Singular minimizers in the calculus of variations
Gratwick, Richard (2011) Singular minimizers in the calculus of variations. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b2569219~S1
This thesis examines the possible failure of regularity for minimizers of onedimensional
variational problems. The direct method of the calculus of variations
gives rigorous assurance that minimizers exist, but necessarily admits the possibility
that minimizers might not be smooth. Regularity theory seeks to assert some extra
smoothness of minimizers.
Tonelli's partial regularity theorem states that any absolutely continuous
minimizer has a (possibly infinite) classical derivative everywhere, and this derivative
is continuous as a function into the extended real line. We examine the limits of
this theorem. We find an example of a reasonable problem where partial regularity
fails, and examples where partial regularity holds, but the infinite derivatives of
minimizers permitted by the theorem occur very often, in precise senses.
We construct continuous Lagrangians, strictly convex and superlinear in the
third variable, such that the associated variational problems have minimizers nondifferentiable on dense second category sets. Thus mere continuity is an insufficient
smoothness assumption for Tonelli's partial regularity theorem.
Davie showed that any compact null set can occur as the singular set of
a minimizer to a problem given via a smooth Lagrangian with quadratic growth.
The proof relies on enforcing the occurrence of the Lavrentiev phenomenon. We
give a new proof of the result, but constructing also a Lagrangian with arbitrary
superlinear growth, and in which the Lavrentiev phenomenon does not occur in the
Universal singular sets record how often a given Lagrangian can have minimizers
with infinite derivative. Despite being negligible in terms of both topology
and category, they can have dimension two: any compact purely unrectifiable set
can lie inside the universal singular set of a Lagrangian with arbitrary superlinearity.
We show this also to be true of Fσ purely unrectifiable sets, suggesting a possible
characterization of universal singular sets.
|Item Type:||Thesis or Dissertation (PhD)|
|Subjects:||Q Science > QA Mathematics|
|Library of Congress Subject Headings (LCSH):||Calculus of variations|
|Official Date:||April 2011|
|Institution:||University of Warwick|
|Theses Department:||Mathematics Institute|
|Extent:||v, 147 leaves|
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