Archive for Rational Mechanics and Analysis

, Volume 202, Issue 1, pp 177–211

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


DOI: 10.1007/s00205-011-0413-3

Cite this article as:
Gratwick, R. & Preiss, D. Arch Rational Mech Anal (2011) 202: 177. doi:10.1007/s00205-011-0413-3


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.

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Mathematics Institute, Zeeman BuildingUniversity of WarwickCoventryUK