Shape differentiability of Helmholtz scattering problems via explicit shape calculus

TL;DR

This paper investigates how small shape perturbations affect solutions to Helmholtz scattering problems, providing sharp differentiability results using explicit shape calculus and boundary value problem analysis.

Contribution

It extends explicit shape calculus to Helmholtz scattering, deriving new differentiability results for solutions under various boundary conditions.

Findings

Sharp differentiability results for domain-to-solution mappings

Application to both classic and limit Sobolev regularity cases

Comprehensive analysis of shape sensitivity in scattering problems

Abstract

We consider the time-harmonic scalar wave scattering problems with Dirichlet, Neumann, impedance and transmission boundary conditions. Under this setting, we analyze how sensitive diffracted fields and Cauchy data are to small perturbations of a given nominal shape. To this end, we follow [K. Eppler, Int.~J.~Appl. Math. Comput. Sci. 10(3) (2000), pp. 487-516], referred to as explicit shape calculus, and which places great emphasis on the characterization of the domain derivatives as boundary value problems. It consists in combining elliptic regularity theorems, shape calculus and functional analysis, allowing to deduce sharp differentiability results for the domain-to-solution and domain-to-Cauchy data mappings. The technique is applied to both classic and limit Sobolev regularity cases, leading to new and comprehensive differentiability results.