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  1. Abstract We consider the 2D incompressible Euler equation on a corner domain Ω with angle νπ with 1 2 < ν < 1 . We prove that if the initial vorticity ω 0 ∈ L 1 (Ω) ∩ L ∞ (Ω) and if ω 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 non-constant around the boundary. 
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