Reverse Faber-Krahn inequalities for the Logarithmic potential operator

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 $R>1$.