A one-dimensional variational problem with continuous Lagrangian and singular minimizer

TL;DR

This paper constructs a continuous, convex, superlinear Lagrangian with a Lipschitz minimizer that is non-differentiable on a dense set, challenging assumptions in regularity theory.

Contribution

It provides a counterexample showing that continuity alone does not ensure regularity of minimizers in variational problems.

Findings

Lipschitz minimizer with dense non-differentiability

Upper and lower Dini derivatives differ on a dense set

Continuity is insufficient for Tonelli's regularity theorem

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.