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 $-\Delta u+a(\mathbf{x},u)=0$, where $a(\mathbf{x},u)$ is a general nonlinear term that belongs to a Hölder class. It is assumed that the inhomogeneity of $f(\mathbf{x},u)$ is contained in a bounded domain $D$ in the sense that outside $D$, $a(\mathbf{x},u)=\lambda u$ with $\lambda \in \mathbb{C}$. 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. $a(\mathbf{x},u)$ in $D$) by a single boundary measurement; and determining both $D$ and $a(\mathbf{x},u){|}_D$ by $M$ boundary measurements, where $M\in \mathbb{N}$ signifies the number of unknown coefficients in $a(\mathbf{x},u)$. The mathematical argument is based on microlocally characterising the singularities in the solution $u$ induced by the geometric singularities of $D$, and does not rely on any linearisation technique.