An Overview on the Cheeger Problem

Leonardi, Gian Paolo

doi:10.1007/978-3-319-17563-8_6

Cited by 50 publications

(56 citation statements)

References 30 publications

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“…In recent years the Cheeger constant has attracted an increasing attention: we address the interested reader to the review papers [22,24] and to the numerous references therein. Let us also mention that optimal partition problems for the Cheeger constant have been recently considered in [6], under the form (5), and with the aim of finding bounds for the asymptotics of the same problem for the first Dirichlet eigenvalue.…”

Section: Remarkmentioning

confidence: 99%

On the honeycomb conjecture for a class of minimal convex partitions

Bucur

Fragalà

Velichkov

et al. 2018

Trans. Amer. Math. Soc.

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Abstract. We prove that the planar hexagonal honeycomb is asymptotically optimal for a large class of optimal partition problems, in which the cells are assumed to be convex, and the criterion is to minimize either the sum or the maximum among the energies of the cells, the cost being a shape functional which satisfies a few assumptions. They are: monotonicity under inclusions; homogeneity under dilations; a Faber-Krahn inequality for convex hexagons; a convexity-type inequality for the map which associates with every n ∈ N the minimizers of F among convex n-gons with given area. In particular, our result allows to obtain the honeycomb conjecture for the Cheeger constant and for the logarithmic capacity (still assuming the cells to be convex). Moreover we show that, in order to get the conjecture also for the first Dirichlet eigenvalue of the Laplacian, it is sufficient to establish some facts about the behaviour of λ1 among convex pentagons, hexagons, and heptagons with prescribed area.

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Section: Remarkmentioning

confidence: 99%

On the honeycomb conjecture for a class of minimal convex partitions

Bucur

Fragalà

Velichkov

et al. 2018

Trans. Amer. Math. Soc.

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“…We can exclude the case β < +∞, as we would obtain by maximality that ψ 1 (ρ) → +∞ as ρ → β − , however this would contradict the fact that |ψ ′ (ρ)| ≤ ρ for all ρ > 0. On the other hand, if ρ → +∞ one gets a contradiction with (26). The remaining three cases can be discussed in a similar way.…”

Section: Counterexamples To Quadratic Rigidity In Dimension N ≥mentioning

confidence: 84%

Rigidity and trace properties of divergence-measure vector fields

Leonardi

Saracco

2020

Advances in Calculus of Variations

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We consider a ϕ-rigidity property for divergence-free vector fields in the Euclidean n-space, where ϕ(t) is a non-negative convex function vanishing only at t = 0. We show that this property is always satisfied in dimension n = 2, while in higher dimension it requires some further restriction on ϕ. In particular, we exhibit counterexamples to quadratic rigidity (i.e., when ϕ(t) = ct 2 ) in dimension n ≥ 4. The validity of the quadratic rigidity, which we prove in dimension n = 2, implies the existence of the trace of a divergence-measure vector field ξ on a H 1 -rectifiable set S, as soon as its weak normal trace [ξ · ν S ] is maximal on S. As an application, we deduce that the graph of an extremal solution to the prescribed mean curvature equation in a weakly-regular domain becomes vertical near the boundary in a pointwise sense.2010 Mathematics Subject Classification. Primary: 26B20, 28A75. Secondary: 35L65.Key words and phrases. divergence-measure vector field; weak normal trace; rigidity. G.P. Leonardi and G. Saracco have been partially supported by the GNAMPA project: Variational problems and geometric measure theory in metric spaces (2016).

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“…Cheeger constants play an important role in eigenvalue estimates on Riemannian manifolds (see [4]), whereas in the classical Euclidean case (M, g) = (R N , g eucl ) these notions have applications in the denoising problem in image processing, see e.g. [22,25].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Serrin’s overdetermined problem on the sphere

Fall¹,

Minlend²,

Weth

2017

Calc. Var.

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Abstract. We study Serrin's overdetermined boundary value problemin subdomains Ω of the round unit sphere S N ⊂ R N +1 , where ∆ S N denotes the Laplace-Beltrami operator on S N . A subdomain Ω of S N is called a Serrin domain if it admits a solution of this overdetermined problem. In our main result, we construct Serrin domains in S N , N ≥ 2 which bifurcate from symmetric straight tubular neighborhoods of the equator. Our result provides the first example of Serrin domains in S N which are not bounded by geodesic spheres.

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An Overview on the Cheeger Problem

Cited by 50 publications

References 30 publications

On the honeycomb conjecture for a class of minimal convex partitions

On the honeycomb conjecture for a class of minimal convex partitions

Rigidity and trace properties of divergence-measure vector fields

Serrin’s overdetermined problem on the sphere

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