Sharp isoperimetric upper bounds for planar Steklov eigenvalues
Alexandre Girouard, Mikhail Karpukhin, Jean Lagacé · arXiv (Cornell University) · 2020
We solve the isoperimetric problem for the first and second nonzero Steklov eigenvalues of planar domains, without any assumption on the number of connected components of the boundary. Our approach uses the known sharp upper bounds for the weighted Neumann eigenvalues, and a homogenisation method allowing to approximate these eigenvalues by the Steklov eigenvalues of appropriately chosen perforated subdomains.