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Title: Rearrangements and vortices
Author: Masters, Anthony
ISNI:       0000 0004 5357 3769
Awarding Body: University of Bath
Current Institution: University of Bath
Date of Award: 2014
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Rearrangements are two measurable real-valued functions that have equal measure of pre-images of upper level sets. In this thesis, I will investigate several matters and problems relating to rearrangements: the relationship between assumptions on the measure space and desirable properties of the set of rearrangements, and the validity of rearrangement inequalities; generalising the Mountain Pass Lemma over rearrangements; and applying topological degree theory to boundary value problems involving rearrangements. From suppositions on the measure space, such as the measure space having finite measure and no atoms, it can proved that the set of rearrangements is contractible and locally contractible. The Mountain Pass Lemma over rearrangements can be generalised, so instead of considering continuous paths from the closed unit interval to the set of rearrangements; it will consider the continuous functions from the closed unit disc into the set of rearrangements. Topological degree theory is used to associate admissible triples of functions, sets and points with integers. These methods will be applied to a boundary value problem involving rearrangements, where the domain is almost equal to the union of balls, which has been studied using variational methods, providing new multiplicity results. The minimum number of solutions to this boundary value problem is found to be related exponentially to the number of balls contained in the domain.
Supervisor: Burton, Geoffrey Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Rearrangement ; Fluid dynamics ; Degree Theory ; Fixed Points ; Vorticity