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

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

2010-02-16

arxiv.org/abs/1002.3070v2

Mathematics

Classical Analysis and ODEs

27 pages, second author added, introductory material changed, minor typos corrected, some cross-references re-formatted, some

Scientific paper

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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