Partial Geometric Regularity of Some Optimal Connected Transportation Networks

Stepanov, Eugene

doi:10.1007/s10958-005-0514-3

Cited by 8 publications

(7 citation statements)

References 13 publications

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“…This problem is related to a Monge-Kantorovitch problem with a "free Dirichlet region" which was introduced as a model for an optimal urban traffic network [6,9]. The topological and geometric properties of minimizers of this problem (mostly in its constrained version but also in the easier penalized one) were studied by several authors (see [23] for a review on this problem, and [7,27,31,10,22,29,33] for related results on this and similar problems). Problem 1.1 is much different from the average distance minimization problem, and to certain extent is closer to the Mumford-Shah problem.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Regularity for the Optimal Compliance Problem with Length Penalization

Chambolle¹,

Lamboley²,

Lemenant³

et al. 2017

SIAM J. Math. Anal.

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We study the regularity and topological structure of a compact connected set S minimizing the "compliance" functional with a length penalization. The compliance is, here, the work of the force applied to a membrane which is attached along the set S.This shape optimization problem, which can be interpreted as that of finding the best location for attaching a membrane subject to a given external force, can be seen as an elliptic PDE version of the minimal average distance problem.We prove that minimizers in the given region consist of a finite number of smooth curves which meet only at triple points with angles of 120 degrees, contain no loops, and possibly touch the boundary of the region only tangentially. The proof uses, among other ingredients, some tools from the theory of free discontinuity problems (monotonicty formula, flatness improving estimates, blow-up limits), but adapted to the specific problem of min-max type studied here, which constitutes a notable difference with the classical setting and may be useful also for similar other problems.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Regularity for the Optimal Compliance Problem with Length Penalization

Chambolle¹,

Lamboley²,

Lemenant³

et al. 2017

SIAM J. Math. Anal.

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“…Question III. At last, it is natural to mention here a similar problem introduced in [5] and sometimes called irrigation problem (see [4,6,7,12,13,14]) on minimization of the average distance functional but over compact connected sets of finite length rather than over discrete sets of points like in this paper. The statement of such a problem is obtained by replacing the constraint on cardinality #Σ of the unknown minimizer Σ in the location problem by the similar constraint on the one-dimensional Hausdorff measure H 1 (Σ).…”

Section: Concluding Remarks and Open Problemsmentioning

confidence: 99%

“…Further, several questions then arise about the behavior of Σ(t). For instance, it has been proven in [6,7,13,14] that the solutions of the long-term irrigation problem under suitable conditions on problem data do not contain loops, have a finite number of endpoints and may have only regular tripods (i.e. triple junctions with the branches with infinitesimal angles 120 • between each other) as branching points, which are at most finite in number.…”

Section: Concluding Remarks and Open Problemsmentioning

confidence: 99%

Long-term planningversusshort-term planning in the asymptotical location problem

et al. 2008

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Abstract. Given the probability measure ν over the given region Ω ⊂ R n , we consider the optimal location of a set Σ composed by n points in Ω in order to minimize the average distance Σ → Ω dist (x, Σ) dν (the classical optimal facility location problem). The paper compares two strategies to find optimal configurations: the long-term one which consists in placing all n points at once in an optimal position, and the short-term one which consists in placing the points one by one adding at each step at most one point and preserving the configuration built at previous steps. We show that the respective optimization problems exhibit qualitatively different asymptotic behavior as n → ∞, although the optimization costs in both cases have the same asymptotic orders of vanishing.Mathematics Subject Classification. 90B80, 90B85, 49J45, 46N10, 60K30.

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“…The first one is the average distance problem studied in [10,12,39,32,10,35,9,23,40,24,26,38]. For some Ω ⊂ R 2 , Λ > 0, and f ≥ 0 a positive L 1 density on Ω, the average distance problem is the the following:…”

Section: Introductionmentioning

confidence: 99%

Approximation of Length Minimization Problems Among Compact Connected Sets

Bonnivard¹,

Lemenant²,

Santambrogio³

2015

SIAM J. Math. Anal.

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In this paper we provide an approximationà la Ambrosio-Tortorelli of some classical minimization problems involving the length of an unknown one-dimensional set, with an additional connectedness constraint, in dimension two. We introduce a term of new type relying on a weighted geodesic distance that forces the minimizers to be connected at the limit. We apply this approach to approximate the so-called Steiner Problem, but also the average distance problem, and finally a problem relying on the p-compliance energy. The proof of convergence of the approximating functional, which is stated in terms of Γ-convergence relies on technical tools from geometric measure theory, as for instance a uniform lower bound for a sort of average directional Minkowski content of a family of compact connected sets.

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Partial Geometric Regularity of Some Optimal Connected Transportation Networks

Cited by 8 publications

References 13 publications

Regularity for the Optimal Compliance Problem with Length Penalization

Regularity for the Optimal Compliance Problem with Length Penalization

Long-term planningversusshort-term planning in the asymptotical location problem

Approximation of Length Minimization Problems Among Compact Connected Sets

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