A higher dimensional generalization of Hersch-Payne-Schiffer inequality for Steklov eigenvalues

In this paper, we generalize the Hersch-Payne-Schiffer inequality for Steklov eigenvalues to higher dimensional case by extending the trick used by Hersch, Payne and Schiffer to higher dimensional manifolds. More precisely, we show that, for a compact oriented Riemannian manifold with boundary of dimension n, the multiplication of a Steklov eigenvalue for functions and a Steklov eigenvalue for differential (n−2)-forms can be controlled from above by a certain eigenvalue of the Laplacian operator on the boundary.