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The Lp convergence of Fourier series on triangular domains
Part of
Ryan L. Acosta Babb
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 66 / Issue 2 / May 2023

Published online by Cambridge University Press:

19 June 2023, pp. 453-474
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We prove Lp norm convergence for (appropriate truncations of) the Fourier series arising from the Dirichlet Laplacian eigenfunctions on three types of triangular domains in $\mathbb{R}^2$: (i) the 45-90-45 triangle, (ii) the equilateral triangle and (iii) the hemiequilateral triangle (i.e. half an equilateral triangle cut along its height). The limitations of our argument to these three types are discussed in light of Lamé’s Theorem and the image method.

3 - The Laplacian on a Domain
from Part Two - The Laplace and Schrödinger Operators
Rupert L. Frank, Ludwig-Maximilians-Universität München, Ari Laptev, Imperial College of Science, Technology and Medicine, London, Timo Weidl, Universität Stuttgart
Book:

Schrödinger Operators: Eigenvalues and Lieb–Thirring Inequalities

Published online:

03 November 2022

Print publication:

17 November 2022, pp 167-243
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Summary

We discuss the definition of the Laplace operator on an open subset in Euclidean space as a self-adjoint operator with Dirichlet or Neumann boundary conditions and we derive its basic spectral properties. Among others, we include the spectral inequalities of Faber–Krahn, Hersch, and Friedlander. Then, using the technique of Dirichlet–Neumann bracketing, we derive Weyl's law for the asymptotic distribution of eigenvalues. We supplement this with a discussion of non-asymptotic bounds, including Pólya's conjecture and its proof for tiling domains and domains of product form. We present the sharp eigenvalue bounds of Berezin and Li–Yau. Finally, using separation of variables in spherical coordinates, we discuss the Laplacian on a ball.

Complete asymptotic expansions for eigenvaluesofDirichlet Laplacian in thin three-dimensional rods *
Denis Borisov, Giuseppe Cardone
Journal:

ESAIM: Control, Optimisation and Calculus of Variations / Volume 17 / Issue 3 / July 2011

Published online by Cambridge University Press:

06 August 2010, pp. 887-908

Print publication:

July 2011
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We consider the Dirichlet Laplacian in a thin curvedthree-dimensional rod. The rod is finite. Its cross-section isconstant and small, and rotates along the reference curve in anarbitrary way. We find a two-parametric set of the eigenvalues ofsuch operator and construct their complete asymptotic expansions. Weshow that this two-parametric set contains any prescribed number ofthe first eigenvalues of the considered operator. We obtain thecomplete asymptotic expansions for the eigenfunctions associatedwith these first eigenvalues.

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