Determining the first order perturbation of a bi-harmonic operator on bounded and unbounded domains from partial data

TL;DR

This paper establishes the unique determination of magnetic and electric potentials in a bi-harmonic operator from partial boundary data, applicable to both bounded and unbounded domains, advancing inverse boundary value problem theory.

Contribution

It provides new uniqueness results for inverse boundary problems involving bi-harmonic operators with first order perturbations, using partial data in both slab and bounded domain settings.

Findings

Unique determination of potentials in slabs from partial data.

Unique determination of potentials in bounded domains under boundary neighborhood assumptions.

Results applicable to partial boundary measurements in hyperplane regions.

Abstract

In this paper we study inverse boundary value problems with partial data for the bi-harmonic operator with first order perturbation. We consider two types of subsets of $\mathbb{R}^{n}(n\geq 3)$, one is an infinite slab, the other is a bounded domain. In the case of a slab, we show that, from Dirichlet and Neumann data given either on the different boundary hyperplanes of the slab or on the same boundary hyperplane, one can uniquely determine the magnetic potential and the electric potential. In the case of a bounded domain, we show the unique determination of the magnetic potential and the electric potential from partial Dirichlet and Neumann data under two different assumptions. The first assumption is that the magnetic and electric potentials are known in a neighborhood of the boundary, in this situation we obtain the uniqueness result when the Dirichlet and Neumann data are only given on two arbitrary open subsets of the boundary. The second assumption is that the Dirichlet and Neumann data are known on the same part of the boundary whose complement is a part of a hyperplane, we also establish the unique determination result in this local data case.