Multi-parameter analysis of the obstacle scattering problem

TL;DR

This paper investigates how the acoustic scattering problem's solution and far field pattern depend analytically on obstacle shape, wave number, and boundary data, providing insights into parameter sensitivities in obstacle scattering.

Contribution

It establishes the real analyticity of the solution, far field pattern, and Dirichlet-to-Neumann map with respect to key parameters in obstacle scattering problems.

Findings

Solution and far field pattern depend real analytically on parameters

Analytic dependence established for obstacle shape, wave number, and boundary data

Results apply to Dirichlet-to-Neumann map in scattering analysis

Abstract

We consider the acoustic field scattered by a bounded impenetrable obstacle and we study its dependence upon a certain set of parameters. As usual, the problem is modeled by an exterior Dirichlet problem for the Helmholtz equation $\Delta u +k^2u=0$. We show that the solution $u$ and its far field pattern $u_\infty$ depend real analytically on the shape of the obstacle, the wave number $k$, and the Dirichlet datum. We also prove a similar result for the corresponding Dirichlet-to-Neumann map.