Determining anomalies in a semilinear elliptic equation by a minimal number of measurements

Abstract

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.