2018/06/10 by Miyanishi, Yoshihisa
#47A75 (primary) #58J50 (secondary) #FOS: Mathematics #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.1806.03657
We deduce eigenvalue asymptotics of the Neumann--Poincaré operators in three dimensions. The region Ω is C2, α (α>0) bounded in \mathbf R3 and the Neumann--Poincaré operator \mathcal K∂Ω : L2(∂ Ω) → L2(∂ Ω) is defined by \mathcal K∂Ω[ψ](\bf x) := (1)/(4π) ∫∂ Ω \frac⟨ \bf y-\bf x, \bf n(\bf y) ⟩|\bf x-\bf y|3 ψ(\bf y) dS\bf y where dS\bf y is the surface element and \bf n(\bf y) is the outer normal vector on ∂ Ω. Then the ordering eigenvalues of the Neumann--Poincaré operator λj (\mathcal K∂ Ω) satisfy |λj(\mathcal K∂ Ω)| ∼ \(3W(∂ Ω) - 2πχ(∂ Ω))/(128 π) \1/2 j-1/2 as j → ∞. Here W(∂ Ω) and χ(∂ Ω) denote, respectively, the Willmore energy and the Euler charateristic of the boundary surface ∂Ω. This formula is the so-called Weyl's law for eigenvalue problems of Neumann--Poincaré operators.