Reverse Faber-Krahn inequalities for the Logarithmic potential operator

TL;DR

This paper establishes reverse Faber-Krahn inequalities for the largest eigenvalue of the Logarithmic potential operator in planar domains, exploring symmetrization, obstacle effects, and eigenvalue properties for specific geometries.

Contribution

It introduces reverse inequalities for the eigenvalues of the Logarithmic potential operator and analyzes their behavior under symmetrization, obstacle translation, and domain geometry.

Findings

Reverse Faber-Krahn inequalities for $ au_1(\

Monotonicity of $ au_1(\

Eigenvalue characterizations for domains and balls.

Abstract

For a bounded open set $\Omega \subset \mathbb{R}^2,$ we consider the largest eigenvalue $\tau_1(\Omega)$ of the Logarithmic potential operator $\mathcal{L}$. If $diam(\Omega)\le 1$, we prove reverse Faber-Krahn type inequalities for $\tau_1(\Omega)$ under polarization and Schwarz symmetrization. Further, we establish the monotonicity of $\tau_1(\Omega\setminus\mathcal{O})$ with respect to certain translations and rotations of the obstacle $\mathcal{O}$ within $\Omega$. The analogous results are also stated for the largest eigenvalue of the Riesz potential operator. Furthermore, we investigate properties of the smallest eigenvalue $\tilde{\tau}_1(\Omega)$ for a domain whose transfinite diameter is greater than 1. Finally, we characterize the eigenvalues of $\mathcal{L}$ on $B_R$, including the $\tilde{\tau}_1(B_R)$ when $R>1$.