A New Type of CGO Solutions and its Applications in Corner Scattering

TL;DR

This paper introduces a novel class of complex geometric optics solutions to analyze corner scattering phenomena in a specific PDE, demonstrating that corners on the boundary of material supports always scatter incident waves and discussing non-scattering scenarios.

Contribution

The paper develops a new type of CGO solutions tailored for corner scattering problems, advancing the understanding of wave behavior in piecewise smooth media.

Findings

Corners on the boundary always scatter incident waves.

The new CGO solutions facilitate analysis of corner scattering.

Discussion on conditions for non-scattering waves.

Abstract

We consider corner scattering for the operator $\nabla \cdot \gamma(x)\nabla +k^2\rho(x)$ in $\mathbb{R}^2$, with $\gamma$ a positive definite symmetric matrix and $\rho$ a positive scalar function. A corner is referred to one that is on the boundary of the (compact) support of $\gamma(x)-I$ or $\rho(x)-1$, where $I$ stands for the identity matrix. We assume that $\gamma$ is a scalar function in a small neighborhood of the corner. We show that any admissible incident field will be scattered by such corners, which are allowed to be concave. Moreover, we provide a brief discussion on the existence of non-scattering waves when $\gamma-I$ has a jump across the corner. In order to prove the results, we construct a new type of complex geometric optics (CGO) solutions.