Determining a first order perturbation of the biharmonic operator by partial boundary measurements

TL;DR

This paper proves that a first order perturbation of the biharmonic operator can be uniquely identified using boundary measurements, even on small boundary subsets, highlighting a difference from the Laplacian case.

Contribution

It establishes unique determination of first order perturbations of the biharmonic operator from boundary data, a result not generally true for the Laplacian.

Findings

Unique determination of first order perturbations from boundary measurements.

The result holds even with measurements on small boundary subsets.

Contrasts with the Laplacian case where such determination does not always hold.

Abstract

We consider an operator $\Delta^2 + A(x)\cdot D+q(x)$ with the Navier boundary conditions on a bounded domain in $R^n$, $n\ge 3$. We show that a first order perturbation $A(x)\cdot D+q$ can be determined uniquely by measuring the Dirichlet--to--Neumann map on possibly very small subsets of the boundary of the domain. Notice that the corresponding result does not hold in general for a first order perturbation of the Laplacian.