Existence results for non-coercive variational problems

TL;DR

This paper establishes existence results for a class of one-dimensional, non-convex, non-coercive variational problems using convex analysis tools, expanding the theoretical understanding of such problems.

Contribution

It provides a new existence theorem for non-coercive, non-convex variational problems in one dimension, leveraging convex hull closure techniques.

Findings

Existence of solutions under non-coercive, non-convex conditions

Use of convex hull closure of epigraphs in proofs

Extension of convex case results to more general problems

Abstract

The aim of this paper is to give an existence result for a class of one-dimensional, non-convex, non-coercive problems in the Calculus of Variations. The main tools for the proof are an existence theorem in the convex case and the closure of the convex hull of the epigraph of functions strictly convex at infinity.