Proof of the honeycomb asymptotics for optimal Cheeger clusters

Bucur, Dorin; Fragalà, Ilaria

doi:10.1016/j.aim.2019.04.036

Cited by 10 publications

(10 citation statements)

References 18 publications

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“…Theorem 1.1 also allows to obtain a Faber-Krahn-type inequality for the so-called weighted Cheeger constant, and in turn a lower bound for the first eigenvalue for a degenerate elliptic operators. For similar results see also [9,13,14,19,34,44,47].…”

Section: Introductionsupporting

confidence: 57%

Some isoperimetric inequalities with respect to monomial weights

et al. 2021

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We solve a class of isoperimetric problems on $\mathbb{R}^2_+ $ with respect to monomial weights. Let $\alpha $ and $\beta $ be real numbers such that $0\le \alpha <\beta+1$, $\beta\le 2 \alpha$. We show that, among all smooth sets $\Omega$ in $\mathbb{R} ^2_+$ with fixed weighted measure $\iint_{\Omega } y^{\beta} dxdy$, the weighted perimeter $\int_{\partial \Omega } y^\alpha \, ds$ achieves its minimum for a smooth set which is symmetric w.r.t. to the $y$--axis, and is explicitly given. Our results also imply an estimate of a weighted Cheeger constant and a bound for eigenvalues of some nonlinear problems.

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Section: Introductionsupporting

confidence: 57%

Some isoperimetric inequalities with respect to monomial weights

et al. 2021

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show abstract

“…Theorem 1.1 also allows to obtain a Faber-Krahn -type inequality for the so-called weighted Cheeger constant, and in turn a lower bound for the first eigenvalue for a degenerate elliptic operators. For similar results see also [9], [12], [13], [18], [30], [39], [41]. We define the ratio…”

Section: Introductionmentioning

confidence: 78%

Some isoperimetric inequalities onRNwith respect to weights |x|

Alvino

Brock

Chiacchio

et al. 2017

Journal of Mathematical Analysis and Applications

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We solve a class of isoperimetric problems on R 2 + := (x, y) ∈ R 2 : y > 0 with respect to monomial weights. Let α and β be real numbers such that 0 ≤ α < β +1, β ≤ 2α. We show that, among all smooth sets Ω in R 2 + with fixed weighted measure Ω y β dxdy, the weighted perimeter ∂Ω y α ds achieves its minimum for a smooth set which is symmetric w.r.t. to the y-axis, and is explicitly given. Our results also imply an estimate of a weighted Cheeger constant and a lower bound for the first eigenvalue of a class of nonlinear problems.

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“…One of the main results of this paper is Theorem , where we prove that the

k

th Cheeger constant

h_{k} false(normalΩ false)

can be characterized as

\underset{p \to 1}{trueprefixlim} {frakturL}_{k} false(p; normalΩ false) = h_{k} false(normalΩ false) .

Let us mention that a related spectral partition problem was studied in . In that paper, the author investigated the limit as

p \to 1

of the quantity

{normalΛ}_{k}^{(p)} false(normalΩ false) : = inf \{0true \sum_{i = 1}^{k} λ_{1} false(p; E_{i} false) : E_{i} \subset Ω, false| E_{i} false| > 0 \forall i, E_{i} \cap E_{j} = \emptyset \forall i \neq j\}

and proved its convergence toward

H_{k} false(normalΩ false) : = inf \{0true \sum_{i = 1}^{k} h_{1} false(E_{i} false) : E_{i} \subset Ω, false| E_{i} false| > 0 \forall i, E_{i} \cap E_{j} = \emptyset \forall i \neq j\} .

Existence and qualitative properties of minimizing

k

‐tuples of sets for

H_{k} false(normalΩ false)

were also comprehensively studied in …”

Section: Introductionmentioning

confidence: 98%

On the higher Cheeger problem

Bobkov

Parini

2018

J. London Math. Soc.

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We develop the notion of higher Cheeger constants for a measurable set Ω ⊂ R N . By the kth Cheeger constant we mean the valuewhere the infimum is taken over all k-tuples of mutually disjoint subsets of Ω, and h1(Ei) is the classical Cheeger constant of Ei. We prove the existence of minimizers satisfying additional 'adjustment' conditions and study their properties. A relation between h k (Ω) and spectral minimal k-partitions of Ω associated with the first eigenvalues of the p-Laplacian under homogeneous Dirichlet boundary conditions is stated. The results are applied to determine the second Cheeger constant of radially symmetric planar domains.

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Proof of the honeycomb asymptotics for optimal Cheeger clusters

Cited by 10 publications

References 18 publications

Some isoperimetric inequalities with respect to monomial weights

Some isoperimetric inequalities with respect to monomial weights

Some isoperimetric inequalities onRNwith respect to weights |x|

On the higher Cheeger problem

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