Maximisation and minimisation on classes of rearrangements

Abstract

SynopsisMaximisation and minimisation of the Dirichlet integral of a function vanishing on the boundary of a bounded domain are studied, subject to the constraint that the Laplacean be a rearrangement of a given function. When the Laplacean is two-signed, non-existence of minimisers is proved, and some information on the limits of minimising sequences obtained; this contrasts with the known existence of minimisers in the one-signed case. When the domain is a ball and the Laplacean is one-signed, maximisers and minimisers are shown to be radial and monotone. Existence of maximisers is proved subject additionally to a finite number of linear constraints, with particular reference to ideal fluid flows of prescribed angular momentum in a disc.