Stability for an inverse source problem of the biharmonic operator

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 R 3 . Firstly, to connect the boundary data with the unknown source, we shall consider an eigenvalue problem for the bi-Schr: odinger operator ∆ 2 `V pxq 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 φ and their Laplacian ∆φ corresponding to the bi-Schr: odinger operator. This type of upper bounds was proved by Hassell and Tao for the Schr: odinger operator. Secondly, we investigate the meromorphic continuation of the resolvent of the bi-Schr: odinger operator and prove the existence of a resonance-free region and an estimate of L 2 comp ´L2loc 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.

An inverse random source problem for the biharmonic wave equation

“…Proof. The uniqueness can be proved similarly to the deterministic case given in [21]. It then suffices to show the existence and regularity of the solution.…”

“…The reason is not only the increase of the order which leads to the failure of the methods developed for the second order equations, but also the properties of the solutions for the higher order equations are more sophisticated. Some of the inverse boundary value problems for bi-and poly-harmonic operators can be found in [8,12,13,21,[26][27][28].…”

Direct and inverse problems for the Schrödinger operator in a three-dimensional planar waveguide

“…The first stability result was obtained in [4] for the inverse source problem of the Helmholtz equation by using multi-frequency data. Later on, the increasing stability was studied for the inverse source problems of various wave equations including the acoustic, elastic, and electromagnetic wave equations, as well as the wave equation with the biharmonic Schrödinger operator [5,12,17,18,[22][23][24]. A more recent study on the stability for the inverse medium problem can be found in [6].…”

“…Motivated by [22,24], we consider an eigenvalue problem for the Schrödinger operator in Ω and deduce integral equations, which connect the scattering data u(x, κ)| Γ and the unknown source function f . It is required to obtain a bound of the analytic continuation of the data from the given data to the higher frequency data in order to show the stability.…”

Stability for an inverse source problem of the damped biharmonic plate equation

“…This paper is concerned with the inverse source problem of determining f from the boundary measurements upx, kq, ∇upx, kq, ∆upx, kq, ∇∆upx, kq, x P BB R corresponding to the wavenumber k given in a finite interval. In general, there is no uniqueness for the inverse source problems of the wave equations at a fixed frequency [2,12]. Computationally, a more serious issue is the lack of stability, i.e., a small variation of the data might lead to a huge error in the reconstruction.…”