On Calder\'on's inverse inclusion problem with smooth shapes by a single partial boundary measurement

TL;DR

This paper proves that high-curvature conditions, instead of corner singularities, enable the unique determination of smooth-shaped conductive inclusions within a homogeneous medium using only a single partial boundary measurement.

Contribution

It introduces a novel local uniqueness result for Calderón's inverse problem, relaxing corner singularity requirements to high-curvature conditions for smooth shapes.

Findings

High-curvature conditions suffice for uniqueness

First local uniqueness result for smooth shapes with one boundary measurement

Advances understanding of inverse problems with limited data

Abstract

We are concerned with the Calder\'on inverse inclusion problem, where one intends to recover the shape of an inhomogeneous conductive inclusion embedded in a homogeneous conductivity by the associated boundary measurements. We consider the highly challenging case with a single partial boundary measurement, which constitutes a long-standing open problem in the literature. It is shown in several existing works that corner singularities can help to resolve the uniqueness and stability issues for this inverse problem. In this paper, we show that the corner singularity can be relaxed to be a certain high-curvature condition and derive a novel local unique determination result. To our best knowledge, this is the first (local) uniqueness result in determining a conductive inclusions with general smooth shapes by a single (partial) boundary measurement.