Radial representation of lower semicontinuous envelope

TL;DR

This paper extends the radial representation of lower semicontinuous envelopes to nonconvex functions by introducing radial uniform upper semicontinuity, with applications to relaxation problems in calculus of variations.

Contribution

It introduces radial uniform upper semicontinuity as a nonconvex analogue of convexity, enabling radial representation results for a broader class of functions.

Findings

Established radial representation for nonconvex functions

Applied the theory to relaxation of gradient-constrained integral functionals

Extended classical convex analysis results to nonconvex settings

Abstract

We give an extension to a nonconvex setting of the classical radial representation result for lower semicontinuous envelope of a convex function on the boundary of its effective domain. We introduce the concept of radial uniform upper semicontinuity which plays the role of convexity, and allows to prove a radial representation result for nonconvex functions. An application to the relaxation of multiple integrals with constraints on the gradient is given.