On the continuity of functionals defined on partitions

Ruf, Matthias

doi:10.1515/acv-2016-0061

Cited by 5 publications

(3 citation statements)

References 8 publications

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“…They also addressed the problem of lower semicontinuity which has been further developed over the last years, see, e.g., [11,Section 5.3] or [27,28,29]. Recent advances dealing with density and continuity results [15,56] witness that the study of this class of functionals is of ongoing interest.…”

Section: Introductionmentioning

confidence: 99%

Functionals Defined on Piecewise Rigid Functions: Integral Representation and $$\varGamma $$-Convergence

Friedrich

Solombrino

2020

Arch Rational Mech Anal

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We analyse integral representation and Γ -convergence properties of functionals defined on piecewise rigid functions, i.e., functions which are piecewise affine on a Caccioppoli partition where the derivative in each component is constant and lies in a set without rank-one connections. Such functionals account for interfacial energies in the variational modeling of materials which locally show a rigid behavior. Our results are based on localization techniques for Γ -convergence and a careful adaption of the global method for relaxation [17,18] to this new setting, under rather general assumptions. They constitute a first step towards the investigation of lower semicontinuity, relaxation, and homogenization for free-discontinuity problems in spaces of (generalized) functions of bounded deformation.2010 Mathematics Subject Classification. 49J45, 49Q20, 70G75, 74R10.

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Section: Introductionmentioning

confidence: 99%

Functionals Defined on Piecewise Rigid Functions: Integral Representation and $$\varGamma $$-Convergence

Friedrich

Solombrino

2020

Arch Rational Mech Anal

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“…In the last step we used the continuity assumption on the integrand and a Reshetnyak-type continuity result for functionals defined on partitions that is proven in [31]. Letting B ↓ A we obtain the claim.…”

Section: Convergence Of Boundary Value Problemsmentioning

confidence: 96%

Continuum limit and stochastic homogenization of discrete ferromagnetic thin films

2018

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We study the discrete-to-continuum limit of ferromagnetic spin systems when the lattice spacing tends to zero. We assume that the atoms are part of a (maybe) non-periodic lattice close to a flat set in a lower dimensional space, typically a plate in three dimensions. Scaling the particle positions by a small parameter ε > 0 we perform a Γ-convergence analysis of properly rescaled interfacial-type energies. We show that, up to subsequences, the energies converge to a surface integral defined on partitions of the flat space. In the second part of the paper we address the issue of stochastic homogenization in the case of random stationary lattices. A finer dependence of the homogenized energy on the average thickness of the random lattice is analyzed for an example of magnetic thin system obtained by a random deposition mechanism.

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“…This approach was further developed by subsequent contributions over the last years, see, e.g., [7,Section 5.3] or [14,15,16]. Let us also mention some recent advances dealing with density and continuity results [8,36], witnessing that the study of this class of functionals is of ongoing interest.…”

Section: Introductionmentioning

confidence: 99%

Lower semicontinuity for functionals defined on piecewise rigid functions and on $GSBD$

Friedrich¹,

Perugini²,

Solombrino³

2020

Preprint

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In this work, we provide a characterization result for lower semicontinuity of surface energies defined on piecewise rigid functions, i.e., functions which are piecewise affine on a Caccioppoli partition where the derivative in each component is a skew symmetric matrix. This characterization is achieved by means of an integral condition, called BD-ellipticity, which is in the spirit of BV -ellipticity defined by Ambrosio and Braides [5]. By specific examples we show that this novel concept is in fact stronger compared to its BV analog. We further provide a sufficient condition implying BD-ellipticity which we call symmetric joint convexity. This notion can be checked explicitly for certain classes of surface energies which are relevant for applications, e.g., for variational fracture models. Finally, we give a direct proof that surface energies with symmetric jointly convex integrands are lower semicontinuous also on the larger space of GSBD p functions.

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On the continuity of functionals defined on partitions

Cited by 5 publications

References 8 publications

Functionals Defined on Piecewise Rigid Functions: Integral Representation and $$\varGamma $$-Convergence

Functionals Defined on Piecewise Rigid Functions: Integral Representation and $$\varGamma $$-Convergence

Continuum limit and stochastic homogenization of discrete ferromagnetic thin films

Lower semicontinuity for functionals defined on piecewise rigid functions and on $GSBD$

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