Upper bound of the counting function of Steklov eigenvalues

Abstract: We study the counting function of Steklov eigenvalues on compact manifolds with boundary and obtain its upper bound involving the leading term of Weyl's law. Our estimate can be viewed as a weakened version of P\'{o}lya's Conjecture in the Steklov case on general manifolds. As a byproduct, we also obtain a description about the decay behavior of Steklov eigenfunctions near the boundary.