Continuous Essential Selections and Integral Functionals

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. Keywords: Set-valued and variational analysis Continuous selections Integral functionals Convex duality 1 µ is strictly positive if µ(O) > 0 for every nonempty open O.

Conjugates of integral functionals on continuous functions

“…We now turn to the condition C(D) = cl dom I h in Corollary 6. We say that a function y : Following [Per14], we say that a set-valued mapping S is outer µ-regular if…”

“…Our main theorems sharpen those of [Roc71] and [Per14] by giving necessary and sufficient conditions for the conjugacy of integral functionals on C and functionals on the space of measures with integral representation involving the recession function of the conjugate integrand. For closely related results, we refer to [AB88,BV88] and the references therein.…”

“…On the other hand, when dom I h ⊆ C(D), we have I h + δ C(D) = I h . Sufficient conditions for the above inclusion can be found in [Roc71], [Per14] and Section 4 below.…”

“…Rockafellar [Roc71] and more recently Perkkiö [Per14] gave conditions for the validity of the integral representation…”

Convex duality in nonlinear optimal transport

“…In this appendix we prove Theorem 9. The proof follows the arguments in [12] which in turn are based on those in [15] and [11]. We reproduce the proofs here since we allow for unbounded scaling functions ψ t and we do not assume that S is locally compact.…”