Boundary determination of all scalar conductivity jets

We first make explicit the full physical DN map and the identity on a possibly disconnected $\mathrm{\Omega }$. The compact smooth manifold $\bar{\mathrm{\Omega }}$ has finitely many connected components $M_a={\bar{\mathrm{\Omega }}}_a$, $1\le a\le J$: connected coordinate neighborhoods give an open cover, and a finite subcover meets every component. Each $M_a$ has nonempty boundary; otherwise its same-dimensional inclusion in ${\mathbb{R}}^n$ would make it both open and compact, impossible for a nonempty subset of ${\mathbb{R}}^n$. Thus the connected-domain suppliers [F1] and [F2] apply on every $M_a$. Fixed finite atlases identify $H^1(\mathrm{\Omega })={⨁}_a{H}^1({\mathrm{\Omega }}_a)$ and $H^{1\mathrm{/}2}(\mathrm{\partial }\mathrm{\Omega })={⨁}_a{H}^{1\mathrm{/}2}(\mathrm{\partial }{\mathrm{\Omega }}_a)$, with the corresponding finite sum for the boundary dual. The full physical map is ${\mathrm{\Lambda }}_{\gamma }={⨁}_a{\mathrm{\Lambda }}_{\gamma {\mathrm{\mid }}_{M_a}}$: solve separately on each component, and its weak boundary pairing is the sum of the component weak pairings. This is exactly the global physical flux pairing, not an additional assumption on the conductivities or data. Equality of the global maps implies equality of the component maps by inputs and tests supported on one component. Applying [F2] on each component and summing gives the global Alessandrini identity used below. The quasimode supplier [F3], Proof 5.1, already constructs the exact solutions componentwise. Boundary normal collars and derivatives are componentwise as well. Consequently all later localization and normal-jet steps apply without a connectedness restriction.

Fix $x_0\in \mathrm{\partial }\mathrm{\Omega }$ and a sufficiently small patch. Choose $\chi \in C_c^{\mathrm{\infty }}(\mathrm{\partial }\mathrm{\Omega })$ supported in the patch and equal to one near $x_0$, and use [F3] to construct approximate beams for ${\gamma }_1,{\gamma }_2$ with the same boundary trace. Let $u_{\lambda ,j}$ be their exact conductivity solutions. Equality of the DN maps and [F2] give ${\int }_{\mathrm{\Omega }}({\gamma }_1-{\gamma }_2)\mathrm{\nabla }{u}_{\lambda ,1}\cdot \overline{\mathrm{\nabla }{u}_{\lambda ,2}}dx=0.$ Replacing each exact solution by its approximate beam changes this integral by $O({\lambda }^{-N})$: the coefficient is bounded, the beam $H^1$ norms are uniformly bounded, and each exact correction is $O({\lambda }^{-N})$ in $H^1$ by [F3].

Set $k=0$ in [F3] and let $\lambda \to \mathrm{\infty }$ in step 1.2. Since $c_0>0$, ${\int }_{\mathrm{\partial }\mathrm{\Omega }}({\gamma }_1-{\gamma }_2)\mathrm{\mid }\chi {\mathrm{\mid }}^{2}dS=0.$ If ${\gamma }_1-{\gamma }_2$ were positive or negative at any boundary point, continuity would give a patch on which it has that strict sign; choosing $\chi$ supported in that patch and equal to one near the point would make this integral nonzero. Thus ${\gamma }_1={\gamma }_2$ on $\mathrm{\partial }\mathrm{\Omega }$.

Fix an integer $k\ge 1$ and suppose inductively that ${\mathrm{\partial }}_{x_n}^j({\gamma }_1-{\gamma }_2)=0$ on a boundary patch for every $j<k$. Taylor's formula with integral remainder in the inward normal coordinate gives ${\gamma }_1(y,x_n)-{\gamma }_2(y,x_n)=x_n^k{F}_k(y,x_n),F_k(y,0)=\frac{{\mathrm{\partial }}_{x_n}^k({\gamma }_1-{\gamma }_2)(y,0)}{k!},$ for a smooth $F_k$. Apply [F2] to the exact beams from [F3], replace them by their approximate beams, and multiply the resulting integral identity by ${\lambda }^k$. The replacement error is $O({\lambda }^{k-N})$, since the weight $x_n^k$ is bounded and the corrections are $O({\lambda }^{-N})$ in $H^1$. Choose $N>k$ and apply the weighted limit in [F3]; then $c_k{\int }_{\mathrm{\partial }\mathrm{\Omega }}{F}_k(y,0)\mathrm{\mid }\chi (y){\mathrm{\mid }}^{2}dS(y)=0.$ Since $c_k>0$, the same localized sign argument as in step 2.1 gives $F_k(y,0)=0$ throughout the patch. Therefore the $k$th inward normal derivatives agree there.

Step 2.1 is the base case for the induction. Step 3.1 proves each subsequent order on a patch that may be chosen around any prescribed boundary point. Thus every boundary point has a neighborhood on which the order-$k$ normal derivative difference is zero for every integer $k\ge 0$, proving the full boundary-jet assertion.