Local Neumann-to-Dirichlet Map

• Local Neumann-to-Dirichlet Map is a boundary operator that maps mean-free current densities to measured electric potentials in elliptic PDEs.
• It utilizes bisweep data from opposing delta currents with a holomorphic extension, allowing complete reconstruction from localized measurements.
• Exploiting high-order derivatives at a single boundary point yields local uniqueness in conductivity determination, reducing experimental complexity in EIT.

The Local Neumann-to-Dirichlet Map is a fundamental boundary operator arising in the analysis of elliptic partial differential equations, particularly in the context of the inverse conductivity problem and the Calderón problem. It associates a prescribed, mean-free boundary current density to the resulting boundary electric potential, up to an additive constant, under the assumption that the conductivity is homogeneous (equal to 1) near the boundary. A central advance in the characterization of the map is the use of bisweep data—measuring relative potential differences between boundary points under injected delta currents—which uniquely determines the map. The analytic properties of this bisweep data, notably its holomorphic extension in two dimensions, enable the reconstruction of the entire Neumann-to-Dirichlet map from local measurements at a single boundary point, leading to a truly local uniqueness result for the inverse conductivity problem.

1. Definition and Formal Properties

The Neumann-to-Dirichlet (N-to-D) map, denoted as $A_\sigma$, is defined for a smooth, bounded, simply connected domain $D$ as the operator taking a mean-free boundary current $f$ (supported on $\partial D$) to the potential $u|_{\partial D}$, where $u$ solves the conductivity equation: $$\nabla \cdot (\sigma \nabla u) = 0 \quad \text{in } D$$ with Neumann boundary condition: $$\frac{\partial u}{\partial \nu} = f \quad \text{on } \partial D$$ under the constraint that $u$ is defined only up to an additive constant.

A crucial assumption is that $\sigma \equiv 1$ in a neighborhood of $\partial D$, so $(\sigma - 1)$ is compactly supported within $D$. This ensures that $A_\sigma$ behaves as a pseudodifferential operator with the same principal symbol as that for the unit conductivity and enables analytic and factorization techniques.

2. Bisweep Data and Boundary Measurement

The bisweep data $S_\sigma(x, y)$ captures the response of the medium when delta currents of opposite signs are injected at two boundary points $x, y \in \partial D$. Its definition is: $$S_\sigma(x, y) = \left\langle \delta_x - \delta_y, (A_\sigma - A_1)(\delta_x - \delta_y) \right\rangle_{\partial D}$$ Here, $\delta_z$ is the Dirac delta distribution at $z \in \partial D$, and $A_1$ is the N-to-D map for $\sigma \equiv 1$.

Experimentally, bisweep data corresponds to potential differences generated by applying currents via two small electrodes—justifying its physical relevance. This data encapsulates the effect of internal inhomogeneities (regions where $\sigma \neq 1$) as manifested on the boundary under this two-electrode paradigm.

3. Holomorphic Extension of Bisweep Data

In two dimensions, the bisweep function $S_\sigma(z_1, z_2)$ can be extended holomorphically with respect to both variables in some neighborhood $U \subset \mathbb{C}$ of the boundary, i.e., $S_\sigma$ is analytic on $U \times U$.

This analytic property implies that $S_\sigma$ admits a two-variable Taylor expansion, and thus all the information of the boundary response is encoded in its derivatives at any fixed point. The holomorphic extension is a key ingredient in establishing reconstruction strategies via local data.

4. Reconstruction via Point Measurements

Exploiting the analytic structure, the paper establishes that it suffices to know the derivatives of $S_\sigma$ at a single boundary point, parametrized angularly, to determine the full bisweep function and thereby the entire N-to-D map. Specifically, derivatives of the bisweep data at a point—corresponding to distributional current densities localized at the same boundary point—uniquely determine the full boundary operator. The polarization identity allows one to relate these derivatives to inner products: $\langle f, (A_\sigma - A_1) g \rangle_{\partial D}$ for mean-free distributions $f, g$ supported at the boundary point.

Practically, this result affirms that high-order local measurements of boundary potential under point-supported currents suffice for operator reconstruction, which is established in Theorem 2.2.

5. Local Uniqueness in the Calderón Problem

This pointwise determination leads to a new, truly local uniqueness result for the inverse conductivity problem, and specifically the Calderón problem, in the plane. For any $z \in \partial D$, the complete map is uniquely determined by the set

$$\left\{ \langle f, (A_\sigma - A_1) f \rangle_{\partial D} \ : \ f \in \mathcal{D}_z \right\}$$

where $\mathcal{D}_z$ denotes the mean-free distributions supported at $z$.

The N-to-D map, up to natural isomorphisms, is equivalent to the Dirichlet-to-Neumann map, and prior work (Astala–Päivärinta) asserts that the latter uniquely determines an isotropic conductivity in two dimensions. Therefore, measurements restricted to a single boundary point (and of all necessary derivatives) can determine the conductivity uniquely—except in the anisotropic case, where uniqueness holds up to boundary-fixing diffeomorphisms.

6. Practical Implications and Experimental Paradigms

In inverse problems and imaging scenarios such as electrical impedance tomography (EIT), measurement accessibility is often restricted to local (pointwise) data. The main result—that the full boundary operator can be reconstructed from data at a single point (with knowledge of boundary homogeneity)—dramatically reduces the experimental burden.

The physical correspondence between bisweep data and two-electrode measurements is direct and further validates prevalent experimental approaches. The methodology underpins global uniqueness from localized measurements, impacting the design and interpretation of EIT and related modalities.

Moreover, it enhances the theoretical understanding of inverse boundary value problems and offers a foundation for reconstructive techniques in medical and geophysical imaging, where minimal and localized data acquisition is often essential.

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Summary Table: Key Constructs in Local Neumann-to-Dirichlet Map

Construct	Definition	Role in Uniqueness/Reconstruction
N-to-D Map $A_\sigma$	Operator from mean-free boundary current to boundary potential under $\nabla\cdot(\sigma\nabla u)=0$	Main boundary operator
Bisweep Data $S_\sigma(x, y)$	Relative potential for opposite delta currents at boundary points $x$ and $y$	Encodes response to internal inhomogeneity
Point Measurement Set	Derivatives of bisweep data at a boundary point, or inner products of $(A_\sigma-A_1)$ with mean-free distributions	Sufficient for full operator recovery

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The analytic properties of bisweep data, together with the local measurement paradigm, establish deep connections between experimental boundary measurements and theoretical reconstruction of interior conductivities, marking a significant refinement in the paper of local uniqueness and minimal data recovery for inverse boundary value problems.