Occurrence of gap for one-dimensional scalar autonomous functionals with one end point condition

TL;DR

This paper investigates the Lavrentiev phenomenon in one-dimensional scalar variational problems, providing conditions that prevent its occurrence and illustrating with a specific example where it does occur.

Contribution

It introduces a new criterion based on the behavior of the Lagrangian on two functions' graphs that ensures the absence of the Lavrentiev phenomenon, weakening previous boundedness conditions.

Findings

Example of Lagrangian with Lavrentiev phenomenon

New criterion prevents the phenomenon under weaker conditions

Criterion depends only on behavior on two functions' graphs

Abstract

Let $L:\mathbb R\times \mathbb R\to [0, +\infty[\,\cup\{+\infty\}$ be a Borel function. We consider the problem \begin{equation}\tag{P}\min F(y)=\int_0^1L(y(t), y'(t))\,dt: y(0)=0,\, y\in W^{1,1}([0,1],\mathbb R).\end{equation} We give an example of a real valued Lagrangian $L$ for which the Lavrentiev phenomenon occurs. We state a condition, involving only the behavior of $L$ on the graph of two functions, that ensures the non-occurrence of the phenomenon. Our criterium weakens substantially the well-known condition, that $L$ is bounded on bounded sets.