Relaxation of isotropic functionals with linear growth defined on manifold constrained Sobolev mappings

Abstract

Abstract. In this paper we study the lower semicontinuous envelope with respect to the L 1 -topology of a class of isotropic functionals with linear growth defined on mappings from the n-dimensional ball into R N that are constrained to take values into a smooth submanifold Y of R N .

Homogenization of variational problems in manifold valued BV-spaces

“…We present in this appendix a relaxation result already proved in [2] for M = S d−1 , and in [36] for isotropic integrands. The proof can be obtained following the one of [2, Theorem 3.1] replacing the standard projection on the sphere (used in Lemma 5.2, Proposition 6.2 and Lemma 6.4 of [2]) by the projection on M of [35] as in Proposition 2.1.…”

“…6 where the proof of the theorem is completed. Finally we state in the Appendix a relaxation result for general manifolds and integrands which extends [2,36].…”

“…In view of the studies performed in [31,36], the domain is exactly given by BV ( ; M). Under the additional (standard) assumption, (H 4 ) there exist C > 0 and 0 < q < 1 such that f (y, ξ) − f ∞ (y, ξ) ≤ C 1 + |ξ | 1−q 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].…”