The inverse conductivity problem with one measurement: uniqueness for convex polyhedra

Abstract

Let Ω \Omega denote a smooth domain in R n {R^n} containing the closure of a convex polyhedron D. Set χ D {\chi _D} equal to the characteristic function of D. We find a flux g so that if u is the nonconstant solution of div ( ( 1 + χ D ) ∇ u ) = 0 \operatorname {div}\;((1 + {\chi _D})\nabla u) = 0 in Ω \Omega with ∂ u ∂ n = g \frac {{\partial u}}{{\partial n}} = g on ∂ Ω \partial \Omega , then D is uniquely determined by the Cauchy data g and f ≡ u / ∂ Ω f \equiv u/\partial \Omega .