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

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.