Bounds to Eigenvalues of Rhombical Membranes

Stadter, James T.

doi:10.1137/0114029

Cited by 16 publications

(3 citation statements)

References 6 publications

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“…The exact values for the "square" eigenfunctions are )3 2r and the double eigenvalue s 9 5r. The calculated values of * agree with these exact values to the number of decimal places used in the [11] by other methods. The difficulty stems from the existence of two corners with nonintegral values of a.…”

Section: ) Thensupporting

confidence: 76%

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Approximations and Bounds for Eigenvalues of Elliptic Operators

Fox¹,

Henrici²,

Moler³

1967

SIAM J. Numer. Anal.

201

148

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Section: ) Thensupporting

confidence: 76%

“…Difference methods in combination with collocation and conformal mapping have been used by Reid [14]) gives upper and lower bounds for certain domains. For example, Stadter [11] uses the method for rhombical domains.…”

mentioning

confidence: 99%

Approximations and Bounds for Eigenvalues of Elliptic Operators

Fox¹,

Henrici²,

Moler³

1967

SIAM J. Numer. Anal.

201

148

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“…After Hooker and Protter's paper, several methods were applied to obtaining accurate approximations for the first eigenvalue of the rhombus. However, and as opposed to the approach in both [6] and the present paper, these did not provide an explicit expression in terms of the parameter involved-see, for instance, [8,10].…”

Section: Introductionmentioning

confidence: 86%

Precise bounds and asymptotics for the first Dirichlet eigenvalue of triangles and rhombi

Freitas

2007

Journal of Functional Analysis

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We study the asymptotic expansion of the first Dirichlet eigenvalue of certain families of triangles and of rhombi as a singular limit is approached. In certain cases, which include isosceles and right triangles, we obtain the exact value of all the coefficients of the unbounded terms in the asymptotic expansion as the angle opening approaches zero, plus the constant term and estimates on the remainder. For rhombi and other triangle families such as isosceles triangles where now the angle opening approaches π , we have the first two terms plus bounds on the remainder. These results are based on new upper and lower bounds for these domains whose asymptotic expansions coincide up to the orders mentioned. Apart from being accurate near the singular limits considered, our lower bounds for the rhombus improve upon the bound by Hooker and Protter for angles up to approximately 22 • and in the range (31 • , 54 • ). These results also show that the asymptotic expansion around the degenerate case of the isosceles triangle with vanishing angle opening depends on the path used to approach it.

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Zur Einschliessung von Eigenwerten

Velte¹

1981

Numerische Behandlung Von Differentialgleichungen Band 3

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Bounds to Eigenvalues of Rhombical Membranes

Cited by 16 publications

References 6 publications

Approximations and Bounds for Eigenvalues of Elliptic Operators

Approximations and Bounds for Eigenvalues of Elliptic Operators

Precise bounds and asymptotics for the first Dirichlet eigenvalue of triangles and rhombi

Zur Einschliessung von Eigenwerten

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