Rearrangements of functions in analysis and PDE theory:
In the general area of rearrangements of functions and their applications
to variational problems in analysis and hydrodynamics, some possible
thesis topics are as follows:
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To prove stability of some known examples of steady vortices in the
plane and 3-space, using transport theory and variational problems on
isovortical surfaces;
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To establish criteria for a smooth vector-valued function to have a
polar factorisation, that is, a representation as the composition of a
gradient of a convex function with a measure-preserving map;
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To develop a theory of transport along trajectories of non-smooth vector
fields, in the context of DiPerna-Lions theory of transport equations;
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To apply minimax methods, beyond the Mountain-Pass Lemma, to prove
existence of steady states on an isovortical surface.
[G.R. Burton's homepage]