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.
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Communicated by L. Ambrosio
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Gratwick, R., Preiss, D. A One-Dimensional Variational Problem with Continuous Lagrangian and Singular Minimizer. Arch Rational Mech Anal 202, 177–211 (2011). https://doi.org/10.1007/s00205-011-0413-3
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DOI: https://doi.org/10.1007/s00205-011-0413-3