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

Juutinen, Petri; Lindqvist, Peter

doi:10.1007/s00526-004-0295-4

Cited by 71 publications

(92 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We interpret this assertion as a parabolic analog of a theorem of P. Juutinen, P. Lindqvist and J. Manfredi [18]. We also encourage the reader to compare this Theorem 1.3 with the results of [20]. (Ω × (0, ∞)) has a subsequence (v p k ) k∈N that converges locally uniformly to a continuous function v on Ω × (0, ∞).…”

Section: Large P Limitmentioning

confidence: 73%

A doubly nonlinear evolution for the optimal Poincaré inequality

Hynd

Lindgren

2016

Calc. Var.

View full text Add to dashboard Cite

We study the large time behavior of solutions of the PDE |v t | p−2 v t = ∆ p v. A special property of this equation is that the Rayleigh quotient Ω |Dv(x, t)| p dx/ Ω |v(x, t)| p dx is nonincreasing in time along solutions. As t tends to infinity, this ratio converges to the optimal constant in Poincaré's inequality. Moreover, appropriately scaled solutions converge to a function for which equality holds in this inequality. An interesting limiting equation also arises when p tends to infinity, which provides a new approach to approximating ground states of the infinity Laplacian.

show abstract

Section: Large P Limitmentioning

confidence: 73%

A doubly nonlinear evolution for the optimal Poincaré inequality

Hynd

Lindgren

2016

Calc. Var.

View full text Add to dashboard Cite

show abstract

“…Also, if |E| = 0 we use the convention P (E)/|E| = +∞. We denote by Λ 1 (Ω) the inverse of the radius r 1 of the largest ball included in Ω, while Λ 2 (Ω) will denote the inverse of the largest positive number r 2 such that there exist two disjoint balls of radius r 2 contained in Ω: it is remarkable to notice that Λ 1 and Λ 2 are indeed two eigenvalues, precisely they coincide with the first two eigenvalues of the ∞−Laplacian (see [22]). …”

Section: Extremal Cases: P = 1 and P = ∞mentioning

confidence: 99%

“…The first fact is proven in [15] and [29], respectively. For the second, one can consult [22] and the references therein.…”

Section: Extremal Cases: P = 1 and P = ∞mentioning

confidence: 99%

“…where in the last inequality we used the (sharp) quantitative stability estimate 1 for Λ 1 (see [22], equation (2.6)). Then one could bravely guess that for p "very large", inequality (4.1) has to hold with the exponent κ 1 (N, p) replaced by p, which is strictly small if N ≥ 3: this would prove that 1 The relation between the Fraenkel asymmetry α(Ω) as defined in [22] and our definition is given by A(Ω) = 2 α(Ω).…”

Section: Extremal Cases: P = 1 and P = ∞mentioning

confidence: 99%

See 1 more Smart Citation

On the Hong–Krahn–Szego inequality for the p-Laplace operator

Brasco

Franzina

2012

manuscripta math.

View full text Add to dashboard Cite

Abstract.Given an open set Ω, we consider the problem of providing sharp lower bounds for λ2(Ω), i.e. its second Dirichlet eigenvalue of the p−Laplace operator. After presenting the nonlinear analogue of the Hong-Krahn-Szego inequality, asserting that the disjoint unions of two equal balls minimize λ2 among open sets of given measure, we improve this spectral inequality by means of a quantitative stability estimate. The extremal cases p = 1 and p = ∞ are considered as well. IntroductionIn this paper, we are concerned with Dirichlet eigenvalues of the p−Laplace operatorwhere 1 < p < ∞. For every open set Ω ⊂ R N having finite measure, these are defined as the real numbers λ such that the boundary value problemhas non trivial (weak) solutions. In particular, we are mainly focused on the following spectral optimization problemwhere c > 0 is a given number, λ 2 (·) is the second Dirichlet eigenvalue of the p−Laplacian and | · | stands for the N −dimensional Lebesgue measure. We will go back on the question of the well-posedness of this problem in a while, for the moment let us focus on the particular case p = 2: in this case we are facing the eigenvalue problem for the usual Laplace operator and, as it is well known, Dirichlet eigenvalues form a discrete nondecreasing sequence of positive real numbers 0 < λ 1 (Ω) ≤ λ 2 (Ω) ≤ λ 3 (Ω) ≤ . . . , going to ∞. In particular, it is meaningful to speak of a second eigenvalue so that problem (1.1) is well-posed and we know that its solution is given by any disjoint union of two balls having measure c/2. Moreover, these are the only sets which minimize λ 2 under volume constraint. Using the scaling properties both of the eigenvalues of −∆ and of the Lebesgue measure, we can reformulate the previous result in scaling invariant form as follows

show abstract

“…The limit as p → ∞ of the eigenvalue problem was studied in [13], [12] and an anisotropic version in [4]. In those papers the authors prove that…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2013

View full text Add to dashboard Cite

Abstract. In this paper we study the dependence of the first eigenvalue of the infinity Laplace with respect to the domain. We prove that this first eigenvalue is continuous under some weak convergence conditions that are fulfilled when a sequence of domains converges in Hausdorff distance. Moreover, it is Lipschitz continuous but not differentiable when we considers deformations obtained via a vector field. Our results are illustrated with simple examples.

show abstract

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

Cited by 71 publications

References 31 publications

A doubly nonlinear evolution for the optimal Poincaré inequality

A doubly nonlinear evolution for the optimal Poincaré inequality

On the Hong–Krahn–Szego inequality for the p-Laplace operator

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

Contact Info

Product

Resources

About