On the trivial solutions for the rotating patch model

In this paper we study the clockwise simply connected rotating patches for Euler equations. By using the moving plane method we prove that Rankine vortices are the only solutions to this problem in the class of {\it slightly} convex domains. We discuss in the second part of the paper the case where the angular velocity $\Omega=\frac12$ and we show without any geometric condition that the set of the V-states is trivial and reduced to the Rankine vortices.