Stable determination of polygonal inclusions in Calder\'on's problem by a single partial boundary measurement

TL;DR

This paper proves that the shape and location of polygonal inclusions inside a body can be uniquely identified and stably estimated from a single partial boundary measurement, advancing inverse boundary value problem theory.

Contribution

It introduces a logarithmic stability estimate and a uniqueness result for polygonal inclusions in Calderón's problem using minimal boundary data.

Findings

Logarithmic stability estimate for polygonal inclusions

Uniqueness of support determination in nested polygonal geometries

Potential applicability to other inverse boundary problems

Abstract

We are concerned with the Calder\'on problem of determining an unknown conductivity of a body from the associated boundary measurement. We establish a logarithmic type stability estimate in terms of the Hausdorff distance in determining the support of a convex polygonal inclusion by a single partial boundary measurement. We also derive the uniqueness result in a more general scenario where the conductivities are piecewise constants supported in a nested polygonal geometry. Our methods in establishing the stability and uniqueness results have a significant technical initiative and a strong potential to apply to other inverse boundary value problems.