==== TABLE OF CONTENTS ====
Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem//analytic-estimates.tex
Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem//boundary-reduction.tex
Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem//coefficient-recovery.tex
Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem//introduction.tex
Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem//main.tex
Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem//physical-amplitudes.tex
Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem//transport-matching.tex
Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem//references.bib
===========================
==== BEGIN Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem//analytic-estimates.tex ====
\section{An augmented system and analytic estimates}
\label{sec:analytic-estimates}
This section constructs a scalar-Laplacian augmented operator and a
supported estimate that will control differences of transferred physical
displacement--divergence pairs.
Adjoining the displacement divergence to obtain a system with Laplacian
principal part goes back to Weck \cite{Weck1969}; see the account of
Imanuvilov and Yamamoto \cite[Section~1]{ImanuvilovYamamoto2004Carleman}.
The identities below give the formulation used in this proof.
We first work with one of the smoothly extended coefficient pairs from
Lemma~\ref{lem:common-extension}, and suppress its subscript. Set
\[
m=\mu,\qquad k=\lambda+\mu,\qquad \ell=\lambda+2\mu=k+m.
\]
The energy assumptions imply $m>0$, $k>m/3$, and $\ell>4m/3$.
For an independent vector--scalar pair $U=(u,d)$, define
\begin{equation}
\label{eq:augmented-operators}
\begin{aligned}
R(u,d)_i
& =m\Delta u_i+k\partial_i d
+\sum_{j=1}^3(\partial_jm)(\partial_ju_i+\partial_iu_j)
+(\partial_i\lambda)d,\\
S(u,d)
& =\ell\Delta d+2\nabla k\cdot\nabla d
+2\nabla m\cdot\Delta u
+2\sum_{i,j=1}^3(\partial_i\partial_jm)\partial_iu_j
+(\Delta\lambda)d,\\
P&=m\Delta+\nabla m\cdot\nabla.
\end{aligned}
\end{equation}
The scalar $d$ will equal $\diverg u$ for physical solutions, but keeping
it independent gives the compatibility identity needed below.
\begin{lemma}
\label{lem:augmented-system}
For every smooth pair $(u,d)$,
\begin{equation}
\label{eq:augmented-identity}
Lu=R(u,\diverg u),\qquad
\diverg R(u,d)-S(u,d)=P(\diverg u-d).
\end{equation}
In particular, $Lu=0$ implies $R(u,\diverg u)=S(u,\diverg u)=0$.
The normalized augmented operator
\begin{equation}
\label{eq:normalized-augmented}
\Lcal U=
\left(m^{-1}R(u,d),\,
\ell^{-1}\bigl(S(u,d)-2m^{-1}\nabla m\cdot R(u,d)\bigr)\right)
\end{equation}
has the form
\begin{equation}
\label{eq:laplace-principal-form}
\Lcal=\Delta\Id_4+\sum_{j=1}^3 A_j(x)\partial_j+A_0(x)
\end{equation}
with smooth matrix coefficients. Consequently, the difference of the
normalized augmented operators for the two coefficient pairs has order
at most one and coefficients supported in $\overline\Omega$.
\end{lemma}
\begin{proof}
The first identity is the coordinate expansion of $\diverg\sigma(u)$.
To verify the second independently of the constraint, write
$v=\diverg u$. Differentiating \eqref{eq:augmented-operators} gives
\[
\begin{aligned}
\diverg R(u,d)
={}&m\Delta v+\nabla m\cdot\nabla v+k\Delta d
+(\nabla k+\nabla\lambda)\cdot\nabla d\\
&+2\nabla m\cdot\Delta u
+2\sum_{i,j=1}^3(\partial_i\partial_jm)\partial_iu_j
+(\Delta\lambda)d.
\end{aligned}
\]
Subtracting $S$ and using $k=\lambda+m$ gives
$m\Delta(v-d)+\nabla m\cdot\nabla(v-d)$.
For the principal part in \eqref{eq:normalized-augmented}, the only
second derivatives of $u$ in its scalar numerator cancel. More
explicitly, if
\[
C_m(u)_i=\sum_{j=1}^3(\partial_jm)
(\partial_ju_i+\partial_iu_j),
\]
then this numerator is
\[
\begin{aligned}
S-2m^{-1}\nabla m\cdot R
={}&\ell\Delta d+
(2\nabla k-2km^{-1}\nabla m)\cdot\nabla d\\
&+2\sum_{i,j=1}^3(\partial_i\partial_jm)\partial_iu_j
-2m^{-1}\nabla m\cdot C_m(u)\\
&+\bigl(\Delta\lambda-2m^{-1}\nabla m\cdot\nabla\lambda\bigr)d.
\end{aligned}
\]
This proves \eqref{eq:laplace-principal-form}. The support assertion follows
from equality of the extended coefficients off $\overline\Omega$.
\end{proof}
The order-one difference of the two augmented operators is the forcing
term to be estimated after physical transfer in
Section~\ref{sec:transport-matching}.
Fix a nonzero $\theta\in\C^3$ with $\theta\cdot\theta=0$, and write
\begin{equation}
\label{eq:phase-notation}
D_\theta=\theta\cdot\nabla,\qquad
E_\tau(x)=e^{\tau\theta\cdot x},\qquad h=\tau^{-1}.
\end{equation}
For $s\in\R$, the semiclassical Sobolev norm on $\R^3$ is
\[
\norm{v}_{H_h^s}^2
=\int_{\R^3}(1+h^2\abs{\xi}^2)^s
\abs{\widehat v(\xi)}^2\,\dd\xi.
\]
For vectors, we sum the component norms. These norms and all Hilbert
space adjoints below use the Hermitian structure, whereas products
such as $\theta\cdot\theta$ remain complex bilinear. Constants in the
following estimates may depend on the coefficients, the fixed ball,
and $\theta$, but not on sufficiently large $\tau$.
\begin{proposition}[Supported Carleman estimate]
\label{prop:carleman}
Let $B$ be a bounded ball. For $U\in C_c^\infty(B;\C^4)$ and all
sufficiently small $h>0$,
\begin{equation}
\label{eq:augmented-carleman}
\norm{E_\tau^{-1}U}_{H_h^1}
\le Ch\norm{E_\tau^{-1}\Lcal U}_{H_h^{-1}}.
\end{equation}
\end{proposition}
\begin{proof}
Put $\varphi(x)=-\operatorname{Re}(\theta\cdot x)$. A nonzero complex
null vector has nonzero real part, so $\varphi$ is a nonconstant linear
limiting Carleman weight for the Laplacian. For
$\varphi_\varepsilon=\varphi+h\varphi^2/(2\varepsilon)$,
\cite[Lemma~2.1]{SaloTzou2009}, with Sobolev index $s=-1$, gives
\begin{equation}
\label{eq:convexified-carleman}
\frac{h}{\sqrt\varepsilon}\norm{v}_{H_h^1}
\le C\norm{e^{\varphi_\varepsilon/h}h^2\Delta
e^{-\varphi_\varepsilon/h}v}_{H_h^{-1}},
\qquad v\in C_c^\infty(B),
\end{equation}
for $h\ll\varepsilon\ll1$, with $C$ independent of sufficiently small
$\varepsilon$. Its statement has a gain of two derivatives and permits
every real $s$. The sign of the Laplacian is immaterial here.
Apply \eqref{eq:convexified-carleman} componentwise. A first-order term
in \eqref{eq:laplace-principal-form} contributes
\[
e^{\varphi_\varepsilon/h}h^2A_j\partial_j
e^{-\varphi_\varepsilon/h}v
=hA_j(h\partial_jv)-hA_j(\partial_j\varphi_\varepsilon)v.
\]
Its $H_h^{-1}$ norm is bounded by $C_1h\norm{v}_{H_h^1}$,
uniformly when $h/\varepsilon$ is small. The zeroth-order terms have
the same bound. Choose $\varepsilon>0$ so small that these terms are
absorbed by the left side of \eqref{eq:convexified-carleman}, and then
choose $h$ sufficiently small relative to this fixed $\varepsilon$.
We obtain
\[
\norm{e^{\varphi_\varepsilon/h}U}_{H_h^1}
\le C_\varepsilon h
\norm{e^{\varphi_\varepsilon/h}\Lcal U}_{H_h^{-1}}.
\]
The factor $e^{\varphi^2/(2\varepsilon)}$ is now fixed. It and its
inverse, smoothly localized around $\overline B$, are bounded
multipliers on $H_h^1$, uniformly for $0<h\le1$, by the product rule.
They are bounded on $H_h^{-1}$ by duality. Removing this factor gives
the same estimate with $\varphi$ in place of $\varphi_\varepsilon$.
The multiplier constants may depend on $\varepsilon$; no subsequent
absorption involves them.
Finally, write $\beta=\operatorname{Im}\theta$.
Multiplication by $e^{-i\beta\cdot x/h}$ shifts the semiclassical
frequency $h\xi$ by the fixed vector $-\beta$. The weights
$1+\abs{h\xi}^2$ and $1+\abs{h\xi-\beta}^2$ are comparable,
so this multiplier and its inverse are uniformly bounded on
$H_h^{\pm1}$. Since
$E_\tau^{-1}=e^{\varphi/h}e^{-i\beta\cdot x/h}$, this proves
\eqref{eq:augmented-carleman}.
\end{proof}
\begin{corollary}
\label{cor:physical-solvability}
For the same $B$ and $\theta$, and all sufficiently large $\tau$,
every $w\in C_c^\infty(B;\C^3)$
satisfies
\begin{equation}
\label{eq:physical-test-estimate}
\norm{E_\tau^{-1}w}_{L^2(B)}
\le C\norm{E_\tau^{-1}Lw}_{L^2(B)}.
\end{equation}
Moreover, for every $f\in L^2(B;\C^3)$ there exists
$z\in L^2(B;\C^3)$ satisfying, distributionally,
\begin{equation}
\label{eq:physical-solvability}
E_\tau^{-1}L(E_\tau z)=f\quad\text{in }B,
\qquad \norm{z}_{L^2(B)}\le C\norm{f}_{L^2(B)}.
\end{equation}
\end{corollary}
\begin{proof}
Apply Proposition~\ref{prop:carleman} to $(w,\diverg w)$.
By \eqref{eq:augmented-identity}, its augmented forcing is formed
from $Lw$ and $\diverg(Lw)$. If $F=E_\tau^{-1}Lw$, then
\[
E_\tau^{-1}\diverg(Lw)=\diverg F+\tau\theta\cdot F.
\]
The Fourier definition of the norms gives
$\norm{\partial_jF}_{H_h^{-1}}\le h^{-1}\norm{F}_{L^2}$.
Smooth coefficient multipliers, localized near $\overline B$, are
uniformly bounded on $H_h^{-1}$. Hence
\[
\norm{E_\tau^{-1}\Lcal(w,\diverg w)}_{H_h^{-1}}
\le Ch^{-1}\norm{F}_{L^2}.
\]
Taking the displacement component on the left side of
\eqref{eq:augmented-carleman} proves
\eqref{eq:physical-test-estimate}.
Let $A_\theta=E_\tau^{-1}LE_\tau$ on test functions in $B$.
Substituting $w=E_\tau v$ in
\eqref{eq:physical-test-estimate} gives
$\norm{v}_{L^2}\le C\norm{A_\theta v}_{L^2}$.
Formal self-adjointness of $L$ gives
\[
A_\theta^*
=e^{\tau\overline\theta\cdot x}L
e^{-\tau\overline\theta\cdot x}
=A_{-\overline\theta}.
\]
The same estimate for the null direction $-\overline\theta$ thus
holds for $A_\theta^*$. Use the $L^2$ inner product linear in its
first argument. The functional
\[
A_\theta^*v\longmapsto (v,f)_{L^2(B)},
\qquad v\in C_c^\infty(B;\C^3),
\]
is well-defined and has norm at most $C\norm{f}_{L^2}$. Extend it by
the Hahn--Banach Theorem and represent it as $(\,\cdot\,,z)_{L^2}$.
Then $(A_\theta^*v,z)=(v,f)$ for every test function $v$, which is
the distributional equation in \eqref{eq:physical-solvability}.
The norm of the representing vector gives the asserted bound.
\end{proof}
\begin{lemma}[Scaled interior estimate]
\label{lem:scaled-interior}
Let $B'\Subset B$. There exists $h_{B'}>0$ such that, whenever
$0<h=\tau^{-1}<h_{B'}$ and $z,f\in L^2(B;\C^3)$ satisfy
$E_\tau^{-1}L(E_\tau z)=f$ distributionally in $B$,
\begin{equation}
\label{eq:scaled-interior}
\sum_{j=0}^2 h^j\norm{\nabla^jz}_{L^2(B')}
\le C_{B'}\bigl(\norm{z}_{L^2(B)}
+h^2\norm{f}_{L^2(B)}\bigr).
\end{equation}
If $f$ is smooth, then $z$ is smooth in $B$ for each fixed $\tau$.
\end{lemma}
\begin{proof}
The positive principal coefficient form of $L$ is
\[
m\abs{\xi}^2\abs{v}^2+k\abs{\xi\cdot v}^2,
\qquad \xi\in\R^3,\quad v\in\C^3.
\]
Thus its ellipticity is uniform on $\overline B$. Conjugation leaves
the principal part unchanged and introduces coefficients of size
$O(h^{-1})$ and $O(h^{-2})$ in orders one and zero. On a ball
$B(x_0,2h)\subset B$, set $x=x_0+hy$ and multiply the equation by
$h^2$. The resulting operator on the fixed ball $B(0,2)$ has a
uniformly strongly elliptic principal part, while its coefficients
and their derivatives in $y$ are uniformly bounded. As
$(x_0,h)$ ranges over $\overline B'\times[0,h_0]$, these operators
form a compact smooth coefficient family with a common ellipticity
bound. The local parametrix estimate persists under a small
coefficient perturbation, so a finite cover gives a uniform constant.
Interior $H^2$ regularity for distributional $L^2$ solutions of smooth
elliptic systems is given by
\cite[Theorem~15.1 and Remark~15.2]{Dyatlov2026}. The same parametrix
argument with matrix symbols gives the localized estimate for systems
corresponding to the scalar Proposition~15.6 of that source. With the
uniform constants just obtained, it yields
\[
\sum_{j=0}^2 h^j\norm{\nabla^jz}_{L^2(B(x_0,h))}
\le C\bigl(\norm{z}_{L^2(B(x_0,2h))}
+h^2\norm{f}_{L^2(B(x_0,2h))}\bigr).
\]
For small $h$, cover $B'$ by such inner balls with uniformly bounded
overlap of the doubled balls. Squaring and summing proves
\eqref{eq:scaled-interior}. Iterated interior elliptic regularity,
with $\tau$ fixed, proves the final assertion.
\end{proof}
==== END Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem//analytic-estimates.tex ====
==== BEGIN Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem//boundary-reduction.tex ====
\section{Boundary reduction and physical transfer}\label{sec:boundary-reduction}
Throughout the proof, the two coefficient pairs satisfy the hypotheses
of Theorem~\ref{thm:main} and have equal displacement-to-traction maps.
\begin{lemma}[Common exterior extension]\label{lem:common-extension}
The two pairs admit real smooth extensions to $\R^3$ that satisfy
\eqref{eq:energy-positivity} everywhere, coincide on
$\R^3\setminus\Omega$, and equal a common constant admissible pair
outside a ball.
\end{lemma}
\begin{proof}
By Theorem~1.2 of Tan and Liu \cite{TanLiu2023}, equality of the elastic
Dirichlet-to-Neumann maps determines all boundary partial derivatives
of both Lam\'e moduli. Our inequalities imply
$\lambda+\mu>\mu/3>0$, so their positivity condition holds.
Their definition of the boundary operator is
$\lambda(\diverg u)n+\mu(\nabla u+(\nabla u)^T)n$, so its Euclidean
specialization agrees with \eqref{eq:physical-dn}.
Consequently the two pairs have identical jets at every point of
$\partial\Omega$. Lin and Nakamura also give local boundary
reconstruction, with curved boundaries discussed in Section~4 of
\cite{LinNakamura2017}.
Smoothly extend the first pair to a neighborhood of
$\overline\Omega$. By compactness and strict positivity, shrink the
neighborhood so that \eqref{eq:energy-positivity} still holds there.
Choose a smooth cutoff equal to one on a smaller neighborhood of
$\overline\Omega$ and supported in the first neighborhood. Interpolate
with any constant pair satisfying \eqref{eq:energy-positivity}.
The admissible set in the $(\lambda,\mu)$ plane is convex, so this
produces a smooth admissible global extension of the first pair, constant
outside a compact set.
For the second pair use its prescribed values in $\Omega$ and the first
extension on the complement. Equality of all jets makes this piecewise
definition smooth across each boundary component. Positivity holds on
both sides and on the boundary. This proves the assertion, including
when the complement of $\Omega$ has bounded components.
\end{proof}
Fix these extensions and retain the notation $\lambda_j,\mu_j$ and
$L_j=L_{\lambda_j,\mu_j}$ for them. Choose $r>0$ so that
$\overline\Omega\subset B_r$ and the extensions are constant outside
$B_r$. Here $B_R$ is the open ball of radius $R$ centered at the origin.
\begin{proposition}[Transfer of physical solutions]\label{prop:physical-transfer}
Let $B$ be a ball containing $\overline B_r$. Every smooth solution
$u_1$ of $L_1u_1=0$ on $B$ has a smooth counterpart $u_2$ satisfying
$L_2u_2=0$ on $B$ and
\[
u_2=u_1\quad\text{on }B\setminus\overline\Omega.
\]
In particular, $u_2-u_1$ and
$\diverg u_2-\diverg u_1$ are smooth and supported in
$\overline\Omega$.
\end{proposition}
\begin{proof}
Solve the second Dirichlet problem in $\Omega$ with trace
$u_1|_{\partial\Omega}$, and use $u_1$ outside $\overline\Omega$.
Matching traces give an $H^1_{\mathrm{loc}}(B)$ field $u_2$.
The restrictions of the two interior solutions have identical boundary
tractions because their Dirichlet values and boundary maps agree.
For a smooth test vector compactly supported in $B$, the weak boundary
term contributed by the interior therefore agrees with that for $u_1$.
The coefficients and displacements coincide on the exterior, where the
same test contributes the opposite interface term. Hence the glued
field satisfies $L_2u_2=0$ weakly on $B$.
The extended operator is smooth and strongly elliptic. Interior
elliptic regularity, applied also across $\partial\Omega$, gives
$u_2\in C^\infty(B;\C^3)$. The difference is zero off
$\overline\Omega$, as are all its derivatives there. Its support and
the support of its divergence are thus contained in
$\overline\Omega\Subset B_r$.
\end{proof}
==== END Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem//boundary-reduction.tex ====
==== BEGIN Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem//coefficient-recovery.tex ====
\section{Recovery of the coefficients}\label{sec:coefficient-recovery}
The holomorphic comparison from Proposition~\ref{prop:transport-matching}
is constrained by the geometry of its polarization bundle. We first show
that this comparison is the identity, and then recover the coefficients from
the transport matrices.
\subsection{The bundle on the null conic}
Let $H\to\mathcal C$ be the holomorphic bundle with fiber $H_\theta$, and
let $\mathcal N\subset H$ be the null line subbundle with fiber
$\C\theta$. Thus $\Ecal=H\oplus\mathcal O$, where $\mathcal O$ denotes
the trivial holomorphic line bundle on $\mathcal C$.
On $\mathbb P^1(\C)$, let $\mathcal O(-1)$ denote the tautological line
bundle and $\mathcal O(-2)$ its tensor square.
\begin{lemma}\label{lem:conic-bundle}
The bundle $H$ has no nonzero global holomorphic sections, and
$H/\mathcal N$ is holomorphically trivial. More precisely, under the
identification $\mathcal C\simeq\mathbb P^1(\C)$,
\[
H\simeq\mathcal O(-1)^{\oplus2},\qquad
\mathcal N\simeq\mathcal O(-2),
\]
and their quotient sequence is
\begin{equation}\label{eq:conic-quotient}
0\longrightarrow\mathcal O(-2)
\xrightarrow{\ (z_0,z_1)^T\ }
\mathcal O(-1)^{\oplus2}
\xrightarrow{\ (-z_1,z_0)\ }
\mathcal O\longrightarrow0.
\end{equation}
Here $[z_0:z_1]$ are homogeneous coordinates on $\mathbb P^1(\C)$.
\end{lemma}
\begin{proof}
A homogeneous parametrization of the conic is
\begin{equation}\label{eq:conic-parametrization}
\theta(z_0,z_1)
=(z_0^2-z_1^2,\ i(z_0^2+z_1^2),\ 2z_0z_1).
\end{equation}
It identifies $\mathbb P^1(\C)$ with $\mathcal C$: on the first coordinate
chart its inverse is
$z_1/z_0=\theta_3/(\theta_1-i\theta_2)$, and on the second it is
$z_0/z_1=\theta_3/(-\theta_1-i\theta_2)$. The denominators cannot both
vanish at a point of $\mathcal C$.
Consider the two degree-one columns
\[
h_1=(z_0,iz_0,z_1),\qquad
h_2=(-z_1,iz_1,z_0).
\]
Direct calculation, with the complex bilinear product, gives
\begin{equation}\label{eq:conic-frame}
\theta\cdot h_1=\theta\cdot h_2=0,\qquad
h_1\times h_2=i\theta,\qquad
\theta=z_0h_1+z_1h_2.
\end{equation}
To interpret these homogeneous formulas as bundle maps, write
$\mathcal O(-1)_{[z]}=\C z$ for $z=(z_0,z_1)$.
The assignment
\[
(\xi_1z,\xi_2z)\longmapsto \xi_1h_1(z)+\xi_2h_2(z)
\]
is independent of the representative: replacing $z$ by $cz$ replaces
$\xi_j$ by $c^{-1}\xi_j$ and $h_j(z)$ by $ch_j(z)$.
Its local polynomial formulas are holomorphic. Since the images lie in
$H$ and are linearly independent at every point, it is an isomorphism
$\mathcal O(-1)^{\oplus2}\to H$. Likewise,
$\xi z^{\otimes2}\mapsto\xi\theta(z)$ is well defined because
$\theta(cz)=c^2\theta(z)$, and identifies $\mathcal O(-2)$ with
$\mathcal N$. The final identity in \eqref{eq:conic-frame} gives the
first map in \eqref{eq:conic-quotient}.
The expression $-z_1\xi_1+z_0\xi_2$ is also unchanged under the same
representative scaling, so its local polynomial formulas define a
holomorphic map to the trivial line bundle $\mathcal O$. The row
$(-z_1,z_0)$ is everywhere surjective, and its kernel is the image of
$(z_0,z_1)^T$.
This proves the quotient assertion, including at both coordinate-chart
endpoints. It is also consistent with the induced bilinear form: for
$p=\xi_1 h_1+\xi_2 h_2$ in a local frame,
\[
p\cdot p=(-z_1\xi_1+z_0\xi_2)^2.
\]
Finally, any holomorphic section of $H$ has holomorphic ambient components
on the compact conic, so those components are constant. Such a constant
vector is orthogonal to every null vector. The vectors
$\theta(1,0)$, $\theta(0,1)$ and $\theta(1,1)$ span $\C^3$, so the
section is zero.
\end{proof}
\begin{remark}
The exact sequence \eqref{eq:conic-quotient} does not split holomorphically:
a splitting would give a nonzero global section of $H$. On the other
hand, the displayed splitting of $H$ itself gives nonscalar holomorphic
endomorphisms of $H$. The next argument uses the transport equation to
restrict those endomorphisms.
\end{remark}
\begin{proposition}\label{prop:comparison-rigidity}
The comparison endomorphism of
Proposition~\ref{prop:transport-matching} is $G=\Id$. Consequently, for
every $x\in\R^3$ and every nonzero null vector $\theta$,
\begin{equation}\label{eq:equal-restricted-transports}
M_{1,\theta}=M_{2,\theta}\quad\text{on }\Ecal_\theta.
\end{equation}
\end{proposition}
\begin{proof}
Fix $x$. Relative to $\Ecal=H\oplus\mathcal O$, the upper-right block
of the holomorphic endomorphism $G(x,\cdot)$ is a global holomorphic
section of $H$, and therefore vanishes by Lemma~\ref{lem:conic-bundle}.
Its lower-right block is a holomorphic scalar function on the compact
conic, and hence is independent of direction. We may thus write
\[
G=\begin{pmatrix}F&0\\ A&f\end{pmatrix},\qquad f=f(x).
\]
All these blocks are smooth in $x$. Taking the upper-right block of
\eqref{eq:comparison-transport}, and using
\eqref{eq:transport-matrix}, gives
\[
0=\theta f-F\theta.
\]
It follows that $F-f\Id_H$ vanishes on $\mathcal N$. This holomorphic
bundle morphism consequently factors through $H/\mathcal N$. By
Lemma~\ref{lem:conic-bundle}, its factor is a morphism
$\mathcal O\to H$, which must vanish. Thus $F=f\Id_H$.
The upper-left block of \eqref{eq:comparison-transport} now reads
\begin{equation}\label{eq:comparison-rank}
(D_\theta f)\Id_{H_\theta}=\theta A.
\end{equation}
The right side has rank at most one, while $H_\theta$ has dimension two.
Therefore $D_\theta f=0$. Equation~\eqref{eq:comparison-rank} then gives
$A=0$, since $\theta\ne0$. As null vectors span $\C^3$, we obtain
$\nabla f=0$. The exterior identity value of $G$ fixes $f=1$, so $G=\Id$.
Substitution in \eqref{eq:comparison-transport} proves
\eqref{eq:equal-restricted-transports}.
\end{proof}
\subsection{A finite-type uniqueness lemma}
We will show that equality of the restricted bottom-left transport blocks
leaves at most a scalar-identity ambiguity, then use the following
elementary lemma to remove it. Indices
in the lemma range from $1$ to $n$; all sums are written explicitly.
\begin{lemma}\label{lem:finite-type}
Let $U\subset\R^n$ be connected and open, with $n\ge2$, and let
$\eta,q$ and $B_{ij}^{\ a}$ be smooth, real- or complex-valued functions
on $U$ satisfying
\begin{equation}\label{eq:finite-type-hypothesis}
\partial_i\partial_j\eta
=\sum_{a=1}^n B_{ij}^{\ a}\partial_a\eta+\delta_{ij}q.
\end{equation}
Then $(\nabla\eta,q)$ satisfies a homogeneous first-order system with
smooth coefficients. In particular, if $\eta$ vanishes on a nonempty
open subset of $U$, then $\eta=q=0$ on $U$.
\end{lemma}
\begin{proof}
Put $v_a=\partial_a\eta$. For each $s$, choose one index $i=i(s)$ with
$i\ne s$, without summing over $i$. Equation~\eqref{eq:finite-type-hypothesis}
and commutation of third derivatives give
\begin{align*}
\partial_s q
&=\partial_i\left(\sum_{a=1}^n B_{si}^{\ a}v_a\right)
-\partial_s\left(\sum_{a=1}^n B_{ii}^{\ a}v_a\right)\\
&=\sum_{a=1}^n
(\partial_i B_{si}^{\ a}-\partial_s B_{ii}^{\ a})v_a
+\sum_{a=1}^n
(B_{si}^{\ a}\partial_i v_a-B_{ii}^{\ a}\partial_s v_a).
\end{align*}
There is no derivative of $q$ on the right because the off-diagonal
entry $\partial_s\partial_i\eta$ in
\eqref{eq:finite-type-hypothesis} has $\delta_{si}=0$.
Substituting that equation for the derivatives of $v$ yields
\begin{equation}\label{eq:finite-type-system}
\begin{aligned}
\partial_s v_a&=\sum_{b=1}^n B_{as}^{\ b}v_b+\delta_{as}q,\\
\partial_s q&=\sum_{b=1}^n C_s^{\ b}v_b+d_s q,
\end{aligned}
\end{equation}
where the smooth coefficients are explicitly
\begin{equation}\label{eq:finite-type-coefficients}
\begin{aligned}
C_s^{\ b}
&=\partial_i B_{si}^{\ b}-\partial_s B_{ii}^{\ b}
+\sum_{a=1}^n
\bigl(B_{si}^{\ a}B_{ai}^{\ b}-B_{ii}^{\ a}B_{as}^{\ b}\bigr),\\
d_s&=B_{si}^{\ i}-B_{ii}^{\ s}.
\end{aligned}
\end{equation}
This is the claimed homogeneous system for $v$ and $q$.
If $\eta=0$ on an open subset, then $v=0$ there, and a diagonal entry of
\eqref{eq:finite-type-hypothesis} also gives $q=0$ there. Along any
piecewise smooth path in $U$, Equation~\eqref{eq:finite-type-system}
restricts to a homogeneous linear ordinary differential equation for
$(v,q)$. Uniqueness with zero initial data propagates its vanishing along
the path. An open connected subset of $\R^n$ is path connected, so
$v=q=0$ on $U$. Finally, $\eta$ is constant on $U$, and its value on the
initial open subset is zero.
\end{proof}
\subsection{Completion of the coefficient recovery}
\begin{proof}[Proof of Theorem~\ref{thm:main}]
Suppose that the two displacement-to-traction maps agree, and use the
common exterior extensions furnished by Lemma~\ref{lem:common-extension}.
Propositions~\ref{prop:transport-matching} and
\ref{prop:comparison-rigidity} give equality of the transport matrices
on $\Ecal_\theta$ for every nonzero null $\theta$. In particular,
\eqref{eq:transport-coefficients} gives
\[
D_\theta\log(k_1/\ell_1)
=D_\theta\log(k_2/\ell_2).
\]
Since null vectors span $\C^3$, the difference of these logarithms has
zero gradient. Its exterior value is zero. Hence $k_1/\ell_1=k_2/\ell_2$
on $\R^3$, and the functions
\begin{equation}\label{eq:common-ratios}
c=\frac{m_1}{\ell_1}=\frac{m_2}{\ell_2}>0,
\qquad b_* =\log\frac{k_1}{m_1}=\log\frac{k_2}{m_2}
\end{equation}
are well defined and smooth. Indeed, $m_j/\ell_j=1-k_j/\ell_j$, and
$k_j/m_j=(k_j/\ell_j)/(1-k_j/\ell_j)$.
Equality of the bottom-left blocks on $\Ecal_\theta$ means
\[
(Q_{1,\theta}-Q_{2,\theta})\cdot p=0
\qquad(p\in H_\theta).
\]
The annihilator of $H_\theta=\theta^\perp$ for the nondegenerate
ambient bilinear form is $\C\theta$. Since each $Q_{j,\theta}$ is
linear in $\theta$, there is a smooth ambient matrix $K(x)$ such that
\begin{equation}\label{eq:null-line-preservation}
Q_{1,\theta}-Q_{2,\theta}=K(x)\theta\in\C\theta
\qquad(\theta\cdot\theta=0).
\end{equation}
For fixed $x$, the proportionality factor in
\eqref{eq:null-line-preservation} is a holomorphic function on $\mathcal C$.
To see this at every point, choose a local holomorphic representative
$\theta$ and a component $\theta_a\ne0$; the factor is
$(K\theta)_a/\theta_a$. These expressions agree on overlapping charts
and are unchanged by rescaling the representative. Compactness of
$\mathcal C$ makes the factor constant in direction. The spanning
property of null vectors then gives
\begin{equation}\label{eq:scalar-ambient-difference}
K(x)=\kappa(x)\Id,\qquad \kappa(x)=\tfrac13\tr K(x).
\end{equation}
In particular, $\kappa$ is smooth.
Set
\[
\eta=\log m_1-\log m_2,\qquad v=\nabla\eta=g_1-g_2,
\qquad \beta=\nabla b_*,\qquad \alpha=\frac{c-1/2}{c},
\]
where $g_j=\nabla\log m_j$. Because
$\nabla\log k_j=g_j+\beta$, the expression for $Q$ in
\eqref{eq:transport-coefficients} becomes
\[
2Q_{j,\theta}
=\bigl((c-1/2)g_jg_j^T+c g_j\beta^T
-c\nabla^2\log m_j\bigr)\theta.
\]
Subtract these identities, use
$g_1g_1^T-g_2g_2^T=g_1v^T+vg_2^T$, and apply
\eqref{eq:scalar-ambient-difference}. Defining the smooth scalar
$q=-2\kappa/c$, we obtain
\[
\nabla^2\eta
=\alpha(g_1v^T+vg_2^T)+v\beta^T+q\Id.
\]
Equivalently,
\begin{equation}\label{eq:coefficient-hessian}
\begin{aligned}
\partial_i\partial_j\eta
&=\sum_{a=1}^3 B_{ij}^{\ a}\partial_a\eta+\delta_{ij}q,\\
B_{ij}^{\ a}
&=\alpha(g_1)_i\delta_{ja}
+\bigl(\alpha(g_2)_j+\beta_j\bigr)\delta_{ia}.
\end{aligned}
\end{equation}
The coefficients $B_{ij}^{\ a}$ are smooth on $\R^3$. They are fixed
functions determined by the two coefficient pairs under comparison.
The common exterior extension gives $\eta=0$ on a nonempty exterior open
set. Lemma~\ref{lem:finite-type}, applied to
\eqref{eq:coefficient-hessian}, yields $\eta=0$ throughout $\R^3$.
Thus $m_1=m_2$. Equation~\eqref{eq:common-ratios} then gives
$k_1=m_1e^{b_*}=m_2e^{b_*}=k_2$, and therefore
\[
\mu_1=\mu_2,\qquad
\lambda_1=k_1-m_1=k_2-m_2=\lambda_2.
\]
Restricting to $\Omega$ proves the theorem.
\end{proof}
==== END Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem//coefficient-recovery.tex ====
==== BEGIN Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem//introduction.tex ====
\section{Introduction}\label{sec:introduction}
Let $\Omega\subset\R^3$ be a bounded connected domain with smooth boundary.
For real functions $\lambda,\mu\in C^\infty(\overline\Omega)$ satisfying
\begin{equation}\label{eq:energy-positivity}
\mu>0,\qquad 3\lambda+2\mu>0
\quad\text{on }\overline\Omega,
\end{equation}
define the strain, stress, and static elasticity operator by
\begin{equation}\label{eq:physical-operator}
e(u)=\frac{\nabla u+(\nabla u)^T}{2},\qquad
\sigma_{\lambda,\mu}(u)=\lambda(\diverg u)I+2\mu e(u),\qquad
L_{\lambda,\mu}u=\diverg\sigma_{\lambda,\mu}(u).
\end{equation}
Here $\mu$ is the shear modulus and $\lambda+2\mu/3$ is the bulk modulus.
The inequalities in \eqref{eq:energy-positivity} give uniformly positive
elastic energy. For each $f\in H^{1/2}(\partial\Omega;\R^3)$ there is a
unique $u_f\in H^1(\Omega;\R^3)$ with $L_{\lambda,\mu}u_f=0$ and trace $f$.
The displacement-to-traction map is defined weakly by
\begin{equation}\label{eq:physical-dn}
\langle\Lambda_{\lambda,\mu}f,g\rangle
=\int_\Omega\left[
\lambda(\diverg u_f)(\diverg v_g)+2\mu e(u_f):e(v_g)
\right]\dd x,
\end{equation}
where $v_g\in H^1(\Omega;\R^3)$ has trace $g$, and $A:B=\tr(A^TB)$.
The weak equation makes this expression independent of the extension.
For smooth boundary data it is the traction
$\sigma_{\lambda,\mu}(u_f)n$, with $n$ the outward normal.
\begin{theorem}\label{thm:main}
Let $\Omega\subset\R^3$ be any bounded connected domain with
$C^\infty$ boundary. For $j=1,2$, let
$\lambda_j,\mu_j\in C^\infty(\overline\Omega;\R)$ satisfy
$\mu_j>0$ and $3\lambda_j+2\mu_j>0$ on $\overline\Omega$.
If
\[
\Lambda_{\lambda_1,\mu_1}
=\Lambda_{\lambda_2,\mu_2}
\colon H^{1/2}(\partial\Omega;\R^3)
\longrightarrow H^{-1/2}(\partial\Omega;\R^3),
\]
then $\lambda_1=\lambda_2$ and $\mu_1=\mu_2$ throughout $\Omega$.
\end{theorem}
Thus Theorem~\ref{thm:main} gives a positive resolution of the smooth
three-dimensional isotropic elastic Calder\'on uniqueness problem under
\eqref{eq:energy-positivity}. It uses the full zero-frequency boundary
operator in fixed Euclidean coordinates. The coefficients need not be
analytic or close to constants, and their agreement near the boundary is
not an additional hypothesis.
\paragraph{History and significance.}
The inverse problem asks whether static boundary measurements determine
both spatially varying elastic moduli.
The scalar conductivity problem posed by Calder\'on
\cite{Calderon1980} and the complex geometric optics uniqueness theorem
of Sylvester and Uhlmann \cite{SylvesterUhlmann1987} are important
antecedents, but elasticity couples two moduli through vector
displacements. Nakamura and Uhlmann recovered the elastic boundary
Taylor series \cite{NakamuraUhlmann1995Boundary}. Their original global
claim \cite{NU1994} was corrected in \cite{NU2003}. The erratum identifies
two defects: the prescribed initial-value problem for a planar matrix
transport need not be solvable, and substitution of the corrected complex
geometric optics solutions yields a pseudodifferential equation where a
partial differential equation had been asserted. Its corrected theorem
proves uniqueness when both $\|\nabla\mu_j\|_{C^m}$ are sufficiently
small.
Eskin and Ralston developed higher-order matrix complex geometric optics
expansions using invertible planar transport frames, together with
restricted elasticity uniqueness results
\cite{EskinRalston2002Elasticity,EskinRalston2004}. A direct inspection of
the differential map from their auxiliary vector--scalar fields to
physical displacement \cite[Section~3]{EskinRalston2004} shows that it is
noninjective. The reduction alone therefore does not transfer equality
of the physical boundary maps to equality of unrestricted auxiliary
Cauchy data. The proof here transfers exact physical solutions while
retaining their actual divergence.
In two dimensions, Imanuvilov and Yamamoto proved global uniqueness for
smooth coefficients satisfying $\mu>0$ and $\lambda+\mu>0$, without a
smallness assumption \cite{ImanuvilovYamamoto2015Elasticity}.
In three dimensions, Lin and Nakamura gave local boundary reconstruction
at finite regularity \cite{LinNakamura2017}; Tan and Liu obtained boundary
jets on smooth Riemannian manifolds and global recovery under their
analytic hypotheses
\cite{TanLiu2023}. A recent preprint of Chen, Jiang, Liu, and Tao treats
structured shear moduli under axisymmetry, separation, or directional
quasianalyticity in the specified geometries
\cite{ChenJiangLiuTao2026Elasticity}.
Theorem~\ref{thm:main} treats the full smooth class under
\eqref{eq:energy-positivity}.
\paragraph{Conventions.}
All differential identities are complexified by linearity. Dot products
of complex vectors, including the null condition and orthogonality, use
the complex bilinear extension of the Euclidean product. Sobolev norms
and Hilbert-space adjoints use the usual Hermitian structure.
Equality of the real boundary maps extends complex linearly to the
complex boundary data used in the proof.
\paragraph{Proof strategy.}
The proof converts equality of the boundary maps into an exact exterior
comparison of physical solutions. Boundary determination first gives
matching boundary jets, from which we construct common smooth extensions
of the coefficients outside $\Omega$.
Given a solution in the first medium, equality of displacement and
traction then lets us glue a solution in the second medium to it across
$\partial\Omega$. The two solutions, and their actual divergences,
coincide in the exterior. Section~\ref{sec:boundary-reduction} establishes
this transfer.
Fix a nonzero complex null vector $\theta$, so
$\theta\cdot\theta=0$, and write $D_\theta=\theta\cdot\nabla$.
The real and imaginary parts of $\theta$ span a plane on which
$D_\theta$ is a nonzero multiple of a Cauchy--Riemann operator; the
remaining real coordinate is transverse to these planes.
For solutions with exponential phase $e^{\tau\theta\cdot x}$,
the leading displacement--divergence pair $(a,b)$ has
$a\in H_\theta=\{a\in\C^3:\theta\cdot a=0\}$ and $b\in\C$.
The resulting transport space
$\Ecal_\theta=H_\theta\oplus\C$ has dimension three.
The displacement--divergence reduction to a system with Laplacian
principal part is classical in Lam\'e unique continuation
\cite{Weck1969,ImanuvilovYamamoto2004Carleman}. The formulation here uses
an independent vector--scalar pair $(u,d)$, retains that principal part
after normalization, and satisfies a compatibility identity for the
defect $\diverg u-d$.
Its supported Carleman estimate, together with a compatible high-order
transport recursion and a correction using the physical elasticity
operator, realizes a full local frame in $\Ecal_\theta$ by exact
displacements. The construction controls both the leading pair and its
weighted first derivatives. Sections~\ref{sec:analytic-estimates}
and~\ref{sec:physical-amplitudes} prove these facts.
The leading equations have a useful normalization. For each coefficient
pair, set
\[
p=\sqrt\mu\,a,\qquad
s=-\frac{\lambda+\mu}{2\sqrt\mu}\,b
-\frac12\nabla\log\mu\cdot p.
\]
This is an invertible change of variables on $\Ecal_\theta$, since
$\mu>0$ and $\lambda+\mu>0$. In the $j$th medium, the normalized
transport takes the form
\[
D_\theta\binom{p}{s}
=M_{j,\theta}\binom{p}{s},\qquad
M_{j,\theta}=
\begin{pmatrix}0&\theta\\Q_{j,\theta}\cdot&T_{j,\theta}\end{pmatrix}.
\]
Here $Q_{j,\theta}\cdot$ is the covector
$p\mapsto Q_{j,\theta}\cdot p$. The lower coefficients are smooth in $x$
and linear in $\theta$;
Lemma~\ref{lem:transport-normal-form} gives their formulas.
The upper equation $D_\theta p=\theta s$ is the same for both media.
Transport comparison followed by holomorphic variation of the complex
direction has a close precedent in Ceki\'c's work on connection Laplacians
\cite{Cekic2025}, whose parameter argument follows Eskin's analysis of
Yang--Mills potentials \cite{Eskin2001}. Ceki\'c treats full
connection-system boundary data on a trivial vector bundle. The comparison
needed here must be obtained from physical elasticity data on the
constrained fibers $\Ecal_\theta$.
On a cylinder given by a planar disk times a transverse interval, let
$Y_1:\C^3\to\Ecal_\theta$ be the matrix of normalized columns of one
local frame realized in the first medium. Transfer its exact physical
solutions to the second
medium. The two augmented operators have the same principal part, so the
forcing for the difference of the physical pairs has only first order.
The derivative bound and the supported Carleman estimate make the
transferred leading pairs bounded in $L^2$. Weak limits supply a matrix
$Y_2:\C^3\to\Ecal_\theta$ of three second-medium transport columns.
The two normalizations agree in
the exterior because the coefficients do, so $Y_2=Y_1$ there.
The local comparison $G=Y_2Y_1^{-1}$ is therefore an endomorphism of
$\Ecal_\theta$ equal to the identity on the part of that cylinder
outside $\overline\Omega$.
Only the original columns $Y_1$ need to form an invertible frame.
The initial comparison is obtained in a weak sense on transverse
cylinders. Section~\ref{sec:transport-matching} proves uniqueness for
supported planar transport equations and an estimate on fixed supported
Sobolev spaces. These results assemble the local comparisons into a
unique smooth field $G(x,[\theta])\in\operatorname{End}(\Ecal_\theta)$
on $\R^3\times\mathcal C$, where
$\mathcal C=\{[\theta]\in\C\mathrm P^2:\theta\cdot\theta=0\}$ is the
projective null conic. For a fixed ball $\mathcal B\subset\R^3$
independent of $[\theta]$, it satisfies
\[
D_\theta G=M_{2,\theta}G-GM_{1,\theta},
\qquad G(x,[\theta])=\Id\quad(x\notin\mathcal B).
\]
Differentiating this equation in a conjugate conic coordinate while
holding $x$ fixed gives a supported homogeneous transport equation.
Its uniqueness makes $G(x,\cdot)$ holomorphic.
Finally, the planes $H_\theta$ form a holomorphic bundle $H$ on
$\mathcal C$. It has no nonzero global holomorphic sections, while its
quotient by the null line subbundle with fiber $\C\theta$ is trivial.
These facts and the universal upper transport equation force the
displacement block of $G$ to be
scalar. The remaining upper block equation uses $\dim H_\theta=2$ to
make that scalar constant in $x$ and remove the remaining off-diagonal
block; the exterior value then gives $G=\Id$.
Equality of the transports first determines
$(\lambda+\mu)/(\lambda+2\mu)$. The remaining equations for
$\eta=\log\mu_1-\log\mu_2$ have the form
$\nabla^2\eta=B(x)\nabla\eta+q(x)I$, where
$B(x):\R^3\to\R^{3\times3}$ and the scalar $q(x)$ are smooth.
One differentiation closes a homogeneous first-order system for
$(\nabla\eta,q)$, whose zero exterior values propagate by uniqueness
for ordinary differential equations along paths.
Section~\ref{sec:coefficient-recovery} gives this rigidity and recovers
both coefficients.
==== END Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem//introduction.tex ====
==== BEGIN Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem//main.tex ====
\newcommand{\DoNotLoadEpstopdf}{}
\documentclass[11pt]{article}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
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\usepackage{amsmath,amssymb,amsthm,mathtools}
\usepackage{enumitem}
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\pdfinfoomitdate=1
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\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
\newcommand{\N}{\mathbb N}
\newcommand{\dd}{\mathrm d}
\newcommand{\Id}{\mathrm{Id}}
\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\diverg}{div}
\newcommand{\norm}[1]{\left\lVert #1\right\rVert}
\newcommand{\abs}[1]{\left\lvert #1\right\rvert}
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\newtheorem{remark}[theorem]{Remark}
\numberwithin{equation}{section}
\title{Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem}
\author{OpenAI}
\date{September 24, 2026}
\hypersetup{pdftitle={Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem},pdfauthor={OpenAI}}
\begin{document}
\maketitle
\begin{abstract}
We prove that the full static displacement-to-traction map uniquely determines both real smooth Lam\'e moduli on every bounded connected smooth domain in $\R^3$, provided $\mu>0$ and $3\lambda+2\mu>0$ on the closure. This resolves the smooth three-dimensional isotropic elastic Calder\'on uniqueness problem under these positivity assumptions.
\end{abstract}
\tableofcontents
\medskip
\input{sections/introduction}
\input{sections/boundary-reduction}
\input{sections/analytic-estimates}
\input{sections/physical-amplitudes}
\input{sections/transport-matching}
\input{sections/coefficient-recovery}
\begingroup
\small
\bibliographystyle{plain}
\bibliography{references}
\endgroup
\end{document}
==== END Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem//main.tex ====
==== BEGIN Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem//physical-amplitudes.tex ====
\section{Physical solutions realizing local transport frames}
\label{sec:physical-amplitudes}
We now construct exact elasticity solutions with controlled leading
behavior for both the displacement and its actual divergence. We begin
by identifying the equations obeyed by these leading terms.
Fix the null vector $\theta$ from \eqref{eq:phase-notation}, and
write $D=D_\theta$ when convenient. Since
$\theta\cdot\theta=0$, conjugating the operators in
\eqref{eq:augmented-operators} gives
\[
E_\tau^{-1}R(E_\tau a,E_\tau b)=R(a,b)+\tau R^\theta(a,b),
\qquad
E_\tau^{-1}S(E_\tau a,E_\tau b)=S(a,b)+\tau S^\theta(a,b),
\]
where
\begin{equation}
\label{eq:leading-operators}
\begin{aligned}
R^\theta(a,b)
&=2mDa+(Dm)a+\theta(kb+\nabla m\cdot a),\\
S^\theta(a,b)
&=2\ell Db+2(Dk)b+4\nabla m\cdot Da
+2(D\nabla m)\cdot a.
\end{aligned}
\end{equation}
For smooth vector amplitudes $a_0,a_1$, the first two terms of a trial
displacement give
\[
E_\tau^{-1}\diverg\bigl(E_\tau(a_0+\tau^{-1}a_1)\bigr)
=\tau\theta\cdot a_0+
\bigl(\diverg a_0+\theta\cdot a_1\bigr)
+\tau^{-1}\diverg a_1.
\]
An $O(1)$ leading divergence therefore requires $\theta\cdot a_0=0$,
while its scalar term also contains $\theta\cdot a_1$ and need not equal
$\diverg a_0$. We retain that scalar as the component $b$ of the leading
pair.
Define the fixed complex vector spaces
\begin{equation}
\label{eq:constrained-fibers}
H_\theta=\{a\in\C^3:\theta\cdot a=0\},
\qquad \Ecal_\theta=H_\theta\oplus\C.
\end{equation}
The transport equation on $\Ecal_\theta$ is
\begin{equation}
\label{eq:constrained-transport}
R^\theta(a,b)=0,\qquad S^\theta(a,b)=0,
\qquad (a,b)\in\Ecal_\theta.
\end{equation}
Indeed, putting $P^\theta=2mD+(Dm)$, we have
\begin{equation}
\label{eq:normal-transport-component}
\theta\cdot R^\theta(a,b)=P^\theta(\theta\cdot a).
\end{equation}
Thus $R^\theta$ takes values in $H_\theta$ when $a$ does. On
$\Ecal_\theta$, the coefficient multiplying $D(a,b)$ in the two
transport equations is
\[
\begin{pmatrix}
2m\Id_{H_\theta}&0\\
4\nabla m\cdot&2\ell
\end{pmatrix}.
\]
It is invertible, so \eqref{eq:constrained-transport} is a square
rank-three system with principal part $D$.
For clarity, write $\theta=\alpha+i\beta$ with
$\alpha,\beta\in\R^3$. The null condition says
$\abs\alpha=\abs\beta=\rho>0$ and $\alpha\cdot\beta=0$.
Let $(y_1,y_2,t)$ be positively oriented orthonormal coordinates in
the directions $\alpha/\rho$, $\beta/\rho$, and their cross product.
Then, for $z=y_1+iy_2$,
\[
D=\rho(\partial_{y_1}+i\partial_{y_2})
=2\rho\partial_{\overline z}.
\]
We denote the planar disk $\{y\in\R^2:\abs y<R\}$ by $D_R^2$.
For each chosen transverse coordinate $t_0$, we will select a smooth
frame of \eqref{eq:constrained-transport} for $t$ near $t_0$ and then
realize its three columns by physical solutions.
Earlier systems complex geometric optics constructions also use
invertible planar transport frames \cite[Section~1]{EskinRalston2004}.
The next lemma records the local parameter dependence and inhomogeneous
solvability with transverse support needed here.
\begin{lemma}[Planar frames and inhomogeneous transports]
\label{lem:planar-frames}
Let $I\subset\R$ be an open interval and let
$A\in C^\infty(D_R^2\times I;\C^{n\times n})$.
For each fixed $t\in I$, the equation
\begin{equation}
\label{eq:planar-frame-equation}
\partial_{\overline z}V=A(z,t)V
\end{equation}
has a smooth invertible matrix solution on $D_R^2$.
For every $R'<R$ and $t_0\in I$, there are an interval
$I_0\Subset I$ containing $t_0$ and a smooth invertible solution
$V$ on $D_{R'}^2\times I_0$.
Given this $V$, a radius $R''<R'$, and
$F\in C^\infty(D_{R'}^2\times I_0;\C^n)$, the equation
\begin{equation}
\label{eq:planar-inhomogeneous}
\partial_{\overline z}v=Av+F
\end{equation}
has a smooth solution on $D_{R''}^2\times I_0$. If
$F(\,\cdot\,,t)=0$ for $t\notin K$, where $K\Subset I_0$ is
compact, the solution can be chosen with the same transverse
support property. These assertions also hold with $D$ in place of
$\partial_{\overline z}$, and for systems obtained by multiplying
$D$ by an invertible smooth matrix.
\end{lemma}
\begin{proof}
We first construct frames locally at a fixed parameter. For a
bounded function $f$ supported in a disk of radius $\delta$, the
Cauchy transform
\[
(\mathcal Tf)(z)=\frac1\pi
\int_{\C}\frac{f(w)}{z-w}\,\dd y_1(w)\,\dd y_2(w)
\]
satisfies $\partial_{\overline z}\mathcal Tf=f$ in distributions
and $\norm{\mathcal Tf}_{L^\infty}\le C\delta\norm f_{L^\infty}$.
The latter estimate follows by integrating $\abs{z-w}^{-1}$ over
the support disk. Choose a smooth cutoff $\chi$ supported in a
sufficiently small disk and equal to one on a smaller disk. On
bounded matrix functions, the equation
\[
V=\Id_n+\mathcal T(\chi A V)
\]
is then solved by a convergent Neumann series. Shrinking the disk
until the operator norm is less than $1/3$ gives
$\norm{V-\Id_n}_{L^\infty}<1/2$, hence invertibility. The
distributional equation and interior elliptic regularity for
$\partial_{\overline z}$ show that $V$ is smooth on the smaller
disk. This proves local smooth invertible solvability.
If $V_i,V_j$ are two such local frames, then
$\partial_{\overline z}(V_i^{-1}V_j)=0$ on their overlap.
Their transition functions therefore define a holomorphic vector
bundle on $D_R^2$, whose underlying smooth bundle is trivial.
Its holomorphic frame bundle has fiber $\mathrm{GL}_n(\C)$.
The disk is Stein and contractible, so a continuous section of
this frame bundle exists. The Oka principle for sections of holomorphic
fiber bundles with complex homogeneous fibers, in the form of
\cite[Theorem~1.1, p.~1018]{Forstneric2009}, supplies a holomorphic
section; this homogeneous-fiber case goes back to Grauert
\cite{Grauert1958}. In the original smooth trivialization it is
a smooth invertible matrix solving
\eqref{eq:planar-frame-equation} throughout $D_R^2$.
To obtain local parameter dependence, choose such a frame $V_0(z)$
at $t_0$. After writing $V=V_0W$, the equation becomes
\[
\partial_{\overline z}W=B(z,t)W,
\qquad
B(z,t)=V_0(z)^{-1}\bigl(A(z,t)-A(z,t_0)\bigr)V_0(z).
\]
Choose a cutoff $\chi\in C_c^\infty(D_R^2)$ equal to one near
$\overline D_{R'}^2$. On its compact support, $B(\,\cdot\,,t)$
tends uniformly to zero as $t\to t_0$. On a sufficiently small
interval $I_0\Subset I$, solve
\[
W=\Id_n+\mathcal T(\chi B(\,\cdot\,,t)W)
\]
by a Neumann series with norm less than $1/3$. This gives an
invertible solution on $D_{R'}^2\times I_0$. The integral operator
depends smoothly on $t$ in operator norm on $L^\infty$; its
inverse does also. Hence all parameter derivatives of $W$ exist
in $L^\infty$. Differentiating its equation and applying interior
elliptic regularity successively gives smoothness jointly in
$(z,t)$, with bounds on compact subsets. Thus $V=V_0W$ has the
claimed parameter regularity.
Finally, choose $\chi\in C_c^\infty(D_{R'}^2)$ equal to one near
$\overline D_{R''}^2$. The formula
\[
v=V\mathcal T(\chi V^{-1}F)
\]
solves \eqref{eq:planar-inhomogeneous} on the smaller disk.
The transform acts only in the planar variable, so it preserves
the specified support in $t$. Dividing a system with invertible
derivative coefficient by that coefficient, and using
$D=2\rho\partial_{\overline z}$, proves the final assertion.
\end{proof}
To carry a leading frame through physical transfer, we need control of
both the displacement--divergence pair and its derivatives.
Corollary~\ref{cor:physical-solvability} gives physical solvability with
an $L^2$ estimate independent of $\tau$. We will make the
correction small enough after taking a divergence by first constructing
a compatible expansion of truncation order $N\ge4$. The proof below
obtains a physical residual of size $O(\tau^{1-N})$ and a conjugated
divergence correction of size $O(\tau^{2-N})$. The resulting strong
convergence identifies the first medium's leading columns and, by exterior
agreement, the exterior values of the transferred columns. The weighted
$O(\tau)$ derivative bound controls the first-order forcing obtained by
applying the difference of the two normalized augmented operators to
the first physical pair.
\begin{proposition}[Physical realization of local transport frames]
\label{prop:physical-amplitudes}
For every nonzero null vector $\theta$ and every
$t_0\in(-2r,2r)$, there exist an open interval
$J\subset(-2r,2r)$ containing $t_0$ and three smooth solutions
$(a_0^{\nu},b_0^{\nu})$, $1\le\nu\le3$, of
\eqref{eq:constrained-transport} on a neighborhood of
$\overline B_{5r}$, with the following properties.
The three columns form a frame of $\Ecal_\theta$ at every point
of $D_{2r}^2\times J$. For each $\nu$ and all sufficiently large
$\tau$, there is $u_\tau^{\nu}\in C^\infty(B_{5r};\C^3)$ with
$Lu_\tau^{\nu}=0$ such that, on putting
$U_\tau^{\nu}=(u_\tau^{\nu},\diverg u_\tau^{\nu})$,
\begin{equation}
\label{eq:physical-leading-limit}
\begin{aligned}
E_\tau^{-1}U_\tau^{\nu}
&\longrightarrow (a_0^{\nu},b_0^{\nu})
&&\text{in }L^2(B_{4r}),\\
\norm{E_\tau^{-1}\nabla U_\tau^{\nu}}_{L^2(B_{4r})}
&=O(\tau).
\end{aligned}
\end{equation}
\end{proposition}
\begin{proof}
We construct compatible formal amplitudes and then correct the
truncated displacement using the physical operator.
Fix an integer $N\ge4$ and radii
$R_0>R_1>\cdots>R_N>5r$. The coefficients are smooth on all of
$\R^3$. Apply Lemma~\ref{lem:planar-frames} to the rank-three
system \eqref{eq:constrained-transport}, starting on a disk
strictly larger than $D_{R_0}^2$. It supplies a smooth
invertible frame
\[
\mathcal F(y,t):\C^3\longrightarrow\Ecal_\theta
\quad\text{on }D_{R_0}^2\times I_0,
\]
where $I_0\Subset(-2r,2r)$ contains $t_0$. Choose an interval
$J$ containing $t_0$ with $\overline J\subset I_0$, and choose
$\chi\in C_c^\infty(I_0)$ equal to one on $J$. Multiply each
column of $\mathcal F$ by $\chi(t)$ and extend it by zero in
$t$ to obtain $(a_0^{\nu},b_0^{\nu})$ on
$D_{R_0}^2\times\R$. Since $D\chi=0$, these columns still
solve \eqref{eq:constrained-transport}. They are a frame on
$D_{2r}^2\times J$, and all have transverse support in the
fixed compact set $K=\supp\chi\Subset I_0$.
We construct a physical solution for any one of these columns
and suppress $\nu$. Higher-order transport expansions for systems appear
in Eskin and Ralston
\cite[Section~2, equations~(2.2)--(2.4)]{EskinRalston2004}.
The recursion here must cancel the vector residual
while making the conjugated actual divergence have leading term $b_0$.
Its scalar equation makes the next normal constraint compatible with
the vector equation;
the final correction will be made only in the physical displacement
equation. For $1\le j\le N$, we seek smooth coefficients $(a_j,b_j)$
on successively smaller disks such that
\begin{equation}
\label{eq:physical-recursion}
\begin{aligned}
R^\theta(a_j,b_j)&=-R(a_{j-1},b_{j-1}),\\
S^\theta(a_j,b_j)&=-S(a_{j-1},b_{j-1}),\\
\theta\cdot a_j&=-(\diverg a_{j-1}-b_{j-1}).
\end{aligned}
\end{equation}
Throughout this recursion, write $q_j=\diverg a_j-b_j$ and set
$a_{-1}=b_{-1}=q_{-1}=0$.
The last equation in \eqref{eq:physical-recursion} is the
displacement--divergence constraint at order $j$.
We first verify the compatibility of these three equations.
Conjugating $P$ gives $E_\tau^{-1}PE_\tau=P+\tau P^\theta$.
Comparing coefficients of $\tau$ in the conjugated identity
\eqref{eq:augmented-identity} yields, for arbitrary $(a,b)$,
\begin{equation}
\label{eq:transport-compatibility}
\diverg R^\theta(a,b)+\theta\cdot R(a,b)-S^\theta(a,b)
=P(\theta\cdot a)+P^\theta(\diverg a-b).
\end{equation}
Suppose the preceding recursive equations have been solved.
Apply \eqref{eq:transport-compatibility} to $(a_{j-1},b_{j-1})$.
For $j\ge2$, the preceding vector and scalar equations, followed by
\eqref{eq:augmented-identity}, give
\[
\begin{aligned}
\diverg R^\theta(a_{j-1},b_{j-1})
-S^\theta(a_{j-1},b_{j-1})
&=-\diverg R(a_{j-2},b_{j-2})+S(a_{j-2},b_{j-2})\\
&=-Pq_{j-2}.
\end{aligned}
\]
The preceding normal constraint gives the same term on the other side:
\[
P(\theta\cdot a_{j-1})=-Pq_{j-2}.
\]
For $j=1$, both expressions are zero by the homogeneous leading
equations and $q_{-1}=0$. In every case these two terms cancel in
\eqref{eq:transport-compatibility}, leaving
\begin{equation}
\label{eq:recursion-normal-compatibility}
\theta\cdot R(a_{j-1},b_{j-1})=P^\theta q_{j-1}.
\end{equation}
Choose a constant vector $n_\theta\in\C^3$ with
$\theta\cdot n_\theta=1$, and set
$a_j^{\mathrm p}=-q_{j-1}n_\theta$, $b_j^{\mathrm p}=0$.
After subtracting this particular pair from the desired
$(a_j,b_j)$, the unknown takes values in $\Ecal_\theta$.
Equations \eqref{eq:normal-transport-component} and
\eqref{eq:recursion-normal-compatibility} show that the residual
vector source belongs to $H_\theta$. Its scalar source is
unrestricted. We are therefore left with an inhomogeneous
square system on $\Ecal_\theta$ with invertible derivative
coefficient. Use the uncut frame $\mathcal F$ and the
inhomogeneous part of Lemma~\ref{lem:planar-frames} to solve it
on $D_{R_j}^2\times I_0$. All source terms, including the
particular pair and its derivatives, have transverse support
in $K$; the constructed solution retains that support and
extends smoothly by zero in $t$. This proves the induction.
After finitely many steps, every coefficient is smooth on
$D_{R_N}^2\times\R$, hence on a neighborhood of
$\overline B_{5r}$.
Set
\[
a^{[N]}=\sum_{j=0}^N\tau^{-j}a_j,
\qquad b^{[N]}=\sum_{j=0}^N\tau^{-j}b_j.
\]
The first equation of \eqref{eq:physical-recursion} telescopes
to
\begin{equation}
\label{eq:truncated-vector-residual}
E_\tau^{-1}R(E_\tau a^{[N]},E_\tau b^{[N]})
=\tau^{-N}R(a_N,b_N).
\end{equation}
The normal constraints telescope separately, giving
\begin{equation}
\label{eq:truncated-divergence}
E_\tau^{-1}\diverg(E_\tau a^{[N]})
=b^{[N]}+\tau^{-N}q_N.
\end{equation}
Using the dependence of $R$ on its scalar argument, these
identities give the physical residual explicitly:
\begin{equation}
\label{eq:physical-truncated-residual}
\begin{aligned}
f_\tau
&\coloneqq E_\tau^{-1}L(E_\tau a^{[N]})\\
&=\tau^{-N}\bigl[R(a_N,b_N)+k\nabla q_N
+(\nabla\lambda)q_N+\tau k\theta q_N\bigr].
\end{aligned}
\end{equation}
All coefficients on the right are fixed smooth functions.
Consequently $\norm{f_\tau}_{L^2(B_{5r})}=O(\tau^{1-N})$.
Corollary~\ref{cor:physical-solvability} gives a correction
$r_\tau\in L^2(B_{5r};\C^3)$ satisfying
\[
E_\tau^{-1}L(E_\tau r_\tau)=-f_\tau,
\qquad \norm{r_\tau}_{L^2(B_{5r})}=O(\tau^{1-N}).
\]
It is smooth in $B_{5r}$ by Lemma~\ref{lem:scaled-interior}.
The same Lemma, on $B_{4r}$, gives
\begin{equation}
\label{eq:physical-remainder-two-derivatives}
\sum_{j=0}^2h^j\norm{\nabla^jr_\tau}_{L^2(B_{4r})}
=O(\tau^{1-N}).
\end{equation}
Define its conjugated physical divergence by
\[
d_\tau=E_\tau^{-1}\diverg(E_\tau r_\tau)
=\diverg r_\tau+\tau\theta\cdot r_\tau.
\]
It follows directly from
\eqref{eq:physical-remainder-two-derivatives} that
\begin{equation}
\label{eq:physical-divergence-remainder}
\norm{d_\tau}_{L^2(B_{4r})}
+h\norm{\nabla d_\tau}_{L^2(B_{4r})}
\le C\tau\sum_{j=0}^2h^j
\norm{\nabla^jr_\tau}_{L^2(B_{4r})}
=O(\tau^{2-N}).
\end{equation}
Now put $u_\tau=E_\tau(a^{[N]}+r_\tau)$. It is a smooth exact
solution of $Lu_\tau=0$ on $B_{5r}$, and its physical pair
satisfies
\[
E_\tau^{-1}(u_\tau,\diverg u_\tau)
=\bigl(a^{[N]}+r_\tau,
b^{[N]}+\tau^{-N}q_N+d_\tau\bigr).
\]
The smooth finite expansions and the remainder estimates imply
the first assertion of \eqref{eq:physical-leading-limit}.
They also show that, if the last displayed pair is denoted
by $\mathcal A_\tau$, then
\[
\norm{\mathcal A_\tau}_{L^2(B_{4r})}
+h\norm{\nabla\mathcal A_\tau}_{L^2(B_{4r})}=O(1).
\]
Since, for $j=1,2,3$,
$E_\tau^{-1}\partial_j(E_\tau\mathcal A_\tau)
=\partial_j\mathcal A_\tau+\tau\theta_j\mathcal A_\tau$,
the second assertion of \eqref{eq:physical-leading-limit}
follows. Repeating the construction for the three initial
columns proves the Proposition.
\end{proof}
==== END Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem//physical-amplitudes.tex ====
==== BEGIN Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem//transport-matching.tex ====
\section{Matching the leading transports}\label{sec:transport-matching}
Proposition~\ref{prop:transport-matching} constructs the unique holomorphic
comparison of the two leading transports. We first derive a transport normal
form and prove a supported estimate for parameter dependence.
Throughout this section, a dot
between complex vectors denotes the complex bilinear Euclidean product.
\begin{lemma}[Transport normal form]\label{lem:transport-normal-form}
For one coefficient pair, set $g=\nabla\log m$. The change of variables
\begin{equation}\label{eq:transport-change}
p=\sqrt m\,a,
\qquad
s=-\frac{k}{2\sqrt m}\,b-\frac12 g\cdot p
\end{equation}
is a smooth invertible map of $\Ecal_\theta$ to itself. It transforms
\eqref{eq:constrained-transport} into
\begin{equation}\label{eq:transport-matrix}
D_\theta\binom{p}{s}
=M_\theta\binom{p}{s},
\qquad
M_\theta=
\begin{pmatrix}
0&\theta\\
Q_\theta\cdot&T_\theta
\end{pmatrix}
\quad\text{on }H_\theta\oplus\C,
\end{equation}
where
\begin{equation}\label{eq:transport-coefficients}
\begin{aligned}
T_\theta&=D_\theta\log(k/\ell),\\
Q_\theta&=\frac12\left[
\left(\frac m\ell D_\theta\log k
-\frac12\theta\cdot g\right)g
-\frac m\ell D_\theta g\right].
\end{aligned}
\end{equation}
In particular, the bottom-left entry in \eqref{eq:transport-matrix}
is the covector $p\mapsto Q_\theta\cdot p$. Both $D_\theta$ and
$M_\theta$ depend linearly and holomorphically on $\theta$.
\end{lemma}
\begin{proof}
The inverse of \eqref{eq:transport-change} is
\[
a=m^{-1/2}p,
\qquad
b=-\frac{2\sqrt m}{k}\left(s+\frac12g\cdot p\right).
\]
It is well defined because $m,k>0$, and multiplication by $\sqrt m$
preserves $H_\theta$. Write $D=D_\theta$. Substitution into the first
equation in \eqref{eq:leading-operators} gives
\[
0=2\sqrt m\,Dp+\theta(kb+\sqrt m\,g\cdot p),
\qquad\text{hence}\qquad Dp=\theta s.
\]
Dividing the second equation by two and using
$\nabla m=mg$ yields
\[
0=\ell Db+(Dk)b+2\nabla m\cdot Da+(D\nabla m)\cdot a.
\]
Here
\[
\begin{aligned}
2\nabla m\cdot Da
&=2\sqrt m\,g\cdot Dp
-\sqrt m\,(D\log m)g\cdot p,\\
(D\nabla m)\cdot a
&=\sqrt m\,\bigl((D\log m)g+Dg\bigr)\cdot p.
\end{aligned}
\]
The terms containing $(D\log m)g\cdot p$ cancel. Thus
\begin{equation}\label{eq:transformed-divergence-transport}
\ell Db=-(Dk)b
-\sqrt m\bigl(2(\theta\cdot g)s+(Dg)\cdot p\bigr).
\end{equation}
Differentiate the definition of $s$ and substitute
\eqref{eq:transformed-divergence-transport}, the expression for $b$,
and $Dp=\theta s$. With $c=m/\ell$ and $\gamma=D\log m=\theta\cdot g$,
the result is
\[
Ds=c(D\log k-\gamma)s
+\frac12\bigl[(cD\log k-\gamma/2)g-cDg\bigr]\cdot p.
\]
Finally, since $\ell=k+m$,
\[
D\log(k/\ell)
=D\log k-\frac{Dk+Dm}{\ell}
=\frac m\ell(D\log k-D\log m).
\]
This proves the formulas. The top row takes values in $H_\theta$
because $\theta\cdot\theta=0$. All the direction dependence displayed
in the formulas is linear.
\end{proof}
The comparison equation can now be described in terms of the columns
that the physical construction must supply. Fix $\theta$ and a cylinder
$D_{2r}^2\times J$. Suppose for the moment that
$Y_1:\C^3\to\Ecal_\theta$ is a smooth invertible matrix of normalized
transport columns for the first medium, and that
$Y_2:\C^3\to\Ecal_\theta$ has locally $L^2$ columns satisfying the
second transport equation in distributions:
\[
D_\theta Y_j=M_{j,\theta}Y_j,\qquad j=1,2.
\]
The columns in each $Y_j$ are normalized by that medium's own change
of variables \eqref{eq:transport-change}. These changes agree outside
$\overline\Omega$, where the extended coefficients agree. Thus exterior
agreement of the corresponding physical leading columns gives
$Y_2=Y_1$ there.
Under this exterior agreement, define
\[
G=Y_2Y_1^{-1}\in\operatorname{End}(\Ecal_\theta).
\]
Only $Y_1$ is required to be invertible. Multiplication by its smooth
inverse is legitimate for locally $L^2$ columns, and the product rule
in distributions gives
\[
D_\theta G=M_{2,\theta}G-GM_{1,\theta},
\qquad G=\Id\quad\text{outside }\overline\Omega
\]
on the cylinder. Consequently $W=G-\Id$ is locally $L^2$ and supported
in $\overline\Omega$ within that cylinder.
For a transverse coordinate $t$, restrict the coefficients to the
corresponding plane and use a basis of $\Ecal_\theta$ fixed in $x$.
On planar endomorphism-valued unknowns define
\begin{equation}\label{eq:planar-comparison-operator}
\Pcal_{\theta,t}W
=D_\theta W-M_{2,\theta}W+WM_{1,\theta},
\qquad
f_{\theta,t}=M_{2,\theta}-M_{1,\theta}.
\end{equation}
The supported planar equation required for the comparison is
\begin{equation}\label{eq:inhomogeneous-comparison}
\Pcal_{\theta,t}W=f_{\theta,t}.
\end{equation}
Matrices here may be regarded as vectors of their nine entries.
For almost every $t$, the $L^2$ planar restriction of the local $W$
is supported in $\overline{D_r^2}$ because $\overline\Omega\subset B_r$.
The physical transfer and weak-limit argument below will supply the
second columns $Y_2$. Slicing their distributional cylinder equation
will initially give \eqref{eq:inhomogeneous-comparison} only for almost
every $t$. We therefore need both uniqueness for supported planar
solutions and a way to pass from this almost-everywhere construction
to smooth dependence on every parameter. The next lemma provides these
facts on fixed Hilbert spaces, which will also allow the planes to vary.
\begin{lemma}[Supported planar transport]\label{lem:supported-transport}
Let $\mathcal U\subset\R^d$ be open, let $n\geq1$, and suppose that
\[
\Pcal_\rho
=\gamma(\rho)(\partial_{y_1}+i\partial_{y_2})
+A(y,\rho),
\qquad \rho\in\mathcal U,
\]
where $\gamma$ is smooth and nonzero and $A$ is a smooth complex
$n\times n$ matrix on a neighborhood of
$\overline{D_{2r}^2}\times\mathcal U$. Then the following statements hold.
\begin{enumerate}[label=\textup{(\roman*)}]
\item A distributional solution of $\Pcal_\rho W=0$ on $D_{2r}^2$
whose support is a compact subset of that disk is zero.
\item For every integer $j\geq0$ and every
$W\in H^{j+1}(D_{2r}^2;\C^n)$ supported in $\overline{D_r^2}$,
\begin{equation}\label{eq:supported-transport-estimate}
\norm{W}_{H^{j+1}(D_{2r}^2)}
\leq C_j\norm{\Pcal_\rho W}_{H^j(D_{2r}^2)},
\qquad
\supp W\subset\overline{D_r^2}.
\end{equation}
The constants are locally uniform in $\rho$.
\item Suppose that $f(y,\rho)$ is smooth on a neighborhood of
$\overline{D_{2r}^2}\times\mathcal U$. If the equation
$\Pcal_\rho W=f(\cdot,\rho)$ has an $L^2$ solution supported in
$\overline{D_r^2}$ for a dense set of parameters $\rho$, then it has
such a solution for every $\rho$. These solutions are unique and form
a smooth function of $(y,\rho)$.
\end{enumerate}
The same conclusions apply to matrix-valued unknowns, after their
entries are regarded as a vector.
\end{lemma}
\begin{proof}
For a fixed parameter, $\Pcal_\rho$ is elliptic in the two real planar
variables: its principal symbol is
$\gamma(\rho)(i\xi_1-\xi_2)$, which is nonzero for
$\xi\in\R^2\setminus\{0\}$. On each sufficiently small disk,
Lemma~\ref{lem:planar-frames} gives a smooth invertible matrix $F$
satisfying $\Pcal_\rho F=0$. If $\Pcal_\rho W=0$, then
\[
(\partial_{y_1}+i\partial_{y_2})(F^{-1}W)=0
\]
in distributions. Its components are therefore holomorphic functions
of $y_1+iy_2$. A homogeneous solution that vanishes on an open set
vanishes on every overlapping frame disk by the holomorphic identity
theorem. Connectedness propagates this conclusion throughout
$D_{2r}^2$. A compactly supported solution vanishes near the boundary
of the disk, proving \textup{(i)}.
We give the estimate on the fixed supported space. Extending $W$ by
zero to $\R^2$ is harmless because its support lies in $\overline{D_r^2}$.
The Fourier symbol just computed gives
\[
\norm{W}_{H^{j+1}}
\leq C\bigl(
\norm{\gamma(\rho)(\partial_{y_1}+i\partial_{y_2})W}_{H^j}
+\norm{W}_{L^2}\bigr).
\]
Here and below in this proof the norms can equivalently be taken on
$D_{2r}^2$, since the functions under consideration are supported in
the smaller disk. Smooth multiplication and, for $j\geq1$, the
interpolation estimate
$\norm{W}_{H^j}\leq\varepsilon\norm{W}_{H^{j+1}}
+C_\varepsilon\norm{W}_{L^2}$ give
\begin{equation}\label{eq:supported-elliptic-preestimate}
\norm{W}_{H^{j+1}}
\leq C\bigl(\norm{\Pcal_\rho W}_{H^j}
+\norm{W}_{L^2}\bigr).
\end{equation}
The coefficients can be cut off outside a neighborhood of the smaller
disk for this calculation. We will also use local regularity for an
$L^2$ distributional solution with smooth right side. In a local frame
$F$, its equation becomes a scalar Cauchy--Riemann equation componentwise
with smooth right side. Subtracting a smooth local particular solution,
as supplied by Lemma~\ref{lem:planar-frames}, leaves holomorphic
components. The original solution is therefore smooth in the disk.
To remove the last term in \eqref{eq:supported-elliptic-preestimate},
suppose that \eqref{eq:supported-transport-estimate} fails at a fixed
parameter. There would then be a sequence $W_\nu$, supported in
$\overline{D_r^2}$, such that
\[
\norm{W_\nu}_{H^{j+1}}=1,
\qquad
\norm{\Pcal_\rho W_\nu}_{H^j}\longrightarrow0.
\]
After passing to a subsequence, compactness of the embedding into
$L^2(D_{2r}^2)$ gives a strong $L^2$ limit $W$. This limit is supported
in $\overline{D_r^2}$ and solves $\Pcal_\rho W=0$ distributionally.
Part~\textup{(i)} gives $W=0$. Equation
\eqref{eq:supported-elliptic-preestimate} now contradicts the
normalization of $W_\nu$. This proves \textup{(ii)} at a fixed parameter.
For the parameter assertions, equip the fixed complex Hilbert spaces
\[
\begin{aligned}
X_j&=\{W\in H^{j+1}(D_{2r}^2;\C^n):
\supp W\subset\overline{D_r^2}\},\\
Z_j&=H^j(D_{2r}^2;\C^n)
\end{aligned}
\]
with their usual Hermitian Sobolev inner products. The subspace $X_j$
is closed. The map $\rho\mapsto\Pcal_\rho$ is smooth in
$\mathcal L(X_j,Z_j)$. The estimate at $\rho_0$ persists in a
neighborhood: absorb
$\norm{(\Pcal_\rho-\Pcal_{\rho_0})W}_{Z_j}$ into its left side when
the operator norm of the difference is small. This proves local
uniformity in \textup{(ii)}.
Write $\Pcal_\rho^*:Z_j\to X_j$ for the Hilbert space adjoint with
respect to these fixed inner products. In particular, this is not an
adjoint in the complex bilinear pairing used for the null vectors.
The lower bound in \textup{(ii)} implies that
$\Pcal_\rho^*\Pcal_\rho:X_j\to X_j$ is coercive and invertible.
Consequently
\begin{equation}\label{eq:supported-left-inverse}
T_{\rho,j}
=(\Pcal_\rho^*\Pcal_\rho)^{-1}\Pcal_\rho^*
:Z_j\longrightarrow X_j
\end{equation}
is a bounded left inverse. It depends smoothly on $\rho$, since
adjoints and inversion of bounded invertible operators do so locally.
This argument uses a lower bound for $\Pcal_\rho$; it requires no
surjectivity onto $Z_j$.
For each $j$, set $W_j(\rho)=T_{\rho,j}f(\cdot,\rho)$. An $L^2$
supported solution, whenever it exists, is smooth by interior
ellipticity and hence belongs to every $X_j$. It must equal
$W_j(\rho)$ by the left-inverse identity. Thus the smooth
$Z_j$-valued residual
\[
\Pcal_\rho W_j(\rho)-f(\cdot,\rho)
\]
vanishes on the assumed dense set, and continuity makes it zero for
every $\rho$. Uniqueness in \textup{(i)} identifies the solutions for
different $j$. Smooth dependence with values in every $H^{j+1}$,
together with Sobolev embedding in the planar variables, gives joint
smoothness in $(y,\rho)$. This proves \textup{(iii)}.
\end{proof}
Let
\[
\mathcal C=\{[\theta]\in\C\mathrm P^2:
\theta\cdot\theta=0\}
\]
be the projective null conic. The fibers
$\Ecal_\theta=H_\theta\oplus\C$ form a holomorphic subbundle
$\Ecal$ of the trivial bundle $\mathcal C\times\C^4$. For example,
on a chart where $\theta_i\ne0$, the vectors
$e_j-(\theta_j/\theta_i)e_i$, $j\ne i$, give a holomorphic frame
of $H_\theta$; adjoining the scalar component gives one of $\Ecal$.
Ceki\'c constructs a transport comparison for connection Laplacians
that equals the identity in the outer exterior component
\cite[Theorem~3.4 in the arXiv version]{Cekic2025}, using full
connection-system boundary data on a trivial vector bundle. The comparison
below is obtained from physical elastic transfer and weak limits on the
constrained fibers $\Ecal_\theta$.
\begin{proposition}[Matching of the transports]\label{prop:transport-matching}
For the two extended coefficient pairs of
Lemma~\ref{lem:common-extension}, equality of the physical
displacement-to-traction maps implies that there is a unique smooth
field
\[
G(x,[\theta])\in\operatorname{End}(\Ecal_\theta),
\qquad (x,[\theta])\in\R^3\times\mathcal C,
\]
satisfying, for every nonzero representative $\theta$,
\begin{equation}\label{eq:comparison-transport}
\begin{aligned}
D_\theta G&=M_{2,\theta}G-GM_{1,\theta},\\
G(x,[\theta])&=\Id\qquad\text{if }|x|>2r.
\end{aligned}
\end{equation}
For each fixed $x$, this field is a holomorphic section of
$\operatorname{End}(\Ecal)$ over $\mathcal C$. In fact, for each
fixed direction the first equation has at most one smooth solution whose
difference from the identity has compact spatial support.
\end{proposition}
\begin{proof}
\emph{Physical transfer and weak limits.}
Fix a nonzero null vector $\theta$ and $t_0\in(-2r,2r)$. Apply
Proposition~\ref{prop:physical-amplitudes} to the first coefficient
pair. It supplies three exact smooth solutions on $B_{5r}$, with the
limits and estimates in \eqref{eq:physical-leading-limit}, and an
interval $J$ containing $t_0$ on which their leading columns form a
frame on $D_{2r}^2\times J$. We perform the following construction
for each column, temporarily suppressing its index.
By Proposition~\ref{prop:physical-transfer}, the first solution
$u_{1,\tau}$ has a smooth transferred solution $u_{2,\tau}$ on
$B_{5r}$, with
$u_{2,\tau}=u_{1,\tau}$ outside $\overline\Omega$. Define the
physical augmented vectors
\[
V_{j,\tau}=(u_{j,\tau},\diverg u_{j,\tau}),\qquad j=1,2.
\]
The difference $V_{2,\tau}-V_{1,\tau}$ is smooth and compactly
supported in $\overline\Omega\subset B_r$. By
Lemma~\ref{lem:augmented-system},
\begin{equation}\label{eq:physical-transfer-forcing}
\Lcal_2(V_{2,\tau}-V_{1,\tau})
=(\Lcal_1-\Lcal_2)V_{1,\tau}.
\end{equation}
Both normalized systems have principal part $\Delta\Id_4$.
Their difference is therefore an operator of order at most one,
whose coefficients are supported in $\overline\Omega$.
The weighted derivative bound in
\eqref{eq:physical-leading-limit} implies
\[
\begin{aligned}
\norm{E_\tau^{-1}(\Lcal_1-\Lcal_2)V_{1,\tau}}_{L^2}
&\leq C\left(
\norm{E_\tau^{-1}V_{1,\tau}}_{L^2(B_{4r})}
+\norm{E_\tau^{-1}\nabla V_{1,\tau}}_{L^2(B_{4r})}
\right)\\
&=O(\tau).
\end{aligned}
\]
The function on the left is extended by zero off its compact support
when a norm on $\R^3$ is used. Apply
Proposition~\ref{prop:carleman} to the difference in
\eqref{eq:physical-transfer-forcing}. Since
$\norm{F}_{H_h^{-1}}\leq\norm{F}_{L^2}$ and $h=\tau^{-1}$, we obtain
\begin{equation}\label{eq:bounded-transferred-amplitudes}
\begin{aligned}
\norm{E_\tau^{-1}(V_{2,\tau}-V_{1,\tau})}_{H_h^1}
&\leq Ch\norm{E_\tau^{-1}(\Lcal_1-\Lcal_2)V_{1,\tau}}_{H_h^{-1}}\\
&\leq C.
\end{aligned}
\end{equation}
Thus $E_\tau^{-1}V_{2,\tau}$ is bounded in $L^2(B_{4r})$.
After passage to a subsequence, chosen simultaneously for the three
columns, it converges weakly to a vector $(\widetilde a,\widetilde b)$.
The strong convergence of the first amplitudes and the exterior
agreement of the physical solutions give
\begin{equation}\label{eq:exterior-leading-agreement}
(\widetilde a,\widetilde b)=(a_0,b_0)
\quad\text{almost everywhere on }
B_{4r}\setminus\overline\Omega.
\end{equation}
These weak limits satisfy the constrained transport for the second
pair. To see this directly, write
$(a_\tau,b_\tau)=E_\tau^{-1}V_{2,\tau}$. The exact equations and
the null condition give
\[
\begin{aligned}
R_2^\theta(a_\tau,b_\tau)
+\tau^{-1}R_2(a_\tau,b_\tau)&=0,\\
S_2^\theta(a_\tau,b_\tau)
+\tau^{-1}S_2(a_\tau,b_\tau)&=0.
\end{aligned}
\]
When tested against a fixed compactly supported smooth function,
the terms multiplied by $\tau^{-1}$ tend to zero: all their
derivatives can be transferred to that test function, while the
amplitudes remain bounded in $L^2$. Weak convergence passes the
remaining terms to the limit. Moreover, the physical divergence
identity says
\[
\theta\cdot a_\tau
=\tau^{-1}(b_\tau-\diverg a_\tau),
\]
so $\theta\cdot\widetilde a=0$ distributionally. We have proved
\[
R_2^\theta(\widetilde a,\widetilde b)=0,
\qquad S_2^\theta(\widetilde a,\widetilde b)=0,
\qquad (\widetilde a,\widetilde b)\in\Ecal_\theta.
\]
No derivative estimate on the transferred amplitudes, beyond their
$L^2$ bound, is needed for this passage to the limit.
The physical transfer has therefore produced the second transport
columns, but only as weak limits. We next obtain a supported comparison
on almost every transverse plane and use supported uniqueness to extend
it to every plane.
\emph{A comparison on almost every plane.}
Apply \eqref{eq:transport-change} to the three leading columns for
each pair and denote the resulting maps by
$Y_j:\C^3\to\Ecal_\theta$. For $j=2$, these columns are initially
locally $L^2$; they satisfy the equations distributionally. Since
$J\subset(-2r,2r)$, the cylinder on which the first columns form a frame
satisfies
\[
D_{2r}^2\times J\subset B_{\sqrt8\,r}\subset B_{4r}.
\]
Thus the weak limits constructed above are defined throughout this
cylinder, where
\[
D_\theta Y_j=M_{j,\theta}Y_j,
\]
and $Y_1$ is smooth and invertible. The change of variables agrees
for the two pairs outside $\overline\Omega$. Define
\[
G=Y_2Y_1^{-1}.
\]
The calculation preceding Lemma~\ref{lem:supported-transport} now applies:
$G$ is a locally $L^2$ endomorphism on the cylinder, equals the identity
outside $\overline\Omega$, and satisfies the first equation of
\eqref{eq:comparison-transport} distributionally. With $W=G-\Id$, this
is the cylinder version of \eqref{eq:inhomogeneous-comparison}.
The operator $D_\theta$ differentiates only the planar variables.
Testing the cylinder equation with products of a planar test
function and a transverse test function, and then using a countable
dense collection of planar tests, gives
\eqref{eq:inhomogeneous-comparison} on $D_{2r}^2$ for almost every
$t\in J$. For these $t$, $W(\cdot,t)$ belongs to $L^2(D_{2r}^2)$
and has support in $\overline{D_r^2}$. Indeed, $W=0$ when $|y|>r$,
and on its remaining support multiplication by $Y_1^{-1}$ is locally
bounded uniformly in $t$. These assertions can first be made on
compact subintervals of $J$ and then exhausted over $J$.
\emph{Existence on every plane.}
In the orthonormal coordinates associated with the fixed $\theta$,
\[
D_\theta=|\operatorname{Re}\theta|
(\partial_{y_1}+i\partial_{y_2}).
\]
Thus \eqref{eq:planar-comparison-operator}, as an operator on matrix
entries, has the form of Lemma~\ref{lem:supported-transport}.
Its coefficients and the source $f_{\theta,t}$ are smooth in $(y,t)$.
The set of parameters for which a supported solution has just been
obtained has full measure in $J$, and hence is dense. Part~\textup{(iii)}
of that lemma extends existence to every $t\in J$ and makes $W$
smooth in $(y,t)$. It also makes the choice unique, so the results
from different intervals $J$ agree on their overlaps.
Since $t_0$ was arbitrary, this constructs $G$ for every
$-2r<t<2r$. Extend $W=G-\Id$ by zero in the planar variables
outside $D_{2r}^2$. This extension is smooth because $W$ is already
zero for $r<|y|<2r$. The equation holds across this extension, and
it holds farther out because $f_{\theta,t}=0$ there. If $|t|>r$,
the entire plane misses $\overline\Omega$, so $f_{\theta,t}=0$.
Part~\textup{(i)} of Lemma~\ref{lem:supported-transport} then gives
$W=0$ on that plane. Extending by zero also for $|t|\geq2r$ is
therefore smooth. For this fixed direction we have constructed a
global smooth comparison with
\begin{equation}\label{eq:comparison-uniform-support}
\supp(G(\cdot,\theta)-\Id)
\subset\{(y,t):|y|\leq r,\ |t|\leq r\}
\subset\overline{B_{2r}}.
\end{equation}
The final containing ball is independent of the direction.
For later use, uniqueness holds even if a larger compact support is
allowed. The difference of two such comparisons solves the
homogeneous equation on every plane and has compact support there.
The local-frame and holomorphic-continuation argument in
Lemma~\ref{lem:supported-transport}, applied on a disk containing
that support, makes the difference zero on each plane.
\emph{Smooth dependence on the direction.}
The preceding construction gives a unique comparison separately for
every nonzero null vector $\theta$. If $c\in\C\setminus\{0\}$,
then $\Ecal_{c\theta}=\Ecal_\theta$, and
\[
D_{c\theta}=cD_\theta,
\qquad M_{j,c\theta}=cM_{j,\theta}.
\]
The equations for $\theta$ and $c\theta$ are consequently identical
after division by $c$. Compact-support uniqueness gives
$G(x,c\theta)=G(x,\theta)$, so the field is already well defined
on projective directions as a pointwise family.
Choose a local smooth representative $\theta(v)$ of those directions,
where $v$ ranges over an open subset of $\R^2$, and a smooth local
frame of $\Ecal$. Define the orthonormal vectors
\[
e_1(v)=\frac{\operatorname{Re}\theta(v)}
{|\operatorname{Re}\theta(v)|},
\quad
e_2(v)=\frac{\operatorname{Im}\theta(v)}
{|\operatorname{Im}\theta(v)|},
\quad
e_3(v)=e_1(v)\times e_2(v).
\]
They vary smoothly: the real and imaginary parts of a nonzero null
vector are nonzero, perpendicular, and of equal length. In the
coordinates
\[
x=y_1e_1(v)+y_2e_2(v)+te_3(v),
\]
and the chosen bundle frame, \eqref{eq:inhomogeneous-comparison}
is a smooth family of operators of the form in
Lemma~\ref{lem:supported-transport}, with parameters $(t,v)$ on
the fixed disk $D_{2r}^2$. For every value of these parameters a
solution supported in $\overline{D_r^2}$ has already been
constructed. The smooth left inverse
\eqref{eq:supported-left-inverse} therefore expresses this unique
solution as a smooth function of $(y,t,v)$. Returning to physical
coordinates gives joint smoothness in $(x,v)$.
This reasoning uses the separate fixed-direction existence results
and their uniqueness. In particular, none of the earlier weakly
convergent subsequences has to be chosen simultaneously for varying
directions.
\emph{Holomorphic dependence at fixed physical coordinates.}
A close precedent for this parameter argument is Ceki\'c
\cite[Theorem~3.4 and Lemma~4.1 in the arXiv version]{Cekic2025}, whose
parameter argument follows Eskin's complex-parameter method \cite{Eskin2001}.
Now let $v$ be a complex local coordinate on $\mathcal C$. Choose
a holomorphic nonzero representative $\theta(v)$ and a holomorphic
frame of $\Ecal$ on this chart. This frame depends only on $v$.
In this frame the matrices $M_{j,\theta(v)}(x)$ are holomorphic in
$v$ for each fixed $x$: the ambient formulas
\eqref{eq:transport-matrix}--\eqref{eq:transport-coefficients} are
holomorphic in $\theta$, and passage to a holomorphic bundle frame
preserves that property.
Represent $G$ in the same frame and differentiate the first equation
of \eqref{eq:comparison-transport} by $\partial_{\bar v}$,
\emph{holding $x\in\R^3$ fixed}. Joint smoothness just established
justifies this differentiation. Since the representative and the
frame are holomorphic and the frame is independent of $x$, the
result is exactly
\[
D_{\theta(v)}(\partial_{\bar v}G)
-M_{2,\theta(v)}\partial_{\bar v}G
+(\partial_{\bar v}G)M_{1,\theta(v)}=0.
\]
The derivative here is a matrix in a holomorphic trivialization,
hence represents an endomorphism of the same fiber $\Ecal_{\theta(v)}$.
The common support bound \eqref{eq:comparison-uniform-support}
implies that this derivative is zero for $|x|>2r$. Compact-support
uniqueness on each plane gives $\partial_{\bar v}G=0$. The smooth
moving coordinates were used to establish joint regularity; the
equation was differentiated only after returning to fixed physical
coordinates. We conclude that $G(x,\cdot)$ is holomorphic on
$\mathcal C$ for every $x$, completing the proof.
\end{proof}
==== END Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem//transport-matching.tex ====
==== BEGIN Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem//references.bib ====
@article{NU1994,
author = {Nakamura, Gen and Uhlmann, Gunther},
title = {Global uniqueness for an inverse boundary problem arising in elasticity},
journal = {Inventiones Mathematicae},
volume = {118},
year = {1994},
pages = {457--474},
doi = {10.1007/BF01231541},
note = {\href{https://doi.org/10.1007/BF01231541}{doi:10.1007/BF01231541}; see the erratum below}
}
@article{NU2003,
author = {Nakamura, Gen and Uhlmann, Gunther},
title = {Erratum: {Global} uniqueness for an inverse boundary value problem arising in elasticity},
journal = {Inventiones Mathematicae},
volume = {152},
number = {1},
year = {2003},
pages = {205--207},
doi = {10.1007/s00222-002-0276-1},
note = {\href{https://doi.org/10.1007/s00222-002-0276-1}{doi:10.1007/s00222-002-0276-1}}
}
@article{LinNakamura2017,
author = {Lin, Yi-Hsuan and Nakamura, Gen},
title = {Boundary determination of the {Lam\'{e}} moduli for the isotropic elasticity system},
journal = {Inverse Problems},
volume = {33},
number = {12},
year = {2017},
pages = {125004},
doi = {10.1088/1361-6420/aa942d},
note = {\href{https://doi.org/10.1088/1361-6420/aa942d}{doi:10.1088/1361-6420/aa942d}}
}
@article{TanLiu2023,
author = {Tan, Xiaoming and Liu, Genqian},
title = {Determining {Lam\'{e}} coefficients by the elastic {Dirichlet-to-Neumann} map on a {Riemannian} manifold},
journal = {Inverse Problems},
volume = {39},
number = {9},
year = {2023},
pages = {095004},
doi = {10.1088/1361-6420/ace649},
note = {\href{https://doi.org/10.1088/1361-6420/ace649}{doi:10.1088/1361-6420/ace649}}
}
@article{SaloTzou2009,
author = {Salo, Mikko and Tzou, Leo},
title = {Carleman estimates and inverse problems for {Dirac} operators},
journal = {Mathematische Annalen},
volume = {344},
number = {1},
year = {2009},
pages = {161--184},
doi = {10.1007/s00208-008-0301-9},
note = {\href{https://doi.org/10.1007/s00208-008-0301-9}{doi:10.1007/s00208-008-0301-9}}
}
@incollection{EskinRalston2004,
author = {Eskin, Gregory and Ralston, James},
title = {Inverse boundary value problems for systems of partial differential equations},
booktitle = {Recent Development in Theories \& Numerics},
editor = {Hon, Yiu-Chung and Yamamoto, Masahiro and Cheng, Jin and Lee, June-Yub},
publisher = {World Scientific},
address = {River Edge, NJ},
year = {2003},
pages = {105--113},
doi = {10.1142/9789812704924_0009},
note = {\href{https://doi.org/10.1142/9789812704924_0009}{doi:10.1142/9789812704924\_0009}; \href{https://arxiv.org/abs/math/0403507}{arXiv:math/0403507}}
}
@article{Eskin2001,
author = {Eskin, Gregory},
title = {Global uniqueness in the inverse scattering problem for the {Schr{\"o}dinger} operator with external {Yang--Mills} potentials},
journal = {Communications in Mathematical Physics},
volume = {222},
number = {3},
year = {2001},
pages = {503--531},
doi = {10.1007/s002200100522},
note = {\href{https://doi.org/10.1007/s002200100522}{doi:10.1007/s002200100522}}
}
@article{Cekic2025,
author = {Ceki{\'c}, Mihajlo},
title = {{Calder{\'o}n} problem for systems via complex parallel transport},
journal = {SIAM Journal on Mathematical Analysis},
volume = {57},
number = {4},
year = {2025},
pages = {3524--3560},
doi = {10.1137/24M166574X},
eprint = {2309.09348},
archivePrefix = {arXiv},
primaryClass = {math.AP},
note = {\href{https://doi.org/10.1137/24M166574X}{doi:10.1137/24M166574X}; \href{https://arxiv.org/abs/2309.09348v3}{arXiv:2309.09348v3}}
}
@article{Grauert1958,
author = {Grauert, Hans},
title = {Analytische {Faserungen} {\"u}ber holomorph-vollst{\"a}ndigen {R{\"a}umen}},
journal = {Mathematische Annalen},
volume = {135},
year = {1958},
pages = {263--273},
doi = {10.1007/BF01351803},
note = {\href{https://doi.org/10.1007/BF01351803}{doi:10.1007/BF01351803}}
}
@article{Forstneric2009,
author = {Forstneri{\v c}, Franc},
title = {Oka manifolds},
journal = {Comptes Rendus. Math\'{e}matique},
volume = {347},
number = {17--18},
year = {2009},
pages = {1017--1020},
doi = {10.1016/j.crma.2009.07.005},
note = {\href{https://doi.org/10.1016/j.crma.2009.07.005}{doi:10.1016/j.crma.2009.07.005}}
}
@article{Weck1969,
author = {Weck, Norbert},
title = {Au{\ss}enraumaufgaben in der {Theorie} station{\"a}rer {Schwingungen} inhomogener elastischer {K{\"o}rper}},
journal = {Mathematische Zeitschrift},
volume = {111},
year = {1969},
pages = {387--398},
url = {https://eudml.org/doc/171213},
note = {\href{https://eudml.org/doc/171213}{EuDML record}}
}
@article{ImanuvilovYamamoto2004Carleman,
author = {Imanuvilov, Oleg Yu. and Yamamoto, Masahiro},
title = {Carleman estimate for a stationary isotropic {Lam{\'e}} system and the applications},
journal = {Applicable Analysis},
volume = {83},
number = {3},
year = {2004},
pages = {243--270},
doi = {10.1080/00036810310001632772},
url = {https://www.ms.u-tokyo.ac.jp/preprint/pdf/2003-19.pdf},
note = {\href{https://doi.org/10.1080/00036810310001632772}{doi:10.1080/00036810310001632772}; \href{https://www.ms.u-tokyo.ac.jp/preprint/pdf/2003-19.pdf}{institutional preprint UTMS 2003--19}}
}
@book{McLean2000,
author = {McLean, William},
title = {Strongly Elliptic Systems and Boundary Integral Equations},
publisher = {Cambridge University Press},
address = {Cambridge},
year = {2000},
isbn = {9780521663328}
}
@incollection{Calderon1980,
author = {Calder\'{o}n, Alberto P.},
title = {On an inverse boundary value problem},
booktitle = {Seminar on Numerical Analysis and Its Applications to Continuum Physics},
series = {Cole\c{c}\~{a}o Atas},
number = {12},
publisher = {Sociedade Brasileira de Matem\'{a}tica},
address = {Rio de Janeiro},
year = {1980},
pages = {65--73},
note = {\href{https://sites.math.washington.edu/~reu/papers/reference/calderon.pdf}{original scan}}
}
@article{SylvesterUhlmann1987,
author = {Sylvester, John and Uhlmann, Gunther},
title = {A global uniqueness theorem for an inverse boundary value problem},
journal = {Annals of Mathematics},
volume = {125},
number = {1},
year = {1987},
pages = {153--169},
doi = {10.2307/1971291},
note = {\href{https://doi.org/10.2307/1971291}{doi:10.2307/1971291}}
}
@article{NakamuraUhlmann1995Boundary,
author = {Nakamura, Gen and Uhlmann, Gunther},
title = {Inverse problems at the boundary for an elastic medium},
journal = {SIAM Journal on Mathematical Analysis},
volume = {26},
number = {2},
year = {1995},
pages = {263--279},
doi = {10.1137/S0036141093247494},
note = {\href{https://doi.org/10.1137/S0036141093247494}{doi:10.1137/S0036141093247494}}
}
@article{EskinRalston2002Elasticity,
author = {Eskin, Gregory and Ralston, James},
title = {On the inverse boundary value problem for linear isotropic elasticity},
journal = {Inverse Problems},
volume = {18},
number = {3},
year = {2002},
pages = {907--921},
doi = {10.1088/0266-5611/18/3/324},
note = {\href{https://doi.org/10.1088/0266-5611/18/3/324}{doi:10.1088/0266-5611/18/3/324}}
}
@article{ImanuvilovYamamoto2015Elasticity,
author = {Imanuvilov, Oleg Yu. and Yamamoto, Masahiro},
title = {Global uniqueness in inverse boundary value problems for the {Navier--Stokes} equations and {Lam\'{e}} system in two dimensions},
journal = {Inverse Problems},
volume = {31},
number = {3},
year = {2015},
pages = {035004},
doi = {10.1088/0266-5611/31/3/035004},
note = {\href{https://doi.org/10.1088/0266-5611/31/3/035004}{doi:10.1088/0266-5611/31/3/035004}}
}
@misc{ChenJiangLiuTao2026Elasticity,
author = {Chen, Lu and Jiang, Yan and Liu, Hongyu and Tao, Longyue},
title = {On an inverse boundary problem in elastodynamics},
year = {2026},
note = {\href{https://arxiv.org/abs/2609.20134v1}{arXiv:2609.20134v1}, 17 September 2026}
}
@misc{Dyatlov2026,
author = {Dyatlov, Semyon},
title = {Lecture notes for 18.155: distributions, elliptic regularity, and applications to {PDE}s},
year = {2026},
note = {Lecture notes, version dated 2 October 2026; accessed 3 October 2026; \href{https://math.mit.edu/~dyatlov/18.155/155-notes.pdf}{author-hosted notes}}
}
==== END Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem//references.bib ====