Han Hong

Shape optimization of Steklov eigenvalues in higher dimensions

The optimization of Steklov eigenvalues is an important and interesting topic both in spectral theory and geometric analysis. The classical Weinstock theorem states that the disk uniquely maximizes the first Steklov eigenvalue among all simply-connected planar domains having fixed boundary length. It is known that there is an upper bound on $\sigma_k$ for domains in $\mathbb{R}^n$, $n\geq 3$, but the optimal upper bound and domain are unknown. I will recall some known results in dimension 2 and discuss some new results which show that in dimension $n\geq 3$ certain aspects of the topology of the domain do not have an effect when considering shape optimization questions for Steklov eigenvalues.