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Singular sets and the Lavrentiev phenomenon

Published online by Cambridge University Press:  28 September 2015

Richard Gratwick
Richard Gratwick*
Affiliation:
Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UK, ([email protected])
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Abstract

We show that non-occurrence of the Lavrentiev phenomenon does not imply that the singular set is small. Precisely, given a compact Lebesgue null subset E ⊆ ℝ and an arbitrary superlinearity, there exists a smooth strictly convex Lagrangian with this superlinear growth such that all minimizers of the associated variational problem have singular set exactly E but still admit approximation in energy by smooth functions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 145 , Issue 3 , June 2015 , pp. 513 - 533
Copyright
Copyright © Royal Society of Edinburgh 2015 

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