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 $\Omega$ 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) $ is defined by $$ {\mathcal K}_{\partial\Omega}[\psi]({\bf x}) := \frac{1}{4\pi} \int_{\partial \Omega} \frac{\langle {\bf y}-{\bf x}, {\bf n}({\bf y}) \rangle}{|{\bf x}-{\bf y}|^3} \psi({\bf y})\; dS_{\bf y} $$ 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 $\partial\Omega$. This formula is the so-called Weyl's law for eigenvalue problems of Neumann--Poincar\'e operators.