Variations, Approximation, and Low Regularity in One Dimension

TL;DR

This paper explores the behavior of minimizers in one-dimensional variational problems with minimally smooth Lagrangians, revealing limitations of regularity and proposing necessary conditions under low regularity assumptions.

Contribution

It provides an elementary approximation result and constructs a counterexample to Lipschitz regularity, advancing understanding of minimizers with low regularity.

Findings

Existence of a nowhere locally Lipschitz minimizer.

Approximate continuity and one-sided derivatives are necessary conditions.

Standard Lipschitz variation cannot always be used for approximation.

Abstract

We investigate the properties of minimizers of one-dimensional variational problems when the Lagrangian has no higher smoothness than continuity. An elementary approximation result is proved, but it is shown that this cannot be in general of the form of a standard Lipschitz "variation". Part of this investigation, but of interest in its own right, is an example of a nowhere locally Lipschitz minimizer which serves as a counter-example to any putative Tonelli partial regularity statement. Under these low assumptions we find it nonetheless remains possible to derive necessary conditions for minimizers, in terms of approximate continuity and equality of the one-sided derivatives.