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arXiv · 1002.3070

A one-dimensional variational problem with continuous Lagrangian and singular minimizer

Richard Gratwick·David Preiss
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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.

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Richard Gratwick, David Preiss. 2010-02-16. A one-dimensional variational problem with continuous Lagrangian and singular minimizer. https://doi.org/10.1007/s00205-011-0413-3

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