Sharp Stability of Some Spectral Inequalities

Brasco, Lorenzo; Pratelli, Aldo

doi:10.1007/s00039-012-0148-9

Cited by 34 publications

(60 citation statements)

References 14 publications

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“…In this case the proof runs very similarly to the linear case p = 2 treated in [7]. We start applying the quantitative Faber-Krahn inequality (4.1) to Ω + : if we indicate with B the ball of unit radius, recalling (3.1) and using the definition of δ + , we find…”

Section: The Stability Issuementioning

confidence: 83%

“…The following technical Lemma of geometrical content completes the proof of Theorem 4.2: this is the same as [7,Lemma 3.3] and we omit the proof. …”

Section: The Stability Issuementioning

confidence: 87%

“…we will prove the Hong-Krahn-Szego inequality for the p−Laplacian. The key step in the proof is the following technical result: this is an adaptation of a similar result for the linear case p = 2 (see [7,Lemma 3.1], for example). …”

Section: The Hong-krahn-szego Inequalitymentioning

confidence: 99%

“…In the case of the Hong-Krahn-Szego inequality, the relevant notion of asymmetry is the Fraenkel 2−asymmetry, introduced in [7] A 2 (Ω) = inf…”

Section: The Stability Issuementioning

confidence: 99%

“…We conclude this introduction with the plan of the paper: in order to make the work as selfcontained as possible, Section 2 recalls the basic facts about the first two eigenvalues of −∆ p that we will need in the following; in Section 3 we prove the Hong-Krahn-Szego inequality for λ 2 , while Section 4 provides a quantitative version of the latter, thus extending to the nonlinear case a result recently proven in [7]. In Section 5, for the sake of completeness, we consider the shape optimization problem (1.1) for the "extremal" cases, i.e.…”

Section: Introductionmentioning

confidence: 99%

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

Brasco

Franzina

2012

manuscripta math.

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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

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Section: The Stability Issuementioning

confidence: 83%

“…The following technical Lemma of geometrical content completes the proof of Theorem 4.2: this is the same as [7,Lemma 3.3] and we omit the proof. …”

Section: The Stability Issuementioning

confidence: 87%

Section: The Hong-krahn-szego Inequalitymentioning

confidence: 99%

“…In the case of the Hong-Krahn-Szego inequality, the relevant notion of asymmetry is the Fraenkel 2−asymmetry, introduced in [7] A 2 (Ω) = inf…”

Section: The Stability Issuementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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