Lower semicontinuity in spaces of weakly differentiable functions

Kristensen, Jan

doi:10.1007/s002080050277

Cited by 111 publications

(101 citation statements)

References 32 publications

(1 reference statement)

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“…We note that, as pointed out by the referee, this case also follows from results in [7], without the need for Lemma 2.1.…”

Section: Remark 22 If {U I } Is a Sequence In Bv (ω) With |D S U I supporting

confidence: 65%

On the Representation of Effective Energy Densities

Larsen¹

2000

ESAIM: COCV

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Abstract.We consider the question raised in [1] of whether relaxed energy densities involving both bulk and surface energies can be written as a sum of two functions, one depending on the net gradient of admissible functions, and the other on net singular part. We show that, in general, they cannot. In particular, if the bulk density is quasiconvex but not convex, there exists a convex and homogeneous of degree 1 function of the jump such that there is no such representation.

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“…We note that, as pointed out by the referee, this case also follows from results in [7], without the need for Lemma 2.1.…”

Section: Remark 22 If {U I } Is a Sequence In Bv (ω) With |D S U I supporting

confidence: 65%

On the Representation of Effective Energy Densities

Larsen¹

2000

ESAIM: COCV

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“…We observe that for configurations with uniformly bounded energy E ε (y ε ), the absolute continuous part of the gradient satisfies ∇ y ε ≈ SO(2) as the stored energy density is frame-indifferent and minimized on SO (2). Assuming that y ε → y in L 1 , one can show that ∇ y ∈ SO(2) almost everywhere, applying lower semicontinuity results for SBV functions (see [35]) and using the fact that the quasiconvex envelope of W is minimized exactly on SO(2) (see [45]). …”

Section: The Segmentation Problemmentioning

confidence: 88%

A Derivation of Linearized Griffith Energies from Nonlinear Models

Friedrich

2017

Arch Rational Mech Anal

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We derive Griffith functionals in the framework of linearized elasticity from nonlinear and frame indifferent energies in a brittle fracture via -convergence. The convergence is given in terms of rescaled displacement fields measuring the distance of deformations from piecewise rigid motions. The configurations of the limiting model consist of partitions of the material, corresponding piecewise rigid deformations and displacement fields which are defined separately on each component of the cracked body. Apart from the linearized Griffith energy the limiting functional also comprises the segmentation energy, which is necessary to disconnect the parts of the specimen.

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“…If the sequence z j is in the form z j = Du j , where Ω ⊂ R n is open and bounded, and (u j ) is a bounded sequence in W 1,p (Ω, R N ) for some 1 < p ≤ ∞, then the corresponding Young measure ν x is called p-gradient Young measures (see [10,19,23]). The Young measure is trivial if ν x is a Dirac measure for a.e.…”

Section: Depending Only On E(x) Note That X → H(e(x)) = Q(f (X)) Is mentioning

confidence: 99%

An approximation theorem for sequences of linear strains and its applications

Zhang

2004

ESAIM: COCV

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Abstract.We establish an approximation theorem for a sequence of linear elastic strains approaching a compact set in L 1 by the sequence of linear strains of mapping bounded in Sobolev space W 1,p . We apply this result to establish equalities for semiconvex envelopes for functions defined on linear strains via a construction of quasiconvex functions with linear growth.Mathematics Subject Classification. 26B25, 41A30, 49J45.

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Lower semicontinuity in spaces of weakly differentiable functions

Cited by 111 publications

References 32 publications

On the Representation of Effective Energy Densities

On the Representation of Effective Energy Densities

A Derivation of Linearized Griffith Energies from Nonlinear Models

An approximation theorem for sequences of linear strains and its applications

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