  We consider the 2D incompressible Euler equation on a corner domain $\Omega$
with angle $\nu\pi$ with $\frac{1}{2}<\nu<1$. We prove that if the initial
vorticity $\omega_0 \in L^{1}(\Omega)\cap L^{\infty}(\Omega)$ and if $\omega_0$
is non-negative and supported on one side of the angle bisector of the domain,
then the weak solutions are unique. This is the first result which proves
uniqueness when the velocity is far from Lipschitz and the initial vorticity is
nontrivial around the boundary.
