Reconstruction of the Defect by the Enclosure Method for Inverse Problems of the Magnetic Schrödinger Operator

Abstract

This study is based on the paper [1]. We give the formula to extract the position and the shape of the defect D generated in the object (conductor) Ω from the observation data on the boundary ∂Ω for the magnetic Schrödinger operator by using the enclosure method proposed by Ikehata [2]. We show a reconstruction formula of the convex hull of the defect D from the observed data, assuming certain higher regularity for the potentials of the magnetic Schrödinger operator, under the Dirichlet condition or the Robin condition on the boundary ∂D in the two and three dimensional case. Let Ω ⊂ R(n = 2, 3) be a bounded domain where the boundary ∂Ω is C and let D be an open set satisfying D ⊂ Ω and Ω \ D is connected. The defect D consists of the union of disjoint bounded domains {Dj}j=1, where the boundary of D is Lipschitz continuous. First, we define the DN map for the magnetic Schrödinger equation with no defect D in Ω. Here, let D Au := ∑n j=1 DA,j(DA,ju), where DA,j := 1 i ∂j +Aj and A = (A1, A2, · · · , An). Definition 1. Suppose q ∈ L∞(Ω), q ≥ 0, A ∈ C(Ω, R). For a given f ∈ H(∂Ω), we say u ∈ H(Ω) is a weak solution to the following boundary value problem for the magnetic Schrödinger equation { D Au+ qu = 0 in Ω, u = f on ∂Ω, (1.1)