Extremal energies of Laplacian operator: Different configurations for steady vortices

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 deter...

Full description

Saved in:
Bibliographic Details
Published in:Journal of mathematical analysis and applications 2017-04, Vol.448 (1), p.140-155
Main Author: Mohammadi, Seyyed Abbas
Format: Article
Language:English
Subjects:
Laplacian operator
Rearrangement
Shape optimization
Steady vortices
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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 determine a nearly optimal solution which is an approximation of the optimal solution when the problems are in low contrast regime. 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 lead us to a conjecture about the location of the maximizers in the set of rearrangements of a function.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2016.09.011