B2-convexity implies strong and weak lower semicontinuity of partitions of ℝ<sup><i>n</i></sup>

Caraballo, David G.

doi:10.2140/pjm.2011.253.321

Cited by 4 publications

(4 citation statements)

References 35 publications

(53 reference statements)

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“…This helps considerably with establishing regularity results, since such arguments are typically carried out locally, and it also helps with lower semicontinuity and existence results (see, e.g. [30,31]). THEOREM 9 (Local structure) Suppose P ¼ K 1::s 2 P s LS : Then for each x 2 @ tot P there exists an R40 such that…”

Section: Definitionmentioning

confidence: 99%

Uniform local energy estimates and partial regularity for space partitions

Caraballo

2011

Applicable Analysis

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For partitions of space into sets of finite perimeter, and for general convex surface energy density functions, we show that surface energy may be locally linearly approximated, with uniform error bounds which hold even when the interfaces and surface energy density functions are non-smooth. We show that locally simple partitions -those satisfying a quite general density ratio condition -have boundaries which locally almost everywhere consist of a single interface, and have boundary closures which are (n À 1) dimensional. Local simplicity typically holds for variational problems in which surface area, surface energy or a surface plus bulk energy sum are to be minimized. Local simplicity rules out pathological boundary behaviour, helps establish lower semicontinuity, existence and regularity results and facilitates the use of finite perimeter partitions in applications in materials science, image processing and other fields.

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Section: Definitionmentioning

confidence: 99%

Uniform local energy estimates and partial regularity for space partitions

Caraballo

2011

Applicable Analysis

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“…This is the well-known problem of Caccioppoli partitions and here the H 1 -ellipticity reduces to the BV -ellipticity (see (1.6)). Lower semicontinuity for functionals defined on Caccioppoli partitions was studied in [1,2], and several necessary and/or sufficient conditions for lower semicontinuity have been studied in [7,8,31]. For more general functionals defined on BV , lower semicontinuity and relaxation were studied in [3,5,19,29,30].…”

Section: Introductionmentioning

confidence: 99%

Homogenization of vector-valued partition problems and dislocation cell structures in the plane

Conti

Garroni

Müller

2016

Boll Unione Mat Ital

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We consider target-space homogenization for energies defined on partitions parametrized\ud by a discrete lattice B ⊂ RN. For a smallσ > 0, the variable is a piecewise constant\ud function taking values in σB, and the energy depends on the jumps and their orientation. In\ud the limit as σ → 0 we obtain a homogenized functional defined on functions of bounded\ud variation. This result is relevant in the study of dislocation structures in plastically deformed\ud crystals. We review recent literature on the topic and propose our limiting effective energy\ud as a continuum model for strain-gradient plasticity

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“…Moreover, many applications of these properties have been provided, e.g. in [2,4,5,9,10,11,13,14,16].…”

Section: Introductionmentioning

confidence: 99%

A Note on Some Topological Properties of Sets With Finite Perimeter

Delladio¹

2015

Glasgow Math. J.

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Some well-known results about the 2-density topology on ${\mathcal R}$ (in particular in the context of the Lusin–Menchoff property) are extended to τbm, i.e. the m-density topology on ${\mathcal R}$n with m ∈ (n,+∞). Every set of finite perimeter in ${\mathcal R}$n is equivalent (in measure) to a set in τbm0, where m0=n+1+${1\over n-1}$. There exists a set of finite perimeter in ${\mathcal R}$n which is not equivalent (in measure) to any member in the a.e.-modification of τbm, whatever m ∈ [n,+∞).

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B2-convexity implies strong and weak lower semicontinuity of partitions of ℝⁿ

Cited by 4 publications

References 35 publications

Uniform local energy estimates and partial regularity for space partitions

Uniform local energy estimates and partial regularity for space partitions

Homogenization of vector-valued partition problems and dislocation cell structures in the plane

A Note on Some Topological Properties of Sets With Finite Perimeter

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B2-convexity implies strong and weak lower semicontinuity of partitions of ℝn

Cited by 4 publications

References 35 publications

Uniform local energy estimates and partial regularity for space partitions

Uniform local energy estimates and partial regularity for space partitions

Homogenization of vector-valued partition problems and dislocation cell structures in the plane

A Note on Some Topological Properties of Sets With Finite Perimeter

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B2-convexity implies strong and weak lower semicontinuity of partitions of ℝⁿ