The Cheeger constant of a Jordan domain without necks

Leonardi, Gian Paolo; Neumayer, Robin; Saracco, Giorgio Maria

doi:10.1007/s00526-017-1263-0

Cited by 28 publications

(45 citation statements)

References 30 publications

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“…Moreover, this property is also satisfied for all

r > (1 - R) / 2

since

Ω^{'}

contains no ball of such radius

r

. Applying now [, Theorem 1.4], we conclude that the Cheeger set of

Ω^{'}

is given by

O_{r_{0}}

, that is,

h_{1} false(Ω^{'} false) = F false(r_{0} false)

. Moreover,

O_{r_{0}}

is a unique minimizer of

h_{1} false(Ω^{'} false)

.…”

Section: Applicationsmentioning

confidence: 64%

“…Proof Evidently, each

O_{r}

with

r \in [0, (1 - R) / 2]

is an admissible set for the minimization problem

h_{1} false(Ω^{'} false)

, that is,

h_{1} false(Ω^{'} false) ⩽ F false(r_{0} false) : = \underset{r \in [0, false(1 - R false) / 2]}{trueprefixmin} F false(r false) .

To show that

h_{1} false(Ω^{'} false) = F false(r_{0} false)

, let us recall the following definition from . It is said that

Ω^{'}

has no necks of radius

r

if for any two balls

B_{r} false(x_{0} false), B_{r} false(x_{1} false) \subset {normalΩ}^{'}

there exists a continuous curve

γ : false[0, 1 false] \to {normalΩ}^{'}

such that

γ false(0 false) = x_{0}, γ false(1 false) = x_{1}, and B_{r} false(γ (t) false) \subset {normalΩ}^{'} for all t \in false[0, 1 false] .

It is not hard to observe that

Ω^{'}

satisfies this property for all

...

…”

Section: Applicationsmentioning

confidence: 99%

See 1 more Smart Citation

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.

show abstract

“…Moreover, this property is also satisfied for all

r > (1 - R) / 2

since

Ω^{'}

contains no ball of such radius

r

. Applying now [, Theorem 1.4], we conclude that the Cheeger set of

Ω^{'}

is given by

O_{r_{0}}

, that is,

h_{1} false(Ω^{'} false) = F false(r_{0} false)

. Moreover,

O_{r_{0}}

is a unique minimizer of

h_{1} false(Ω^{'} false)

.…”

Section: Applicationsmentioning

confidence: 64%

“…Proof Evidently, each

O_{r}

with

r \in [0, (1 - R) / 2]

is an admissible set for the minimization problem

h_{1} false(Ω^{'} false)

, that is,

h_{1} false(Ω^{'} false) ⩽ F false(r_{0} false) : = \underset{r \in [0, false(1 - R false) / 2]}{trueprefixmin} F false(r false) .

To show that

h_{1} false(Ω^{'} false) = F false(r_{0} false)

, let us recall the following definition from . It is said that

Ω^{'}

has no necks of radius

r

if for any two balls

B_{r} false(x_{0} false), B_{r} false(x_{1} false) \subset {normalΩ}^{'}

there exists a continuous curve

γ : false[0, 1 false] \to {normalΩ}^{'}

such that

γ false(0 false) = x_{0}, γ false(1 false) = x_{1}, and B_{r} false(γ (t) false) \subset {normalΩ}^{'} for all t \in false[0, 1 false] .

It is not hard to observe that

Ω^{'}

satisfies this property for all

...

…”

Section: Applicationsmentioning

confidence: 99%

On the higher Cheeger problem

Bobkov

Parini

2018

J. London Math. Soc.

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“…The following result is crucial to our purposes. It can be regarded as a transposition, valid within the class A, of a well-known result for the inner Cheeger set of convex bodies due to Kawohl and Lachand-Robert (see Theorem 1 in [13]); we also refer to [16] for a recent extension to domains 'without necks'.…”

Section: Definition 17mentioning

confidence: 99%

Proof of the honeycomb asymptotics for optimal Cheeger clusters

Bucur

Fragalà

2019

Advances in Mathematics

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We prove that, in the limit as k → +∞, the hexagonal honeycomb solves the optimal partition problem in which the criterion is minimizing the largest among the Cheeger constants of k mutually disjoint cells in a planar domain. As a by-product, the same result holds true when the Cheeger constant is replaced by the first Robin eigenvalue of the Laplacian.2010 Mathematics Subject Classification. 52C20, 51M16, 49Q10.

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“…Moreover, precise estimates for the Cheeger constant and explicit characterisation of the Cheeger set for general strips were given in [9] and further improved in [11]. See also the recent papers [12] (which contains a characterisation of the maximal Cheeger set within a Jordan domain of the Euclidean plane) and [13] (where, motivated by capillarity-related issues, two nontrivial examples of minimal Cheeger sets in the plane are constructed). It is frustrating that already a three-dimensional cube does not admit an explicitly known Cheeger constant and there is no explicit analytical description of its Cheeger set (see [5,Open Problem 1]).…”

Section: Introductionmentioning

confidence: 99%