RECONSTRUCTION OF CONDUCTIVITY AND ITS DERIVATIVES

AT THE BOUNDARY FROM THE LOCALIZED DIRICHLET TO NEUMANN MAP

GEN NAKAMURA AND KAZUMI TANUMA

Let Ω ∈ R^{n} (n ≥ 2) be a bounded domain with Lipschitz boundary ∂Ω. Physically Ω is considered as an isotropic, static and conductive medium with conductivity γ ∈ L^{∞}(Ω). When an electric potential f ∈ H^{1/2}(∂Ω) is applied to the boundary ∂Ω, the potential u solves the Dirichlet problem

∇ · (γ∇u) = 0 in Ω,  u|_{∂Ω} = f.  (1)

Assume that there is a constant δ > 0 such that γ(x) ≥ δ (a.e. x ∈ Ω). Then, there exists a unique weak solution u ∈ H^{1}(Ω) to (1). Define the Dirichlet to Neumann map Λ_{γ} : H^{1/2}(∂Ω) ⟶ H^{−1/2}(∂Ω) by

< Λ_{γ}f, g >= ∫_{Ω} γ∇u · ∇v dx (g ∈ H^{1/2}(∂Ω)),  (2)

where u is the solution to (1), v is any v ∈ H^{1}(Ω) satisfying v|_{∂Ω} = g and < , > is the bilinear pairing between H^{1/2}(∂Ω) and H^{−1/2}(∂Ω). Note that Λ_{γ}f = γ∇u · n when f ∈ H^{3/2}(∂Ω), γ ∈ C^{1}(\overline{Ω}) and ∂Ω is C^{2}, where n is the unit outer normal to ∂Ω. Hence Λ_{γ}f is the current flux across ∂Ω produced by the potential f on ∂Ω.

The problem of determining conductivity of the medium from the measurements of the electric potential on the boundary and the corresponding current flux across the boundary is expressed as Inverse Problem “Determine γ(x) from Λ_{γ}”. Since this problem was posed by A.P.Calderon, many results on uniqueness, stability, reconstruction have been proved.

In this talk, letting x_{0} be any point on ∂Ω, we give a formula for reconstructing γ and its normal derivative at x_{0} ∈ ∂Ω simultaneously from the localized Λ_{γ} around x_{0}. In our new formula we need only some regularity assumptions on γ and on ∂Ω around x_{0}, and we take as the Dirichlet data the highly oscillating functions which are compactly supported in a small neighborhood of x_{0} on ∂Ω.