Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem

Proof. By Theorem 1.2 of Tan and Liu [18], equality of the elastic Dirichlet-to-Neumann maps determines all boundary partial derivatives of both Lamé 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 (1.3). 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 [12].

Smoothly extend the first pair to a neighborhood of \overline\Omega. By compactness and strict positivity, shrink the neighborhood so that (1.1) 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 (1.1). 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.□

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.□

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 (3.1) gives

Subtracting S and using k=\lambda+m gives m\Delta(v-d)+\nabla m\cdot\nabla(v-d).

For the principal part in (3.3), the only second derivatives of u in its scalar numerator cancel. More explicitly, if

then this numerator is

This proves (3.4). The support assertion follows from equality of the extended coefficients off \overline\Omega.□

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), [16, Lemma 2.1], with Sobolev index s=-1, gives

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 (3.7) componentwise. A first-order term in (3.4) contributes

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 (3.7), and then choose h sufficiently small relative to this fixed \varepsilon. We obtain

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 (3.6).□

Proof. Apply Proposition 3.2 to (w,\diverg w). By (3.2), its augmented forcing is formed from Lw and \diverg(Lw). If F=E_\tau^{-1}Lw, then

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

Taking the displacement component on the left side of (3.6) proves (3.8).

Let A_\theta=E_\tau^{-1}LE_\tau on test functions in B. Substituting w=E_\tau v in (3.8) gives \norm{v}_{L^2}\le C\norm{A_\theta v}_{L^2}. Formal self-adjointness of L gives

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

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 (3.9). The norm of the representing vector gives the asserted bound.□

Proof. The positive principal coefficient form of L is

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 [4, Theorem 15.1 and Remark 15.2]. 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

For small h, cover B' by such inner balls with uniformly bounded overlap of the doubled balls. Squaring and summing proves (3.10). Iterated interior elliptic regularity, with \tau fixed, proves the final assertion.□

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

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

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 [8, Theorem 1.1, p. 1018], supplies a holomorphic section; this homogeneous-fiber case goes back to Grauert [9]. In the original smooth trivialization it is a smooth invertible matrix solving (4.5) 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

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

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

solves (4.6) 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.□

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 4.1 to the rank-three system (4.3), starting on a disk strictly larger than D_{R_0}^2. It supplies a smooth invertible frame

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 (4.3). 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 [7, Section 2, equations (2.2)–(2.4)]. 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

Throughout this recursion, write q_j=\diverg a_j-b_j and set a_{-1}=b_{-1}=q_{-1}=0. The last equation in (4.8) 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 (3.2) yields, for arbitrary (a,b),

Suppose the preceding recursive equations have been solved. Apply (4.9) to (a_{j-1},b_{j-1}). For j\ge2, the preceding vector and scalar equations, followed by (3.2), give

The preceding normal constraint gives the same term on the other side:

For j=1, both expressions are zero by the homogeneous leading equations and q_{-1}=0. In every case these two terms cancel in (4.9), leaving

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 (4.4) and (4.10) 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 4.1 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

The first equation of (4.8) telescopes to

The normal constraints telescope separately, giving

Using the dependence of R on its scalar argument, these identities give the physical residual explicitly:

All coefficients on the right are fixed smooth functions. Consequently \norm{f_\tau}_{L^2(B_{5r})}=O(\tau^{1-N}).

Corollary 3.3 gives a correction r_\tau\in L^2(B_{5r};\C^3) satisfying

It is smooth in B_{5r} by Lemma 3.4. The same Lemma, on B_{4r}, gives

Define its conjugated physical divergence by

It follows directly from (4.14) that

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

The smooth finite expansions and the remainder estimates imply the first assertion of (4.7). They also show that, if the last displayed pair is denoted by \mathcal A_\tau, then

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 (4.7) follows. Repeating the construction for the three initial columns proves the Proposition.□

Proof. The inverse of (5.1) is

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 (4.1) gives

Dividing the second equation by two and using \nabla m=mg yields

Here

The terms containing (D\log m)g\cdot p cancel. Thus

Differentiate the definition of s and substitute (5.4), the expression for b, and Dp=\theta s. With c=m/\ell and \gamma=D\log m=\theta\cdot g, the result is

Finally, since \ell=k+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.□

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 4.1 gives a smooth invertible matrix F satisfying \Pcal_\rho F=0. If \Pcal_\rho W=0, then

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 (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

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

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 4.1, leaves holomorphic components. The original solution is therefore smooth in the disk.

To remove the last term in (5.8), suppose that (5.7) fails at a fixed parameter. There would then be a sequence W_\nu, supported in \overline{D_r^2}, such that

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 (i) gives W=0. Equation (5.8) now contradicts the normalization of W_\nu. This proves (ii) at a fixed parameter.

For the parameter assertions, equip the fixed complex Hilbert spaces

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 (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 (ii) implies that \Pcal_\rho^*\Pcal_\rho:X_j\to X_j is coercive and invertible. Consequently

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

vanishes on the assumed dense set, and continuity makes it zero for every \rho. Uniqueness in (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 (iii).□

Proof. Physical transfer and weak limits. Fix a nonzero null vector \theta and t_0\in(-2r,2r). Apply Proposition 4.2 to the first coefficient pair. It supplies three exact smooth solutions on B_{5r}, with the limits and estimates in (4.7), 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 2.2, 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

The difference V_{2,\tau}-V_{1,\tau} is smooth and compactly supported in \overline\Omega\subset B_r. By Lemma 3.1,

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 (4.7) implies

The function on the left is extended by zero off its compact support when a norm on \R^3 is used. Apply Proposition 3.2 to the difference in (5.11). Since \norm{F}_{H_h^{-1}}\leq\norm{F}_{L^2} and h=\tau^{-1}, we obtain

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

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

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

so \theta\cdot\widetilde a=0 distributionally. We have proved

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.

A comparison on almost every plane. Apply (5.1) 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

Thus the weak limits constructed above are defined throughout this cylinder, where

and Y_1 is smooth and invertible. The change of variables agrees for the two pairs outside \overline\Omega. Define

The calculation preceding Lemma 5.2 now applies: G is a locally L^2 endomorphism on the cylinder, equals the identity outside \overline\Omega, and satisfies the first equation of (5.10) distributionally. With W=G-\Id, this is the cylinder version of (5.6). 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 (5.6) 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.

Existence on every plane. In the orthonormal coordinates associated with the fixed \theta,

Thus (5.5), as an operator on matrix entries, has the form of Lemma 5.2. 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 (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 (i) of Lemma 5.2 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

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 5.2, applied on a disk containing that support, makes the difference zero on each plane.

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

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

They vary smoothly: the real and imaginary parts of a nonzero null vector are nonzero, perpendicular, and of equal length. In the coordinates

and the chosen bundle frame, (5.6) is a smooth family of operators of the form in Lemma 5.2, 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 (5.9) 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.

Holomorphic dependence at fixed physical coordinates. A close precedent for this parameter argument is Cekić [2, Theorem 3.4 and Lemma 4.1 in the arXiv version], whose parameter argument follows Eskin’s complex-parameter method [5]. 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 (5.2)–(5.3) 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 (5.10) by \partial_{\bar v}, 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

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 (5.14) 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.□

Proof. A homogeneous parametrization of the conic is

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

Direct calculation, with the complex bilinear product, gives

To interpret these homogeneous formulas as bundle maps, write \mathcal O(-1)_{[z]}=\C z for z=(z_0,z_1). The assignment

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 (6.3) gives the first map in (6.1).

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,

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.□

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 6.1. Its lower-right block is a holomorphic scalar function on the compact conic, and hence is independent of direction. We may thus write

All these blocks are smooth in x. Taking the upper-right block of (5.10), and using (5.2), gives

It follows that F-f\Id_H vanishes on \mathcal N. This holomorphic bundle morphism consequently factors through H/\mathcal N. By Lemma 6.1, its factor is a morphism \mathcal O\to H, which must vanish. Thus F=f\Id_H.

The upper-left block of (5.10) now reads

The right side has rank at most one, while H_\theta has dimension two. Therefore D_\theta f=0. Equation (6.5) 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 (5.10) proves (6.4).□

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 (6.6) and commutation of third derivatives give

There is no derivative of q on the right because the off-diagonal entry \partial_s\partial_i\eta in (6.6) has \delta_{si}=0. Substituting that equation for the derivatives of v yields

where the smooth coefficients are explicitly

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 (6.6) also gives q=0 there. Along any piecewise smooth path in U, Equation (6.7) 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.□

Proof of Theorem 1.1. Suppose that the two displacement-to-traction maps agree, and use the common exterior extensions furnished by Lemma 2.1. Propositions 5.3 and 6.3 give equality of the transport matrices on \Ecal_\theta for every nonzero null \theta. In particular, (5.3) gives

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

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

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

For fixed x, the proportionality factor in (6.10) 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

In particular, \kappa is smooth.

Set

where g_j=\nabla\log m_j. Because \nabla\log k_j=g_j+\beta, the expression for Q in (5.3) becomes

Subtract these identities, use g_1g_1^T-g_2g_2^T=g_1v^T+vg_2^T, and apply (6.11). Defining the smooth scalar q=-2\kappa/c, we obtain

Equivalently,

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 6.4, applied to (6.12), yields \eta=0 throughout \R^3. Thus m_1=m_2. Equation (6.9) then gives k_1=m_1e^{b_*}=m_2e^{b_*}=k_2, and therefore

Restricting to \Omega proves the theorem.□