We are concerned with the inverse boundary problem of determining anomalies associated with a semilinear elliptic equation of the form −∆u + a(x, u) = 0, where a(x, u) is a general nonlinear term that belongs to a Hölder class. It is assumed that the inhomogeneity of a(x, u) is contained in a bounded domain D in the sense that outside D, a(x, u) = λu with λ ∈ R. We establish novel unique identifiability results in several general scenarios of practical interest. These include determining the support of the inclusion (i.e. D) independent of its content (i.e. a(x, u) in D) by a single boundary measurement; and determining both D and a(x, u)|D by M boundary measurements, where M ∈ N signifies the number of unknown coefficients in a(x, u). The mathematical argument is based on microlocally characterising the singularities in the solution u induced by the geometric singularities of D.
The purpose of the paper is twofold. First, we show that partialdata transmission eigenfunctions associated with a conductive boundary condition vanish locally around a polyhedral or conic corner in R n , n = 2, 3. Second, we apply the spectral property to the geometrical inverse scattering problem of determining the shape as well as its boundary impedance parameter of a conductive scatterer, independent of its medium content, by a single far-field measurement. We establish several new unique recovery results. The results extend the relevant ones in [30] in two directions: first, we consider a more general geometric setup where both polyhedral and conic corners are investigated, whereas in [30] only polyhedral corners are concerned; second, we significantly relax the regularity assumptions in [30] which is particularly useful for the geometrical inverse problem mentioned above. We develop novel technical strategies to achieve these new results.
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