Sharp bounds for the first two eigenvalues of an exterior Steklov eigenvalue problem

Let $U\subset {\mathbb{R}}^n$ ($n\ge 3$) be an exterior Euclidean domain with smooth boundary $\mathrm{\partial }U$. We consider the Steklov eigenvalue problem on $U$. First we derive a sharp lower bound for the first eigenvalue in terms of the support function and the distance function to the origin of $\mathrm{\partial }U$. Second under various geometric conditions on $\mathrm{\partial }U$ we obtain sharp upper bounds for the first eigenvalue. Along the proof, we get a sharp upper bound for the capacity of $\mathrm{\partial }U$ when $n=3$ and $\mathrm{\partial }U$ is connected. Last we also discuss an upper bound for the second eigenvalue.