What one boundary patch can reveal about an unknown interior

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Family 365 · Partial differential equations
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-26

1. The question

Suppose an object is inaccessible inside. You can impose voltages on its surface and measure the resulting currents. Can those measurements determine how well each interior point conducts electricity?

Calderón’s inverse problem asks this mathematically. Its geometric version allows conductivity to depend on direction, so the unknown resembles a Riemannian metric. A further version includes a connection, which describes how vector-valued quantities change when transported through the object.

Family 365 contains three manuscripts with complementary conclusions. M2 claims that any open boundary patch determines an arbitrary smooth metric, up to the unavoidable freedom to change coordinates. M1 adds simultaneous recovery of a smooth unitary connection on a trivial rank-two bundle. M3 claims that, with only bounded measurable scalar conductivity, even measurements on the entire boundary can fail to distinguish different interiors.

The contrast concerns regularity as well as the type of unknown. It does not say that the same class of conductivities is both recoverable and unrecoverable.

2. Background and essential definitions

Voltages, currents and boundary response

For a positive scalar conductivity $\gamma(x)$ in a domain $\Omega\subset\mathbb R^n$, a stationary voltage $u$ satisfies

$$\operatorname{div}(\gamma\nabla u)=0.$$

The equation expresses conservation of electric current. A prescribed boundary voltage $f$ determines an interior solution. The Dirichlet-to-Neumann map, abbreviated DN map, sends $f$ to its outward conductivity flux $\gamma\partial_\nu u$. Knowing this map means knowing the response to every admissible voltage, not just a finite set of experiments.

Its weak formulation uses energy:

$$\langle\Lambda_\gamma f,h\rangle =\int_\Omega\gamma\nabla u_f\cdot\nabla u_h\,dx.$$

Here $u_f,u_h$ solve the equation with boundary values $f,h$. This definition still works for bounded measurable coefficients. Uniform positivity means fixed constants satisfy $0<c\le\gamma\le C<\infty$ almost everywhere.

For a boundary patch $\Gamma$, both $f$ and $h$ are supported in $\Gamma$; voltages vanish on the rest of the boundary. The measurement is therefore restricted on both the input and observation sides.

Geometry and its unavoidable ambiguity

A metric $g$ gives lengths and angles on a manifold $M$. Its harmonic functions solve $\Delta_g u=0$, and their boundary response is defined by

$$\langle\Lambda_{g,\Gamma}f,h\rangle =\int_M\langle du_f,du_h\rangle_g\,dV_g.$$

The output includes the boundary density. This matters when the boundary metric is itself unknown: the definition does not silently assume a known surface measure.

A diffeomorphism is a smooth invertible change of coordinates with smooth inverse. If $\Phi$ fixes the measured boundary points, the metrics $g$ and $\Phi^*g$ give identical measurements. The notation $\Phi^*g$ means that lengths are computed after applying $\Phi$. Recovery can at best be up to this coordinate freedom.

For $n\ge3$, scalar Euclidean conductivity corresponds to

$$g=\gamma^{2/(n-2)}g_{\mathrm{Eucl}}.$$

This power makes the metric energy equal the conductivity energy. General anisotropic conductivities correspond to more general metrics.

Connections and gauge freedom

In M1, each point carries a copy of $\mathbb C^2$, with its usual Hermitian inner product. A connection replaces differentiation by

$$d_Au=du+Au,$$

where $A$ is a matrix-valued one-form. Its matrices are skew-Hermitian, $A^*=-A$, so parallel transport preserves the inner product. The connection Laplacian is $L_{g,A}=d_A^*d_A$; the star here denotes the adjoint with respect to the metric and Hermitian structure.

A unitary matrix $U(x)$ changes the basis in each fiber. This is a gauge transformation, producing

$$A\longmapsto U^{-1}AU+U^{-1}dU.$$

If $U$ equals the identity on the measured patch, this basis change is invisible to the boundary experiment. Both coordinate and gauge freedoms must therefore appear in a correct uniqueness statement.

3. Development history and previous partial results

Calderón posed the conductivity problem in 1980 and studied its linearization. Sylvester and Uhlmann established full-boundary uniqueness for smooth scalar conductivities in dimensions at least three in 1987. Their special complex-valued solutions became a central tool in inverse problems.

The anisotropic problem is harder because the geometry is unknown. Lee and Uhlmann obtained boundary determination and real-analytic recovery under additional hypotheses. Lassas and Uhlmann subsequently recovered real-analytic manifolds from a boundary patch without the earlier topological restrictions. Real analyticity supplies strong continuation properties that arbitrary smooth functions need not have.

Partial-data methods then recovered coefficients under particular arrangements of the measurement regions, reflection symmetries, known boundary neighborhoods or special product geometries. Smooth Einstein metrics supplied another important geometric class. Recent results near prescribed backgrounds use smallness, ordering or foliation assumptions. M2 targets arbitrary smooth metrics and an arbitrary common patch without those restrictions.

The Poisson embedding approach of Lassas, Liimatainen and Salo represents an interior point by its values on boundary-generated harmonic functions. This supplies a geometric language for reconstruction; in the general higher-dimensional setting, their global continuation result uses real analyticity.

For connections, previous work recovered connections and potentials on known surfaces or suitable known higher-dimensional geometries. Cekić obtained same-patch recovery for Yang–Mills connections on a fixed smooth metric. Gabdurakhmanov and Kokarev obtained joint recovery of geometry, bundle and connection in the real-analytic category. M1’s distinction is simultaneous smooth recovery without prescribing the metric or imposing the Yang–Mills equation.

Regularity has a different history for scalar conductivities. Astala and Päivärinta proved planar uniqueness with merely bounded measurable, uniformly positive conductivity. In higher dimensions, uniqueness extends below smoothness: Haberman treated specified Sobolev classes in dimensions three and four, and Caro and Rogers treated positive Lipschitz conductivities in all dimensions at least three. M3 concerns the larger three-dimensional measurable class, without derivative control.

4. What the new manuscripts claim

M2: smooth geometry from an arbitrary patch

Let $M$ be a compact connected smooth manifold of dimension $n\ge3$, with nonempty smooth boundary. Let $\Gamma$ be any nonempty proper relatively open boundary subset. For smooth positive-definite metrics, including smoothness up to the boundary, M2 claims

$$\Lambda_{g_1,\Gamma}=\Lambda_{g_2,\Gamma} \quad\Longrightarrow\quad g_2=\Phi^*g_1,\qquad \Phi|_\Gamma=\mathrm{Id}.$$

The diffeomorphism extends smoothly through the boundary. Neither orientability nor a connected boundary is required. The inaccessible boundary need not be fixed pointwise.

Full-boundary data yield a diffeomorphism fixing the whole boundary. A scalar corollary states exact equality $\gamma_1=\gamma_2$ for smooth positive conductivities on a bounded connected smooth Euclidean domain from any common open patch. No conductivity values are assumed known near the inaccessible boundary.

M1: simultaneous metric and connection recovery

Under the same geometric hypotheses, take arbitrary smooth unitary connections $A_1,A_2$ on $M\times\mathbb C^2$. Equality of their zero-frequency local energy forms for all smooth vector-valued inputs supported in $\Gamma$ is claimed to imply

$$\begin{aligned} g_2&=\Phi^*g_1,\\ A_2&=U^{-1}(\Phi^*A_1)U+U^{-1}dU, \end{aligned}$$

with $\Phi|_\Gamma=\mathrm{Id}$ and $U|_\Gamma=I$. No field equation is imposed on $A_j$. The bundle is specifically trivial and of complex rank two; the theorem is not stated here as an arbitrary-bundle result.

M3: rough scalar conductivities with identical full data

On $B(0,3)\subset\mathbb R^3$, M3 constructs two real scalar conductivities $\gamma_0,\gamma_1$, uniformly positive and bounded, both equal to one near the boundary, such that

$$\Lambda_{\gamma_0}=\Lambda_{\gamma_1}, \qquad |\{x:\gamma_0(x)\ne\gamma_1(x)\}|>0.$$

The equality is for the entire weak DN operator. The coefficients are exactly indistinguishable, rather than merely having responses that approach one another in a limit.

A localization corollary places independent copies inside any finite collection of disjoint interior balls of a bounded connected Lipschitz domain in $\mathbb R^3$. With $m$ balls, there are $2^m$ distinct conductivities sharing one full boundary response. That common response is not asserted to equal the response of the constant conductivity one.

5. Why the advance matters

Importance: 5/5. If correct, the smooth results resolve central forms of the anisotropic Calderón uniqueness problem and extend geometric recovery to coupled transport data. The rough counterexample simultaneously identifies a genuine failure of unrestricted scalar recovery in three dimensions.

Together, these claims would sharpen what boundary measurements can encode. An arbitrarily small open patch can contain enough information for smooth uniqueness, while the entire boundary can fail in a broader coefficient class.

This is information-theoretic uniqueness, however, rather than a promise of practical imaging accuracy. An infinite collection of exact boundary measurements can determine an unknown even when reconstruction from finitely many noisy measurements is extremely unstable.

6. The main idea

Continue a geometric match through the interior

For M2, boundary measurements first determine all boundary derivatives of the metric in suitable coordinates. The argument attaches a small exterior region where the two geometries agree. Equality of boundary response then supplies equality of Green functions there. A Green function describes the response to a concentrated interior source.

Solutions generated from that exterior region distinguish interior points and provide enough harmonic functions to form local coordinates. The task becomes extending an existing match across any proposed frontier.

The manuscript constructs a mixed Green kernel, satisfying one metric’s equation in one variable and the other metric’s equation in the second. Its flux through a small moving sphere transfers harmonic functions from one geometry to the other. The transfer reproduces constants and coordinate functions, so their contributions cancel in a Taylor expansion.

The crucial claimed estimate remains uniform as the spheres shrink. Finite harmonic tests with sufficiently many independent second derivatives convert this control into an improved bound for the difference of normalized inverse metrics: a provisional bound of order $r$ improves to order $r^{3/2}$, where $r$ is sphere radius. With a single spatial rescaling held fixed, the radius can tend to zero, forcing agreement on an actual neighborhood.

The argument retains second-order terms caused by moving the spheres; treating all geometric errors as lower-order terms would miss the needed estimate. Harmonic coordinates then remove the remaining conformal factor. Global source evaluations prevent ambiguous identifications and eliminate interior frontiers.

Handle matrix-valued transport as well

M1 develops the analogous mechanism for vector-valued solutions. A harmonic frame simplifies the operator, but the transfer still has a matrix-valued first moment. Unlike scalar coordinate displacement, that moment cannot simply be normalized away.

The manuscript organizes its nonscalar components into a differential system related to conformal Killing fields, infinitesimal transformations preserving angles. In dimension at least three, this system has a finite-type property: sufficiently high derivatives are determined by finitely many lower derivatives. This improves the first-moment estimate enough to close the shrinking-sphere argument for both metric and connection.

The scalar companion supplies analytic methods, not a shortcut that applies a scalar uniqueness theorem directly to matrix measurements.

Make every rough harmonic solution share a hidden average

M3 uses a different mechanism. It first constructs a block with one parent and two child torus boundaries. Two selected voltage solutions control all current triples whose sum is zero. An exact scalarization step replaces an anisotropic tensor by one positive scalar coefficient while preserving both selected voltage-and-flux responses.

Nested copies of the block force every global harmonic function $u$ to have the same weighted average $u_*$ on the relevant internal surfaces. The construction produces a nonzero bounded function $w$, vanishing near the exterior boundary, with a special source-annihilation identity involving every such $u$.

For small nonzero $\delta$, put

$$\rho=1+\delta w,\qquad \widetilde\gamma=\rho^2\gamma,\qquad \widetilde u=u_*+\frac{u-u_*}{\rho}.$$

The identity makes $\widetilde u$ solve the new conductivity equation. Since $\rho=1$ in an outer collar, voltage and flux there stay unchanged. Thus every boundary response agrees even though the interior coefficient changes.

The intermediate scalarization preserves only two designated responses. The nested construction and the identity for every harmonic $u$ are what promote that finite-input statement to equality of the full operator. Mere approximation by rapidly oscillating scalar materials would not establish this exact conclusion.

7. Scope, limitations, and open questions

The smooth results assume exact zero-frequency data, arbitrary smoothness up to a smooth boundary, and dimension at least three. They give uniqueness modulo the stated symmetries, without a quantitative noise-stability estimate or a finite-measurement reconstruction algorithm.

M1 concerns a fixed trivial rank-two Hermitian bundle. More general bundles, other ranks, additional unknown matrix potentials and lower coefficient regularity require separate statements.

M3 gives a three-dimensional counterexample with some finite ellipticity contrast $C/c$. It does not prescribe arbitrarily small contrast, prove nonuniqueness in every higher dimension, or locate a sharp regularity threshold between known uniqueness classes and bounded measurability. Its conclusion also does not contradict planar measurable uniqueness.

The family therefore leaves substantial questions about intermediate regularity, quantitative reconstruction and the extension of joint recovery to more general geometric systems.

8. Sources and further reading

The primary historical papers linked above trace smooth scalar uniqueness, analytic geometric recovery, the Poisson embedding approach, connection recovery and low-regularity conductivity theory. For background terminology, Wikipedia’s articles on Dirichlet-to-Neumann maps and connections on vector bundles provide optional complements to the definitions here.

The three manuscripts below are read together as one family. Their conclusions are reported as stated; the distinction between smooth recovery and measurable nonuniqueness is essential to interpreting them.

Source manuscripts

• M1. OpenAI, Determination of a metric and a unitary connection from one boundary patch (October 5, 2026).
• M2. OpenAI, Smooth anisotropic uniqueness in the Calderón problem from one boundary patch (September 24, 2026).
• M3. OpenAI, Nonuniqueness for bounded measurable scalar conductivities in three dimensions (September 23, 2026).

Sources consulted on 26 October 2026 at repository revision 1581d4c. Download the manuscript citations.