Relaxation and Integral Representation for Functionals of Linear Growth on Metric Measure spaces

Abstract

Abstract:This article studies an integral representation of functionals of linear growth on metric measure spaces with a doubling measure and a Poincaré inequality. Such a functional is de ned via relaxation, and it de nes a Radon measure on the space. For the singular part of the functional, we get the expected integral representation with respect to the variation measure. A new feature is that in the representation for the absolutely continuous part, a constant appears already in the weighted Euclidean case. As an application we show that in a variational minimization problem involving the functional, boundary values can be presented as a penalty term.

Tensorization of Cheeger energies, the space H1,1 and the area formula for graphs

“…In the next remark, instead, we compare with the relaxation point of view of [16]. If we denote by μ a weak limit of |∇f n |, we obviously have μ ≥ |Df | on open sets A, hence μ ≥ |∇f | * ,1 m. From Lemma 5.2 we obtain that μ m and that |∇f n | → |∇f | * ,1 in L 1 (X, m).…”

“…In the presence of doubling and (1, 1)-Poincaré inequality for locally Lipschitz functions, however, (ii) is sufficient, see the proof of the implication from (iii) to (i) in Theorem 4.3 which only uses |Df | m (see also [16,Theorem 4.6]). We use the notation |∇f | * ,1 because, at this level of generality, we expect that weak gradients depend on the integrability exponent, even for Lipschitz functions, see [11] for examples compatible even with the doubling assumption.…”

“…In the metric measure setting, the functional A has been an object of investigation in the recent paper [17], see also [16] for generalizations. In the classical theory, it is well known that A corresponds to the surface area of the subgraph…”

“…In general equality does not hold in (4.9), not even under the doubling and Poincaré assumptions: in[16], a metric measure space is built endowing X = [0, 1] with the Euclidean distance and the weighted measure m = ωL 1 ∈ P(X), with ω ≡ 1 on a "fat" Cantor set K ⊂ (0, 1) (i.e. a compact totally disconnected set with positive Lebesgue measure) and ω = 1/2, say, on (0, 1) \ K. It is clear that (X, d, m), being comparable to the standard Euclidean structure, satisfies the doubling and Poincaré assumptions.…”

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.…”