Calibrations for minimal networks in a covering space setting

Carioni, Marcello; Pluda, Alessandra

doi:10.48550/arxiv.1707.01448

Cited by 3 publications

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“…The map f Σ in (2.36) is an isometry between (M H , d M H ) and (Y Σ , d Y Σ ). 9 Here by [[x, x ]] we mean the path corresponding to the segment from x to x , for every…”

Section: We Denote By [γ] H the Equivalence Class Of [γ] ∈ M Induced ...mentioning

confidence: 99%

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Covers, soap films and BV functions

Bellettini¹,

Paolini²,

Pasquarelli³

et al. 2018

Geometric Flows

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In this paper we review the double covers method with constrained BV functions for solving the classical Plateau's problem. Next, we carefully analyze some interesting examples of soap films compatible with covers of degree larger than two: in particular, the case of a soap film only partially wetting a space curve, a soap film spanning a cubical frame but having a large tunnel, a soap film that retracts onto its boundary, hence not modelable with the Reifenberg method, and various soap films spanning an octahedral frame.

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“…The map f Σ in (2.36) is an isometry between (M H , d M H ) and (Y Σ , d Y Σ ). 9 Here by [[x, x ]] we mean the path corresponding to the segment from x to x , for every…”

Section: We Denote By [γ] H the Equivalence Class Of [γ] ∈ M Induced ...mentioning

confidence: 99%

“…We conclude this introduction by mentioning that calibrations, applied to the covering space method, have been considered in [7], [8] and, more recently, in [9] in connection with the BV approach in dimension two.…”

Section: Introductionmentioning

confidence: 99%

Covers, soap films and BV functions

Bellettini¹,

Paolini²,

Pasquarelli³

et al. 2018

Geometric Flows

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“…In [16] Marchese and Massaccesi rephrase the Steiner Problem as a mass minimization for 1-rectifiable currents with coefficients in a group and this leads to a suitable definition of calibrations (see also [6]). Finally reviving the approach via covering space by Brakke [7] (see [2] for the existence theory) another notion of calibrations has been produced [8].…”

Section: Introductionmentioning

confidence: 99%

“…The second part of the paper has a different focus and it can be seen as a completion of [8] as we restrict our attention to calibrations on coverings. In Theorem 4.2 we prove that the existence of a calibration for a constrained set E in a covering Y implies the minimality of E not only among (constrained) finite perimeter sets, but also in the larger class of finite linear combinations of characteristic functions of finite perimeter sets (satisfying a suitable constraint).…”

Section: Introductionmentioning

confidence: 99%

“…They have been introduced in the context of minimal surfaces [4,11,14] and adapted to the case of the Steiner problem in several variants. Our goal is to compare the different notions of calibrations for the Steiner Problem and for planar minimal partitions appearing in [8,15,16]. The paper is then complemented with remarks on the convexification of the problem, on non-existence of calibrations and on calibrations in families.…”

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On different notions of calibrations for minimal partitions and minimal networks in $\mathbb{R}^2$

Carioni,

Pluda

2018

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Calibrations are a possible tool to validate the minimality of a certain candidate. They have been introduced in the context of minimal surfaces [4,11,14] and adapted to the case of the Steiner problem in several variants. Our goal is to compare the different notions of calibrations for the Steiner Problem and for planar minimal partitions appearing in [8,15,16]. The paper is then complemented with remarks on the convexification of the problem, on non-existence of calibrations and on calibrations in families.

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The oriented mailing problem and its convex relaxation

Carioni¹,

Marchese²,

Massaccesi³

et al. 2019

Preprint

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In this note we introduce a new model for the mailing problem in branched transportation in order to allow the cost functional to take into account the orientation of the moving particles. This gives an effective answer to [1, Problem 15.9]. Moreover we define a convex relaxation in terms of rectifiable currents with group coefficients. With such approach we provide the problem with a notion of calibration. Using similar techniques we define a convex relaxation and a corresponding notion of calibration for a variant of the Steiner tree problem in which a connectedness constraint is assigned only among a certain partition of a given set of finitely many points.

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Calibrations for minimal networks in a covering space setting

Cited by 3 publications

References 0 publications

Covers, soap films and BV functions

Covers, soap films and BV functions

On different notions of calibrations for minimal partitions and minimal networks in $\mathbb{R}^2$

The oriented mailing problem and its convex relaxation

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