The Payne conjecture for Dirichlet and Buckling eigenvalues

We prove the long-standing Payne conjecture that the $k^{text{th}}$ eigenvalue in the buckling problem for a clamped plate is not less than the ${k+1}^{text{st}}$ eigenvalue for the membrane of the same shape which is fixed on the boundary. Moreover, we show that the Payne conjecture is still true for $n$-dimensional case ($nge 2)$.