Constrained BV functions on covering spaces for minimal networks and Plateau’s type problems

Amato, Stefano; Bellettini, Giovanni; Paolini, Maurizio

doi:10.1515/acv-2015-0021

Cited by 8 publications

(35 citation statements)

References 21 publications

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“…As a further consequence of Lemma 2.6, the boundary datum S is attained by any constrained function on the cover, in the following sense. If 2 ≤ n ≤ 7 and u is a minimizer, it is possible to show that equality holds in (2.24) [2].…”

Section: 3mentioning

confidence: 99%

“…Indeed, let ρ be the closed curve going from x 0 to x following γ x , and then backward from x to x 0 along λ x . Recalling that link 2…”

Section: 4mentioning

confidence: 99%

“…The relations between a constrained double-cover solution and other notions of solution to the Plateau problem can be found in [2]. Proof.…”

Section: 4mentioning

confidence: 99%

“…• when n = 2, one can model, among others, the Steiner minimal graph problem connecting a finite number k ≥ 3 of points in the plane [2]; • when n = 3, one can consider configurations with singularities (triple curves, quadruple points etc. ), in particular when S is the one-dimensional skeleton of a polyhedron; • choosing carefully the cover, it is possible to model soap films with higher topological genus, as in the example of the one-skeleton of a tetrahedron 10 discussed in [4]: the resulting soap film seems not to be modelable using the Reifenberg approach [17].…”

Section: Covers Of Degree Larger Than Twomentioning

confidence: 99%

“…Recenlty, a slightly different approach has been proposed in [2]; it is based on the minimization of the total variation for functions defined on a single covering space and satisfying a suitable contraint on the fibers. Also this method does not impose any topological restriction on the solutions.…”

Section: Introductionmentioning

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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Section: 3mentioning

confidence: 99%

“…Indeed, let ρ be the closed curve going from x 0 to x following γ x , and then backward from x to x 0 along λ x . Recalling that link 2…”

Section: 4mentioning

confidence: 99%

“…The relations between a constrained double-cover solution and other notions of solution to the Plateau problem can be found in [2]. Proof.…”

Section: 4mentioning

confidence: 99%

Section: Covers Of Degree Larger Than Twomentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Covers, soap films and BV functions

Bellettini¹,

Paolini²,

Pasquarelli³

et al. 2018

Geometric Flows

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Calibrations in families for minimal networks

Carioni

Pluda

2018

Proc Appl Math and Mech

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The Steiner Problem in its classical formulation reads as follow: given a finite collection S = {p 1 . . . , p m } of points in the plane find a connected set with minimal length that contains S. It is well known that minimizers are finite union of segments that meet in triple junctions forming angles of 120 degrees. Finding explicitly a solution, it is however much more challenging (even numerically). In 1995 Brakke [3] and more recently Amato, Bellettini and Paolini [1] introduced an alternative approach to the Steiner problem rephrasing it in a covering space setting: minimizing the perimeter among constrained sets in a suitable covering space of R 2 \ S is equivalent to minimize the length among all connected planar networks that contain the given m points. The covering, here denoted by Y , can be constructed by a cut and paste procedure that we sketchy describe here (see [1] for details). Consider a network in the plane that contains S and another Lipschitz curve Σ that connects the points and does not intersect the network. The union of these two (the network and the curve) creates a partition in m regions of the plane. Consider m copies of R 2 \ S and lift each of these regions to one of the copies. Moreover introduce an equivalence relation that identifies the points along the curve of the different copies in such a way that the m regions can be seen as one set E in the covering space Y (given by the union of the copies together with the equivalence relation). Then the perimeter of the set in Y is twice the length of the network. The set E has some special features: it is a set of finite perimeter in Y such that for almost every x in the base space there exists exactly one point y of E such that p(y) = x, where p is the projection onto the base space R 2 \ S. We denote by P constr (Y ) the space of all sets in Y satisfying the previous properties. Then we look forWe underline that this quantity is independent on the choice of the curve Σ in the construction of Y .Our first goal in [5] was to introduce a theory of calibrations for Problem (1).Definition Given E ∈ P constr (Y ) a calibration for E is a (sufficiently regular) vector field Φ : Y → R 2 such thatwhere with Φ i we denote the restriction of Φ on the i-th sheet of the covering Y . The main purpose of searching for a calibration is to show easily the minimality of a certain candidate. Indeed we have the following:Theorem If Φ : Y → R 2 is a calibration for E ∈ P constr (Y ), then E is a minimizer among all sets in P constr (Y ).Finding a calibration is not easy. In [5] we exhibit a calibration only in the case S is composed of 3 and 4 points, but for general configuration of points it seems to be an hard task. In particular it is a long standing open problem to find a calibration when S is composed of points lying at the vertices of a regular polygon [4]. Indeed, even if different notions of calibrations are present in the literature (see for instance [7,8]) and despite the effort of more than one author, this problem has never been addressed. This...

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Calibrations for minimal networks in a covering space setting

Carioni

Pluda

2020

ESAIM: COCV

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In this paper we define a notion of calibration for an approach to the classical Steiner problem in a covering space setting and we give some explicit examples. Moreover we introduce the notion of calibration in families: the idea is to divide the set of competitors in a suitable way, defining an appropriate (and weaker) notion of calibration. Then, calibrating the candidate minimizers in each family and comparing their perimeter, it is possible to find the minimizers of the minimization problem. Thanks to this procedure we prove the minimality of the Steiner configurations spanning the vertices of a regular hexagon and of a regular pentagon.

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Constrained BV functions on covering spaces for minimal networks and Plateau’s type problems

Cited by 8 publications

References 21 publications

Covers, soap films and BV functions

Covers, soap films and BV functions

Calibrations in families for minimal networks

Calibrations for minimal networks in a covering space setting

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