Estimation of the eigenvalues and the integral of eigenfunctions of the Newtonian potential operator

We consider the problem of estimating the eigenvalues and the integral of the corresponding eigenfunctions, associated to the Newtonian potential operator, defined in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^d,$ where $d=2,3$, in terms of the maximum radius of $\mathrm{\Omega }$. We first provide these estimations in the particular case of a ball and a disc. Then we extend them to general shapes using a, derived, monotonicity property of the eigenvalues of the Newtonian operator. The derivation of the lower bounds are quite tedious for the 2D-Logarithmic potential operator. Such upper/lower bounds appear naturally while estimating the electric/acoustic fields propagating in ${\mathbb{R}}^d$ in the presence of small scaled and highly heterogeneous particles.