A One-Dimensional Variational Problem with Continuous Lagrangian and Singular Minimizer

Richard Gratwick*, David Preiss

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We construct a continuous Lagrangian, strictly convex and superlinear in the third variable, such that the associated variational problem has a Lipschitz minimizer which is non-differentiable on a dense set. More precisely, the upper and lower Dini derivatives of the minimizer differ by a constant on a dense (hence second category) set. In particular, we show that mere continuity is an insufficient smoothness assumption for Tonelli's partial regularity theorem.

Original languageEnglish
Pages (from-to)177-211
Number of pages35
JournalArchive for Rational Mechanics and Analysis
Volume202
Issue number1
Early online date22 Mar 2011
DOIs
Publication statusPublished - 31 Oct 2011

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