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

We deduce eigenvalue asymptotics of the Neumann–Poincar'e operators in three dimensions. The region $\mathrm{\Omega }$ is $C^{2,\alpha }$ ($\alpha >0$) bounded in ${\mathbf{R}}^3$ and the Neumann–Poincar'e operator ${\mathcal K}{\partial\Omega} : L^2(\partial \Omega) \rightarrow L^2(\partial \Omega) $isdefinedby$$ {\mathcal K}{\partial\Omega}[\psi]({\bf x}) := \frac{1}{4\pi} \int_{\partial \Omega} \frac{\langle {\bf y}-{\bf x}, {\bf n}({\bf y}) \rangle}{|{\bf x}-{\bf y}|^3} \psi({\bf y}); dS_{\bf y} where $dS_{\bf y}$ is the surface element and ${\bf n}({\bf y})$ is the outer normal vector on $\partial \Omega$. Then the ordering eigenvalues of the Neumann--Poincar\'e operator $\lambda_j ({\mathcal K}_{\partial \Omega})$ satisfy |\lambda_j({\mathcal K}_{\partial \Omega})| \sim \Big{\frac{3W(\partial \Omega) - 2\pi \chi(\partial \Omega)}{128 \pi} \Big}^{1/2} j^{-1/2}\quad $\text{as}$\ j \rightarrow \infty. $$Here$W(\partial \Omega)$and$\chi(\partial \Omega)$ denote, respectively, the Willmore energy and the Euler charateristic of the boundary surface $\mathrm{\partial }\mathrm{\Omega }$. This formula is the so-called Weyl’s law for eigenvalue problems of Neumann–Poincar'e operators.