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

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.