A global uniqueness theorem for an inverse boundary value problem. (English) Zbl 0625.35078

Sylvester, John; Uhlmann, Gunther

Ann. Math. (2) 125, 153-169 (1987).

The authors prove that the single smooth coefficient, \(\gamma\), of the elliptic operator \(L_{\gamma}=\nabla \cdot \gamma \nabla\) in a bounded region \(\Omega \leq {\mathbb{R}}^ n\) (n\(\geq 3)\) can be recovered from the map which sends the boundary values of a \(\gamma\)-harmonic function u \((L_{\gamma}u=0\) in \(\Omega)\) to the boundary values of its conormal derivative \(\gamma\) (\(\partial u/\partial \nu)\). This shows that the isotropic conductivity of a body can, in principal, be recovered from voltage to current measurement at the boundary.

MSC:

35R30	Inverse problems for PDEs
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
31A25	Boundary value and inverse problems for harmonic functions in two dimensions
31A05	Harmonic, subharmonic, superharmonic functions in two dimensions

Keywords:

global uniqueness; Dirichlet to Neumann map; inverse boundary value problem; smooth coefficient; harmonic function; isotropic conductivity; body; recovered from voltage