Revisited convexity notions for $L^\infty$ variational problems

TL;DR

This paper explores convexity concepts related to weak* lower semicontinuity of supremal functionals and their approximation, aiming to establish foundational understanding for $L^ Infty$ variational problems.

Contribution

It provides a detailed analysis of convexity notions specific to $L^ Infty$ variational problems and their relation to approximation methods, filling a gap in the theoretical foundation.

Findings

Clarified the role of convexity in $L^ Infty$ lower semicontinuity

Connected supremal functional convexity to approximation techniques

Established groundwork for future research in $L^ Infty$ calculus of variations

Abstract

We address a deep study of the convexity notions that arise in the study of weak* lower semicontinuity of supremal functionals as well as those raised by the power-law approximation of such functionals. Our quest is motivated by the knowledge we have on the analogous integral functionals and aims at establishing a solid groundwork to ease any research in the $L^\infty$ context.