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Title: Variational problems : perturbations and optimal sets
Author: Iversen , Mette
Awarding Body: University of Bristol
Current Institution: University of Bristol
Date of Award: 2012
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The calculus of variations looks in general at finding critical points of a given functional, and shape optimisation is specifically concerned with variational problems, where the unknown is the set on which the problem is defined. This thesis attempts to navigate through the topic of optimal sets, drawing on both the early beginnings of such investigations, and the work of the very active current research community. The behaviour under perturbations of quantities given by variational characterisations is a recurring theme. The starting point is a renewed look at the classical question of minimising individual eigenvalues of the Dirichlet. Laplacian over domains in Rn subject to geometric constraints, with particular focus on bounding the number of components of optimal sets. The results obtained give rise to questions concerning analogous results for minimisers of functions of eigenvalues. This thesis looks specifically at the corresponding properties for convex combinations of the first three eigenvalues, a study which in turn feeds back interesting information on the optimal sets for individual eigenvalues. Research on eigenvalue optimisation leads naturally to the study of the behaviour of the eigenvalues themselves and various other quantities under perturbations of a given domain. This thesis includes an investigation into bounds on the spectra of sets in terms of the principal frequency of their difference. Finally, a quantity of interest besides the eigenvalues is the torsional rigidity, and we conclude with an estimate of its behaviour under perturbations of a ball in Euclidean space Rn.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available