A One-Dimensional Variational Problem with Continuous Lagrangian and Singular Minimizer
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- Gratwick, R. & Preiss, D. Arch Rational Mech Anal (2011) 202: 177. doi:10.1007/s00205-011-0413-3
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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.