Inverse conductivity problem with one measurement: Error estimates and approximate identification for perturbed disks

This paper studies the global uniqueness and stability questions of the inverse conductivity problem to determine the unknown object D entering div((1 + (k - 1)_XD)∇u) = 0 in Ω and ∂u/∂v = g on ∂Ω from the boundary measurement ∧(g) = u|∂Ω. The results of this paper are fourfold. We first obtain a Hölder stability estimate for disks. Second, a uniform stability estimate for the direct problem is obtained. Third, we obtain the stability estimates \D\ \D̄2| + |D2 \D̄1| ≤ C(∥∧_D1(g) - ∧_D2(g)∥^α_L∞ _(∂Ω) + ∈) for some α > 0 when g satisfies some condition if D₁ and D₂ are ∈-perturbations of two disks. We then drop the condition on g and show that if ∧_D1 (g) = ∧_D2 (g) on ∂Ω, then the two domains must be very close.