Local uniqueness for an inverse boundary value problem with partial data

TL;DR

This paper proves a local uniqueness result for Schrödinger potentials in dimensions three and higher, showing that partial boundary data can distinguish different potentials under certain local conditions.

Contribution

It introduces a novel local uniqueness theorem for inverse boundary value problems with partial data for Schrödinger equations in higher dimensions.

Findings

Potentials with positive essential infima can be uniquely identified locally.

Partial boundary data suffices for potential distinction under specific local inequalities.

The result applies to potentials in L-infinity with certain positivity conditions.

Abstract

In dimension $n\geq 3$, we prove a local uniqueness result for the potentials $q$ of the Schr\"odinger equation $-\Delta u+qu=0$ from partial boundary data. More precisely, we show that potentials $q_1,q_2\in L^\infty$ with positive essential infima can be distinguished by local boundary data if there is a neighborhood of a boundary part where $q_1\geq q_2$ and $q_1\not\equiv q_2$.