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 Ω is C2,α (α>0) bounded in R3 and the Neumann–Poincaré operator K∂Ω:L2(∂Ω)→L2(∂Ω) is defined byK∂Ω[ψ](x):=14π∫∂Ω〈y−x,n(y)〉|x−y|3ψ(y)dSy where dSy is the surface element, n(y) is the outer normal vector on ∂Ω, and the integral is understood in the principal value sense. Then the ordered eigenvalues λj(K∂Ω) of the Neumann–Poincaré operator satisfy|λj(K∂Ω)|∼{3W(∂Ω)−2πχ(∂Ω)128π}1/2j−1/2asj→∞. Here W(∂Ω) and χ(∂Ω) denote the Willmore energy and the Euler characteristic of the boundary surface ∂Ω. This formula is the so-called Weyl's law for eigenvalue problems of Neumann–Poincaré operators.