  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.
