  In this paper we prove symmetry of compactly supported steady solutions of
the 2D Euler equations. Assuming that $\Omega = \{x \in \mathbb{R}^2:\ u(x)
\neq 0\}$ is an annular domain, we prove that the streamlines of the flow are
circular. We are also able to remove the topological condition on $\Omega$ if
we impose regularity and nondegeneracy assumptions on $u$ at $\partial \Omega$.
The proof uses that the corresponding stream function solves an elliptic
semilinear problem $-\Delta \phi = f(\phi)$ with $\nabla \phi=0$ at the
boundary. One of the main difficulties in our study is that $f$ is not
Lipschitz continuous near the boundary values. However, $f(\phi)$ vanishes at
the boundary values and then we can apply a local symmetry result of F. Brock
to conclude.
  In the case $\partial_{\nu} u \neq 0$ at $\partial \Omega$ this argument is
not possible. In this case we are able to use the moving plane scheme to show
symmetry, despite the possible lack of regularity of $f$. We think that such
result is interesting in its own right and will be stated and proved also for
higher dimensions. The proof requires the study of maximum principles, Hopf
lemma and Serrin corner lemma for elliptic linear operators with singular
coefficients.
