Faber Krahn inequality of Robin eigenvalue of the Weighted Laplacian

Abstract

In this paper, we prove two Faber-Krahn-type inequalities for the first eigenvalue of the weighted Laplacian with Robin boundary conditions on $\mathbb{R}^{n}$ and $\mathbb{H}^{n}$. In both settings, the optimal domain is shown to be a ball centered at the origin.