  In this paper, we study the vortex patch problem in an ideal fluid in a
planar bounded domain. By solving a certain minimization problem and studying
the limiting behavior of the minimizer, we prove that for any harmonic function
$q$ corresponding to a nontrivial irrotational flow, there exists a family of
steady vortex patches approaching the set of extremum points of $q$ on the
boundary of the domain. Furthermore, we show that each finite collection of
strict extreme points of $q$ corresponds to a family of steady multiple vortex
patches approaching it.

Analysis of PDEs