Direct and inverse scattering problems for quasi-linear biharmonic operator in 3D

• Main Author: Kultima, J. (Jaakko)
• Format: Dissertation
• Language: English
• Published: University of Oulu 2019
• Online Access: http://jultika.oulu.fi/Record/nbnfioulu-201908272824

Abstract. We consider direct and inverse scattering problems for three-dimensional biharmonic operator \(Hu = ∆^2u + Vu\), where \(∆\) is the Laplacian and \(V\) is a scalar valued perturbation. The scattering problem for this operator is given as a partial differential equation \(Hu = k^4u\), with a parameter \(k\). In the direct scattering problem, our goal is to find the solution \(u\) while the perturbation (V\) is known. We also assume that the solution \(u\) can be written as a sum of two functions \(u_{0}\) and \(u_{sc}\), where \(u_{0}\) is a plane wave and \(u_{sc}\) is an outgoing wave in the sense that it satisfies to the Sommerfeld radiation conditions at the infinity. Our approach in this text is to first modify the partial differential equation into an integral equation by using the fundamental solution. Next, we show that this integral equation is solvable, and it has a unique solution. Finally, we prove two main results of this text; an asymptotic formula for the solution with large values of \(x ∈ \mathbb{R}^3\) and Saito’s formula. The asymptotic behaviour of the solution leads us to defining the scattering amplitude. In the inverse scattering problem, the goal is to gather some information about the unknown perturbation V while the behaviour of the function u is known. With Saito’s formula we obtain two corollaries regarding the inverse scattering problem, namely uniqueness and a representation formula for the function \(V(x, 1)\), when the scattering amplitude is known. We end the text by first defining the inverse Born approximation for both full scattering data and backscattering data. We also discuss some results that have been obtained previously with this approach.