On the Cheeger sets in strips and non-convex domains

Leonardi, Gian Paolo; Pratelli, Aldo

doi:10.1007/s00526-016-0953-3

Cited by 38 publications

(68 citation statements)

References 24 publications

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“…In the general case, for a Cheeger set Ω − of Ω \ Ω s , few results are available [36]: In the general case, for a Cheeger set Ω − of Ω \ Ω s , few results are available [36]:…”

Section: Application Examplesmentioning

confidence: 99%

Critical Yield Numbers of Rigid Particles Settling in Bingham Fluids and Cheeger Sets

Frigaard¹,

Iglesias²,

Mercier³

et al. 2017

SIAM J. Appl. Math.

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We consider the fluid mechanical problem of identifying the critical yield number Yc of a dense solid inclusion (particle) settling under gravity within a bounded domain of Bingham fluid, i.e., the critical ratio of yield stress to buoyancy stress that is sufficient to prevent motion. We restrict ourselves to a two-dimensional planar configuration with a single antiplane component of velocity. Thus, both particle and fluid domains are infinite cylinders of fixed cross-section. We then show that such yield numbers arise from an eigenvalue problem for a constrained total variation. We construct particular solutions to this problem by consecutively solving two Cheeger-type set optimization problems. Finally, we present a number of example geometries in which these geometric solutions can be found explicitly and discuss general features of the solutions. 639 of the fluid. Plug regions may occur either within the interior of a flow or may be attached to the wall. In general, as the applied forcing decreases, the plug regions increase in size and the velocity decreases in magnitude. It is natural that at some critical ratio of the driving stresses to the resistive yield stress of the fluid, the flow stops altogether. This critical yield ratio or yield number is the topic of this paper.Critical yield numbers are found for even the simplest one-dimensional (1D) flows, such as Poiseuille flows in pipes and plane channels or uniform film flows, e.g., paint on a vertical wall. These limits have been estimated and calculated exactly for flows around isolated particles, such as the sphere [8] (axisymmetric flow) and the circular disc [46, 48] (two-dimensional (2D) flow). Such flows have practical application in industrial non-Newtonian suspensions, e.g., mined tailings transport, cuttings removal in drilling of wells, etc.The first systematic study of critical yield numbers was carried out by Mosolov and Miasnikov [40,41], who considered antiplane shear flows, i.e., flows with velocity u = (0, 0, w(x 1 , x 2 )) in the x 3 -direction along ducts (infinite cylinders) of arbitrary cross-section Ω. These flows driven by a constant pressure gradient only admit the static solution (w(x 1 , x 2 ) = 0) if the yield stress is sufficiently large. Amongst the many interesting results in [40,41] the key contributions relate to exposing the strongly geometric nature of calculating the critical yield number Y c . First, they show that Y c can be related to the maximal ratio of area to perimeter of subsets of Ω. Second, they develop an algorithmic methodology for calculating Y c for specific symmetric Ω, e.g., rectangular ducts. This methodology is extended further by [29].Critical yield numbers have been studied for many other flows, using analytical estimates, computational approximations, and experimentation. Critical yield numbers to prevent bubble motion are considered in [18,50]. Settling of shaped particles is considered in [31,45]. Natural convection is studied in [32,33]. The onset of landslides is studied in [28,30,26] (where the...

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Section: Application Examplesmentioning

confidence: 99%

Critical Yield Numbers of Rigid Particles Settling in Bingham Fluids and Cheeger Sets

Frigaard¹,

Iglesias²,

Mercier³

et al. 2017

SIAM J. Appl. Math.

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“…The analyticity of Ω ∩ ∂ * C β , the closedness and the estimate on the dimension of Ω ∩ (∂C β \ ∂ * C β ) follow from classical regularity results, see for instance [67] or [58]. We refer the reader to the latter reference for a proof of the tangentiality condition on ∂ * C β ∩ ∂ * Ω.…”

Section: Theorem 31 Let ψ Be the Euclidean Norm Then ω ∩ ∂ * C β Imentioning

confidence: 91%

“…One can prove [58] the following facts: P θ admits a unique (hence maximal) Cheeger subset Ch(P θ ) (as in Figure 6(a)); moreover, there exists a unique θ 0 ∈ (0, π 2 ) such that P θ0 is Cheeger. Our idea is to construct an optimal selection, solving (1.4) in Ch(P θ ) (for θ ≠ θ 0 ), and then foliate the remaining part of P θ with arcs of circles, taking as vector field the outward unit normal to the arcs.…”

Section: Examples Of Optimal Selections In Non ϕ C -Calibrable Facetsmentioning

confidence: 98%

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Anisotropic mean curvature on facets and relations with capillarity

Amato¹,

Bellettini²,

Tealdi³

2015

Geometric Flows

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Abstract:Given an anisotropy ϕ on R 3 , we discuss the relations between the ϕ-calibrability of a facet F ⊂ ∂E of a solid crystal E, and the capillary problem on a capillary tube with base F. When F is parallel to a facet︀ B F ϕ of the unit ball of ϕ, ϕ-calibrability is equivalent to show the existence of a ϕ-subunitary vector field in F, with suitable normal trace on ∂F, and with constant divergence equal to the ϕ-mean curvature of F. Assuming E convex at F,̃︀ B F ϕ a disk, and F (strictly) ϕ-calibrable, such a vector field is obtained by solving the capillary problem on F in absence of gravity and with zero contact angle. We show some examples of facets for which it is possible, even without the strict ϕ-calibrability assumption, to build one of these vector fields. The construction provides, at least for convex facets of class C 1,1 , the solution of the total variation flow starting at 1 F .

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“…In particular, in [LP16], the Cheeger set of a strip is shown to satisfy (1.2) and (1.3) as well. In the same paper, the inner Cheeger formula for a strip of width 2 and length L is used to provide a first order expansion of the Cheeger constant in terms of 1/L, as L → +∞ (see Theorem 3.2 in [LP16]). One should not expect a characterization of the type (1.2) to hold in general.…”

Section: Introductionmentioning

confidence: 98%

The Cheeger constant of a Jordan domain without necks

2017

Self Cite

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We show that the maximal Cheeger set of a Jordan domain Ω without necks is the union of all balls of radius r=h(Ω)^−1 contained in Ω. Here, h(Ω) denotes the Cheeger constant of Ω, that is, the infimum of the ratio of perimeter over area among subsets of Ω, and a Cheeger set is a set attaining the infimum. The radius r is shown to be the unique number such that the area of the inner parallel set Ωr is equal to π r^2. The proof of the main theorem requires the combination of several intermediate facts, some of which are of interest in their own right. Examples are given demonstrating the generality of the result as well as the sharpness of our assumptions. In particular, as an application of the main theorem, we illustrate how to effectively approximate the Cheeger constant of the Koch snowflake

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On the Cheeger sets in strips and non-convex domains

Cited by 38 publications

References 24 publications

Critical Yield Numbers of Rigid Particles Settling in Bingham Fluids and Cheeger Sets

Critical Yield Numbers of Rigid Particles Settling in Bingham Fluids and Cheeger Sets

Anisotropic mean curvature on facets and relations with capillarity

The Cheeger constant of a Jordan domain without necks

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