On the structure of the second eigenfunctions of the $p$-Laplacian on a ball

Let B 1 be a ball in R N centred at the origin and B 0 be a smaller ball compactly contained in B 1 . For p ∈ (1, ∞), using the shape derivative method, we show that the first eigenvalue of the p-Laplacian in annulus B 1 \ B 0 strictly decreases as the inner ball moves towards the boundary of the outer ball. The analogous results for the limit cases as p → 1 and p → ∞ are also discussed. Using our main result, further we prove the nonradiality of the eigenfunctions associated with the points on the first nontrivial curve of the Fučik spectrum of the p-Laplacian on bounded radial domains. Mathematics Subject Classification (2010): 35J92, 35P30, 35B06, 49R05. Ω |∇u| p dx : u ∈ W 1,p 0 (Ω) \ {0} with u p = 1 .In this article we consider Ω of the formdenotes the open ball of radius r > 0 centred at z ∈ R N . Since the p-Laplacian is invariant under orthogonal transformations, it can be easily seen that λ 1 (B R 1 (x) \ B R 0 (y)) = λ 1 (B R 1 (0) \ B R 0 (se 1 ))

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“…In either case, we have two disjoint domains Ω 1 and Ω 2 such that λ 1 (Ω 1 ) < t + c(t) and λ 1 (Ω 2 ) < c(t).…”

Section: Application To the Fučik Spectrummentioning

confidence: 99%

“…We obtain the above result as a simple consequence of Theorem 1.1. Moreover, Theorem 1.3 gives a generalization and a simpler proof for Theorem 1.1 of [1] which states the nonradiality of second eigenfunctions of the p-Laplacian on a ball.…”

Section: Introductionmentioning

confidence: 99%

On the strict monotonicity of the first eigenvalue of the 𝑝-Laplacian on annuli

Anoop

Bobkov

Sasi

2018

Trans. Amer. Math. Soc.

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“…Being proved by entirely different arguments, our Theorem 1.2 represents the direct generalization of these results for the nonlinear settings under the additional assumptions (O 3 ) and (O 4 ). In the resonant case with the p-Laplacian, the validity of the Payne conjecture was proposed as an open problem even in the case of a ball, see [3,Remark 4.2]. Easily, a ball satisfies (O 1 ) − (O 4 ), and hence Theorem 1.2 applies.…”

Section: The Model Case Of the Nonlinearity Which Satisfiesmentioning

confidence: 99%

On a property of the nodal set of least energy sign-changing solutions for quasilinear elliptic equations

Bobkov

Kolonitskii

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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In this note we prove the Payne-type conjecture about the behaviour of the nodal set of least energy sign-changing solutions for the equation −∆pu = f (u) in bounded Steiner symmetric domains Ω ⊂ R N under the zero Dirichlet boundary conditions. The nonlinearity f is assumed to be either superlinear or resonant. In the latter case, least energy sign-changing solutions are second eigenfunctions of the zero Dirichlet p-Laplacian in Ω. We show that the nodal set of any least energy sign-changing solution intersects the boundary of Ω. The proof is based on a moving polarization argument.

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“…where Γ 0 and Γ # 0 are the outer boundaries of Ω and Ω # respectively. We call the inequality in (2), as the reverse Faber-Krahn (R-F-K) inequality. A similar inequality for the second Neumann eigenvalue (the first non-zero eigenvalue), namely µ 2 (Ω) ≤ µ 2 (Ω * ) is obtained in [26] by Szegö for planar domains, and in [27] for higher dimensions by Weinberger.…”

mentioning

confidence: 99%

“…The proof of Szegö mainly rely on the conformal mapping, and the proof of Weinberger based on construction of a test function using a radial function together with a suitable translation of the origin. In [19], for proving the inequality (2), authors used the interior parallels and an isoperimetric inequality, deduced from an inequality for the interior parallels due to B. Sz. Nagy [25].…”

mentioning

confidence: 99%

On reverse Faber-Krahn inequalities

Anoop

Kumar

2020

Journal of Mathematical Analysis and Applications

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Payne-Weinberger showed that 'among the class of membranes with given area A, free along the interior boundaries and fixed along the outer boundary of given length L 0 , the annulus Ω # has the highest fundamental frequency,' where Ω # is a concentric annulus with the same area as Ω and the same outer boundary length as L 0 .We extend this result for the higher dimensional domains and p-Laplacian with p ∈ (1, ∞), under the additional assumption that the outer boundary is a sphere. As an application, we prove that the nodal set of the second eigenfunctions of p-Laplacian (with mixed boundary conditions) on a ball and a concentric annulus cannot be a concentric sphere.Keywords Reverse Faber-Krahn inequality · Interior parallels · Isoperimetric inequality · Nagy's inequalities · First eigenvalue of p-Laplacian · Mixed boundary value problems · Elasticity problems · Non-radiality IntroductionLet us first recall the famous conjecture by Lord Rayleigh from his book 'The theory of sound' [23] published in 1877. He conjectured that 'among all planar domains Ω of fixed area, the disk is the domain that minimises the first Dirichlet eigenvalue λ 1 (Ω) of the Laplacian.' This conjecture was open for a very long time. The first proof of this conjecture was published in 1923 by Faber [8]. In 1925, Krahn [14] gave an independent proof for this conjecture, and later he extended the same to the higher dimension. Now this result is collectively known as the Faber-Krahn inequality, and it states that:with the equality if and only if Ω is a ball (up to a measure zero set), where Ω * is a ball of the same measure as Ω.For more on Faber-Krahn inequality and related results, we refer to [18] and [21].

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On the structure of the second eigenfunctions of the $p$-Laplacian on a ball

Cited by 16 publications

References 17 publications

On the strict monotonicity of the first eigenvalue of the 𝑝-Laplacian on annuli

On the strict monotonicity of the first eigenvalue of the 𝑝-Laplacian on annuli

On a property of the nodal set of least energy sign-changing solutions for quasilinear elliptic equations

On reverse Faber-Krahn inequalities

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