Rigidity for homogeneous solutions to the two-dimensional Euler equations in sector-type domains

We study the rigidity problem for $(-\alpha )$-homogeneous solutions to the two-dimensional incompressible stationary Euler equations in sector-type domains ${\mathrm{\Omega }}_{a,b,{\theta }_0}:=\{(r,\theta ):a<r<b,0<\theta <{\theta }_0\}$, where $\alpha \in \mathbb{R}$, $0⩽a<b⩽+\mathrm{\infty }$ and $0<{\theta }_0⩽2\pi$. For each type of domains, depending on whether $a=0$ or $a>0$, and $b=+\mathrm{\infty }$ or $b<+\mathrm{\infty }$, we show that if a solution satisfies some homogeneity assumptions on the boundary of ${\mathrm{\Omega }}_{a,b,{\theta }_0}$ and if the radial or angular component of the velocity does not vanish in $\overline{{\mathrm{\Omega }}_{a,b,{\theta }_0}}\setminus \{\mathbf{0}\}$, then it must be homogeneous throughout $\overline{{\mathrm{\Omega }}_{a,b,{\theta }_0}}\setminus \{\mathbf{0}\}$.