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

TL;DR

This paper provides bounds on the eigenvalues and integrals of eigenfunctions of the Newtonian potential operator in bounded domains, extending from simple shapes to general geometries using monotonicity properties.

Contribution

It introduces new estimations for eigenvalues and eigenfunctions of the Newtonian potential operator, applicable to general shapes based on initial results for simple geometries.

Findings

Derived bounds for eigenvalues in 2D and 3D

Extended estimations from balls and discs to general shapes

Applicable to electric and acoustic field estimations

Abstract

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 $\Omega \subset \mathbb{R}^{d},$ where $d = 2, 3$, in terms of the maximum radius of $\Omega$. 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.