Scattering problems from slightly perturbed periodic surfaces: Part I. regularity of the Bloch transformed fields

TL;DR

This paper investigates the regularity of solutions to scattering problems from slightly perturbed periodic surfaces using the Bloch transform and perturbation theory, establishing high regularity results under certain conditions.

Contribution

It introduces a novel analysis combining Bloch transform and perturbation theory to prove high regularity of solutions for perturbed periodic surface scattering problems.

Findings

Solution regularity depends on the right-hand side space.

The problem can be transformed into a modified problem with high regularity.

Established a relationship between Dirichlet-to-Neumann maps for perturbed surfaces.

Abstract

Rough surface scattering problems are always very challenging both theoretically and numerically. In this paper, we adopt the Bloch transform and the perturbation theory to investigate a special case, i.e., when the rough surface is a slight perturbation of a periodic one. Based on known results from the Bloch transform, the problem could be written into an equivalent bounded variational problem in higher dimensional spaces. The first step is to consider the non-perturbed problem. From the perturbation theory, the solution could be written as a Neumann series. The second step is to consider the slightly perturbed problems. The key point in the process is the relationship between the Dirichlet-to-Neumann map. With the study between these two operators, it is proved that when the right hand side belongs to a certain space, the problem is equivalent to a modified one. Thus a high regularity result is obtained.