Eigenvalue pinching on convex domains in space forms

Aubry, Erwann; Bertrand, Jérôme; Colbois, Bruno

doi:10.1090/s0002-9947-08-04775-2

Cited by 5 publications

(8 citation statements)

References 17 publications

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“…Bucur and Henrot further proved existence for noncompactly supported densities; see Corollary 6 and the remarks following it. Densities with noncompact support were treated earlier by Aubry, Bertrand and Colbois [3,Lemma 4.11].…”

Section: Introductionmentioning

confidence: 99%

Well-Posedness of Weinberger’s Center of Mass by Euclidean Energy Minimization

Laugesen

2020

J Geom Anal

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Section: Introductionmentioning

confidence: 99%

Well-Posedness of Weinberger’s Center of Mass by Euclidean Energy Minimization

Laugesen

2020

J Geom Anal

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“…This permits to reduce the proof of Theorem 6.7 to the case of convex sets satisfying the hypothesis (6.16) of the previous result. A similar statement was contained in the original paper by Melas (this is essentially [63,Proposition 3.1]), but the proof in [9] is quicker and simpler. We reproduce it here, with some minor modifications.…”

Section: Thus We Getmentioning

confidence: 65%

“…Finally, one also needs the following interesting result, whose proof can be found in [9,Section 6.1]. This permits to reduce the proof of Theorem 6.7 to the case of convex sets satisfying the hypothesis (6.16) of the previous result.…”

Section: Case 1 Let Us Suppose Thatmentioning

confidence: 99%

“…In [10, Theorem 0.3] the aforementioned Xu's result is extended to a sufficiently narrow polar cap contained in S N + . In [9] Aubry, Bertrand and Colbois prove pinching results for the Faber-Krahn and Ashbaugh-Benguria inequalities for convex sets in S N , R N and H N . These results show that if the relevant spectral deficit is small, then the set is close to a ball in the Hausdorff metric.…”

Section: Case 1 Let Us Suppose Thatmentioning

confidence: 99%

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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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“…Ávila established analogues of Melas' aforementioned results for convex subsets of S 2 and H 2 in [9], following quantitative estimates for other spectral inequalities in [44]. Qualitative stability results for the Faber-Krahn inequality and other spectral inequalities on space forms is established in [8]. To our knowledge, Theorem 1.1 is the first (sharp or nonsharp) quantitative stability result for Faber-Krahn inequalities on the round sphere and hyperbolic space that is valid for all open bounded sets Ω.…”

Section: Introductionmentioning

confidence: 99%

Sharp quantitative Faber-Krahn inequalities and the Alt-Caffarelli-Friedman monotonicity formula

Allen¹,

Kriventsov²,

Neumayer³

2021

Preprint

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The objective of this paper is two-fold. First, we establish new sharp quantitative estimates for Faber-Krahn inequalities on simply connected space forms. In these spaces, geodesic balls uniquely minimize the first eigenvalue of the Dirichlet Laplacian among all sets of a fixed volume. We prove that the gap between the first eigenvalue of a given set Ω and that of the ball quantitatively controls both the L 1 distance of this set from a ball and the L 2 distance between the corresponding eigenfunctions:where B denotes the nearest geodesic ball to Ω with |B| = |Ω| and u Ω denotes the first eigenfunction with suitable normalization. On Euclidean space, this extends a result of Brasco-De Phillipis-Velichkov; the eigenfunction control largely builds upon on new regularity results for minimizers of critically perturbed Alt-Cafarelli type functionals in our companion paper. On the round sphere and hyperbolic space, the present results are the first sharp quantitative results with respect to any distance; here the local portion of the analysis is based on new implicit spectral analysis techniques.Second, we apply these sharp quantitative Faber-Krahn inequalities in order to establish a quantitative form of the Alt-Caffarelli-Friedman (ACF) monotonicity formula. A powerful tool in the study of free boundary problems, the ACF monotonicity formula is nonincreasing with respect to its scaling parameter for any pair of admissible subharmonic functions, and is constant if and only if the pair comprises two linear functions truncated to complementary half planes. We show that the energy drop in the ACF monotonicity formula from one scale to the next controls how close a pair of admissible functions is from a pair of complementary half-plane solutions. In particular, when the square root of the energy drop summed over all scales is small, our result implies the existence of tangents (unique blowups) of these functions.

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Eigenvalue pinching on convex domains in space forms

Abstract: Abstract. In this paper, we show that the convex domains of H n which are almost extremal for the Faber-Krahn or the Payne-Polya-Weinberger inequalities are close to geodesic balls. Our proof is also valid in other space forms and allows us to recover known results in R n and S n .

Cited by 5 publications

References 17 publications

Well-Posedness of Weinberger’s Center of Mass by Euclidean Energy Minimization

Well-Posedness of Weinberger’s Center of Mass by Euclidean Energy Minimization

7 Spectral inequalities in quantitative form

Sharp quantitative Faber-Krahn inequalities and the Alt-Caffarelli-Friedman monotonicity formula

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