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Title: On the regularity of holonomically constrained minimisers in the calculus of variations
Author: Hopper, Christopher Peter
ISNI:       0000 0004 4692 169X
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2014
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This thesis concerns the regularity of holonomic minimisers of variational integrals in the context of direct methods in the calculus of variations. Specifically, we consider Sobolev mappings from a bounded domain into a connected compact Riemannian manifold without boundary, to which such mappings are said to be holonomically constrained. For a general class of strictly quasiconvex integral functionals, we give a direct proof of local C1,α-Hölder continuity, for some 0 < α < 1, of holonomic minimisers off a relatively closed 'singular set' of Lebesgue measure zero. Crucially, the proof constructs comparison maps using the universal covering of the target manifold, the lifting of Sobolev mappings to the covering space and the connectedness of the covering space. A certain tangential A-harmonic approximation lemma obtained directly using a Lipschitz approximation argument is also given. In the context of holonomic minimisers of regular variational integrals, we also provide bounds on the Hausdorff dimension of the singular set by generalising a variational difference quotient method to the holonomically constrained case with critical growth. The results are analogous to energy-minimising harmonic maps into compact manifolds, however in this case the proof does not use a monotonicity formula. We discuss several applications to variational problems in condensed matter physics, in particular those concerning the superfluidity of liquid helium-3 and nematic liquid crystals. In these problems, the class of mappings are constrained to an orbit of 'broken symmetries' or 'manifold of internal states', which correspond to a sub-group of residual symmetries.
Supervisor: Kristensen, Jan Sponsor: Engineering & Physical Sciences Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Mathematics ; Partial differential equations ; Calculus of variations and optimal control ; holonomic ; quasiconvexity ; variational methods ; regularity ; harmonic maps