Elastic scattering by locally rough interfaces

TL;DR

This paper establishes the first well-posedness results for elastic scattering by locally rough interfaces in 2D and 3D, introducing new identities and methods to prove existence and uniqueness of solutions.

Contribution

It provides the first well-posedness proof for elastic scattering by rough interfaces, including new identities, Green's tensor analysis, and solution existence via variational and boundary integral methods.

Findings

First well-posedness result for elastic scattering by rough interfaces.

Derived explicit Green's tensor for the problem.

Proved uniqueness and existence of solutions across all frequencies.

Abstract

In this paper, we present the first well-posedness result for elastic scattering by locally rough interfaces in both two and three dimensions. Inspired by the Helmholtz decomposition, we discover a fundamental identity for the stress vector, revealing an intrinsic relationship among the generalized stress vector, the Lame constants and certain tangential differential operators. This identity leads to two key limits for surface integrals involving scattered solutions, from which we deduce the first uniqueness result of direct problem for all frequencies. Through a detailed analysis, applying the steepest descent method, subsequently we derive the existence and uniqueness of the corresponding two-layered Green's tensor along with its explicit expression when the transmission coefficient equals 1. Finally, by leveraging properties of the Green's tensor, we establish the existence of solutions via the variational method and the boundary integral equation, thereby achieving the first well-posedness result for elastic scattering by rough interfaces.