')\,
\frac{\sin {\sqrt {\lambda _{k}} (t-t' )}}{\sqrt {\lambda _{k}}}
\phi _{k}(x)\,dS_{g}\,dt'.
 This,  together with Proposition 4.2, implies the following.

\begin{theorem9}  Given $\Gamma \subset \pa M$ and $t>0$,
the boundary spectral data determine the subspace
\begin{equation*}{\bL }(\Gamma ,t) = \mathcal{F}L^{2}(M( \Gamma , \,t))\subset \ell ^{2}.

Now let $h\in C(\p M)$. We can ask if $h$ is the boundary distance
function for some $x \in M$.
In fact, use of Corollary 4.3 allows one to answer this question.
It follows that the boundary spectral data
determine the image in $L^{\infty }(\p M)$ of the
boundary distance representation
$R$. Here, $R:\Mbar \to C(\p M)$ is defined by
\begin{equation*}R(x) = r_{x}(\cdot ), \quad r_{x}(z) = \text{dist}(x,z), \quad z \in \p M.
(Compare \cite{KKL} and \cite{Ku}.)
Clearly, the map $R$ is Lipschitz continuous. Moreover, under the assumptions
of Theorem 4.1 it is injective.
 This follows from Osgood's theorem applied to geodesics normal
to $\partial M$.

Since $\Mbar $ is compact, injectivity and continuity imply that $R$ is a
homeomorphism, i.e., $R(\Mbar )$ with the distance inherited from
$L^{\infty }(\p M)$ and $(\Mbar ,g)$ are homeomorphic, and thus
$R(\Mbar )$ can be identified with $\Mbar $ as a topological manifold.
We hence have the following.

\begin{theorem10} Assume $(\Mbar _{1},g_{1})$ and $(\Mbar _{2},g_{2})$
satisfy the hypotheses of Theorem 4.1.  If they have identical boundary
spectral data, including $\pa M_{1}=\pa M_{2}=X$, as $C^{2}$ manifolds,
then there is a natural correspondence of $R(\Mbar _{1})$ and
$R(\Mbar _{2})\subset C(X)$, producing a uniquely defined homeomorphism
\begin{equation*}\chi :\Mbar _{1}\longrightarrow \Mbar _{2}.

We refer to \cite{AK2LT} for a demonstration that $\chi $ in (4.7)
is a $C^{2}$-diffeomorphism, preserving the metric tensors, which proves
the uniqueness result of Theorem 4.1.

We next consider stabilization of inverse problems using
geometric convergence results and apply them to the Gel'fand problem.
The basic thrust of our argument provides an illustration of a general
``stabilization principle for inverse problems.''

Let us set up some notation.
Denote by $\mathcal{M}_{X}(C^{2}_{*})$ the set of compact, connected manifolds
$\Mbar $ with nonempty boundary $X$, endowed with a metric tensor in
$C^{2}_{*}(\Mbar )$.  Given $(\Mbar ,g)\in \mathcal{M}_{X}(C^{2}_{*})$, set
\begin{equation*}\mathcal{D}(\Mbar ,g)=\{\lambda _{j},\phi _{j}|_{X}\}_{j=1}^{\infty },
the right side denoting the boundary spectral data of $(\Mbar ,g)$.
We have
\begin{equation*}\mathcal{D}:\mathcal{M}_{X}(C^{2}_{*})\longrightarrow \mathcal{B}_{X},
where $\mathcal{B}_{X}$ denotes the set of sequences $\{\mu _{j},\psi _{j}:j\ge 1\}$,
with $\mu _{j}\in \RR ^{+},\ \mu _{j}\nearrow +\infty $, and $\psi _{j}\in L^{2}(X)$,
modulo an equivalence relation, which can be described as follows.
We say $\{\mu _{j},\psi _{j}\}\sim \{\mu _{j},\tilde {\psi }_{j}\}$
if $\psi _{j}(x)=\alpha _{j} \tilde {\psi }_{j}(x)$
for some $\alpha _{j}\in \CC , |\alpha _{j}|=1$.
More generally, if $\mu _{k_{0}}=\cdots =\mu _{k_{1}}$, we allow
\begin{equation*}\psi _{j}(x)=\sum \limits _{k=k_{0}}^{k_{1}} \alpha _{jk} \tilde {\psi }_{k}(x),
\quad j = k_{0}, \dots , k_{1},
for a unitary $l\times l$ matrix $(\alpha _{jk}),\ l=k_{1}-k_{0}+1$.
The content of Theorem 4.1 is that the map (4.9) is one-to-one.

There are natural topologies one can put on the sets in (4.9).
Furthermore, it follows from standard techniques of perturbation
theory (cf.~\cite{K}) that $\mathcal{D}$ is continuous in (4.9).

Now the map (4.9) is  by no means invertible,
giving rise to the
phenomenon of ill-posedness.  One wants to ``stabilize'' the inverse
problem, showing that certain a priori hypotheses on the domain
$(\Mbar ,g)$ put it in a subset $K\subset \mathcal{M}_{X}(C^{2}_{*})$
with $\mathcal{D}^{-1}$ acting
continuously on the image of $K$.
The results of \S {3} provide a tool to accomplish this.

Recall the class $\mathcal{M}(R_{0},i_{0},S_{0},d_{0})$ defined in \S {3}.\,\,Given a
boundary $X$, denote by $\mathcal{M}_{X}(R_{0},$ $i_{0},S_{0},d_{0})$ the set of such
manifolds with boundary $X$.  It follows from Theorem 3.1 that
$\overline{\mathcal{M}_{X}(R_{0},i_{0},S_{0},d_{0})}$ is compact in the $C^{r}$
topology, for any $r\in (1,2)$, and is contained in $\mathcal{M}_{X}(C^{2}_{*})$.
We hence give $\overline{\mathcal{M}_{X}(R_{0},i_{0},S_{0},d_{0})}$ the $C^{r}$
topology, and we see this is independent of $r$, for $r\in (1,2)$.

Combined with Theorem 4.1, these observations yield the following
conditional stability \pagebreak of the Gel'fand inverse problem.

\begin{theorem11} Given $R_{0},i_{0},S_{0},d_{0}\in (0,\infty )$,
\begin{equation*}\mathcal{D}:\overline{\mathcal{M}_{X}(R_{0},i_{0},S_{0},d_{0})}\longrightarrow \mathcal{B}_{X}
is a homeomorphism of $\overline{\mathcal{M}_{X}(R_{0},i_{0},S_{0},d_{0})}$
onto its range, $\mathcal{B}_{X}(R_{0},i_{0},S_{0},d_{0})$; hence
\begin{equation*}\mathcal{D}^{-1}:\mathcal{B}_{X}(R_{0},i_{0},S_{0},d_{0})\longrightarrow \overline{\mathcal{M}_{X}(R_{0},i_{0},S_{0},d_{0})}
is continuous.


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