Isoperimetric Inequalities in Potential Theory

We study the higher eigenvalues and eigenfunctions for the so-called ∞-eigenvalue problem. The problem arises as an asymptotic limit of the nonlinear eigenvalue problems for the p-Laplace operators and is very closely related to the geometry of the underlying domain. We are able to prove several properties that are known in the linear case p = 2 of the Laplacian, but are unknown for other values of p. In particular, we establish the validity of the Payne-Pólya-Weinberger conjecture regarding the ratio of the first two eigenvalues and the Payne nodal conjecture, which deals with the zero set of a second eigenfunction.

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“…[21,22], measures how close a set is to being a ball. Moreover, due to (2.4) it is easy to analyze the stability of Λ 1 under domain variations.…”

Section: Theorem 22mentioning

confidence: 99%

On the higher eigenvalues for the $\infty$ -eigenvalue problem

Juutinen

Lindqvist

2005

Calc. Var.

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“…The latter has been already proved by the first author and De Philippis in [5, Theorem 2.10], by adapting the idea of Hansen and Nadirashvili contained in [22].…”

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confidence: 77%

“…Starting with the works of Hansen & Nadirashvili [22] and Melas [29], there has been a surge of interest towards the stability issue for the Faber-Krahn inequality. In other words, one seeks for quantitative enhancements of (1.1), containing remainder terms measuring the deviation of a set Ω from spherical symmetry.…”

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confidence: 99%

A quantitative stability estimate for the fractional Faber-Krahn inequality

Brasco

Cinti

Vita

2020

Journal of Functional Analysis

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We prove a quantitative version of the Faber-Krahn inequality for the first eigenvalue of the fractional Dirichlet-Laplacian of order s. This is done by using the so-called Caffarelli-Silvestre extension and adapting to the nonlocal setting a trick by Hansen and Nadirashvili. The relevant stability estimate comes with an explicit constant, which is stable as the fractional order of differentiability goes to 1.2010 Mathematics Subject Classification. 47A75, 49Q20, 35R11.

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“…Some pioneering stability results. In this part we recall the quantitative estimates for the Faber-Krahn inequality by Hansen & Nadirashvili [48] and Melas [63]. First of all, as the proof of the Faber-Krahn inequality is based on the Pólya-Szegő principle (2.1), it is better to recall how (2.1) can be proved.…”

Section: Stability For the Faber-krahn Inequalitymentioning

confidence: 99%

“…The quest for quantitative improvements of spectral inequalities has attracted an increasing interest in the last years. To the best of our knowledge, such a quest started with the papers [48] by Hansen and Nadirashvili and [63] by Melas. Both papers concern the Faber-Krahn inequality, which is indeed the most studied case.…”

Section: Introductionmentioning

confidence: 99%

7 Spectral inequalities in quantitative form

Brasco¹,

Philippis²

2017

Shape Optimization and Spectral Theory

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We review some results about quantitative improvements of sharp inequalities for eigenvalues of the Laplacian. 27 4.2. A two-dimensional result by Nadirashvili 29 4.3. The Szegő-Weinberger inequality in sharp quantitative form 32 4.4. Checking the sharpness 35 5. Stability for the Brock-Weinstock inequality 40 5.1. A quick overview of the Steklov spectrum 40 5.2. Weighted perimeters 41 5.3. The Brock-Weinstock inequality in sharp quantitative form 43 5.4. Checking the sharpness 45 6. Some further stability results 46 6.

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Isoperimetric Inequalities in Potential Theory

Cited by 26 publications

References 10 publications

On the higher eigenvalues for the $\infty$ -eigenvalue problem

On the higher eigenvalues for the $\infty$ -eigenvalue problem

A quantitative stability estimate for the fractional Faber-Krahn inequality

7 Spectral inequalities in quantitative form

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