A one-dimensional variational problem with continuous Lagrangian and singular minimizer
Classical Analysis and ODEs
2015-05-18 v2
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.
Cite
@article{arxiv.1002.3070,
title = {A one-dimensional variational problem with continuous Lagrangian and singular minimizer},
author = {Richard Gratwick and David Preiss},
journal= {arXiv preprint arXiv:1002.3070},
year = {2015}
}
Comments
27 pages, second author added, introductory material changed, minor typos corrected, some cross-references re-formatted, some references added