Interchange of infimum and integral

Abstract

We prove a new interchange theorem of infimum and integral. Its distinguishing feature is, on the one hand, to establish a general framework to deal with interchange problems for nonconvex integrands and nondecomposable sets, and, on the other hand, to link the theorems of Rockafellar and Hiai-Umegaki with the one of Bouchitté-Valadier. We give an application to relaxation of nonconvex geometric integrals of Calculus of Variations.

Quasiconvexification of geometric integrals

“…For the convenience of the reader we include the main ideas of the proof (for more details we refer to [3]). (Ω; M m×N ).…”

“…and the first inequality in(3), it is easy to see that |∇u n | p dµ < +∞ resp. lim inf n→+∞ Y |∇v n | p dµ < +∞ .…”

Relaxation of nonconvex unbounded integrals with general growth conditions in Cheeger–Sobolev spaces

“…The following result comes from Hajłasz and Koskela (see [34, (2) where C d 1 is given by the inequality (2.1). If p > κ then for every r > 0 and everȳ …”

“…Such a relaxation problem in such a metric measure setting was studied for the first time in [10] (see also [14,39,2,27,3,40,43,35] and the references therein) when L has p-growth, i.e., there exist α, β > 0 such that for every x ∈ X and every ξ ∈ M, Our motivation for developing relaxation, and more generally calculus of variations, in the setting of metric measure spaces comes from applications to hyperelasticity. In fact, the interest of considering a general measure is that its support can be interpretated as a hyperelastic structure together with its singularities like for example thin dimensions, corners, junctions, etc.…”

“…Generally speaking, in this paper our main contribution (see §2.1, §2.2 and §2.3 for more details and more precisely Theorem 2.7) is to prove that for p > κ, with κ := ln(C d ) ln (2) …”