Rigidity of stationary and uniformly rotating planar Euler flows in the $C^1$ Yudovich class

We prove a rigidity theorem for stationary and uniformly rotating solutions of the two-dimensional incompressible Euler equation with vorticity ${\omega }_0\in C^1({\mathbb{R}}^2)\cap L^1({\mathbb{R}}^2)\cap L^{\mathrm{\infty }}({\mathbb{R}}^2)\mathrm{.}$ If the angular velocity $\mathrm{\Omega }$ satisfies $$\mathrm{\Omega }\le \frac{1}{2}\underset{{\mathbb{R}}^2}{\inf }{\omega }_0\text{or}\mathrm{\Omega }\ge \frac{1}{2}\underset{{\mathbb{R}}^2}{\sup }{\omega }_0,$$ then ${\omega }_0$ is radially symmetric, in particular, every stationary vorticity of one sign in the class $C^1({\mathbb{R}}^2)\cap L^1({\mathbb{R}}^2)\cap L^{\mathrm{\infty }}({\mathbb{R}}^2)$ must be radially symmetric about some point. Furthermore, for $\mathrm{\Omega }\mathrm{\ne }0$, the center is necessarily the origin; The proof is based on the analysis for level-set geometry of a normalized stream function. A componentwise Bernoulli formula, together with the isoperimetric inequality and Pohozaev identities, yields a nonnegative defect measure encoding both geometric defects and topological branching. An analysis at infinity forces this measure to vanish, reducing the problem to a semilinear elliptic equation with a bounded nonnegative Borel nonlinearity. Radial symmetry then follows from a theorem of P.-L.\ Lions.