The Steklov spectrum of hyperrectangles


While the computation of the spectrum of the Laplacian on a hyperrectangle in Euclidean space is a standard exercise for PDE students, understanding the Steklov spectrum on such a domain is more subtle. I will present recent results, joint work with Alexandre Girouard, Iosif Polterovich and Alessandro Savo, about both high energy asymptotics and bottom of the spectrum characterisation. More precisely, I will give a complete characterisation of the eigenfunction and the eigenvalues then show that we have two term asymptotics for the counting function of the eigenvalues that determine the (d−1)-volume of the boundary of the hyperrectangle and the (d−2)-volume of 2-boundary. I will also show that the hypercube maximises the first eigenvalue when normalised either by d-volume of the hyperrectangle or (d−1)-volume of the boundary.