Local uniqueness in the inverse conductivity problem with one measurement

Abstract

We prove local uniqueness of a domain D D entering the conductivity equation div ( ( 1 + χ ( D ) ) ∇ u ) = 0 {\text {div}}((1 + \chi (D))\nabla u) = 0 in a bounded planar domain Ω \Omega given the Cauchy data for u u on a part of ∂ Ω \partial \Omega . The main assumption is that ∇ u \nabla u has zero index on ∂ Ω \partial \Omega which is easy to guarantee by choosing special boundary data for u u . To achieve our goals we study index of critical points of u u on ∂ Ω \partial \Omega .