Revisiting the Lavrentiev Phenomenon in One Dimension

TL;DR

This paper revisits the Lavrentiev phenomenon in one dimension, clarifying previous results, correcting inconsistencies, and providing a complete, concise proof with additional details.

Contribution

It offers a corrected, simplified proof of the absence of the Lavrentiev phenomenon in one dimension, addressing issues in the original theorem and proof.

Findings

Identifies inconsistencies in the original proof

Provides a counterexample to previous claims

Supplies a new, complete proof with supplementary details

Abstract

We clarify and extend insights from Lavrentiev's seminal paper. We examine the original theorem dealing with the absence of the Lavrentiev phenomenon, a cornerstone issue in the calculus of variations. We point out some inconsistencies in the original proof by providing a counterexample and supply the result with a new, concise, and complete reasoning. In the appendix, we also provide additional details to supplement the original proof.