Optimal stability estimate in the inverse boundary value problem for periodic potentials with partial data

TL;DR

This paper establishes a logarithmic stability estimate for the inverse boundary value problem involving periodic potentials in an infinite domain, using partial boundary data, advancing understanding of inverse problems with partial measurements.

Contribution

It provides the first optimal stability estimate for the inverse boundary value problem with periodic potentials and partial boundary data in an infinite domain.

Findings

Log-type stability estimate proven for the inverse problem.

Applicable to domains with flat or spherical boundary portions.

Advances stability analysis in inverse boundary value problems with partial data.

Abstract

We consider the inverse boundary value problem for operators of the form $-\triangle+q$ in an infinite domain $\Omega=\mathbb{R}\times\omega\subset\mathbb{R}^{1+n}$, $n\geq3$, with a periodic potential $q$. For Dirichlet-to-Neumann data localized on a portion of the boundary of the form $\Gamma_1=\mathbb{R}\times\gamma_1$, with $\gamma_1$ being the complement either of a flat or spherical portion of $\partial\omega$, we prove that a log-type stability estimate holds.