Continuous essential selections and integral functionals

TL;DR

This paper characterizes certain convex-valued mappings related to integral functionals, extending results in variational analysis and optimization, with applications to functions of bounded variation.

Contribution

It provides a new characterization of inner semicontinuous convex-valued mappings that include continuous and lower semicontinuous mappings, extending existing convex conjugate results.

Findings

Characterization of inner semicontinuous convex-valued mappings.

Extension of convex conjugate results to broader classes.

Application to integral functionals on functions of bounded variation.

Abstract

Given a strictly positive measure, we characterize inner semicontinuous solid convex-valued mappings for which continuous functions which are selections almost everywhere are selections. This class contains continuous mappings as well as fully lower semicontinuous closed convex-valued mappings that arise in variational analysis and optimization of integral functionals. The characterization allows for extending existing results on convex conjugates of integral functionals on continuous functions. We also give an application to integral functionals on left continuous functions of bounded variation.