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Unique determination of inhomogeneity in an elliptic equation

We consider the inverse problem of determining the shape and location of inhomogeneity in an elliptic equation with a Neumann boundary condition where D̄⊂ Ω and χD is the characteristic function of a subdomain D. For the determination of D, we measure Dirichlet data u on ∂Ω. We prove the uniqueness in this inverse problem within some classes of subdomains of Ω .