\documentclass[JEP,XML,SOM]{cedram}
\datereceived{2015-05-08}
\dateaccepted{2015-07-16}
\dateepreuves{2015-08-06}

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\DeclareMathOperator{\divv}{div}\let\div\divv

\begin{document}
\frontmatter

\title{On a counter-example to quantitative Jacobian bounds}

\author[\initial{Y.} \lastname{Capdeboscq}]{\firstname{Yves} \lastname{Capdeboscq}}

\address{Mathematical Institute,
University of Oxford,
Andrew Wiles Building\\
Woodstock Road,
Oxford OX2 6GG, UK}

\email{yves.capdeboscq@maths.ox.ac.uk}
\urladdr{https://people.maths.ox.ac.uk/capdeboscq/}

\begin{abstract}
This note provides a counter-example to the local positivity of the Jacobian determinant for solutions of the conductivity equation in dimension $3$. It shows that the sign of the determinant cannot be imposed by an a priori choice of boundary data in $H^{{1}/{2}}(\partial\Omega)$ depending only on the upper and lower bound of the conductivity, even locally. The argument uses a scalar two-phase conductivity constructed by Briane, Milton \& Nesi \cite{BRIANE-MILTON-NESI-04,BRIANE-MILTON-2009}.
\end{abstract}

\subjclass{35J55, 35R30, 35B27}

\keywords{Rad\'o-Kneser-Choquet Theorem, hybrid inverse problems, impedance tomography, homogenization}

\alttitle{Sur un contre-exemple aux bornes quantitatives du jacobien}

\altkeywords{Théorème de Rad\'o-Kneser-Choquet, problèmes inverses hybrides, tomographie d'impédance, homogénéisation}

\begin{altabstract}
Cette note fournit un contre-exemple à la positivité locale du déterminant jacobien des solutions de l'équation de conduction en dimension $3$. On montre que le signe du déterminant ne peut pas être imposé par un choix a priori de données au bord dans $H^{{1}/{2}}(\partial\Omega)$ dépendant seulement des bornes inférieure et supérieure de la conductivité, même localement. L'argument utilise une conductivité scalaire à deux phases construite par Briane, Milton \& Nesi \cite{BRIANE-MILTON-NESI-04,BRIANE-MILTON-2009}.
\end{altabstract}

\maketitle
\tableofcontents
\mainmatter

\section{Introduction}

Let $B_{1}$ be the unit open ball centred at the origin in $\mathbb{R}^{d}$.
Given $\gamma\in L^{\infty}\left(\mathbb{R}^{d},\mathbb{R}\right)$
such that $1\leq\gamma(x)\leq\beta$ for a.e. $x\in B_{1}$, consider
$U=\left(u^{1},\ldots,u^{d}\right)$ whose components are solutions
of the Dirichlet boundary value problems
\begin{align*}
-\div\left(\gamma Du^{i}\right) & = 0\quad\text{in }B_{1},\\
u^{i} & = \phi^{i}\quad\text{on }\partial B_{1},
\end{align*}
where $\phi=\left(\phi^{1},\ldots,\phi^{d}\right)$ is an homeomorphism
from $\partial B_{1}$ onto a Jordan curve $\Gamma$ which is the
boundary of a bounded convex domain. The vector valued map $U$ is
called the $\gamma$-harmonic extension of $\phi$.

When $d=2$ it is known that without further assumption on $\gamma$
this implies that $\det DU>0$ almost everywhere in $B_{1}$ -- for
more precise bounds see \cite{ALESSANDRINI-NESI-01,ALESSANDRINI-NESI-2009}.
When $\phi$ is also a $C^{1,\alpha}$ diffeomorphism from $B_{\rho}$
with $\rho>1$ and $\gamma\in C^{0,\alpha}$ then $\det DU$ is bounded
below by a positive constant on $B_{1}$, see \cite{Bauman-Marini-Nesi-2001}.
For harmonic maps, this fact follows from the Rad\'o-Kneser-Choquet
Theorem \cite[Chapter 3.1]{DUREN-2004}.

The Rad\'o-Kneser-Choquet Theorem (and its extensions) does not
hold when $d=3$, see \cite{WOOD-1991,LAUGESEN-96,BRIANE-MILTON-NESI-04,DUREN-2004}.
The recent paper \cite{ALESSANDRINI-NESI-2015} contains a review
of the numerous pathologies and open problems that appear in this
case. All currently available counter-examples are based on specific
boundary data : this is natural, as for example the boundary data
$x=(x^{1},x^{2},x^{3})$ has an harmonic extension (which is itself)
of determinant $1$ in $B_{1}$. In any dimension $d\geq3$, for a
fixed sufficiently regular conductivity $\gamma$, it is known that
there exists a set of Dirichlet data $(\phi^{1},\ldots,\phi^{N})$
with $N\in\left\{ d,d+1,2d+1\right\} $ such that the rank of $[Du^{1},\ldots,Du^{N}]$
is $d$ over all the whole domain, and a positive determinant constraint
is satisfied locally by $d$ of the associated $\gamma$-harmonic
maps $(u^{1},\ldots,u^{N})$, see section~\ref{sec:multidata}.

In the context of coupled, or hybrid, inverse problems, it is desirable
to be able to choose the Dirichlet data independently of the conductivity
$\gamma$, which is an unknown of the problem.

When $d=3$, extrapolating from the existence results mentioned above
for fixed conductivities, one could think that given a priori bounds
on the conductivity, \hbox{$1\leq\gamma(x)\leq\beta$} in $B_{1}$, and possibly
assuming that $\gamma$ is sufficiently regular, if a large enough
variety of boundary conditions is used, the positivity of the determinant
can be guaranteed locally for any such $\gamma$. Indeed, this is
true if $\beta-1$ is small enough by a perturbation argument.

This note shows that if $\beta$ is larger than some universal constant
$\delta_{0}$ defined in Lemma~\ref{Chap5.lem:Two} then in any $C^{1}$
bounded open domain $\Omega\subset\mathbb{R}^{3}$, no Dirichlet data
in $H^{\sfrac{1}{2}}(\partial\Omega)^{3}$ can enforce a local determinant
constraint for every real two-phase constant conductivities (or all
real $C^{\infty}$ conductivities) satisfying $1\leq\gamma(x)\leq\beta$.

The proof is constructive. It uses a function $\gamma$ introduced
in \cite{BRIANE-MILTON-NESI-04} which is defined over $[0,1]^{3}$
and repeated periodically, together with a regularity result in the
theory of homogenization proved in \cite{LI-NIRENBERG-03}. The argument
is that any Dirichlet data whose harmonic extension would satisfy
locally a given positivity determinant constraint has a $\gamma(n\cdot)$-harmonic
extension whose determinant changes sign locally, for $n$ large enough.
The frequency $n$ depends on the positive lower bound for the determinant
and the size of the open set where the constraint is imposed, but
not on the boundary condition.

\subsubsection*{Acknowledgement}
The author is grateful to the referee who made him aware of the results
of Greene \& Wu~\cite{Greene-Wu-1975a,Greene-Wu-1975b}, and to both referees for their valuable comments that improved the presentation
of this note.

\section{Main result}

Let $\Omega\subset\mathbb{R}^{3}$ be an open bounded domain with
$C^{1}$ boundary. Let $\Omega^{\prime}$ a sub-domain of $\Omega$,
such that $d\left(\Omega^{\prime},\partial\Omega\right)>0$.

Given $\phi=\left(\phi^{1},\phi^{2},\phi^{3}\right)$, with $\phi^{i}\in H^{\sfrac{1}{2}}\left(\partial\Omega;\mathbb{R}\right)$
for $i=1,2,3,$ let $U_{0}=(u_{0}^{1},u_{0}^{2},u_{0}^{3})\in H^{1}(\Omega;\mathbb{R}^{3})$
be the harmonic extension of $\phi$, that is, for $i=1,2,3$, $u_{0}^{i}\in H^{1}(\Omega)$
is the unique solution of
\begin{equation}
\label{Chap5.eq:laplace}
\begin{split}
\Delta u_{0}^{i} & = 0\quad\text{in }\Omega,\\
u_{0}^{i} & = \phi^{i}\quad\text{on }\partial\Omega.
\end{split}
\end{equation}
Let $\gamma$ be a piecewise constant $1$-periodic function $\gamma$
such that in $Y=[0,1]^{3}$ we have
\[
\gamma\left(y\right)=1+\left(\delta-1\right)\chi_{Q}(y)\quad\text{for all }y\in Y,
\]
where $\chi_{Q}$ is the $Y$-periodic characteristic function $Q$,
made of rotations and translations of a scaled copy of the tori (that
is, circular annuli whose cross-section is a disk), with cubic symmetry.
An illustration of one such $Q$ is given in Figure~\ref{Chap5.fig:fig1}.

\begin{figure}
\begin{centering}
\includegraphics[width=0.25\columnwidth]{YXZ}\quad\includegraphics[width=0.25\columnwidth]{XZY}\quad\includegraphics[width=0.25\columnwidth]{3D-2}
\par\end{centering}

\caption{\label{Chap5.fig:fig1}Sketch of the periodic set. The various colors
were added for readability, to highlight which elements correspond
to the same ring, once the cell is repeated periodically. $Q$ is
the union of all rings.}
\end{figure}

This type of construction was originally introduced in \cite{BRIANE-MILTON-NESI-04};
the variant presented here was introduced in \cite{BRIANE-MILTON-2009}
(see also \cite{Kadic-Schittny-Buckmann-Kern-Wegener-2015}). The
value of $\delta$ is fixed, and decided by Lemma~\ref{Chap5.lem:Two}
below.

Write $U_{n}=(u_{n}^{1},u_{n}^{2},u_{n}^{3})$ where for $i=1,2,3$,
$u_{n}^{i}\in H^{1}(\Omega;\mathbb{R}^{3})$ is the solution
of
\begin{equation}
\label{Chap5.eq:hom}
\begin{split}
\div\bigl(\gamma\left(nx\right)Du_{n}^{i}\bigr) & = 0\quad\text{in }\Omega,\\
u_{n}^{i} & = \phi^{i}\quad\text{on }\partial\Omega.
\end{split}
\end{equation}
Note that due to cubic symmetry the corresponding effective (or homogenized)
conductivity is a positive real number and therefore as $n\to\infty$,
\[
U_{n}\rightharpoonup U_{0}\quad\text{weakly in }H^{1}(\Omega).
\]

Given $\rho>0$, $x_{0}\in\Omega^{\prime}$ such that $B_{\rho}(x_{0})\subset\Omega^{\prime}$,
and $\lambda>0$ let
\begin{equation}
A(\phi,x_{0},\rho,\lambda):=\left\{ \phi\in H^{\sfrac{1}{2}}(\partial\Omega;\mathbb{R}^{3})\,:\,\det(DU_{0})>\lambda\left\Vert \phi\right\Vert _{H^{\sfrac{1}{2}}\left(\partial\Omega\right)}^{3}\text{ in }B_{\rho}(x_{0})\right\} ,\label{Chap5.eq:Aphi}
\end{equation}
where $U_{0}$ is the harmonic extension of $\phi$ given by \eqref{Chap5.eq:laplace}.
The set $A(\phi,x_{0},\rho,\lambda)$ contains all boundary data whose
harmonic extensions in $\Omega$ satisfy the stated lower determinant
bound in $B_{\rho}(x_{0}).$ We write $\left|X\right|$ the Lebesgue
measure of the set $X$.

The main result is the following.
\begin{thm}
\label{Chap5.thm:One} Given $\rho>0$, $x_{0}\in\Omega^{\prime}$
such that $B_{\rho}(x_{0})\subset\Omega^{\prime}$, and $\lambda>0$,
let $A(\phi,x_{0},\rho,\lambda)$ be defined by \eqref{Chap5.eq:Aphi}.
There exist $n$, depending on $\rho$, $\Omega$, $\Omega^{\prime}$
and $\lambda$ only, a universal constant $\tau>0$ and two open subsets
$B_{+}$ and $B_{-}$ of $B_{\rho}(x_{0})$ such that
\[
\left|B_{+}\right|>\tau\left|B_{\rho}(x_{0})\right|\quad\text{and}\quad\left|B_{-}\right|>\tau\left|B_{\rho}(x_{0})\right|,
\]
and for any $\phi\in A(\phi,x_{0},\rho,\lambda)$,
\[
\det\left(DU_{n}\right)(x)<-\tau\lambda\left\Vert \phi\right\Vert _{H^{\sfrac{1}{2}}\left(\partial\Omega\right)}^{3}\quad\text{on }B_{-},
\]
and
\[
\det\left(DU_{n}\right)(x)>\tau\lambda\left\Vert \phi\right\Vert _{H^{\sfrac{1}{2}}\left(\partial\Omega\right)}^{3}\quad\text{on }B_{+},
\]
where $U_{n}$ is the $\gamma\left(n\cdot\right)$-harmonic extension
of $\phi$ given by \eqref{Chap5.eq:hom}.
\end{thm}

In other words, there is no a priori choice of boundary data which
can ensure a quantitative lower bound of the Jacobian determinant
for piecewise constant scalar conductivities without additional a
priori information, as any given boundary condition would fail either
for harmonic maps or the two-phase composite $\gamma(n\cdot)$ at
a fixed scale $n$. This result is an application of two existing
results in the literature. The first key result is a part of Theorem~3
in \cite{BRIANE-MILTON-NESI-04}.
\begin{lem}
[see Theorem 3 in \cite{BRIANE-MILTON-NESI-04}]\label{Chap5.lem:Two}
There exist $\delta_{0}>0$, $\tau>0,$ $Y_{+}$ and $Y_{-}$ open
subsets of $Y$ both of positive measure $2\tau$ such that the periodic
corrector matrix $P=\nobreak D\zeta$, where $\zeta$ is the solution of
\begin{align*}
\div\left(\gamma D\zeta\right) & = 0\text{ in }\mathbb{R}^{3},\\
\zeta(y)-y &\in H_{\#}^{1}(Y),
\end{align*}
satisfies
\[
\det\left(P\right)(y)>2\tau\text{ in }Y_{+}\quad\text{and}\quad\det\left(P\right)(y)<-2\tau\text{ in }Y_{-}.
\]
\end{lem}

The second key result is a regularity result. Because the conductivity
$\gamma$ is piecewise constant (and therefore piecewise smooth),
and because the set $Q$ has $C^{\infty}$ smooth boundaries (and
therefore $C^{1,\alpha}$ smooth boundaries), the regularity results
given in~\cite{LI-VOGELIUS-00} and \cite{LI-NIRENBERG-03} show that
$U_{n}$ is also piecewise $C^{1,\beta}$ for some $\beta>0$, up
to the boundary of the set $Q$ in $\Omega^{\prime}$. In fact, this
provides uniform $W^{1,\infty}$ estimates for $U_{n}$, independently
of~$n$, see \cite{LI-NIRENBERG-03}. This result has been successfully
expanded to provide error estimates for $DU_{n}$ see \cite{BENHASSEN-BONNETIER-06,Lipton-Mengesha-2012}.
\begin{lem}
[See Theorem 3.4 in \cite{LI-NIRENBERG-03}, Theorem 3.6 in \cite{BENHASSEN-BONNETIER-06} or Theorem 4.2 in \cite{Lipton-Mengesha-2012}]
\label{Chap5.lem:three} There exists a constant $C$ depending on $\Omega$, $\Omega^\prime$, $\delta$ and $Q$ only such that
\begin{align*}
\|DU_{n}\|_{L^{\infty}(\Omega^{\prime})} & \leq C\|\phi\|_{H^{\sfrac{1}{2}}\left(\partial\Omega,\mathbb{R}^{3}\right)},\\
\|P\left(nx\right)\|_{L^{\infty}(\Omega^{\prime})} & \leq C,\\
\tag*{and}
\|DU_{n}-P\left(nx\right)DU_{0}\|_{L^{\infty}(\Omega^{\prime})}&\leq\frac{C}{n^{1/3}}\|\phi\|_{H^{\sfrac{1}{2}}\left(\partial\Omega,\mathbb{R}^{3}\right)}.
\end{align*}
\end{lem}

Another variant of this result (also based on \cite{LI-VOGELIUS-00,LI-NIRENBERG-03})
is given in \cite{AMMARI-BONNETIER-CAPDEBOSCQ-09}. Combining these
two ingredients, we obtain our result.
\begin{proof}
[Proof of Theorem~\ref{Chap5.thm:One}] In $\Omega^{\prime}$, we
have
\begin{align*}
\det\left(DU_{n}\right) & = \det\left(P\left(nx\right)DU_{0}\right)+R_{n},\\
& = \det\left(P\left(nx\right)\right)\det\left(DU_{0}\right)+R_{n}
\end{align*}
with
\begin{multline*}
\left\Vert R_{n}\right\Vert _{L^{\infty}(\Omega^{\prime})}\leq C\|DU_{n}-P\left(nx\right)DU_{0}\|_{L^{\infty}(\Omega^{\prime})}\\
{}\cdot\max\Bigl(\|DU_{n}\|_{L^{\infty}(\Omega^{\prime})}^{2},\left\Vert P\left(nx\right)DU_{0}\right\Vert _{L^{\infty}(\Omega^{\prime})}^{2}\Bigr),
\end{multline*}
as for any two real $d\times d$ matrices,
\[
\left|\det(A)-\det(B)\right|\leq C(d)\|A-B\|_{\infty}\max\Bigl(\|A\|_{^{\infty}}^{d-1},\left\Vert B\right\Vert _{^{\infty}}^{d-1} \Bigr).
\]
Thanks to Lemma~\ref{Chap5.lem:three},
\[
R_{n}\leq\frac{C}{n^{1/3}}\,\|\phi\|_{H^{\sfrac{1}{2}}\left(\partial\Omega,\mathbb{R}^{3}\right)}^{3}.
\]
In the ball $B(x_{0},\rho)$, let
\[
B_{\pm}=\bigl\{x\in B\left(x_{0},\rho\right)\,:\exists\, p\in\mathbb{Z}^{3},\, nx-p\in Y_{\pm}\bigr\}.
\]
For $n$ large enough, $\left|B_{\pm}\right|>\tau\left|B(x_{0},\rho)\right|$.

In $B_{\pm}$, we have thanks to Lemma~\ref{Chap5.lem:Two}, $\pm\det\left(P\left(nx\right)\right)>2\tau$,
therefore
\[
\pm\det(DU_{n})>2\tau\det(DU_{0})-\frac{C}{n^{1/3}}\,\|\phi\|_{H^{\sfrac{1}{2}}\left(\partial\Omega,\mathbb{R}^{3}\right)}^{3}.
\]
With $\phi\in A(\phi,x_{0},\rho,\lambda)$, this implies
\[
\pm\det(DU_{n})>\Bigl(2\tau\lambda-\frac{C}{n^{1/3}} \Bigr)\|\phi\|_{H^{\sfrac{1}{2}}\left(\partial\Omega,\mathbb{R}^{3}\right)}^{3}>\tau\lambda\|\phi\|_{H^{\sfrac{1}{2}}\left(\partial\Omega,\mathbb{R}^{3}\right)}^{3},
\]
for $n^{1/3}\tau\lambda\geq C$.
\end{proof}

\begin{rem*}
Any larger $n$ would lead to the same conclusion, with the same bound.
Note that the above argument shows that for a given radius $\rho$,
and $\lambda$ sufficiently small, one can choose $n=C\lambda^{-3},$
with a constant $C$ depending $\Omega$ and $\Omega^{\prime}$ only.
\end{rem*}

The fact that the conductivity coefficient $\gamma$ is not smooth
is not required for Theorem~\ref{Chap5.thm:One} to hold. This choice
was made in \cite{BRIANE-MILTON-NESI-04} as it corresponds to realisable
composites. Consider as before the periodic function $\gamma$ with
\[
\gamma\left(y\right)=1+\left(\delta-1\right)\chi_{Q}(y)\text{ for all }y\in Y,
\]
with $\delta$ chosen via Lemma~\ref{Chap5.lem:Two}. Introducing
the standard mollifier $\eta\in C^{\infty}(\mathbb{R^{d}})$ given
by $\eta(x)=c_{\eta}\exp(-1/(1-\left|x\right|^{2}))$ for $|x|<1$
and $\eta(x)=0$ otherwise, where $c_{\eta}$ is chosen so that $\|\eta\|_{L^{1}(\mathbb{R^{d}})}=1$,
write for some $M>0$ to be chosen later $\eta_{M}=\eta\left(M\cdot\right)$.
The function $\tilde{\gamma}=\eta_{M}\star\gamma$ is smooth, $Y$-periodic,
and enjoys the same symmetries as $\gamma$ because $\eta$ is radial.
Note that the open subsets $Y_{\pm}$ given in Lemma~\ref{Chap5.lem:Two}
are located in parts of the periodic cell where $P$ is smooth, that
is, away from $\delta Q$ (see Theorem~3 in~\cite{BRIANE-MILTON-NESI-04}).
Thus, as solutions of elliptic boundary value problems depend smoothly
on their coefficients, we can set $M$ large enough so that Lemma~\ref{Chap5.lem:Two}
applies to $\tilde{\gamma}$ (for another universal constant $\tau$).

Write $\tilde{U}_{n}=(\tilde{u}_{n}^{1},\tilde{u}_{n}^{2},\tilde{u}_{n}^{3})$
where for $i=1,2,3$, $\tilde{u}_{n}^{i}\in H^{1}(\Omega;\mathbb{R}^{3})$
is the solution of
\begin{equation}
\label{Chap5.eq:hom-1}
\begin{split}
\div\left(\tilde{\gamma}\left(nx\right)D\tilde{u}_{n}^{i}\right) & = 0\quad\text{in }\Omega,\\
\tilde{u}_{n}^{i} & = \phi^{i}\quad\text{on }\partial\Omega.
\end{split}
\end{equation}
The proof of Theorem~\ref{Chap5.thm:One} is then easily adapted.
Since $\tilde{\gamma}$ is smooth the error estimates corresponding
to Lemma~\ref{Chap5.lem:three} are classical and the rate of decay
of the error is then $n^{-1}$, see \cite{BENSOUSSAN-LIONS-PAPANICOLAOU-78}.
We obtain the following corollary.
\begin{cor}
\label{Chap5.cor:Four} Given $\rho>0$, $x_{0}\in\Omega^{\prime}$
such that $B_{\rho}(x_{0})\subset\Omega^{\prime}$, and $\lambda>0$,
let $A(\phi,x_{0},\rho,\lambda)$ be defined by \eqref{Chap5.eq:Aphi}.
There exist $n$, depending on $\rho$, $\Omega$, $\Omega^{\prime}$
and $\lambda$ only, a universal constant $\tau>0$ and two open subsets
$B_{+}$ and $B_{-}$ of $B_{\rho}(x_{0})$ such that
\[
\left|B_{+}\right|>\tau\left|B_{\rho}(x_{0})\right|\quad\text{and}\quad\left|B_{-}\right|>\tau\left|B_{\rho}(x_{0})\right|,
\]
and for any $\phi\in A(\phi,x_{0},\rho,\lambda)$,
\[
\det(D\tilde{U}_{n})(x)<-\tau\lambda\left\Vert \phi\right\Vert _{H^{\sfrac{1}{2}}\left(\partial\Omega\right)}^{3}\quad\text{on }B_{-},
\]
and
\[
\det(D\tilde{U}_{n})(x)>\tau\lambda\left\Vert \phi\right\Vert _{H^{\sfrac{1}{2}}\left(\partial\Omega\right)}^{3}\quad\text{on }B_{+},
\]
where $\tilde{U}_{n}$ is the $\tilde{\gamma}\left(n\cdot\right)$-harmonic
extension of $\phi$ given by \eqref{Chap5.eq:hom-1}.
\end{cor}

\section{Positive Jacobian bounds using more than \texorpdfstring{$d$}{d} boundary data in~dimension~\texorpdfstring{$d\geq3$}{d3}\mbox{ }}\label{sec:multidata}

Given a sufficiently smooth conductivity $\gamma$ defined in $\mathbb{R}^{d}$,
with $d\geq3$ and satisfying, for some $s\geq0$, and $\beta>1$,
\begin{equation}
\gamma\in C^{s}\left(\mathbb{R}^{d}\right)\quad\text{such that}\quad1\leq\gamma(x)\leq\beta\text{ in }\Omega,\label{eq:globalreg}
\end{equation}
one can ask whether using $N\geq d$ boundary conditions would provide
$N$-tuples such that the Jacobian matrix of the corresponding
solutions has maximal rank.

A construction of boundary conditions $F_{N}=\left(\phi^{1},\ldots,\phi^{N}\right)\in H^{\sfrac{1}{2}}(\partial\Omega)$
ensuring that the solutions $U=(u^{1},\ldots,u^{N})$ of
\begin{equation}
-\div\left(\gamma Du^{i}\right)=0\text{ in }\Omega,\quad u^{i}=\phi^{i}\text{ on }\partial\Omega,\label{eq:multidata}
\end{equation}
are such that everywhere on the domain at least one $d$-tuples of
such solutions satisfy a positive Jacobian constraint is provided
in \cite{BAL-BONNETIER-MONARD-TRIKI-2013,BAL-MONARD-2012}. Their
approach relies on Complex Geometric Optics solutions \cite{CALDERON-1980,SYLVESTER-UHLMANN-87},
adapted for hybrid inverse problems in \cite{BAL-UHLMANN-2010}.

\begin{prop}[See Lemma 3.3 in \cite{BAL-BONNETIER-MONARD-TRIKI-2013}, Lemma~2.1 in \cite{BAL-MONARD-2012}]\label{prop:GCO}
Given $\gamma\in H^{\frac{d}{2}+3+\varepsilon}\left(\mathbb{R}^{3}\right)$
satisfying \eqref{eq:globalreg}. Let $N=2\left\lfloor \frac{d+1}{2}\right\rfloor $.
There exists a non-empty open set $\mathbf{F}_{N}\subset(H^{\sfrac{1}{2}}(\partial\Omega))^{N}$
of N-tuples of illuminations such that for any $\left(\phi^{1},\ldots,\phi^{N}\right)\in\mathbf{F}_{N}$
there exists a constant $c_{0}>0$ so that $\left(u^{1},\ldots,u^{N}\right)$
given by \eqref{eq:multidata} satisfy

If $d$ is even, $\det\left(Du^{1},\ldots,Du^{d}\right)\ge c_{0}$
in $\Omega$.

If $d$ is odd, on an open cover of $\Omega$ of the form $\{\Omega_{2i-1},\Omega_{2i}\}_{1\le i\le M}$,
\begin{align*}
\left|\det\bigl(Du^{1},\ldots,Du^{d-1},Du^{d+1} \bigr)\right|&\ge c_{0} \quad\text{on }\Omega_{2i-1}\\
\tag*{and}\left|\det \bigl(Du^{1},Du^{d-1},Du^{d} \bigr)\right|&\ge c_{0} \quad\text{on }\Omega_{2i}.
\end{align*}

\end{prop}
When this note was submitted, one of the referee suggested an earlier
result \cite{Greene-Wu-1975a,Greene-Wu-1975b}. It is shown in these
articles that on any connected non compact Riemannian manifold of
dimension $d$, there exists $2d+1$ harmonic functions which taken
together give a proper embedding of the manifold in the Euclidean
space $\mathbb{R}^{2d+1}$. Translated in the language and context
of this paper, it shows that given $\gamma\in C^{\infty}\left(\mathbb{R}^{d}\right)$
satisfying \eqref{eq:globalreg}, one can choose $N=2d+1$ to satisfy
the positivity constraint. The main property used to obtain this result
is the unique continuation property of second order linear elliptic
PDE. As this property holds for Lipschitz conductivities \cite{KOCH-TATARU-2001},
this result can be extended from $C^{\infty}$ to $C^{0,1}$, and
is in that sense more general than Proposition~\ref{prop:GCO}, at
the cost of a slightly larger number of boundary conditions.

Both results apply to $\tilde{\gamma}\left(n\cdot\right)$ for each
$n\geq1$. Corollary~\ref{Chap5.cor:Four} shows that the Dirichlet
data must changes with $n$ and that the lower bound on the determinant
tends to nought as $n$ grows if the $H^{\sfrac{1}{2}}(\partial\Omega)$
norm of the Dirichlet data is bounded a priori. The decay of the lower
bound is exponential in $n$ in the proof of Proposition~\ref{prop:GCO}.

In dimension 3, as the set of possible septuplet given in \cite{Greene-Wu-1975a,Greene-Wu-1975b}
is very large, it is possible that there exists a good choice of boundary
conditions for all conductivities satisfying a priori Lipschitz bounds.
Considering $G_{\beta,L}$ given by
\[
G_{\beta,L}=\left\{ \gamma\in C^{0,1}\left(\mathbb{R}^{3}\right)\,:\,0<1\leq\gamma\leq\beta\text{ and }\frac{\left|\gamma(x)-\gamma(y)\right|}{|x-y|}<L\text{ in }\Omega\right\} ,
\]
there could exist $N\in\mathbb{N}$ depending on $\beta,L$ and $(\phi^{1},\ldots,\phi^{N})$
such that for any $\gamma\in\nobreak G_{\beta,L},$
\[
\mathrm{rank}\left[Du^{1},\dots,Du^{N}\right]=3\text{ on }\Omega.
\]
What Theorem~\ref{Chap5.thm:One} and Corollary~\ref{Chap5.cor:Four}
indicate is that an a priori constraint on the oscillations of $\gamma$
is unavoidable.

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