Global uniqueness for a two-dimensional inverse boundary value problem. (English) Zbl 0857.35135

Nachman, Adrian I.

Ann. Math. (2) 143, No. 1, 71-96 (1996).

The author gives a complete solution of a long standing and fundamental problem of uniqueness in the plane inverse conductivity problem. It consists of determining the conductivity coefficient \(\gamma\) of the elliptic equation \(\text{div} (\gamma\nabla u)=0\) in a domain \(\Omega\subset \mathbb{R}^2\) from the Dirichlet-to-Neumann map \(g\to h\) where \(g\) is the Dirichlet data \(u\) on \(\partial\Omega\) and \(h=\nu\cdot\gamma\nabla u\) is the Neumann data. The three-dimensional problem was solved by J. Sylvester and G. Uhlmann [Ann. Math., II. Ser. 125, 153-169 (1987; Zbl 0625.35078)] by using complex geometrical optics. They assumed \(\gamma\in C^2(\Omega)\) and is scalar.
The author assumes \(\gamma\in W^{2,p}(\Omega)\) for scalar \(\gamma\) and \(\gamma\in C^3(\Omega)\) for a general symmetric positive matrix. He proves uniqueness of scalar \(\gamma\) and uniqueness up to a fixed boundary diffeomorphism for general \(\gamma\). The basic method stems from (complex) scattering theory: \(\partial\)-equation, complex scattering amplitude and the related Lippmann-Schwinger equation. One of the crucial steps is to show absence of exceptional points where the solution of this integral equation might not be unique. It is done by factorizing the Schrödinger equation in the plane and using generalized analytic functions whose behavior at infinity is very well understood. The methods of proofs are constructive and, in our opinion, quite promising. The paper contains an extensive bibliography.

Reviewer: V.Isakov (Wichita)

MSC:

35R30	Inverse problems for PDEs
35J25	Boundary value problems for second-order elliptic equations

Keywords:

uniqueness; plane inverse conductivity problem; Dirichlet-to-Neumann map; scattering theory; Lippmann-Schwinger equation; Schrödinger equation

Citations:

Zbl 0625.35078