  We consider the 2D incompressible Euler equation on a bounded simply
connected domain $\Omega$. We give sufficient conditions on the domain $\Omega$
so that for all initial vorticity $\omega_0 \in L^{\infty}(\Omega)$ the weak
solutions are unique. Our sufficient condition is slightly more general than
the condition that $\Omega$ is a $C^{1,\alpha}$ domain for some $\alpha>0$,
with its boundary belonging to $H^{3/2}(\mathbb{S}^1)$. As a corollary we prove
uniqueness for $C^{1,\alpha}$ domains for $\alpha >1/2$ and for convex domains
which are also $C^{1,\alpha}$ domains for some $\alpha >0$. Previously
uniqueness for general initial vorticity in $L^{\infty}(\Omega)$ was only known
for $C^{1,1}$ domains with possibly a finite number of acute angled corners.
The fundamental barrier to proving uniqueness below the $C^{1,1}$ regularity is
the fact that for less regular domains, the velocity near the boundary is no
longer log-Lipschitz. We overcome this barrier by defining a new change of
variable which we then use to define a novel energy functional.
