Removal of the log factor in the asymptotic estimates of polygonal membrane eigenvalues

Bailey, Paul B.; Brownell, F. H.

doi:10.1016/0022-247x(62)90052-5

Cited by 26 publications

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“…This question has been studied by many authors including [1,3,7, 11,12,14,15,17,19]. One approach to this problem proceeds by considering the trace of the heat semigroup on D,…”

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confidence: 99%

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The trace of the heat kernel in Lipschitz domains

Brown¹

1993

Trans. Amer. Math. Soc.

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Abstract. We establish the existence of an asymptotic expansion as t -► 0+ for the trace of the heat kernel for the Neumann Laplacian in a bounded Lipschitz domain. The proof of an asymptotic expansion for the heat kernel for the Dirichlet Laplacian is also sketched. The treatment of the Dirichlet Laplacian extends work of Brossard and Carmona who obtained the same result in C1 -domains.A classical question in analysis asks: What is the relationship between the spectrum of the Laplacian on a domain and the geometry of the domain? This question has been studied by many authors including [1,3,7, 11,12,14,15,17,19]. One approach to this problem proceeds by considering the trace of the heat semigroup on D,where g is the heat kernel on D and 0 < Ai < A2 < A3•• • is the sequence of eigenvalues for -A on D.One attempts to give an asymptotic expansion for tx(g) near t = 0 whose coefficients are quantities associated with the geometry of D. Thus one can deduce geometrical information about D from knowledge of the spectrum.In this paper, we will address this question under minimal smoothness assumptions on the boundary of D. In particular, we will obtain asymptotic expansions for tx(g) when the order of smoothness of the boundary corresponds exactly to the order of accuracy of the asymptotic expansion.In order to state our main theorem, we need one definition: By a Lipschitz domain, we mean a bounded connected open set whose boundary may be locally described as the graph of a Lipschitz function. Our main result is Main Theorem. Suppose that D c R" is a Lipschitz domain, then the heat kernel for the Laplacian with Dirichlet boundary conditions on 3D satisfies tx(g)(t) = (4ntyn'2 (\D\ -^^n~x(dD) + o(txl2)\ .

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“…This question has been studied by many authors including [1,3,7, 11,12,14,15,17,19]. One approach to this problem proceeds by considering the trace of the heat semigroup on D,…”

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confidence: 99%

“…Brossard and R. Carmona [5]. When the domain is smooth or is a polygon, then results of this type are well known, see [1,4,12,15]. These authors either use explicit calculations to study polygons or results on boundary value problems which fail in nonsmooth domains.…”

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confidence: 99%

The trace of the heat kernel in Lipschitz domains

Brown¹

1993

Trans. Amer. Math. Soc.

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“…I( j ~ We observe that the remainder term here does not contain ~ ~ --a result obtained earlier for a polygon (see [6] and [7]) and also for a polyhedron. *…”

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confidence: 65%

Problem of focusing and the asymptotics of the spectral function of the Laplace-Beltrami operator. I

Babich¹,

Levitan²

1983

J Math Sci

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The connection between the asymptotics of the spectral function and the formal shortwave expansion of the solution of the problem of the asymptotics of the Green function near a geodesically concave boundary of a two-dimensional surface is considered in the paper.By means of theorems of Tauberian type the asymptotic behavior of the spectral function of elliptic differential operators reduces to the study of stationary and nonstationary Green functions of wave problems, i.e., to questions with which the mathematical theo1~ of diffraction is concerned.In the theory of diffraction there are many results and techniques which are little known among specialists in the spectral theory of differential operators.Use of the methods of diffraction theory makes it possible, in particular, under certain additional conditions to extend the classical asymptotic formulas for the spectral function of the Laplace operator to a neighborhood of the boundary. This is the topic of our paper. A brief exposition of our results is contained in [i].

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“…This problem is quite similar to the well-known Kac problem for the Laplacian on a domain (the Kac question asks: is it possible to hear the shape of a domain just by "hearing" all of the eigenvalues of the Dirichlet Laplacian? see [51,9,11,25,47,56,7,37,38,60,6,72,24,[21][22][23]70] and the references therein). In order to explain the main method of this paper, we briefly review the historical background for the case of Dirichlet Laplacian on domains.…”

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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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Removal of the log factor in the asymptotic estimates of polygonal membrane eigenvalues

Cited by 26 publications

References 1 publication

The trace of the heat kernel in Lipschitz domains

The trace of the heat kernel in Lipschitz domains

Problem of focusing and the asymptotics of the spectral function of the Laplace-Beltrami operator. I

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

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