Singular Sets and the Lavrentiev Phenomenon

TL;DR

This paper demonstrates that the absence of the Lavrentiev phenomenon does not necessarily mean the singular set is small, by constructing specific examples with prescribed singular sets.

Contribution

It constructs smooth, convex Lagrangians with prescribed singular sets where minimizers exhibit specific singularities, challenging previous assumptions.

Findings

Singular sets can be large despite no Lavrentiev phenomenon

Minimizers can have singular sets exactly equal to any null set

Approximation by smooth functions is still possible despite singularities

Abstract

We show that non-occurrence of the Lavrentiev phenomenon does not imply that the singular set is small. Precisely, given a compact Lebesgue null subset of the line $E$ and an arbitrary superlinearity, there exists a smooth, strictly convex Lagrangian with this superlinear growth, such that all minimizers of the associated variational problem have singular set exactly $E$, but still admit approximation in energy by smooth functions.