University of Birmingham > Talks@bham > Analysis Seminar > Variational problems involving rearrangements of functions

Variational problems involving rearrangements of functions

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If you have a question about this talk, please contact Alessio Martini.

Joint Analysis / Applied Mathematics seminar

Two real functions on a domain in Euclidean space are said to be rearrangements of one another if they have the same decreasing rearrangement, that is, corresponding super-level sets of the two functions always have equal measure.

This concept from real analysis is also relevant to the dynamics of planar incompressible non-viscous fluids, because classically the curves of constant vorticity are convected with the flow and so the areas they enclose remain constant.

We will discuss rearrangements from an analyst’s wiewpoint, but with particular reference to problems from the Calculus of Variations where the set of all rearrangements of one fixed function forms the constraint. In fluids this corresponds to maximizing kinetic energy among “equivortical flows” and solutions represents steady vortices.

Existence of maximisers can be proved by means of convex analysis in conjunction with compactness properties of highly symmetric functions. In order to study stability one needs to prove compactness of all maximizing sequences, which may lack symmetry. This necessitates the use of concentration-compactness.

This talk is part of the Analysis Seminar series.

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