A comparison between Neumann and Steklov eigenvalues

This paper is devoted to a comparison between the normalized first (non-trivial) Neumann eigenvalue $|\Omega| \mu\_1(\Omega)$ for Lipschitz open set $\Omega$ in plane and Steklov $P(\Omega) \sigma\_1(\Omega)$. More precisely, we study ratio $F(\Omega):=|\Omega| \mu\_1(\Omega)/P(\Omega) We prove that this can take arbitrarily small or large values if do not put any restriction on class of sets $\Omega$. Then restrict ourselves convex domains which get explicit bounds. also case thin give more precise The finishes with plot corresponding Blaschke–Santaló diagrams $(x,y)=(|\Omega| \mu\_1(\Omega), P(\Omega) \sigma\_1(\Omega) )$.