Weyl's law for the eigenvalues of the Neumann--Poincaré operators in three dimensions: Willmore energy and surface geometry

Abstract

We deduce eigenvalue asymptotics of the Neumann--Poincaré operators in three dimensions. The region $\Omega$ is $C^{2,\alpha }$ ($\alpha >0$) bounded in ${\mathbf{R}}^3$ and the Neumann--Poincaré operator ${\mathcal{K}}_{\partial \Omega }:L^2(\partial \Omega )\to L^2(\partial \Omega )$ is defined by $${\mathcal{K}}_{\partial \Omega }[\psi ](\mathbf{x}):=\frac{1}{4\pi }{\int }_{\partial \Omega }\frac{⟨\mathbf{y}-\mathbf{x},\mathbf{n}(\mathbf{y})⟩}{|\mathbf{x}-\mathbf{y}{|}^3}\psi (\mathbf{y})d{S}_{\mathbf{y}}$$ where $d{S}_{\mathbf{y}}$ is the surface element and $\mathbf{n}(\mathbf{y})$ is the outer normal vector on $\partial \Omega$. Then the ordering eigenvalues of the Neumann--Poincaré operator ${\lambda }_j({\mathcal{K}}_{\partial \Omega })$ satisfy $$|{\lambda }_j({\mathcal{K}}_{\partial \Omega })|\sim \{\frac{3W(\partial \Omega )-2\pi \chi (\partial \Omega )}{128\pi }{\}}^{1/2}{j}^{-1/2}\text{as}j\to \infty .$$ Here $W(\partial \Omega )$ and $\chi (\partial \Omega )$ denote, respectively, the Willmore energy and the Euler charateristic of the boundary surface $\partial \Omega$. This formula is the so-called Weyl's law for eigenvalue problems of Neumann--Poincaré operators.

15 pages

References in corpus (5)

• Schatten classes on compact manifolds: Kernel conditions
• The essential spectrum of the Neumann--Poincare operator on a domain with corners
• Spectral resolution of the Neumann-Poincaré operator on intersecting disks and analysis of plasmon resonance
• Exponential decay estimates of the eigenvalues for the Neumann-Poincaré operator on analytic boundaries in two dimensions
• Classification of spectra of the Neumann-Poincaré operator on planar domains with corners by resonance

Cited by in corpus (6)

• Spectral structure of the Neumann--Poincaré operator on tori
• Spectral analysis of Neumann-Poincaré operator
• Spectral properties of the Neumann-Poincaré operator in 3D elasticity
• A concavity condition for existence of a negative Neumann-Poincaré eigenvalue in three dimensions
• Convergence rate for eigenvalues of the elastic Neumann--Poincaré operator on smooth and real analytic boundaries in two dimensions
• Surface localization of plasmons in three dimensions and convexity