Quasiconvexification in W1,1 and Optimal Jump Microstructure in BV Relaxation

Larsen, Christopher J.

doi:10.1137/s0036141095295991

Cited by 11 publications

(8 citation statements)

References 14 publications

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“…The following result seems to have appeared first in Lemma 5.1 of [21]; we here give a slightly stronger version with a different proof. For the assertions j j.…”

Section: Quasiconvexitymentioning

confidence: 86%

Lower semicontinuity and Young measures in BV without Alberti's Rank-One Theorem

Rindler¹

2012

Advances in Calculus of Variations

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We give a new proof of sequential weak* lower semicontinuity in BV. I R m / for integral functionals of the formCarathéodory integrand with linear growth at infinity, i.e. jf .x; A/j Ä M.1 C jAj/ for some M 0, and such that the recession function f 1 .x; A/ WD lim x 0 !x;t !1 t 1 f .x 0 ; tA/ exists and is (jointly) continuous. In contrast to the classical proofs by Ambrosio and Dal Maso [J. Funct. Anal. 109 (1992), 76-97] and Fonseca and Müller [Arch. Ration. Mech. Anal. 123 (1993), 1-49], we do not use Alberti's Rank-One Theorem [Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), 239-274], but a rigidity result for gradients. The proof is set in the framework of generalized Young measures and proceeds via establishing Jensen-type inequalities for regular and singular points of Du.

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“…The following result seems to have appeared first in Lemma 5.1 of [21]; we here give a slightly stronger version with a different proof. For the assertions j j.…”

Section: Quasiconvexitymentioning

confidence: 86%

Lower semicontinuity and Young measures in BV without Alberti's Rank-One Theorem

Rindler¹

2012

Advances in Calculus of Variations

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“…for a constant c m,d ≥ 1 that only depends on the dimensions m, d. Since also the set {r ∈ (1, R) : (∂ B r ) > 0} is at most countable, given ∈ (1, R) we can select r ∈ (1, ) such that (28), (29) and (∂ B r ) = 0 hold. For a constant vector a ∈ R m put ϕ = (u j −a)1 B r .…”

Section: Lemmamentioning

confidence: 99%

Relaxation of signed integral functionals in BV

Kristensen

Rindler

2009

Calc. Var.

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“…PROOF: As in [18], our strategy is to analyze S(u) by considering a countable family of level sets for u and noting that S(u) equals the union of pairwise intersections of these sets. The co-area formula then allows us to control the parts of the corresponding level sets for u n that lie outside S(u n ).…”

Section: Transfer Of Jump Setsmentioning

confidence: 99%

Existence and convergence for quasi‐static evolution in brittle fracture

Francfort

Larsen

2003

Comm Pure Appl Math

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162

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This paper investigates the mathematical well-posedness of the variational model of quasi-static growth for a brittle crack proposed by Francfort and Marigo in [15]. The starting point is a time-discretized version of that evolution which results in a sequence of minimization problems of Mumford-Shah-type functionals. The natural weak setting is that of special functions of bounded variation, and the main difficulty in showing existence of the time-continuous, quasi-static growth is to pass to the limit as the time discretization step tends to 0. This is performed with the help of a jump transfer theorem that permits, under weak convergence assumptions for a sequence {u n } of SBV functions to its BV limit u, the transference of the part of the jump set of any test field that lies in the jump set of u onto that of the converging sequence {u n }. In particular, it is shown that the notion of minimizer of a Mumford-Shah-type functional for its own jump set is stable under weak convergence assumptions. Furthermore, our analysis justifies numerical methods used for computing the time-continuous, quasi-static evolution.

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Quasiconvexification in W^1,1 and Optimal Jump Microstructure in BV Relaxation

Cited by 11 publications

References 14 publications

Lower semicontinuity and Young measures in BV without Alberti's Rank-One Theorem

Lower semicontinuity and Young measures in BV without Alberti's Rank-One Theorem

Relaxation of signed integral functionals in BV

Existence and convergence for quasi‐static evolution in brittle fracture

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