  We consider concentrated vorticities for the Euler equation on a smooth
domain $\Omega \subset \mathbf{R}^2$ in the form of \[ \omega = \sum_{j=1}^N
\omega_j \chi_{\Omega_j}, \quad |\Omega_j| = \pi r_j^2, \quad \int_{\Omega_j}
\omega_j d\mu =\mu_j \ne 0, \] supported on well-separated vortical domains
$\Omega_j$, $j=1, \ldots, N$, of small diameters $O(r_j)$. A conformal mapping
framework is set up to study this free boundary problem with $\Omega_j$ being
part of unknowns. For any given vorticities $\mu_1, \ldots, \mu_N$ and small
$r_1, \ldots, r_N\in \mathbf{R}^+$, through a perturbation approach, we obtain
such piecewise constant steady vortex patches as well as piecewise smooth
Lipschitz steady vorticities, both concentrated near non-degenerate critical
configurations of the Kirchhoff-Routh Hamiltonian function. When vortex patch
evolution is considered as the boundary dynamics of $\partial \Omega_j$,
through an invariant subspace decomposition, it is also proved that the
spectral/linear stability of such steady vortex patches is largely determined
by that of the $2N$-dimensional linearized point vortex dynamics, while the
motion is highly oscillatory in the $2N$-codim directions corresponding to the
vortical domain shapes.
