Estimates of the first Dirichlet eigenvalue from exit time moment spectra

Hurtado, Ana; Markvorsen, Steen; Palmer, Vicente

doi:10.1007/s00208-015-1316-7

Cited by 24 publications

(33 citation statements)

References 30 publications

(15 reference statements)

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“…We should also mention that the case p = 1 is naturally of special interest, and has produced a literature too large to describe here; the case of general p has attracted somewhat less interest, nevertheless the reader interested in other results relating the p-th moments of Brownian exit time with the geometry of domains is referred to [1,4,7,9,10,13,14,15,16,20,21,22,23,27,28].…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

On the finiteness of moments of the exit time of planar Brownian motion from comb domains

Boudabra

Markowsky

2021

Ann. Fenn. Math.

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A comb domain is defined to be the entire complex plain with a collection of vertical slits, symmetric over the real axis, removed. In this paper, we consider the question of determining whether the exit time of planar Brownian motion from such a domain has finite p-th moment. This question has been addressed before in relation to starlike domains, but these previous results do not apply to comb domains. Our main result is a sufficient condition on the location of the slits which ensures that the p-th moment of the exit time is finite. Several auxiliary results are also presented, including a construction of a comb domain whose exit time has infinite p-th moment for all p ≥ 1/2.

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Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

On the finiteness of moments of the exit time of planar Brownian motion from comb domains

Boudabra

Markowsky

2021

Ann. Fenn. Math.

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“…. } have been studied in several papers ( [10,11,17,18,20,21]). We point out that T 1 ( ) has also an intepretation in mechanics, being the torsional rigidity of the domain ; isoperimetric inequalities for the functional T 1 ( ) are classical, and were recently extended to the higher exit time moments.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the heat content functional and its critical domains

Savo

2021

Calc. Var.

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We study and classify smooth bounded domains in an analytic Riemannian manifold which are critical for the heat content at all times $$t>0$$ t > 0 . We do that by first computing the first variation of the heat content, and then showing that $$\Omega $$ Ω is critical if and only if it has the so-called constant flow property, so that we can use a previous classification result established in [33] and [34]. The outcome is that $$\Omega $$ Ω is critical for the heat content at time t, for all $$t>0$$ t > 0 , if and only if $$\Omega $$ Ω admits an isoparametric foliation, that is, a foliation whose leaves are all parallel to the boundary and have constant mean curvature. Then, we consider the sequence of functionals given by the exit-time moments $$T_1(\Omega ),T_2(\Omega ),\dots $$ T 1 ( Ω ) , T 2 ( Ω ) , ⋯ , which generalize the torsional rigidity $$T_1$$ T 1 . We prove that $$\Omega $$ Ω is critical for all $$T_k$$ T k if and only if $$\Omega $$ Ω is critical for the heat content at every time t, and then we get a classification as well. The main purpose of the paper is to understand the variational properties of general isoparametric foliations and their role in PDE’s theory; in some respects they generalize the properties of the foliation of $$\mathbf{R}^{n}$$ R n by Euclidean spheres.

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“…In the past half-century, McKean's and Cheng's results have become a continuing source of inspiration concerning the first eigenvalue problem on curved spaces; without seeking completeness, we recall the works of Carroll and Ratzkin [5], Chavel [6], Freitas, Mao and Salavessa [11], Gage [12], Hurtado, Markvorsen and Palmer [16], Li and Wang [20,21], Lott [22], Mao [24], Pinsky [31,32] and Yau [42], where various estimates and rigidity results concerning the equality in (1.5) are established.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…By the differentiation formula (2.3) and (3.14) we have for every γ, x > 0 and integer k ≥ 1 that 15) where by convention the denominator at the right hand side is 1 for k = 1. Let n = 2l, l ∈ N. Due to (3.4) and (3.15), we have 16) where α = α(r, l) is the smallest positive solution to the equation S l (α, r) = 0. As in the odddimensional case, we may assume that αr → Φ as r → ∞ for some Φ > 0; we are going to prove first that α ∼ π r as r → ∞.…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

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New features of the first eigenvalue on negatively curved spaces

Kristály

2020

Advances in Calculus of Variations

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AbstractThe paper is devoted to the study of fine properties of the first eigenvalue on negatively curved spaces. First, depending on the parity of the space dimension, we provide asymptotically sharp harmonic-type expansions of the first eigenvalue for large geodesic balls in the model n-dimensional hyperbolic space, complementing the results of Borisov and Freitas (2017), Hurtado, Markvorsen and Palmer (2016) and Savo (2008); in odd dimensions, such eigenvalues appear as roots of an inductively constructed transcendental equation. We then give a synthetic proof of Cheng’s sharp eigenvalue comparison theorem in metric measure spaces satisfying a Bishop–Gromov-type volume monotonicity hypothesis. As a byproduct, we provide an example of simply connected, non-compact Finsler manifold with constant negative flag curvature whose first eigenvalue is zero; this result is in a sharp contrast with its celebrated Riemannian counterpart due to McKean (1970). Our proofs are based on specific properties of the Gaussian hypergeometric function combined with intrinsic aspects of the negatively curved smooth/non-smooth spaces.

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Estimates of the first Dirichlet eigenvalue from exit time moment spectra

Cited by 24 publications

References 30 publications

On the finiteness of moments of the exit time of planar Brownian motion from comb domains

On the finiteness of moments of the exit time of planar Brownian motion from comb domains

On the heat content functional and its critical domains

New features of the first eigenvalue on negatively curved spaces

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