Homogenization of variational problems in manifold valued BV-spaces

Abstract

International audienceThis paper extends the result of Babadjian and Millot (preprint, 2008) on the homogenization of integral functionals with linear growth defined for Sobolev maps taking values in a given manifold. Through a $\Gamma$-convergence analysis, we identify the homogenized energy in the space of functions of bounded variation. It turns out to be finite for $BV$-maps with values in the manifold. The bulk and Cantor parts of the energy involve the tangential homogenized density introduced in Babadjian and Millot (preprint, 2008), while the jump part involves an homogenized surface density given by a geodesic type problem on the manifold

Homogenization of variational problems in manifold valued Sobolev spaces

“…Indeed, in the case of an integrand with linear growth, the domain of the Γ-limit is obviously larger than the Sobolev space W 1,1 (Ω; M) and the analysis has to be performed in the space of functions of bounded variation. In fact Theorem 1.2 is a first step in this direction and the complete study in BV -spaces can be found in [3].…”

A chromaticity-brightness model for color images denoising in a Meyer’s “u + v” framework

“…where Ω ⊂ R N is an open, bounded set, p ∈ [1, +∞), and M ⊂ R d is a (sufficiently) smooth, m-dimensional manifold. There is a vast literature in this framework (see, for instance, [2,9,10,13,14,18,23,35,41]), motivated, for example, by the study of equilibria for liquid crystals and magnetostrictive materials, where the class of admissible fields is constrained to take values on a certain manifold M (commonly, M = S d−1 , the unit sphere in R d ). As in [2,9,35], the key ingredients in the proof of Theorem 1.2 are the density of smooth functions in W 1,1 (Ω; M) [11,12,29] and a projection technique introduced in [2,30,31].…”

“…There is a vast literature in this framework (see, for instance, [2,9,10,13,14,18,23,35,41]), motivated, for example, by the study of equilibria for liquid crystals and magnetostrictive materials, where the class of admissible fields is constrained to take values on a certain manifold M (commonly, M = S d−1 , the unit sphere in R d ). As in [2,9,35], the key ingredients in the proof of Theorem 1.2 are the density of smooth functions in W 1,1 (Ω; M) [11,12,29] and a projection technique introduced in [2,30,31]. However, new arguments are required as three main features of our problem prevent us from using immediately the relaxation results concerning the constraint case in the BV setting [2,9,35]: unlike [2,35], our starting point cannot be a tangential quasiconvex function as the energy density considered here (see (1.13)) fail always to satisfy such condition (see Remark 4.2); and unlike the general setting in the literature, (i) our manifold, M = [α, β] × S 2 , has boundary, (ii) the recession function f ∞ in our case (see (1.16)) does not satisfy a hypothesis of the type |f (r, s, ξ, η) − f ∞ (r, s, ξ, η)| C(1 + |(ξ, η)| 1−m ) for some C > 0 and m ∈ (0, 1) (for a.e.…”

Relaxation in SBV p (Ω; S d–1)

“…In this context the relaxed energy E has been studied by Alicandro et al [1] when M = S d−1 , the unit sphere in R d , and f has linear growth. This result was later extended by Mucci [41] to general manifolds and for a restricted class of integrands satisfying an isotropy condition, and subsequently by Babadjian and Millot [8], who removed this restriction. Note that the integrands treated and the arguments used in [1,8] fall within the general theory developed for the unconstrained case in [5,13,29].…”

“…The key arguments in [1,8,41] are the density of smooth functions in W 1, p (Ω; M) (see [10,11,37] for the precise statement) and a projection technique introduced in [38,39].…”