On the Fr\'echet derivative in elastic obstacle scattering

TL;DR

This paper studies the Fréchet derivatives of solutions to elastic obstacle scattering problems, providing new proofs and extending results for different boundary conditions using a factorization technique.

Contribution

It introduces new characterizations of Fréchet derivatives in elastic scattering and extends differentiability results to Neumann and impedance boundary conditions.

Findings

Established existence of Fréchet derivatives for elastic scattering solutions.

Provided an alternative proof for Dirichlet boundary condition differentiability.

Proved new differentiability results for Neumann and impedance boundary conditions.

Abstract

In this paper, we investigate the existence and characterizations of the Fr\'echet derivatives of the solution to time-harmonic elastic scattering problems with respect to the boundary of the obstacle. Our analysis is based on a technique - the factorization of the difference of the far-field pattern for two different scatterers - introduced by Kress and Pa\"ivarinta to establish Fr\'echet differentiability in acoustic scattering. For the Dirichlet boundary condition an alternative proof of a differentiability result due to Charalambopoulos is provided and new results are proven for the Neumann and impedance exterior boundary value problems.