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

We consider corner scattering for the operator $\nabla \cdot γ(x)\nabla +k^2ρ(x)$ in $\mathbb{R}^2$, with $γ$ a positive definite symmetric matrix and $ρ$ a positive scalar function. A corner is referred to one that is on the boundary of the (compact) support of $γ(x)-I$ or $ρ(x)-1$, where $I$ stands for the identity matrix. We assume that $γ$ 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 $γ-I$ has a jump across the corner. In order to prove the results, we construct a new type of complex geometric optics (CGO) solutions.

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