A selective review on Mumford–Shah minimizers

Lemenant, Antoine

doi:10.1007/s40574-016-0056-2

Cited by 12 publications

(12 citation statements)

References 58 publications

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“…Applying to (17) and (18) the estimates of Lemma 3.6 for u Σ and u Σ respectively (combined with (20) and (21) respectively), we arrive at (16), concluding the proof.…”

Section: Estimate Of |C(σ) − C(σ )|mentioning

confidence: 55%

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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“…Applying to (17) and (18) the estimates of Lemma 3.6 for u Σ and u Σ respectively (combined with (20) and (21) respectively), we arrive at (16), concluding the proof.…”

Section: Estimate Of |C(σ) − C(σ )|mentioning

confidence: 55%

Regularity for the Optimal Compliance Problem with Length Penalization

Chambolle¹,

Lamboley²,

Lemenant³

et al. 2017

SIAM J. Math. Anal.

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“…Covering each of these segments with balls of radius comparable to δ and centered in a point belonging to the segment itself, we can argue as above and conclude that, for k large enough, S u k ∩ (B 1−δ \ B δ ) consist of three arcs, with uniform C 1,α estimates. Once again the Ascoli-Arzelà Theorem shows that Remarkably, Lemenant [51] extended such a result to the 3-dimensional case with suitable changes in the statement (see also [53] for a sketch of the proof).…”

Section: )mentioning

confidence: 94%

“…However, we shall not enter here into this streamline of results but rather refer for more details on them to the monograph [26] and to the recent review paper [53].…”

Section: The Mumford-shah Conjecturementioning

confidence: 99%

“…This is only due to a matter of taste of the Author and it is also dictated by understandable reasons of space. Even more, it is not possible to be exhaustive in our (short) presentation, therefore we refer to the books by Ambrosio, Fusco and Pallara [7] and David [26] for the proofs of many results we shall only quote, for a more detailed account of the several contributions in literature, for the many connections with other fields and for complete lists of references (see also the recent survey [53] that covers several parts of the regularity theory that are not presented here).…”

Section: Introductionmentioning

confidence: 99%

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Fine regularity results for Mumford-Shah minimizers: porosity, higher integrability and the Mumford-Shah conjecture

Focardi¹

2016

Free Discontinuity Problems

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We review some classical results and more recent insights about the regularity theory for local minimizers of the Mumford and Shah energy and their connections with the Mumford and Shah conjecture. We discuss in details the links among the latter, the porosity of the jump set and the higher integrability of the approximate gradient.In particular, higher integrability turns out to be related with an explicit estimate on the Hausdorff dimension of the singular set and an energetic characterization of the conjecture itself. 2010 MSC. 49J45 ; 49Q20.

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“…This idea translated to the Mumford-Shah functional was exploited in the now classical variational model for crack propagation by Francfort and Marigo [FM98]. We refer to [AFP00,Fus03,Dav05,Foc12,Lem16] for overview references about the Mumford-Shah functional.…”

Section: Introduction and Main Statementsmentioning

confidence: 99%

Stationarity of the crack-front for the Mumford–Shah problem in 3D

Lemenant

Mikayelyan

2018

Journal of Mathematical Analysis and Applications

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In this paper we exhibit a family of stationary solutions of the Mumford-Shah functional in R 3 , arbitrary close to a crack-front. Unlike other examples, known in the literature, those are topologically non-minimizing in the sense of Bonnet [Bon96].We also give a local version in a finite cylinder and prove an energy estimate for minimizers. Numerical illustrations indicate the stationary solutions are unlikely minimizers and show how the dependence on axial variable impacts the geometry of the discontinuity set.A self-contained proof of the stationarity of the crack-tip function for the Mumford-Shah problem in 2D is presented.

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A selective review on Mumford–Shah minimizers

Cited by 12 publications

References 58 publications

Regularity for the Optimal Compliance Problem with Length Penalization

Regularity for the Optimal Compliance Problem with Length Penalization

Fine regularity results for Mumford-Shah minimizers: porosity, higher integrability and the Mumford-Shah conjecture

Stationarity of the crack-front for the Mumford–Shah problem in 3D

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