Reverse Faber-Krahn inequalities for the Logarithmic potential operator

Abstract

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

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