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On uniqueness of recovery of a discontinuous conductivity coefficient. (English) Zbl 0676.35082

The author considers three inverse problems related to elliptic equations. The first of them consists in identifying the coefficient a appearing in the Dirichlet problem

(1)div(a(x)u)=0inΩ n ;(2)u=φonΩ

under the assumption that a is a discontinuous function of the form a=a 0 +χ(Ω * )b, Ω * and b being unknown. Here Ω * is an open subset of Ω, χ(Ω * ) is the characteristic function of Ω * and the functions aC 2 (Ω ¯) and bC 2 (Ω ¯) satisfy the relations: i) 0<a 0 (x) xΩ ¯; ii)0<a 0 (x)+b(x) xΩ * ; iii) b(x)0 xΩ * ·

The author proves the uniqueness of the unknown pair (Ω * ,b) when n=3, under the assumption that the Dirichlet-Neumann map φu(φ)/N is known for any φC 1 (Ω) with Supp φ Γ 0 , Γ 0 being a neighbourhood if Ω. In the case n=2 he obtains a weaker result.

Similar uniqueness results are deduced also in the case where equation (1) is replaced by

(1bis)div(a(y) y u(x,y))=δ(x),y n ,

where δ (x) is the delta function with pole at x and a 0 is constant outside a bounded domain Ω 0 with Ω ¯ 0 Ω, Ω being a convex domain with analytic boundary.

The additional information needed to prove the uniqueness result for (Ω * ,b) is the following: u(x,·) is assigned on Γ 1 for any xΓ 2 , Γ 1 and Γ 2 being two disjoint neighbourhoods in Ω·

The third identification problem is related to the anisotropic equation

(1ter)div(A(x)u)=0inΩ,

where the matrix A(x) admits the decomposition

A(x)=A 0 (x)+χ(Ω * )B(x)xΩ ¯·

Here A 0 (x) and B(x) are known positive definite matrices xΩ ¯, while the open set Ω * is unknown. In this case the additional information is again the Dirichlet-Neumann map as in the first problem: it ensures the uniqueness of the open set Ω * .

Reviewer: A.Lorenzi

MSC:
35R30Inverse problems for PDE
35J25Second order elliptic equations, boundary value problems
35R05PDEs with discontinuous coefficients or data
35A05General existence and uniqueness theorems (PDE) (MSC2000)