A sharp isoperimetric inequality for the Neumann--Poincaré operator in every dimension

Abstract

Let $\Omega \subset {\mathbb{R}}^d$, $d\ge 2$, be a bounded connected domain with boundary of class $C^{1,\alpha }$, where $0<\alpha <1$. For the adjoint Neumann--Poincaré operator $K_{\partial \Omega }^{\ast }$, normalised so that its distinguished eigenvalue is $1/2$, let ${\lambda }_j^{+}(\Omega )$ denote the upper min--max values on the mean-zero energy space. We prove ${\sum }_{j=1}^d{\lambda }_j^{+}(\Omega )\ge \frac{d-2}{2}.$ It follows that ${\lambda }_1^{+}(\Omega )\ge \frac{d-2}{2d},$ with equality if and only if $\Omega$ is a ball. In dimension three this proves the $1/6$-conjecture of Miyanishi and Suzuki. The proof uses the coordinate boundary-charge densities induced by uniform applied fields. Their energy Gram matrix is the perfect-conductor polarization tensor $M_{\infty }$. Positivity of a $2d\times 2d$ Gram matrix yields the endpoint Hashin--Shtrikman inequality $|\Omega |\mathrm{Tr}(M_{\infty }^{-1})\le 1,$ and bounds the trace of the compression of $K_{\partial \Omega }^{\ast }$ to the applied-field space. Equality in the inverse-trace inequality makes the interior Newtonian potential quadratic, so a converse to Newton's theorem identifies $\Omega$ as an ellipsoid. Equality in the spectral estimate also makes the Hessian of this potential isotropic, which forces the ellipsoid to be a ball.