, Volume 202, Issue 1, pp 177-211
Date: 22 Mar 2011

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

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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.

Communicated by L. Ambrosio