Geometric Structures of Laplacian Eigenfunctions

Huaian Diao and Hongyu Liu
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Huaian Diao: Jilin University, School of Mathematics
Hongyu Liu: City University of Hong Kong, Department of Mathematics

Chapter Chapter 2 in Spectral Geometry and Inverse Scattering Theory, 2023, pp 9-61 from Springer

Abstract: Abstract Laplacian eigenvalue problem is arguably the simplest PDE eigenvalue problem, which is stated as finding u ∈ L2(Ω) and λ ∈ ℝ $$\lambda \in \mathbb {R}$$ such that − Δ u = λ u , $$\displaystyle \begin{aligned} {} -\varDelta u=\lambda u, \end{aligned}$$ where Ω is an open set in ℝ n , n = 2 , 3 $$\mathbb {R}^n,n=2,3$$ , under a certain homogeneous boundary condition on ∂Ω, such as the Dirichlet, Neumann or Robin condition. The solution u to (2.1.1) is referred to as a (generalized) Laplacian eigenfunction.

Date: 2023
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DOI: 10.1007/978-3-031-34615-6_2

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