Sharp upper bounds for Steklov eigenvalues of a hypersurface of revolution with two boundary components in Euclidean space

Abstract

We investigate the question of sharp upper bounds for the Steklov eigenvalues of a hypersurface of revolution of the Euclidean space with two boundary components isometric to two copies of S n−1 . For the case of the first non zero Steklov eigenvalue, we give a sharp upper bound B n (L) (that depends only on the dimension n ≥ 3 and the meridian length L > 0) which is reached by a degenerated metric g * , that we compute explicitly. We also give a sharp upper bound B n which depends only on n. Our method also permits us to prove some stability properties of these upper bounds.