  In this paper, we study the radial symmetry properties of stationary and
uniformly-rotating solutions of the 2D Euler and gSQG equations, both in the
smooth setting and the patch setting. For the 2D Euler equation, we show that
any smooth stationary solution with compactly supported and nonnegative
vorticity must be radial, without any assumptions on the connectedness of the
support or the level sets. In the patch setting, for the 2D Euler equation we
show that every uniformly-rotating patch $D$ with angular velocity $\Omega \leq
0$ or $\Omega\geq \frac{1}{2}$ must be radial, where both bounds are sharp. For
the gSQG equation we obtain a similar symmetry result for $\Omega\leq 0$ or
$\Omega\geq \Omega_\alpha$ (with the bounds being sharp), under the additional
assumption that the patch is simply-connected. These results settle several
open questions in [T. Hmidi, J. Evol. Equ., 15(4): 801-816, 2015] and [F. de la
Hoz, Z. Hassainia, T. Hmidi, and J. Mateu, Anal. PDE, 9(7):1609-1670, 2016] on
uniformly-rotating patches. Along the way, we close a question on
overdetermined problems for the fractional Laplacian [R. Choksi, R. Neumayer,
and I. Topaloglu, Arxiv preprint arXiv:1810.08304, 2018, Remark 1.4], which may
be of independent interest. The main new ideas come from a calculus of
variations point of view.
