  In this paper, we study a maximization and a minimization problem associated
with a Poisson boundary value problem. Optimal solutions in a set of
rearrangements of a given function define stationary and stable flows of an
ideal fluid in two dimensions. The main contribution of this paper is to
determine the optimal solutions. At first, we derive the solutions analytically
when the problems are in low contrast regime. Moreover, it is established that
the solutions of both problems are unique. Secondly, for the high contrast
regime, two optimization algorithms are developed. For the minimization
problem, we prove that our algorithm converges to the global minimizer
regardless of the initializer. The maximization algorithm is capable of
deriving all local maximizers including the global one. Numerical experiments
leads us to a conjecture about the location of the maximizers in the set of
rearrangements of a function.
