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

TL;DR

This paper proves new results for uniquely identifying anomalies in a semilinear elliptic equation using minimal boundary measurements, without relying on linearization, by analyzing solution singularities.

Contribution

It establishes novel unique identifiability results for anomalies in semilinear elliptic equations with minimal measurements, including support and coefficient determination.

Findings

Support of the inclusion can be determined from a single measurement.

Both the support and nonlinear coefficients can be identified with multiple measurements.

The approach does not rely on linearization techniques.

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 H\"older 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.