Extremal energies of Laplacian operator: Different configurations for steady vortices

Abstract

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.