  In this paper, we study the incompressible steady Euler flow in a
two-dimensional bounded domain, the stream function of which is the first
eigenfunction of $-\Delta$ with zero Dirichlet data. We give a new variational
characterization of the flow, based on which we are able to prove the nonlinear
stability of the flow in terms of the $L^p$ norm of the vorticity for $p\in
(1,+\infty)$. We also prove a result concerning the nonlinear stability of the
flow in the energy norm.
