[Paper Review] Uniqueness for inverse boundary value problems by Dirichlet-to -Neumann map on subboundaries

Oleg Imanuvilov, Masahiro Yamamoto|arXiv (Cornell University)|Mar 9, 2013

Numerical methods in inverse problems43 references3 citations

TL;DR

This paper establishes uniqueness results for inverse boundary value problems in two and three dimensions by using the Dirichlet-to-Neumann map restricted to subboundaries. The authors employ complex geometric optics solutions constructed via Carleman estimates to prove that the conductivity γ can be uniquely determined from partial boundary measurements, under conditions such as the closure of the union of the measurement and inaccessible boundaries covering the full boundary.

ABSTRACT

We consider inverse boundary value problems for elliptic equations of second order of determining coefficients by Dirichlet-to-Neumann map on subboundaries, that is, the mapping from Dirichlet data supported on $\partialΩ\setminus Γ_-$ to Neumann data on $\partialΩ\setminus Γ_+$. First we prove uniqueness results in three dimensions under some conditions such as $\bar{Γ_+ \cup Γ_-} = \partialΩ$. Next we survey uniqueness results in two dimensions for various elliptic systems for arbitrarily given $Γ_- = Γ_+$. Our proof is based on complex geometric optics solutions which are constructed by a Carleman estimate.

Motivation & Objective

• To establish uniqueness for the inverse conductivity problem when the Dirichlet-to-Neumann map is known only on subboundaries.
• To investigate whether partial boundary measurements suffice to uniquely recover the conductivity γ in both two and three dimensions.
• To extend the Calderón problem to cases where data are collected only on a portion of the boundary, under geometric and regularity conditions.
• To develop a framework based on complex geometric optics solutions and Carleman estimates for handling incomplete data in inverse coefficient problems.

Proposed method

• Construct complex geometric optics solutions to the conductivity equation using a Carleman estimate with a weight function depending on a large parameter τ.
• Utilize the asymptotic behavior of these solutions as τ → ∞ to derive integral identities involving the difference of conductivities.
• Apply integration by parts and boundary trace estimates to control error terms and derive vanishing limits on subboundaries.
• Use holomorphic and antiholomorphic function constructions to model boundary data and exploit the structure of the Dirichlet-to-Neumann map.
• Establish a duality argument via extremal problems in Sobolev spaces to reconstruct boundary data from integral conditions.
• Leverage the fact that if the integral of a boundary function against all holomorphic functions vanishes, then the function must be zero, to deduce uniqueness.

Experimental results

Research questions

• RQ1 Can the conductivity γ be uniquely determined from the Dirichlet-to-Neumann map restricted to a subboundary in three dimensions?
• RQ2 What geometric and regularity conditions on the domain and subboundaries ensure uniqueness in the inverse conductivity problem?
• RQ3 Is uniqueness possible in two dimensions when the Dirichlet-to-Neumann map is known only on a single subboundary Γ₊ = Γ₋?
• RQ4 How does the use of complex geometric optics solutions and Carleman estimates enable recovery from partial boundary data?
• RQ5 Under what conditions does the vanishing of the boundary integral against holomorphic functions imply the function itself is zero?

Key findings

• In three dimensions, uniqueness for the conductivity γ is established when the union of the accessible and inaccessible boundaries covers the full boundary, i.e., $ar{Γ}_+ ∪ Γ_- = \partial\Omega$.
• In two dimensions, uniqueness holds for arbitrary subboundaries Γ₊ = Γ₋, provided the boundary data satisfy a vanishing integral condition against all holomorphic functions.
• The proof relies on constructing complex geometric optics solutions via a Carleman estimate, which allows control of the asymptotic behavior as the large parameter τ → ∞.
• The authors show that if the Dirichlet-to-Neumann data agree on a subboundary, then the difference of conductivities must vanish, implying uniqueness.
• A key technical step involves proving that certain boundary integrals involving the phase function Φ and its conjugate vanish in the limit τ → ∞, which leads to the vanishing of the conductivity difference.
• The existence of a holomorphic function vanishing to high order at finitely many points on the boundary is used to construct solutions that localize the error, enabling the uniqueness argument.