Quasiconvexification of geometric integrals

Abstract

We study the existence of an integral representation for the functionalwhen µ is a positive Radon measure on R N , Ω ⊂ R N is a bounded open set, and f : M m×N → [0, +∞[ is a continuous function not necessarily convex with growth conditions of order p > 1.

Interchange of infimum and integral

“…The problem of finding an integral representation for F µ in absence of condition (d) is in general difficult. Partial results in this direction have been obtained in [4,5] in the case where µ is an Hausdorff measure concentrated over a submanifold of R N . In our context, we have Theorem 7.2.…”

“…The term "normally" refers to the definition of a normal topological space 5. With the notation C 0 c (Ω; [0, 1]) = Cc(Ω; [0, 1]).…”

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

“…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.…”

On the relaxation of variational integrals in metric Sobolev spaces

“…We also find a representation formula for L x . In the setting of euclidean measure spaces, i.e., when X is the closure of a bounded open subset of R N , such representation problems was studied, in the one hand, in the convex case in [BBS97,AHM03,CPZ03,AHM04], and, on the other hand, in the non-convex case in [Man00,Man05] when µ is a "superficial" measure restricted to a smooth manifold. Note also that the study of the lower semicontinuity of variational integrals of type (1.1) was treated in [Fra03] (see also [Moc05]).…”

“…In §2.1, Sobolev spaces with respect to a metric measure space are introduced by using the notion of "normal and tangent space" to a measure as developped in [BBS97,§2], [AHM03,§7] and [Man05,§2] (see also [Zhi96,Zhi00]) in the setting of euclidean measure spaces. In §2.2, we state the main results of the paper, i.e., Theorem 2.14 in §2.2.1 for the convex case and Theorems 2.16, 2.19 and 2.21 in §2.2.2 for the non-convex case.…”