An inequality for the Steklov spectral zeta function of a planar domain
Alexandre Jollivet, V. A. Sharafutdinov · Journal of Spectral Theory · 2018
We consider the zeta function \zeta_\Omega for the Dirichlet-to-Neumann operator of a simply connected planar domain \Omega bounded by a smooth closed curve. We prove that, for a fixed real s satisfying |s|>1 and fixed length L(\partial \Omega) of the boundary curve, the zeta function \zeta_\Omega(s) reaches its unique minimum when \Omega is a disk.This result is obtained by studying the difference \zeta_\Omega(s)-2\big({L(\partial \Omega)\over 2\pi}\big)^s\zeta_R(s) , where \zeta_R stands for the classical Riemann zeta function. The difference turns out to be non-negative for real s satisfying |s|>1 . We prove some growth properties of the difference as s\rightarrow\pm\infty . Two analogs of these results are also provided.