A weighted isoperimetric inequality and applications to symmetrization

Betta, M. F.; Brock, Friedemann; Mercaldo, Anna; Posteraro,

doi:10.1155/s1025583499000375

Cited by 46 publications

(73 citation statements)

References 19 publications

(20 reference statements)

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“…This result has been proved in [6,Theorem 4.2] (see also [7] for a different proof). In other words, we have the following weighted isoperimetric inequality 5) with equality if and only if Ω is a ball centered at the origin.…”

Section: One Step Back: the Linear Casementioning

confidence: 64%

An anisotropic eigenvalue problem of Stekloff type and weighted Wulff inequalities

Brasco¹,

Franzina²

2013

Nonlinear Differ. Equ. Appl.

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Abstract. We study the Stekloff eigenvalue problem for the so-called pseudo p-Laplacian operator. After proving the existence of an unbounded sequence of eigenvalues, we focus on the first nontrivial eigenvalue σ2,p, providing various equivalent characterizations for it. We also prove an upper bound for σ2,p in terms of geometric quantities. The latter can be seen as the nonlinear analogue of the Brock-Weinstock inequality for the first nontrivial Stekloff eigenvalue of the (standard) Laplacian. Such an estimate is obtained by exploiting a family of sharp weighted Wulff inequalities, which are here derived and appear to be interesting in themselves.Mathematics Subject Classification (2010). 35P30, 47A75, 34B15.

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Section: One Step Back: the Linear Casementioning

confidence: 64%

An anisotropic eigenvalue problem of Stekloff type and weighted Wulff inequalities

Brasco¹,

Franzina²

2013

Nonlinear Differ. Equ. Appl.

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“…, d. This means that the term E (0) will have no influence on the determination of the second derivative of the eigenvalue. We will focus only on E (1) and E (2) . (2) : The computations are very technical.…”

Section: Construction Of the Matrix E Of The Second Derivativesmentioning

confidence: 99%

“…We will focus only on E (1) and E (2) . (2) : The computations are very technical. We need first to use a test function φ which is the restriction of a test function Φ defined on a tubular neighborhood of the boundary such that its normal derivative on ∂Ω is zero.…”

Section: Construction Of the Matrix E Of The Second Derivativesmentioning

confidence: 99%

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An extremal eigenvalue problem for the Wentzell–Laplace operator

Dambrine

Kateb

Lamboley

2016

Ann. Inst. H. Poincaré Anal. Non Linéaire

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We consider the question of giving an upper bound for the first nontrivial eigenvalue of the Wentzell-Laplace operator of a domain Ω, involving only geometrical information. We provide such an upper bound, by generalizing Brock's inequality concerning Steklov eigenvalues, and we conjecture that balls maximize the Wentzell eigenvalue, in a suitable class of domains, which would improve our bound. To support this conjecture, we prove that balls are critical domains for the Wentzell eigenvalue, in any dimension, and that they are local maximizers in dimension 2 and 3, using an order two sensitivity analysis. We also provide some numerical evidence.

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“…It is known (see [2] for the general n-dimensinal case ) that if E ⊂ R is a bounded set of finite perimeter, E s is the open interval centered at the origin having the same measure as E then…”

Section: A Weighted Version Of Pólya -Szegö Inequalitymentioning

confidence: 99%