Unique Localization of Unknown Boundaries in a Conducting Medium from Boundary Measurements

Laboratoire de Mathématiques Appliquées, UMR 7641, Université de Versailles, 45 avenue des États-Unis, 78035 Versailles Cedex, France; canuto@math.uvsq.fr.

Received: 3 March 2000
Revised: 12 February 2001

Abstract

We consider the problem of localizing an inaccessible piece I of the boundary of a conducting medium Ω, and a cavity D contained in Ω, from boundary measurements on the accessible part A of ∂Ω. Assuming that g(t,σ) is the given thermal flux for (t,σ) ∈ (0,T) x A, and that the corresponding output datum is the temperature u(T₀,σ) measured at a given time T₀ for σ ∈ A_out ⊂ A, we prove that I and D are uniquely localized from knowledge of all possible pairs of input-output data . The same result holds when a mean value of the temperature is measured over a small interval of time.