Optimization problems for Steklov eigenvalues in higher dimensions.

Dr Han Hong (University of British Columbia) will present his research in Analysis.

The optimization of Steklov eigenvalues is an important and interesting topic both in spectral theory and geometric analysis. In particular, the connection between maximizing metric on surfaces and existence of free boundary minimal surfaces in the unit ball is discovered by Fraser and Schoen. In this talk, I will be mostly focusing optimization problems in Euclidean space. 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.

The talks are held using zoom. Anyone interested in attending (outside the School of Mathematics) should email Ben Sharp at b.g.sharp@leeds.ac.uk to request zoom coordinates