The Dependence of the First Eigenvalue of the Infinity Laplacian With Respect to the Domain

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 ))

show abstract

“…We refer the reader to [20] for related problems on the domain dependence of Λ ∞ . Now we consider the case p = 1.…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

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

Anoop

Bobkov

Sasi

2018

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…For more results concerning the ∞-eigenvalue problem we refer to [4,7,20,27,33] the survey [24] and references therein. According to our knowledge, up to date, there is no investigation on the asymptotic behavior of the Fučík spectrum as p diverges.…”

Section: Introductionmentioning

confidence: 99%

The ∞-Fucík spectrum

Rossi¹,

Salort²,

Silva³

2018

Ann. Acad. Sci. Fenn. Math.

View full text Add to dashboard Cite

show abstract

“…Eigenvalue problems have received an increasing amount of attention along the last decades by many authors, being studied mainly via variational methods. We quote, among many others, [2,3,4,5,6,10,11,13,14,15,17,18,20,21,22,23,25,27,28]. In some of these references the limit as p → ∞ of the eingenvalue problem associated to the classical case, m ≡ 1, was considered.…”

Section: Introductionmentioning

confidence: 99%

The $\infty$-eigenvalue problem with a sign-changing weight

Kaufmann¹,

Rossi²,

Terra³

2018

Preprint

View full text Add to dashboard Cite

Let Ω ⊂ R n be a smooth bounded domain and m ∈ C(Ω) be a sign-changing weight function. For 1 < p < ∞, consider the eigenvalue problem −∆pu = λm(x)|u| p−2 u in Ω, u = 0 on ∂Ω, where ∆pu is the usual p-Laplacian. Our purpose in this article is to study the limit as p → ∞ for the eigenvalues λ k,p (m) of the aforementioned problem. In addition, we describe the limit of some normalized associated eigenfunctions when k = 1.

show abstract

The Dependence of the First Eigenvalue of the Infinity Laplacian With Respect to the Domain

Cited by 9 publications

References 13 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

The ∞-Fucík spectrum

The $\infty$-eigenvalue problem with a sign-changing weight

Contact Info

Product

Resources

About