Inverse problems for semilinear elliptic PDE with measurements at a single point

TL;DR

This paper proves that the potential in a semilinear elliptic PDE can be uniquely identified from boundary measurements taken at a single point or a small boundary subset, even on Riemannian manifolds.

Contribution

It establishes new uniqueness results for inverse boundary value problems with minimal measurement data, including single-point and partial boundary measurements.

Findings

Unique determination of potential from single-point boundary data

Validity of results on Riemannian manifolds

Applicable with measurements on small boundary subsets

Abstract

We consider the inverse problem of determining a potential in a semilinear elliptic equation from the knowledge of the Dirichlet-to-Neumann map. For bounded Euclidean domains we prove that the potential is uniquely determined by the Dirichlet-to-Neumann map measured at a single boundary point, or integrated against a fixed measure. This result is valid even when the Dirichlet data is only given on a small subset of the boundary. We also give related uniqueness results on Riemannian manifolds.