A minimal partition problem with trace constraint in the Grushin plane

Franceschi, Valentina

doi:10.1007/s00526-017-1198-5

Cited by 4 publications

(6 citation statements)

References 28 publications

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“…We emphasize that our main result generalizes the isoperimetric inequality in the Grushin plane proved in [40] (see also [28][29][30]) which corresponds to Theorem 1.1 with α = 0 and β ∈ (−1, 0] (see Rem. 2.15 in Sect.…”

Section: Introductionsupporting

confidence: 66%

“…Remark 2.15. In [40] (see also [28][29][30]) the authors obtain a sharp isoperimetric inequality in the Grushin plane, and they also compute the corresponding isoperimetric sets. Their proof consists of three steps.…”

Section: )mentioning

confidence: 99%

“…We wish to mention that isoperimetric problems w.r.t. non-Euclidean metrics, in particular in the framework of the Grushin space, have received a lot of attention recently, too, see for instance [28][29][30]40] and references therein.…”

Section: Introductionmentioning

confidence: 99%

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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: 66%

Section: )mentioning

confidence: 99%

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Some isoperimetric inequalities with respect to monomial weights

et al. 2021

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“…as k → +∞. Arguing as in the proof of [11,Lemma 2.1], one can prove that ( Ẽk ) k∈N ⊂ S x ∩ S * y . Now let us set E k = Ẽk ∩ H + for each k ∈ N. Since P ( Ẽ) = 2P (E; H + ) and P ( Ẽk ) = 2P (E k ; H + ) by symmetry, from (3.5) we deduce that…”

mentioning

confidence: 84%

Symmetric double bubbles in the Grushin plane

Franceschi

Stefani

2019

ESAIM: COCV

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We address the double bubble problem for the anisotropic Grushin perimeter P α , α ≥ 0, and the Lebesgue measure in R 2 , in the case of two equal volumes. We assume that the contact interface between the bubbles lies on either the vertical or the horizontal axis. We first prove existence of minimizers via the direct method by symmetrization arguments and then characterize them in terms of the given area by first variation techniques. Even though no regularity theory is available in this setting, we prove that angles at which minimal boundaries intersect satisfy the standard 120-degree rule up to a suitable change of coordinates. While for α = 0 the Grushin perimeter reduces to the Euclidean one and both minimizers coincide with the symmetric double bubble found in [10], for α = 1 vertical interface minimizers have Grushin perimeter strictly greater than horizontal interface minimizers. As the latter ones are obtained by translating and dilating the Grushin isoperimetric set found in [19], we conjecture that they solve the double bubble problem with no assumptions on the contact interface.

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“…In [22], the authors characterize isoperimetric sets in this framework. We also refer to [13] for a multidimensional generalization of the isoperimetric problem, and to [12,16] for a first approach to clustering problems in the Grushin plane. Note that in these references the Grushin perimeter is defined in a more general way via De Giorgi's definition allowing for non-Euclidean rectifiable sets.…”

Section: Examplesmentioning

confidence: 99%

On the Steiner property for planar minimizing clusters. The isotropic case

Franceschi,

Pratelli,

Stefani

2021

Preprint

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We consider the isoperimetric problem for clusters in the plane with a double density, that is, perimeter and volume depend on two weights. In this paper we consider the isotropic case, in the parallel paper [14] the anisotropic case is studied. Here we prove that, in a wide generality, minimal clusters enjoy the "Steiner property", which means that the boundaries are made by C 1,γ regular arcs, meeting in finitely many triple points with the 120 • property.

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A minimal partition problem with trace constraint in the Grushin plane

Cited by 4 publications

References 28 publications

Some isoperimetric inequalities with respect to monomial weights

Some isoperimetric inequalities with respect to monomial weights

Symmetric double bubbles in the Grushin plane

On the Steiner property for planar minimizing clusters. The isotropic case

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