  In this paper, we investigate steady Euler flows in a two-dimensional bounded
domain. By an adaption of the vorticity method, we prove that for any
nonconstant harmonic function $q$, which corresponds to a nontrivial
irrotational flow, there exists a family of steady Euler flows with small
circulation in which the vorticity is continuous and supported in a small
neighborhood of the set of maximum points of $q$ near the boundary, and the
corresponding stream function satisfies a semilinear elliptic equation with a
given profile function. Moreover, if $q$ has $k$ isolated maximum points
$\{\bar{x}_1,\cdot\cdot\cdot,\bar{x}_k\}$ on the boundary, we show that there
exists a family of steady Euler flows whose vorticity is continuous and
supported in $k$ disjoint regions of small diameter, and each of them is
contained in a small neighborhood of $\bar{x}_i$, and in each of these small
regions the stream function satisfies a semilinear elliptic equation with a
given profile function.
