The Weyl-type asymptotic formula for biharmonic Steklov eigenvalues on Riemannian manifolds

Liu, Genqian

doi:10.1016/j.aim.2011.07.001

Cited by 36 publications

(39 citation statements)

References 41 publications

(70 reference statements)

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“…In fact, there exists a countable set of eigenvalues. We refer to [9] for a fairly complete study of the spectrum and to [21] for a corresponding Weyl-type asymptotic behaviour. The following result holds…”

Section: Positivity Preservingmentioning

confidence: 99%

The First Biharmonic Steklov Eigenvalue: Positivity Preserving and Shape Optimization

Bucur

Gazzola

2011

Milan J. Math.

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We consider the Steklov problem for the linear biharmonic equation. We survey existing results for the positivity preserving property to hold. These are connected with the first Steklov eigenvalue. We address the problem of minimizing this eigenvalue among suitable classes of domains. We prove the existence of an optimal convex domain of fixed measure.Mathematics Subject Classification (2010). Primary 35P15; Secondary 35J40.

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Section: Positivity Preservingmentioning

confidence: 99%

The First Biharmonic Steklov Eigenvalue: Positivity Preserving and Shape Optimization

Bucur

Gazzola

2011

Milan J. Math.

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“…Denote by ω n the volume of the unit ball in R n . A famous asymptotic formula of L. Sandgren [71] (this formula was first established by Sandgren, and a sharp form was given by the author in [58]) states that N(τ ) = #{k λ k ≤ τ } = ω n vol(∂ ) τ n (2π) n + o(τ n ) as τ → +∞, (1.4) or, what is the same,…”

Section: Introductionmentioning

confidence: 98%

“…The more coefficients of the asymptotic expansion of the heat kernel (associated with Dirichlet-to-Neumann operator map) essentially reflect the more geometric property of the manifold. Computation of spectral invariants is also a challenging problem in spectral geometry (see [41,81,35,58,68,26,27,70,4]). For the coefficients of asymptotic expansion of the trace of the heat kernel associated to the Dirichlet-to-Neumann operator, the first coefficient a 0 (n, x) had been known by (1.5); a 1 (2, x) had been obtained in [29]; the coefficients a 1 (n, x) and a 2 (n, x) were explicitly calculated in [68] and [59] in completely different ways.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic expansion of the trace of the heat kernel associated to the Dirichlet-to-Neumann operator

Liu

2015

Journal of Differential Equations

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For a given bounded domain with smooth boundary in a smooth Riemannian manifold (M, g), by decomposing the Dirichlet-to-Neumann operator into a sum of the square root of the Laplacian and a pseudodifferential operator, and by applying Grubb's method of symbolic calculus for the corresponding pseudodifferential heat kernel operators, we establish a procedure to calculate all the coefficients of the asymptotic expansion of the trace of the heat kernel associated to Dirichlet-to-Neumann operator as t → 0 + . In particular, we explicitly give the first four coefficients of this asymptotic expansion. These coefficients provide precise information regarding the area and curvatures of the boundary of the domain in terms of the spectrum of the Steklov problem.

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“…Consider the set H := [H 2 ∩ H In fact, there exists a countable set of eigenvalues. We refer to [11] for a fairly complete study of the spectrum and to [21] for a corresponding Weyl-type asymptotic behaviour. The first purpose of the present paper is to describe the role of the least Steklov eigenvalue in bounded domains.…”

Section: Introductionmentioning

confidence: 99%

Convex shape optimization for the least biharmonic Steklov eigenvalue

Antunes

Gazzola

2013

ESAIM: COCV

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Abstract. The least Steklov eigenvalue d1 for the biharmonic operator in bounded domains gives a bound for the positivity preserving property for the hinged plate problem, appears as a norm of a suitable trace operator, and gives the optimal constant to estimate the L 2 -norm of harmonic functions. These applications suggest to address the problem of minimizing d1 in suitable classes of domains. We survey the existing results and conjectures about this topic; in particular, the existence of a convex domain of fixed measure minimizing d1 is known, although the optimal shape is still unknown. We perform several numerical experiments which strongly suggest that the optimal planar shape is the regular pentagon. We prove the existence of a domain minimizing d1 also among convex domains having fixed perimeter and present some numerical results supporting the conjecture that, among planar domains, the disk is the minimizer.

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The Weyl-type asymptotic formula for biharmonic Steklov eigenvalues on Riemannian manifolds

Cited by 36 publications

References 41 publications

The First Biharmonic Steklov Eigenvalue: Positivity Preserving and Shape Optimization

The First Biharmonic Steklov Eigenvalue: Positivity Preserving and Shape Optimization

Asymptotic expansion of the trace of the heat kernel associated to the Dirichlet-to-Neumann operator

Convex shape optimization for the least biharmonic Steklov eigenvalue

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