Stability for an inverse source problem of the biharmonic operator

TL;DR

This paper establishes the first stability results for an inverse source problem involving the biharmonic operator with a compactly supported potential in three dimensions, linking boundary data to the source through spectral analysis and resolvent estimates.

Contribution

It introduces a novel stability analysis for the inverse source problem of the biharmonic operator, including Weyl-type bounds and resonance-free regions, extending techniques from Schrödinger operators.

Findings

Proved Weyl-type law for eigenfunction derivatives of the bi-Schrödinger operator.

Established resonance-free regions and resolvent estimates for the operator.

Derived a stability estimate combining data discrepancy and high-frequency tail of the source.

Abstract

In this paper, we study for the first time the stability of the inverse source problem for the biharmonic operator with a compactly supported potential in $\mathbb R^3$. Firstly, to connect the boundary data with the unknown source, we shall consider an eigenvalue problem for the bi-Schr$\ddot{\rm o}$dinger operator $\Delta^2 + V(x)$ on a ball which contains the support of the potential $V$. We prove a Weyl-type law for the upper bounds of spherical normal derivatives of both the eigenfunctions $\phi$ and their Laplacian $\Delta\phi$ corresponding to the bi-Schr$\ddot{\rm o}$dinger operator. This type of upper bounds was proved by Hassell and Tao for the Schr$\ddot{\rm o}$dinger operator. Secondly, we investigate the meromorphic continuation of the resolvent of the bi-Schr$\ddot{\rm o}$dinger operator and prove the existence of a resonance-free region and an estimate of $L^2_{\rm comp} - L^2_{\rm loc}$ type for the resolvent. As an application, we prove a bound of the analytic continuation of the data from the given data to the higher frequency data. Finally, we derive the stability estimate which consists of the Lipschitz type data discrepancy and the high frequency tail of the source function, where the latter decreases as the upper bound of the frequency increases.